On the GIBBS thermodynamic potential of seawater

Prog. Oceanog. Vol. 36, pp. 249-327, 1995 Copyright© 1996 Publishedby ElsevierScience Lid Printedin Great Britain.Allrightsreserved

Pergamon

P I I : S 0 0 7 9 - 6 6 1 1 (96)00001-8

0079- 6611/95$29.00

On the GIBBS thermodynamic potential of seawater RAINER FEISTEL and EBERHARD HAGEN

lnstitut flir Ostseeforschung, Warnemfinde, Germany

Abstract - Free Enthalpy, the GIBBS thermodynamic potential G(S,t,p) of seawater, has been recomputed including the sound speed equation of DEL GROSSO (1974), temperatures of maximum density (TMD) of CALDWELL(1978), freezing point depression measurements of DOHERTY and KESTER (1974), rederived limiting laws and ice properties, and an extended set of dilution heat data of BROMLEY(1968) and MILLERO,HANSENand HOFF (1973). As a new reference state, the standard ocean state has been chosen. The resulting average deviations are 0.0006 kg m -3 for pure water density at 1 atm, 0.002 kg m -3for seawater density at 1 aim, 0.02 m/s for sound speed, 0.01 J k g K "l for heat capacity at 1 aim, 0.4 kJ kg ~ for dilution heats, 0.002°C for freezing points, and 0.04°C for TMDs. Resulting pressuredependent freezing points are in good agreement with experiments and UNESCO (1978) formulas. Enthalpy as thermodynamic potential has been explicitly determined for easy computation of potential temperature, potential density, and sound speed. All functions are expressed in the new International Temperature Scale ITS-90. Copyright © 1996 Published by Elsevier Science Ltd

CONTENTS

1. 2. 3. 4. 5. 6. 7. 8. 9. I 0. 11.

Introduction Heat capacity Density and sound speed Solution theory Freezing points and osmotic pressure Dilution heat Enthalpy as potential function Discussion Acknowledgement References Appendix

250 254 256 263 269 280 289 293 295 295 299

249

2 50

R. FEISTELand E. HAGEN

1. INTRODUCTION

The most compact representation of thermodynamic properties of any substance is the quantitative mathematical tbrmulation of one of its so-called thermodynamic potentials. Such a potential function has been computed for pure water steam in a wide range of values of pressure and temperature (HAAR, GALLAGHERand KELL, 1984). For seawater, however, the UNESCO formulae (FOFONOFFand MILLARD,1983) as described in the International Oceanographic Tables (referred hereafter as IOT-4) for density, heat capacity, and sound speed, represent the widely accepted standard for its equilibrium properties. To compensate mutual inconsistencies between these formulae, in a previous paper (FEISTEL, 1993) we have proposed a polynomial-likefunction for the specific Free Enthalpy G (S,t,p) (referred to as G93 later in this paper) as a function of salinity S, temperature t, and pressure p. Because this function is a thermodynamic potential function expressed in its natural variables, all relevantquantitative thermodynamicproperties can be derived from it by mathematic rules in a completely consistent way. Besides the mechanical ("P-V-T") properies mentioned in IOT-4, in G93 thermochemicalproperties have been included, too; dilution heat measurements of MILLERO,HANSENand HOFF(1973) as well as an expression for the osmotic coefficientderived by MILLEROand LEUNG(1976) from freezing point depression measurements. To maintain compatibility with the present standard, the computation of G93 was kept quantitativelyas close as possible to the IOT-4 formulas, replacing high-pressure density and highpressure heat capacity by new values extracted from the sound speed equation of CHEN and MILLERO(1977) by a method slightly different from that used by those authors. In the present paper, however, we intend to go beyond this state, re-adjusting the previous potential G93 to (i) (ii) (iii) (iv)

(v) (vi)

(vii)

the maximum density data of CALDWELL(1978), the sound speed formula of DEL GROSSO(1974), directly recalculated limiting laws, directly used freezing point measurements, an extended set of dilution heat data, the International Temperature Scale 1990, the standard ocean reference state.

CALDWELL'Smeasurements of the temperature of maximum density show deviations from IOT-4 that amount to 0.3°C in extreme situations such as the winter-time convection in relatively fresh water (humide climate) such as in the Baltic and the Adriatic. Including these data we intend to improve the precision of seawater properties at low temperatures in general, which are important for various convection instabilities and processes of deep water formation at high latitudes. DEL GROSSO's sound velocity differs from IOT-4 speed up to 1 m/s in the deep sea. As recent investigations of long-range travel times (DUSHAW,WORCESTERandCORNUELLE,1993) indicate, DELGROSSO's 1974 equation (DELGROSSO, 1974) is the most reliable one for"Neptunian" waters (within 0.05 m/s), where the CHEN-MILLEROequation (CHEN and MILLERO,1977; UNESCO, 1983) exhibits systematic deviations of about 0.6 m/s. Expecting a growing scientific interest in quantities like entropy, enthalpy, or chemical potential of seawater, even for practical oceanographic applications such as thermohaline process studies, we felt there was a weakness in the data background of the thermochemical properties used for G93, and so we recomputed dilution heat on a larger data set compared with G93, evaluated freezing points directly based on a most recent review on ice properties (YEN, CI-IENGand FUKUSAKO, 1991), and recomputed the limiting law coefficients based on average seawater composition after MILLERO (1982; MILLEROand SOHN, 1992), using atomic weights 1979

The GIBBS thermodynamicpotential of seawater

251

(HOLDEN, 1980) and especially the standard of the dielectric constant of water (WHITE, 1977; HAAR et al.,1984, 1988). Because it is not a priori evident, how a "molecule of seasalt" is to be understood, we have avoided mole-based units for thermodynamic quantities concerning salinity, rather expressing them by particle numbers and masses of the constituents. As a by-product, the thermodynamic potential of ice has been determined in the vicinity of the freezing point_ To obtain quantities like potential temperature or potential density, one may use a temperatureentropy-inversion polynomial (FEISTEL, 1993). An alternative way is to derive enthalpy H(S,o,p) in terms of its natural variables salinity S, entropy o, and pressure p. Then, its f'trst derivatives are chemical potential, potential temperature, and potential density, its second derivatives yield sound speed, and heat capacity. Details are explained in section 7. In 1990, a new practical definition of temperature ITS-90 was released (BLANKE, 1989; PRESTON-THOMAS, 1990; BETTIN and SPIEWECK, 1990; SAUNDERS, 1990; FOFONOFF, 1992), which is closer to the thermodynamic scale. Many quantities like IOT-4 formulas for seawater properties (UNESCO, 1983) are expressed in the former standard IPTS-68 (WAGEN'BRETHand BLANKE, 1971). In the range of interest here, 0-40°C, both can be converted into each other by (fitted with r.m.s.= 0.1 inK, see Fig. I. 1), t68/°C = 1.0002505 Conucrsion

t90/°C. Foz'mulas

(1.1) -

11'S-90

g

O

IPIS-68

.5

" -.5.

-I

0

t

~ 0.2505

~

IO

tgO ~ 0,2400

20

Temper*aCute

30 tgO

in

40 °C

Fig. 1.1. Conversion between temperature scales from 1968, IPTS-68, and 1990, ITS-90. Their difference & = tg0 - t68 can be approximatedby a linear correction, The recommendationof JPOTS, 8t = 0.24. tg0 (lower curve), remains within an error of 0.1 mK between -5 and +20°C, while the formula ~t = 0.2 5 - t90 (upper curve) is correct within 0.2 mK from -10 to +40°C.

252

R. FEISTELand E. HAGEN

Note that the corresponding conversion factor proposed by the Joint Panel on Oceanographic Tables and Standards JPOTS (SAUNDERS, 1990; UNESCO 1991; FOFONOFF 1992) is only 1.00024. In this paper, all thermodynamic functions are formulated in ITS-90, while all data and expressions used for the computations have been evaluated at the corresponding t68 temperatures using the complete conversion formula (BLANKE, 1989). Supposing that Specific Free Enthalpy (GIBBS potential) G(S,t,p), i.e. the Free Enthalpy of 1kg of seawater divided by lkg, is expressed as a function of its natural variables S,t,p and denoting partial derivatives by subscripts, we summarise the most important quantities briefly here (see e.g. LANDAU and LIFSHITZ, 1966; FOFONOFF, 1962; MAMAYEV, 1975,1976; FEISTEL, 1993): Chemical Potential Entropy Specific Volume Heat Capacity Adiabatic Lapse Rate Thermal Expansion Coeff. IsothermalCompressibility Haline Contraction Coeff. Sound Speed U Enthalpy Internal Energy

IX= G s t~ = -G t V=G CP = -PTGtt F = -Gw/Gtt tx = G t IG K = - ~ /~p 13 = -GsP~G (V/U) 2 P= (~2-Gtt Gpp)/Gtt H = G - T G'te E = H - P Gp

(1.2) (1.3) (1.4) (1.5) (1.6)

(1.7) (1.8) (1.9)

where T = t + T ° = t + 273.15 K

(1.10)

is the absolute temperature and p = p + po = p + 0.101325 MPa

(1.11)

is the absolute pressure (P° is atmospheric pressure, p is sea or gauge pressure). In chapter 7 we shall consider similar expressions for the case that enthalpy H(S,o,p) is used as thermodynamic potential. If we express the variables S = 40 PSU • x2, t = 40°C • y, p = 100MPa. z

(1.12)

by dimensionless quantifies x,y,z, we may write, G(S,t,p) = 1J/kg • g(x,y,z)

(1.13)

g(x,y,z) = (gO + gl • y). x21n(x) + E g[i,j,k] x i yJ zk

ijk Groups of coefficients ofg(x,y,z) can be obtained from different measurements, as the following table indicates (FEISTEL, 1993, modified here):

TheGIBBSthermodynamicpotentialof seawater

1. index (left to right) 2. index (downward inside blocks) 3. index (downward block by block)

253

- order in salinity root - order in temperature - order in pressure

Logarithm Terms g0,g1 are taken from ideal solution theory, Coefficientsg ljk are vanishingfor all j and k, Limiting Law Reference Ig000 State gO10

g200 1 ~ g210

CP(0,t,0) Heat Capacity of Water at 1 atm

g020 g030 g040 g050 g050 g060

g220 g230 g240 g250 g250

g320 g330 g340

g001 g011 g021 g031 g041 g051 g061

g201 g211 g221 g231 g241

g301 g311 g321 g331

g401 g411

V(S,t,0) Specific Volume of Seawater at 1 atm,

U(O,t,p) Sound Speed in Water,

g002 g012 g022 g032 g042 g052

g202 g212 g222 g232 g242

g302 g312 g322

g402 g412

U(S,t,p) Sound Speed in Seawater,

TMD(0,p) Points of Maximum Density of Water

g003 g013 g023 g033 g043

g203 g213 g223 g233 g243

g303 g313 g323 g333

g403

TMD(S,p) Points of Maximum Density of Seawater

g004 g014 g024 g034

g204 g214 'g224 g234

g304

g005 g015

g205 g215 g225

V(0.t,0)

Specific Volume of Water at 1 atm,

g400 g 5 0 0 g 6 0 0 ] g410 g 5 1 0 g 6 1 0

FreezingPoint, DilutionHeat CP(S,t,0) Heat Capacity of Seawater at 1 atm

254

R. FEISTELand E. HAGEN

2. HEAT CAPACITY IOT-4 heat capacity (MILLERO, PERRONand DESNOYERS, 1973; UNESCO, 1983) CP83(S,t,p)

has been used at one atmosphere (p=0) for water (S=0) and seawater (S>0) in two subsequent fits. Minimizing the expression (eqs. 1.1, 1.5, 1.12) I dy {~2G(0,t,0)/~t2 + CP83(0,t68(t),0)/(T°+t)}2 = Min! o

by 8-point GAUSS integration we have obtained 5 coefficients g[0,2,0] - g[0,6,0], listed in appendix A. 1. The root mean square (r.m.s.) of the fit was 9.9E-3 J/kgK in CP, or 2.4 ppm. Minimizing the expression

I dx I dy {O2G(S,t,0)/Ot2 + CP83(S,t68(t),0)/(T°+t)}2 = Min! 0

0

by two-dimensional 8-point GAUSS integration we have obtained 7 coefficients, g[2,2,0]-g[2,5,0] and g[3,2,0]-g[3,4,0], as listed in appendix A. 1. The r.m.s, of the fit was 1.1E-2 J/kgK in CP, or 2.6 ppm. The experimental error in CP is believed to be 0.5 J/kgK, or 120 ppm (UNESCO, 1983). Except the use of ITS-90, there is no difference in relation to the previous G93 results. High-pressure heat capacities will be computed on the basis of sound speeds and additional data

as described in the following section. The resulting values deviate from the IOT-4 function increasingly with pressure, up to an r.m.s, of about 5 J/kgK, or 1000 ppm, compare Figs 2.1a-c. Heat Capacity

Deuiation

a t p : 0 flPa

• 05

al

*" ,iq

O-

0 I

10

20 PSU 0 PSU

v ~-.

PSU

h lV.

[,4 0

05

25

PSU

30

PSU

35

PSU

40

PSU

-

0 0

-.10

20

20 Temperature

in

30 °C

40

Fig. 2.1. Difference between specific heats of IOT-4 and this paper, a) (above) at atmospheric pressure, b) (righ0 at salinity S = 35 PSU, c) (righ0 at temperature t = 0°C. The experimental reliability is about 0.5 J/kg/K. Only the differences at p=0 have been minimized. Note the different scales of the ordinates.

The GIBBS thermodynamic potential of seawater Heat C a p a c i t y

Deviation

at

S = 35

255

PSU

30"T

100

HPa

X \

20 \

80

NPa

60

MPa

40

MPa

20

14Pa 14Pa

0 I 10. ¢

0

i i

I11

Ib O.

-10

0

I 10

0

30

20

Te~pePatu~e

Heat C a p a c i t y

in

Deviation

40

eC

at t

= 0 °C

15

100

X \

i

\

] .

MPa

......................

C ,M

~

0 C t

5

,It

....................................................... ~................. S

i

. . . . . . .

/ 80 NPa . . . . . . . . . . . . . . . . . . . . . .

i jf

0

;Si

. . . . . . . . . . . . . . . . . . . . . . . . . . .

.zJ !

v 0 C Ii al

-.

.

0

.

.

.

i 10

.

.

.

.

.

.

.

.

:, 30

20

Salinity

in

PSU

.

.

.

.

40

2 56

R. FEISTELand E. HAGEN

3. DENSITY AND SOUND SPEED

For the recomputation of density we have to adjust the GIBBS potential to three separate quantities simultaneously: (i) densities at atmospheric pressure, (ii) temperatures of maximum density, and (iii) sound speeds. For combining these different sources, the numerical aim is to minimize a mixed mean-square deviation expression, ~{ ~ Wik [Fik - Fk(Si,tl,Pi)]2/Ok2 } = Min!

(3.1) k i It consists of a number of different data groups, denoted by numbers k, like specific volume or sound speed, which we are going to discuss below. Each group enters into the problem with its weight l/Ok2, where o k is the absolute accuracy assumed for this data group. Each group "k" is represented by a number of data values Ftk, labelled by 'T'. These values are either measurements or certain function values as for example from DEL GROSSO's sound speed formula, taken at abscissa triples (Si,ti,Pi). For measured data, these are just the points of measurements; for intervals of a function, these are abscissas of numerical 8-point GAUSS integrations over the interval (with the same number of abscissa points, this integration is about twice as precise as an equidistant summation). Fk is the relevant mathematical expression in G(S,t,p), valid for the group "k", depending on the unknown coefficients, taken at the same abscissas (Si,ti,Pi). Wik is the weight of the i-th data point within the k-th group, normalized to unity, E. Wik = 1, for all k. 1

(3.2)

For measured values, we have taken all data inside one group with equal weights, for functions, Wtk are the weights of the GAUSS integration (ABRAMOWrrZand STEGUN, 1968). There are 15 different groups of data in the problem (3.1), stated for pure water and seawater separately, which we outline briefly now: Seawater density as defined by the International Equation of State (EOS80, UNESCO 1981) has been used as target function at one atmosphere between 0 and 40°C, converted to ITS-90 temperatures. Its pure-water part has been derived from BIGG's formula (BIC,G, 1967) which is the origin of the general technical water standard as well (WAGENBRETHand BLANKE,1971; KEEL, 1975; KELL and WHALLEY, 1975; HAAR et al. 1984, 1988; BETTIN and SPmWECK, 1990). As a result of corrections of density maximum values (UNESCO, 1976, 1981), both differ by typically 10 ppm. Dissolved gases in natural waters may cause deviations of about 4 ppm (BIGNELL,1983) compared to those of air-free standards, and regional deviations in salt composition may have even stronger impacts, up to 50 ppm (BREWERand BRADSHAW, 1975; FOFONOFF, 1985; MILLEROand SOI-IN, 1992) on densities of water samples in oceanographic practice. For the fits (3.1) we have used specific volume of water, V(0,t,0), EOS80 (or IOT-4), normalized by the deviations o k = 3.0E-09 m3/kg (3 ppm), and of seawater, V(S,T,0) with o k = 4.0E-09 m3/kg (4 ppm), corresponding to their estimated experimental error (UNESCO, 1976, 1981).

The GIBBS thermodynamicpotentialof seawater

257

In the neighbourhood of maximum density points, EOS80 is not very reliable (SIEDLERand PETERS, 1986). Temperatures of maximum density (TMD) derived from EOS80 are in good agreement with values measured by CALDWELL(1978)for pure water, but deviate up to 0.3°C for medium salinities. With CALDWELL'Spolynomial (S in PSU, t in °C, p in MPa) TMD = 3.982 -.2229 S -.2004 p (1 + 3.76E-3 S) (1 + 4.02E-3 p)

(3.3)

we have computed table 3.1 for some S and p in the range of CALDWELL'Smeasurements (the measurements themselves are listed in table 5.3). Note that a number of points probably refer to supercooled waters; compare the discussion in section 5. Table 3.1. Difference TMD(EOS80) - TMD(Caldwell) in °C, computed using Eq.(3.3). p/MPa 0 4 8 12 16 20 24 28 32 36 40

S=O -0.00 0.00 0.01 0.02 0.02 0.03 0.03 0.02 0.01 -0.02 -0.06

S=5 S=IO S=15 S=20 S=25 S=30 S=35 0.05 O.I 1 0.15 0.17 0.17 0.13 0.06 0.06 0.12 0.17 0.19 0.19 0.15 0.08 0.07 0.13 0.18 0 . 2 1 0.21 0.18 0.08 0.15 0.21 0.24 0.24 0.21 0.09 0.17 0.23 0.26 0.27 0.10 0.18 0.25 0.29 O.11 0.20 0.27 0 . 1 1 0.21 0.11

S=40 -0.05

PSU

At TMD points, thermal expansion disappears, dWdt = 0, with a slope of dW/dt2= 1.5E-8 m3/ kgKL CALDWELL'STMD data are good within 40 mK, hence we expect IdV/dtl < IdW/dt21 • 40 mK = ak = 6E-10 m3/kgK

(3.4)

over all experimental data points (S~,tl,p~)of CALDWELLin Eq.(3.1). For the thermal expansion coefficient,~ = dln(V)/dt, the uncertainty, (3.4), is equivalent to only 0.6 pprn/K, while originally EOS80 yields about 2 ppm/K for cx at the measured TMDs. (Away from regions of maximum density, o~is typically 100 - 300 ppm/K for water.) High-pressure densities are preferably computed from sound speeds (KELL, 1975; CHENand MILLERO, 1976, 1978). Following the investigations of DUSHOWet al. (1993), the sound speed formula of DEE GROSSO(1974) is the most reliable (within 0.05 m/s) for deep ocean waters with natural combinations of S,t,p, where IOT-4 sound speed U(S,t,p) exhibits systematic deviations of about 0.6 m/s. On the other hand, outside of these windows of "Neptunian waters" in (S,t,p) space, DEL GROSSO's polynomial produces rather useless values. For pure water, some error estimates for sound speed are 0.2 m/s (WILSON, 1959), 0.3 m/s (BARLOWand YAZGAN, 1967, KELLand WHALLEY,1975) and 0.03-0.04 rrds for low pressures (WILLE, 1986). However, because a deviation below 0.02 rrds was achieved in computing sound speed from the GIBBS potential (FEISTEL, 1993), we imposed this value for~k to remain close to the common pure water formulae.

258

R. FEISTELand E. HAGEN

All sound speed formulae have been converted from IFrs-68 to ITS-90. We have combined the "Tables" of DEL GROSSO(1974) with IOT-4 sound speed, for water (table 3.2) and for seawater (table 3.3). DEE GROSSO'S formula uses "kg/cm 2 gauge" as pressure unit, 1 kg/cm 2 gauge Ls 1 kilopond (the weight of 1 kg mass) per cm 2 or 0.980665 bar = 98.0665 kPa (WlLLE, 1986). Table 3.2. Numerical weights for "Tables" of DEL GROSSO (1974) and IOT-4 for water.

Table VII: IOT-4:

S[PSU] 0 0

t[°C] 0-30 0-40

ok[m/s]

p[MPa] 0 0-100

0.02 0.02

Table 3.3. Numerical weights for "Tables" of DEL GROSSO (1974) and IOT-4 for seawater.

Table I: Table II: Table III: Table IVa: Table IVb: Table IVc: Table IVd: Table V: Table VI: IOT-4:

S[PSU] 29-41 33-37 29-43 29-43 29-43 29-43 29-43 33-37 33-37 0-40

t[°C] 0-35 0-35 0-30 0 10 20 30 0 5 0-40

p[MPa] 0 2 0 5 2 1 0.1 0-100 0-100 0-100

~k[rrds] 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.60

All data mentioned so far have been treated together in two runs, one for water and one for seawater. For fresh water, 24 coefficients g[0,j,k], k>0, have been determined as given in table A. 1 of the appendix. The root mean square (r.m.s.) deviations of the fits in the different data groups are, compared to the required accuracies o k (V(S,t,p) specific volume, U(S,t,p) sound speed IOT-4) V(0,t,0): U(0,t,p): Table VII: TMD(0,p):

6k= ok= ok= ok=

3.0E-09, 2.0E-02, 2.0E-02, 4.0E-02,

r.m.s.= r.m.s.= r.m.s.= r.m.s.=

3.7E-10 1.9E-02 8.2E-03 3.4E-02

ma/kg m/s m/s °C

For seawater, 39 coefficients g[i,j,k], i> 1, k>0, have been determined as given in table A. 1. The corresponding error ranges are V(S,t,0): U(S,t,p): Table IV: Table I-III: Table V-VI: TMD(S,p):

ok= 4.0E-09 ok= 6.0E-01 t~k= 5.0E-02 ok= 5.0E-02 Ok----5.0E-02 ok= 4.0E-02

r.m.s.= r.m.s.= r.m.s.= r.m.s.= r.m.s.= r.m.s.=

3.5E-09 4.6E-01 1.3E-02 1.7E-02 1.8E-02 5.5E-02

ma/kg m/s m/s m/s m/s °C

The GIBBS thermodynamic potential of seawater

259

The desired compromise could indeed successfully be achieved, for all data groups the average differences appeared to stay inside the anticipated limits, sometimes even significantly below. The only minor exception is maximum density points of salt water, caused by systematic deviations of all data at salinity 20.1 PSU (if one tentatively skips these data, r.m.s, drops down half the value). The differences between the results of this paper and EOS80 densities are shown in Figs 3. la-c, DEE GROSSOsound speed in Figs 3.2a-b, and IOT-4 sound speed in Figs 3.3a-c. Densit N Deviation

a t p = O MPa

10

25 ZO 30

15 < f

5-

i

.O..PSU

IO 35

:5

.5 "i

PSU PSU PSU PSU

. . . . . . . . . . .

PSU PSU

PSU

Q C

'0

O. ~.ao.

es.

...............

¢

IIJ

~

--5

-I0

!

:

30

40

I

0

10

20

Temperature

in

°C

Fig. 3.1. Differences between IOT-4 density (F_,OS80) and this paper, a) (above) at atmospheric pressure, b) (overleaf) at temperature t = 0°C, c) (overleaf) at salinity S = 35 PSU. The experimental reliability is below 10 g/m3, at atmospheric pressure about 4 g/m 3. At low temperatures and pressures the deviations appear to be the result of the maximum density adjustment, and at higher pressures of the modified sound speed.

260

R. PEISTELand E. HAGEN

D e n s i t y D e v i a t i o n a t t = 0 °C

< Z

30'

C

20,

0 £ I

0 v

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in

Density Deviation at S

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The GIBBS thermodynamic potential of seawater

261

Del Grosso Sound Speed D e v i a t i o n a t t = 0 "C .3-

.2

100

MPa

C .1-

i

o c

I O,

80 60

HPa

40

HPa

20

HPa

HPa

~ -.1 r. Ii

-.2

-.3~ 10

0

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30 in

40

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.2:-

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I

0 C

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O-

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-.aJ

. 0

6Q . r . lt ~ 4 ° Mr. lO

~ 20PWa 20

Tel~lpePmtu~e

30

in

40

*C

Fig. 3.2. Differences between the sound speed formala of DEL GROSSO (1974) and this paper, a) at temperature t = 0°C, b) at salinity S = 35 PSU. DEE GROSSO'S polynomial yields meaningful figures, within about 0.05 m/s, only for natural combinations of S,t,p ("Neptunian waters").

262

R. FEaSTELand E. HAO~

SoundSpeedgeulationat p = O flPa .2

35

lqU

30

PSU

.1

0

PSU PSU

20 PSU 5 PSU 15 iO

PSU PSU

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0

10

20

30

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Sound Speed D e u i a t l o n a t 1.5

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t = 0 "C

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~

-.5 0

£0

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30 in

40

PSU

Fig. 3.3. Differences between the IOT-4 sound speed formula and this paper, a) at atmospheric pressure, b) at temperature t = 0°C, c) (right) at salinity S = 35 PSU. At p = 0, IOT-4 is claimed to be better than DI~. GROSSO's equation, while for deep ocean water its reliability was found to be only 0.6 ntis.

The GIBBS thermodynamicpotentialof seawater

263

Sound Srm~__lPevlation at $ = 35 PSU : 20

40

I

~0

v

o' ¢

NPa

HPa

HPa

60 100

NPa

80

HPa

NPa

-I

o

-2

-3 + 0

20

20

Te~]pt]~atuz~

30

in

40

*C

4. SOLUTION THFX)RY

In the limit of very small concentrations, statistical physics allows for the exact derivation of thermodynamic properties of electrolyte solutions. PLANCK'S ideal solution theory considers solute particles simply as mass points, which carry an additional electrical charge in case of the limiting laws of DEBYE and HUECKEL.At higher densities of the dissolved particles, effects like finite ion size, hydration shell structure or ionic association gain increasing importance for the macroscopic properties of a solution. Then, even for relatively simple electrolytes like aqueous KC1 solution, theory can merely provide rough approximations, and precise data can only be derived from experiments. On the other hand, a number of quantities are very difficult to determine by measurements for extremely low salt concentrations. In such cases, as for some coUigative properties like mixing heat, theoretical limiting laws may be required for the extrapolation of measured data (LEWIS and RANDALL, 1961). In a previous paper (FEISTEL,1993) we adopted a number of coefficients for ideal solution and limiting laws from the formulae of MILLEROand LEUNG (1976) for osmotic coefficients and dilution enthalpies. Tracing these figures back to the background data of their very first computations is a rather winding road; for better transparency we rederive them here directly from theory. The differences which emerge are the result of recently improved physical data as, for example, for the dielectric constant of water. In the case of seawater, the application of solution theory requires knowledge of how many particles of each kind are dissolved in a given sample of seawater with salinity S. Practical salinity

264

R. FEISTELand E. HAGEN

S is merely defined by electrical conductivity (UNESCO, 1981 ), from which no unique conclusion can be drawn about the chemical composition. Unfortunately at present there is no definition of a seawater standard composition, however, one suitable stoichiometry of an "average seawater" has been given by MILLERO( 1982; MILLEROand SOl-IN, 1992), the corresponding salt components (a) and their mass fractions W(a) are listed in table 4.1. Slightly different proposals have been offered by KENNISH(1994), FOFONOFF(1992), WEICHART(1986), ALEKINand LYAKHIN(1984), and MILLERO(1974). For completeness, we mention here the IUPAC 79 mole mass of water itself, A(H20) = 18.0152 g/mol, and of silver, A(Ag) = 107.8680 g/mol. Table 4.1. Average Seawater Composition. Components and mass fractions from MILLERO (1982), mole fractions recomputed with IUPAC 79 standard mole masses (HOLDEN, 1980). a: salt component, Z: ion charge, A: atomic weight, W: weight fraction, X: mole fraction (a)

Z(a)

A(a) g]mol

W(a) ppm

X(a) ppm

Na Mg Ca K Sr

+1 +2 +2 +1 +2

22.9898 24.3050 40.0800 39.0983 87.6200

306566 36500 11717 11348 226

418793 47164 9181 9115 81

C1 SO 4 HCO 3 Br CO 3 B(OH) 4 F

-1 -2 -1 -1 -2 -1 -1

35.4530 96.0576 61.0171 79.9040 60.0092 78.8392 18.9984

550252 77119 3228 1911 331 187 37

487437 25214 1662 751 173 75 61

HsBO 3

0

61.8319

578

293

We have recomputed the mole fractions X(a) on the basis of Standard Atomic Weights 1979 (HOLDEN, 1980), rounded to 4 decimals, A(a), by W(a)/A(a) X(a) =

(4.1) W(b)/A(b)

Exactly to fulfil the normalization condition Z X(a) = 1 and electro-neutrality Z X(a) Z(a) = 0 with integer mole fractions in ppm (part per million), we have rounded up X(HCO 3) and X(B(OH)4) and down X(H3BO3). Following 1VIILLEROand SOHN(1992), salinity S is related to chlorinity CI by the relation (LEWIS and PERKn~, 1981; KENNISH, 1994) S = 1.80655 PSU. Cl/g

(4.2)

The GIBBS thermodynamic potential of seawater

265

and C1 is defined as the mass of silver needed to precipitate CI and Br in 328.5233 g of seawater. If the saltconcentration is denoted by c (absolute salinity, mass of dissolved matter divided by mass of seawater, or the mass fraction of salts, FOFONOFF, 1985) we obtain CI = 328.5233kg • c . (W(CI)/A(C1) + W(Br)/A(Br)) • A(Ag)

(4.3)

and thus the salinity-weight factor (relating Absolute Salinity to Practical Salinity) q = c/S = 1.00488 g/kgPSU,

(4.4)

(MILLERO and LEUNG, 1976). The question remains, however, to which accuracy Practical Salinity, as defined by electrical conductivity, can still be related to seawater stoichiometry by means of Eq.(4.2), if salinity is very different from 35 PSU, as in the case of dilute solutions or brackish waters. So far, however, there is no cheap and easy-to-use alternative to conductivity-based salinity as a measure for the salt concentration (FOFONOFF, 1985). Now we can express the number of particles N(a) of component (a) in m = lkg of seawater with salinity S by N(a) = N A • W(a)/A(a) • q. S- m

(4.5)

with Avogadro's constant N g = 6.0221341 E+23/mol (SEYFRIED, 1989). For later use we define the average mole mass of salt ions, = Z X(a) • A(a) = 31.4058 g/mol,

(4.6)

I/ = Z W(a)/A(a) = 31.8412 mol/kg their mean-square charge (valence factor), = Z X(a) Z(a) 2 = 1.245146,

(4.7)

and the number of salt particles N s per PSU and kg seawater. N s = N a q/ = 1.92688E+22/kgPSU.

(4.8)

The total number of particles, here S. N s, divided by N A, is usually called the mole number, and we can this way formally define (compare FOFONOFF, 1992) M = S Ns/N A = c/ as the "number of moles of seasalt ions" per kg of seawater (which is neither the same as molarity, the number of moles per liter of seawater, nor molality, the number of moles per kg water in seawater). The corresponding "atomic weight of seasalt", , is then about half the value used by MILLEROand LEUNG (1976) for the formulation ofmolal seawater properties. To avoid any kind of confusion, caused only by different definitions of a "seasalt molecule", we prefer to refrain completely from using the deliberate notion of"moles of seasait" in the following.

266

R. FEISVELand E. HAGEN

The Specific Free Enthalpy G of a multi-component dilute aqueous electrolyte solution with mass m is given by (see e.g. LANDAU and LIFSCHITZ, 1966; LEWIS and RANDALL, 1961; FALKENHAGEN, 1971) m G = N(H20) ~t°(T,P) + Z a N ( a ) l k T ln(N(a)/N(I-I20) ) + txa(T,P) }

(4.9)

- { [ Z N(a) Z(a) 2 e2/D(T,P)]3/[36x-v(T,P) N(H20) kT]} in where the summation is carded out over all sorts (a) of dissolved ions and uncharged particles. Further constants and functions used here are N(H20) k e C D(T,P) e°

= = = = = = =

m (l-c) NA/A(H20) 1.380642E-23 J/K 1.602177E-19 As 299792458 m/s 4r~e ° e(t,p) 1.E+7 A m / V s / ( 4 ~ C 2) 8.85418782E-12 As/Vm,

number of H 2 0 molecules Boltzmann constant, - electron charge, light speed, - dielectric constant of water, - vacuum permittivity -

-

-

v(T,P) = volume per water molecule at infinite dilution, lx°(T,P) = energy per water molecule at infinite dilution, Ixa(T,P) = energy per ion of type (a) at infinite dilution. Writing G as a series expansion in salinity using N(a) = X(a) • N s • S • m and v(T,P) • N(H20) = V(0,t,p) m + O(S), G(S,t,p) = G o + G 1 SIn(S) + G 2 S + G 3 S'~S + ...

(4.10)

we fred from (4.9) the coefficients G v G 3 to be Gl(t, p) = N s kT

(4.11)

G3(t, p) = -(2/3){ rc (N s e2/D(t,p)) 3 / (kTV(0,t,p)) }1a G, (t,p) is responsible for ideal solution properties and does not depend on pressure. With T = T°+t, T°= 273.15 K we get from a comparison of the expressions (4.10, 4.11) with G l SIn(S) = I J . (g0+gl • y) x21n(x) + O(x 2)

(4.12)

gO = N s kT ° . 4 0 . 2 /J = 5813.3468 g l = N s k . 4 0 - 4 0 . 2 K/J = 851.3047

(4.13)

the values

G3(t, p) expresses various limiting laws; G3(0,0) the limiting law of freezing point depression, and

The GIBBS thermodynamicpotential of seawater

267

dG~/dt of dilution heat. The biggest uncertainty in the calculation of G 3 is the dielectric constant of water D(T,P) and its dependence on temperature and pressure. Both are needed for the computation of G 3 at p=0 MPa, t=0°C, and its first derivative, dG3/dt = -G 3 [1/T + dln(V)/dt + 3dln(D)/dtl/2.

(4.14)

Many calculations for aqueous electrolytes are based on measurements of OWEN, MILLER, MILNSR AND COGAN (1961), who tabulated the dielectric constant of water from 0-70°C and 1100 MPa. At p=0 MPa and t=0°C, they found e = 87.8956, dv./dt = -.404432/K We have used the lAPS 1977 standard (WHITE, 1977; HAAR etal. 1984, 1988), which has an overall (0-500 MPa, 0-550°C) precision of 0.33, to obtain the values = 87.8163, d~dt = -.387534/K. The differences may serve as rough absolute error estimate. If we assume the error in dD/dt as 3%, the mutual compensation in the expression (D/T+dD/dt), which appears in the limiting law of enthalpy H 3 = G3-T • (dG3/dt), causes an error amplification to about 30% in dilution heat (FALKENHAGEN, 1971). With specific volume V=I.000157 dm3/kg and expansion coefficient dln(V)/dt = -67.9502 ppm/K of water (taken from EOS80 at p=0 MPa and t=0°C) we compare equal powers on both sides of (G3+t dG3/dt) S~/S = 1J. (g[3,0,0]+g[3,1,0] y) x 3

(4.15)

and find the limiting law coefficients g[3,0,0] = -2435.7962 g[3,1,0] = -469.9126.

(4.16)

After treating the terms G 1 and G 3 of Eq. (4.10) we have now to discuss the roles of G Oand G 2. Their pressure derivatives determine density of pure water and haline contraction at infinite dilution, their second temperature derivative provides heat capacity of water and its change with salinity, and all these values can be measured in suitable experiments. Up to G 2 and the terms just mentioned, the f'trst coefficients of G(S,t,p) are G = (CO00 + C010. t) +(CI00 + C l 1 0 . t) S In(S) +(C200 + C210. t) S + higher powers in S,t,p

(4.17)

The coefficients C 100 and C 110 are known (gO, g 1, above), while the four constants CO00, CO 10, C200 and C210, however, have no influence on measurable thermodynamic properties of seawater and cannot be determined from experiments with seawater only (FOFONOFF, 1962). C000 and CO 10 are related to absolute energy and absolute entropy of water molecules, C200 and C210 to absolute energy and absolute entropy of salt particles. One possibility to fix these numbers could be the reference to solid salt, as used for ionic standard entropies (LEWISand RANDALL,1961).

268

R. PEisav_~and E. HAGEN

Very theoretically, energy could be fixed relativistically by mass measurements with more than 13 valid digits, and entropy by the Third Law, if seawater thermodynamics were to be valid down to -273°C, but in practice this is neither possible nor necessary for seawater equilibrium thermodynamics. As a consequence, for use in the oceanographic context all four constants C000, CO 10, C200 and C210 can be chosen quite arbitrarily. We emphasize that their adjustment is not a question of right or wrong, but simply of taste, usefulness, or common agreement. The most natural way to fix them is the definition of one or more reference states (i.e. triples S,t,p), at which certain quantifies like enthalpy or entropy are supposed to have zero value. Changing this "reference frame" can be done at any time by the "transformation" (FOFONOFF, 1962), G' = G + (A+Bt) + (C+Dt)S

(4.18)

with suitable numbers of A,B,C,D, where G' is physically as correct as G, only possessing another reference state. Quantities like density or heat capacity, freezing point or osmotic pressure are "gauge invariant" with respect to this transformation, others like entropy

if' = ff - B - DS,

(4.19)

enthalpy

H' = H + (A-BT °) + (C-DT°)S

(4.20)

or chemical potential

g' = g + C + Dt

(4.21)

are "covariant", i.e. depend on the choice of the free constants. This is to be borne in mind when oceanographic sections with these quantities are discussed, or ff-S diagrams are interpreted, because these graphs will be significantly altered when being transformed. Physical consequences, however, must not depend on the corresponding changes. On the first glimpse it may seem that quantities depending on arbitrarily chosen constants are rather worthless for practical use. Remember, however, the very similar situation in classical mechanics, where the coordinatesr of a mass point obey the GALILEI transformationr'=r +r°+ v°t if the reference frame is changed, and that basic physical laws must be independent of the arbitrary constants r°,v °. For practical use it is desirable to adopt a commonly acceptable reference state. For such a proposal, we f'trst rewrite the lowest order terms of G (Eq. 4.17) as G = CO00 + C 0 1 0 . t +C200'- S + (CI00 + C110. t) S In(S/S°)+ ..o

(4.22)

where, without loss of generality, the term C210 is dropped and a reference salinity S° is introduced instead, and C200' is some modified constant replacing C200. For pure water we propose to set entropy and enthalpy to zero for t=0°C and p=0 MPa: CO00 = 0, C010 = 0

(4.23)

This is compatible with the standard ocean definition, it differs, however, slightly from usual thermodynamic descriptions of water, where the triple point is the reference state (t=0.01°C, P=611.3 Pa, BOLZ and TUVE, 1985; HAAR et ai., 1984, 1988; BROMLEY,DIAMOND, SALAMIand WILKINS, 1970).

The GIBBS thermodynamicpotentialof seawater

269

C200 could be defined using the state of infinite dilution. But, because entropy per salt particle diverges when salinity goes to zero, this is not a possible reference state for the definition of S °. Therefore, we propose to set entropy and enthalpy to zero for the standard ocean state t=0°C, p=0 MPa, S = 35 PSU. This way the values of C200 and C210 depend on the other coefficients of G(S,t,p) and cannot be given yet. (Note that a simple setting C200=0, C210=0 causes the reference state to be dependent on the unit in which S is measured, because of the interrelation between S ° and C210 in Eq. 4.22, which would result in a rather formal convention.) At S=35--40x 2, we have to determine g[2,0,0] and g[2,1,0] from the two linear equations gO x21n(x) + g[O,O,O] + g[2,0,O] x2 + Z g[i,O,O] x i = 0 i>2 (4.24) gl x21n(x) + g[0,1,0] + g[2,1,0] x 2 + 5".g[i,l,0] x i = 0 i>2 The result for g[2,0,0] and g[2,1,0] is given in table A.1. This new definition leads to substantial changes in the "gauge-covariant" thermodynamic functions compared to the previous version (FEISTEL, 1993; FEISTELand HAGEN, 1994). There we had defined the enthalpy per salt particle to vanish at infinite dilution. Entropy per salt particle was obtained implicitely by comparing the series expansion for G with the osmotic coefficient • of MILLEROand LEUNG (1976). However, this was an artefact, caused by an approximate relation between G and ~, since a correct expression of • must not depend on C210. To compare the older GIBBS potential G93 with the present one, a suitable transformation of type (4.18) has to be performed, in other words, the coefficients g[2,0,0] and g[2,1,0] have to be substituted by their values determined here. 5. FREEZINGPOINTS AND OSMOTICPRESSURE Freezing of water is a classical phase transition of first kind, and in (t,p)-space one finds a finite region where both phases of water coexist. Removing heat from water at constant pressure causes its temperature to fall until the freezing point is reached, temperature then remains constant, while the fraction of ice grows until all liquid has become solid. In the presence of salt, things become modified. At the freezing point, which depends additionally on the initial salt concentration, first small ice crystals appear, which do not contain any salt particles, so that the remaining solution gains higher salinity and thus lower freezing temperature. As aresult, the fraction of ice in seawater becomes a function of temperature, andvice versa. Apparent freezing temperature depends on the water-ice ratio, so that seawater freezing is coined by a temperature interval rather than by a sharp value (POUNDER, 1965; HOBBS, 1974). The thermodynamic freezing point of seawater with given salinity is, however, still the temperature when the very first and tiny ice crystals emerge. At the freezing temperature t =O(S,p), ice is in equilibrium with liquid water, i.e. the chemical potential of ice, gl~,, equals the chemical potential of water in seawater, taw: ~w(S,O,p) = ~ ( O , p )

(5.1)

Because specific free enthalpy of ice, G l~, is the chemical potential of solid water by itself, and the chemical potential of liquid water in seawater is given by law = G - ~t S, we may write for (5.1)

270

R. FEISTELand E. HAGEN

G - S OG/OS = G Ice at t = O(S,p).

(5.2)

This relation must hold for any triple (O,S,p) of experimentaUy measured freezing points, and it can be used directly to determine unknown coefficients of G. We return to the execution of this fit later. Once G and G I~ are known, Eq.(5.2) provides a freezing point formula. As long as the freezing point lowering O is small compared to the freezing temperature T ° itself, O/T ° << 1, a Taylor expansion around the freezing point up to the linear or quadratic term is sufficiently precise (LEWIS and RANDALL, 1961): A- O2/2T ° + B . O - C = 0

(5.3)

This quadratic equation for O is easily solved. The coefficients A,B,C are the temperature derivatives of (5.2) taken at the freezing point T °, i.e. at t = 0°C: A = CP~Ce(O,p)- CP(S,O,p) + S BCP(S,O,p)/OS B = ak~(O,p) - o(S,O,p) + S Oa(S,O,p)/OS C = Glee(O,p) - G(S,O,p) + S OG(S,O,p)/bS

(5.4)

where oI°°(t,p) and CPX~(t,p) are entropy and heat capacity of ice. If only the linear terms inO, S, p of Eqs. (5.3) and (5.4) are considered, one obtains the freezing point depression O°(S,p) of an ideal solution in dependence on salinity (RAOULT'Slaw), O°(S,0)/T ° = -N s S kT°/Q

(5.5)

and on pressure (CLAUSIUS-CLAPEYRONequation), O°(O,p)/T ° = -p6WQ

(5.6)

(LEWISand RANDALL,1961; SOMMERFELD,1962; LANDAUand LIFShqTZ,1966; FALKENHAGEN, 1971). Here Q is melting heat (Eq. 5.8), N s the number of dissolved salt particles (Eq. 4.8) andSV the volume excess of ice compared with water (Eq. 5.7). These linear laws present the dominating contributions to freezing point lowering. To make use of (5.2), we have to determine the thermodynamic potential of pure water ice, GXCe(t,p),in the neighbourhood of the melting point, i.e. only in its lowest powers of temperature and pressure. The properties we need to be available are latent heat of fusion, density, heat capacity, thermal expansion, and compressibility of ice. As is obvious from RAOULT's law, the most important quantity for our purpose is melting heat Q. The typical error range for freezing point depression measurements is about 1 ppt (LEWISand RANDALL,1961; DOHERTYand KESTER, 1974; FUJINO,LEWISANDPERKIN,1974; KESTER, 1974; MILLEROand LEUNG, 1973; MILLERO,1978, 1983), therefore melting heat should also be precise to least about 1 ppt. Although such narrow limits are claimed (Q = 6012±4 J/mol by LEWIS and RANDALL,196 l, Q = 6017+4 J/mol by POUNDER, 1965), there is a variety of values between 5999 and 6800 J/mol in the literature on water and ice properties (DORSEY, 1940; LINKE and BAUER, 1970; FALKENHAGEN,1971; DORONINand KHEISIN,1975; ]VIILLEROarid LEUNG, 1976; DORONIN, 1978; YEN, 1981; MILLERO,1983; BOLZ and TUVE, 1985; SIEDLERand PETERS, 1986;YEN et al., 1991, LECHNER, 1992). The value we shall use here (6008 Ymol) was recommended by HOBBS (1974) and is in agreement with the error limit given by LEWIS and RANDALL( 1961).

The GIBBS thermodynamic potential of seawater

271

From a recent review (YEN e t a/.,1991) we have selected the values for pure water ice at atmospheric pressure (all quantities with index "Ice" refer to ice properties): (i) specific volume in m3/kg at t=0°C VlC~ = Gpl~ = (916.71) -1

(5.7)

(ii) fusion enthalpy in kJ/kg at t=0°C

(5.8)

Q = T ° • Gtl~ = 333.5 (iii) specific heat in J/kgK, t in °C CP ice = -(T°+t) • Gt: ¢e = 2096.1 + 7.116. t

(5.9)

(iv) thermal cubic expansion coefficient in pprrd°C, t in °C ~lce = Gtplce/vlce = 158.15 + 0.67. t

(5.10)

(v) isothermal compressibility in ppm/MPa, t in °C K I~ = -GpplCe/V Ice = 232.2 + 0.418 • t

(5.11)

Gt~,Gp~ etc. are the partial derivatives of the GIBBS potential of ice

(5.12)

GIC~(t,p) = 1 J/kg. g(y,z) y = t/40°C, z = p/100MPa. With these data we can determine g(y,z) to be g(y,z) = g00 + g l 0 . y + g20. y2 + g30- y3 + (g01 + g l l -y +g21 • y2). z + (g02 + g 1 2 . y). z 2

(5.13)

with the dimensionless coefficients gik gN

yO

yl

z° z~ z2

0.0 109085.8 -11266.486

48837.64 690.0765 -99.20747

y2 -6139.045 60.65268. 0.0

y3 21.78264 0.0 0.0

At the freezing point t=0°C of pure water at atmospheric pressure we have equal chemical potentials (Eq. 5.2), G(0,0,0) = Gke(0,0), i.e. g[0,0,0] = g00, and enthalpies differing by melting heat Q,

(5.14)

272

R. ~EISTELand E. HAGEN

(5.15)

H(0,0,0) = HI~(0,0) + Q, or, with H = G + To, the entropy difference, o(0,0,0) = oke(0,0) + Q/T °,

(5.16)

i.e. g[0,1,0] = gl0 - Q/T ° • 40°C / 1J/kg. g[0,0,0] and g[0,1,0] are the corresponding free constants of seawater (Eq. 4.23), which we have set to zero. Hence, for any reference state other than the one we use, the coefficients g00, g l 0 of ice are the sums

gOO = g[O,O,O]

(5.17)

gl0 = g[0,1,0] + 48837.64 with the corresponding coefficients of liquid water (Eq. 4.17). After all these necessary preparation steps we can return now to the consideration of freezing point measurements. The minimization problem (Eq. 5.2), E {G - S ~G/~S

- Glee} 2 =

Min! at t = O(S,p),

(5.18)

where the sum is to be extended over all freezing point data, depends on the unknown coefficients g[i,0,0] and g[i,l,0], i>3. However, the influence of g[i,l,0] is relatively weak, so that these coefficients cannot be determined from freezing points alone. The additional data required is dilution heat, which will be discussed in the following chapter. A common fit of freezing points (Eq. 5.18) together with dilution enthalpies (Eqs. 6.3 and 6.6) was carried out, but we discuss the results here already in advance. Using the measurements of DOHERTY and KESTER (1974), referred to here as DK74, for the fit (5.18), we obtained the freezing points after Eq.(5.3) with r.m.s. = 1.6 inK. In table 5.1 the measured freezing points (DK74) in °C are compared with the result of this fit (O(S,0) - DK74) in m K and the corresponding difference between DK74 and MILLERO( 1978), abbreviated as M78. These data, together with the different formulae of DOHERTYand KESTER(1974), FUJ~O, LEWIS and PERKIN (1974) and MILLERO (1978), are shown in Fig.5.1.

The GIBBS thermodynamic potential of seawater

Freezing

Points

a t p : 0 MPa, Data D K 7 4 ( o ) ,

273

FLP74(+)

5-

4.÷

÷

+

3" o.

Z e.

i

o:

÷ 2o

I11

r0 [0

1o

ilu

~.

o: O"

w

I -1-

•

o

. . . . . . . . . . . . . . . . .0. . . . . . . . . . . . . . . . . . . . . . . . . .

] 0 .....................

~

o

O:

...... ""~N?8

0

i o -2-

o ~i i:

Z &

.

.

.

.

.

.

-3-

-4-

) -5, 0

5

10

15

20

Salinity

25

$

in

30

35

40

PSU

Fig. 5.1. Freezing point measurements (circles and crossa~) and formulas (curves) of FUJINO, LEWIS and PERKIN(1974), "FLP74", DOHERTYand KESTER(1974), "DK74", and MILLERO(1978), "M78", drawn relative to the result of this paper, "now". All circles (DK74) have been used for our regression run. DK74 and FLP74 curves are wrong for vanishing salinities because of their intercept at S=0. M78 deviates at small salinities from ours because of a slightly different limiting law (Eq.4.11).

........

2 74

R. FEISTELand E. HAGEN

Table 5.1. Differences between freezing point measurements of DOHERTY and KESTER (1974), "DK74", to this paper, "now", as well as to the formula of MILLERO(1978), "M78". S/PSU 3.78 3.83 6.97 8.38 11.95 11.97 12.33 16.44 16.81 16.99 19.81 20.04 22.01 24.53 24.59 27.01 28.15 30.00 30.22 31.65 32.14 33.00 34.42 34.50 34.90 35.06 35.49 36.18 36.52 36.96 39.75 40.20 r.m.s./mK =

DK74/°C

now/mK

M78/mK

-0.208 -0.213 -0.381 -0.457 -0.648 -0.649 -0.665 -0.886 -0.906 -0.920 - 1.072 - 1.086 - 1.195 -1.330 - 1.333 - 1.471 - 1.535 - 1.637 - 1.652 -1.731 -1.758 - 1.808 - 1.888 - 1.894 - 1.916 -1.925 - 1.950 -1.990 -2.010 -2.032 -2.197 -2.220

0.4 2.7 2.2 2.7 1.9 1.8 -1.6 -2.9 -3.0 1.2 -0.6 0.8 1.7 -2.4 -2.7 0.8 1.1 -0.7 1.9 0.2 -0.5 0.7 -0.3 1.2 0.3 0.1 0.5 0.9 1.3 -2.0 1.2 -2.1

0.2 2.5 1.3 1.6 0.9 0.8 -2.5 -3.3 -3.4 0.9 -0.6 0.9 1.9 -2.0 -2.4 1.2 1.4 -0.5 2.1 0.2 -0.6 0.5 -0.6 0.8 -0.1 -0.3 -0.0 0.3 0.6 -2.8 0.0 -3.3

1.6

1.6

The pressure dependence of freezing points follows automatically from Eq.(5.3). To test its correctness we have compared these predictions with data of FUJINO et al. (1974) as shown in Table 5.2.

The GIBBS thermodynamic po~ntial of seawater

275

Table 5.2. Differences between freezing point measurements under pressure of FUJINO,et al, (1974), "FLP74", to the formula of KESTER (1974), "K74", the formula of M m t ~ o (1978), "M78", and this paper, "now". Note that the latter figures follow from theory, based only on the fit to one-atmosphere freezing point lowering data and high-pressure ice properties after YEN et al. (1991). S PSU

p MPa

27.62 27.62 27.62 27.62 27.62 27.62 27.62 27.62 27.62 27.62

0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0

- 1.553 -1.606 - 1.659 - 1.712 - 1.765 - 1.818 - 1.872 -1.925 -1.978 -2.031

32.76 32.76 32.76 32.76 32.76 32.76 32.76 32.76 32.76 32.76

0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0

35.00 35.00 35.00 35.00 35.00 35.00 35.00 35.00 35.00 35.00 35.00 35.00 35.00 35.00

0.7 1.4 2oI 2.8 3.5 4.2 ,4.9 5.6 6.3 '7.0 '7.7 8.4 9.1 9.8

r.m.s./mK =

FLP74 °C

K74 mK

M78 mK

now mK

-4.3 -3.6 -2.8 -2.0 -1.3 -0.5 0.2 1.0 1.8 2.5

-3.8 -3.4 -2.9 -2.5 -2.1 - 1.6 - 1.2 -0.8 -0.3 0.1

-3.8 -3.1 -2.5 -2.0 - 1.6 -1.3 - 1.1 - 1.1 -1.1 -1.2

-1.844 - 1.897 - 1.950 -2.003 -2.057 -2.110 -2.163 -2.216 -2.269 -2.322

-2.7 -2.0 - 1.2 -0.5 0.3 1.1 1.8 2.6 3.4 4.1

-2.5 -2.1 - 1.6 - 1.2 -0.8 -0.3 0.1 0.5 1.0 1.4

-2.1 - 1.4 -0.9 -0.5 -0.2 -0.0 0.1 0.1 -0.1 -0.3

-1.973 -2.026 -2.079 -2.132 -2.185 -2.238 -2.292 -2.345 -2.398 -2.451 -2.504 -2.557 -2.610 -2.664

- 1.4 -0.6 0.1 0.9 1.7 2.4 3.2 4.0 4.7 5.5 6.2 7.0 7.8 8.5

- 1.7 - 1.3 -0.9 -0.4 -0.0 0.4 0.9 1.3 1.7 2.2 2.6 3.0 3.4 3.9

-1,0 -0.5 0.0 0,4 0.7 0.8 0.9 0.8 0.7 0.4 0.0 -0.5 - 1.0 - 1.7

5.0

2.7

1.9

276

R. F~STELand E. HAGEN

"FLP74" are temperatures computed with the polynomial of FUJINOet al., taken approximately where their measurements were made. They are claimed to be precise within 3 mK. "K74" are their differences to a polynomial proposed by KESTER(1974) and DOHERTYand KESTER(1974), "M78" to the polynomial of MILLERO(1978), and "now" to the values after Eq.(5.3). We see that the agreement is excellent especially at higher pressures. The limit of validity of Eq. (5.3) with respect to pressure is difficult to estimate, it is mainly a result of the ice compressibility formula (5.11), which is based on only a few data (YEN, 1981; YEN et al., 1991). Note that our formula (5.3), for a wider range of salinities and pressures up to 8 MPa, does confirms the assumption of FUJINO et al. (1974) and MILLERO(1978), that the pressure effect on freezing point lowering is almost (within 3 mK) independent of salinity. The recommendation of JPOTS, UNESCO q1978), based on the formula of MILLERO(1978), is expected to be valid within 3 mK up to 5 MPa. Therefore, we show the differences to our results for various salinities and up to 7.5 MPa in Fig.5.2. Because ice compressibility and water density used in this paper remain valid for higher pressures, we believe that our nonlinear freezing point formula (5.3, 5.4) may still hold beyond that limit. CALDWELL(1978) determined maximum densities at low temperatures and high pressures. In his paper he emphasized that supercooling of saline water was impossible, and that all these data have been obtained in the liquid phase. However, if we compute the corresponding freezing points, we find almost all maximum density temperatures below it, marked by "?", in table 5.3. The general relation of freezing points and maximum densities is depicted for various pressuresin Fig. 5.3. Both these temperatures depend almost linearly on salinity and pressure. Freezing Points Under Pressure, COMlmred v i t h IMESCI] 1978 5

3' E

/ - i -n i

i

C

!

!,,t 4

Z

!

-3.

...................................................

-4-

-5 o

•

i

~

~

i

:

i

i. . . . . . . . . . . . .

i .........................

i ...................................................

i

:

~

~ P:~*ssu~

~ r

in

~

"

~

•

io

NPa

Fig. 5.2. C~npatison of hi~-pressure freezing points afterIk41LLERO (1978) and eqs. (5.3, 5.4) of this paper. Both are derived by th~u-modynamic calculations and agree very well with experimental data (see Table 5.2).

The GIBBS thermodynamic potential of seawater

277

D e n s i t y n a x i M and F r e e z i n g P o i n t s from O t o Z5 tlPa

@

e.

1"

.......

i. . . . . . . . . . . . . . . . . . . . . . . . .

@

f4

£ J

O"

-1,

.....................................

-2 ~

-3.

........... ~

~

.

,

.

....................

~

i .............

.

~.~0

MPm

~

5 MPa

~

15 MPa

~

~0

~

i -4 0

5

10

15

~-0

2b

Salinity $ in

30

35

40

PSU

Fig. 5.3. Up to high salinities and pressures of 2000 m depth, maximum density temperatures, TMD, (steeper lines) and freezing points exhibit rather linear dependencies. The TMD lines below freezing temperature (for supercooled water) are suppressed.

MPa

278

R. FF~STELand E. HA~F2~

Table 5.3. Comparison of maximum density temperatures TMD after CALDWELL (1978) with freezing points O(S,p) after Eq. (5.3) and M78 after MILLEaO (1978). TMDs below the freezing 9oint are marked with a "?". S PSU

p MPa

TMD °C

O(S,p) °C

M78 °C

10.45 10.46 10.46 10.45 10.45 10.45 10.46 10.46 10.45

1.93 4.24 4.31 8.58 13.24 16.86 22.96 24.06 32.89

1.33 0.82 0.80 -0.17 -1.21 -2.03 -3.52 -3.80 -6.08

-0.71 -0.88 -0.89 - 1.21 - 1.57 -1.85 -2.33 -2.42 -3.13

-0.71 -0.89 -0.89 - 1.21 -1.56 -1.84 -2.30 -2.38 -3.04

20.10 20.10 20.10 20.10 20.10 20.10 20.10 20.10

.69 1.48 2.45 3.96 7.52 9.62 11.24 12.93

-0.70 -0.85 -1.10 - 1.43 -2.24 -2.71 -3.11 -3.51

-1.14 - 1.20 -1.27 -1.39 -1.65 -1.82 -1.94 -2.07

-1.14 -1.20 -1.27 -1.39 -1.65 -1.81 -1.93 -2.06

25.16 25.16 25.16 25.16 25.16 25.16 25.16

1.14 2.00 3.67 5.69 6.29 7.57 8.63

-1.82 -1.99 -2.40 -2.88 -2.99 -3.32 -3.54

-1.45 -1.52 -1.64 - I. 80 -1.84 -1.94 -2.02

-1.45 -1.52 -1.64 - 1.80 -1.84 -1.94 -2.02

9 9 9 9 9 9

29.84 29.84 29.84 29.84 29.84 29.84 29.84

1.17 2.38 3.86 4.69 5.62 6.24 6.90

-2.88 -3.21 -3.55 -3.74 -3.96 -4.13 -4.31

-1.72 -1.81 -1.92 -1.98 -2.05 -2.10 -2.15

-1.72 -1.81 -1.92 -1.98 -2.05 -2.10 -2.15

9 9 9 9 9 9 9

This obvious contradiction implies that either one of CALDWELL'S observations or highpressure free zing points must be in error. Only further experiments can resolve this paradox. There are no data available for densities of seawater at low temperatures, and a polynomial of high order may exhibit large errors when extrapolated beyond its fitted range (CALDWELL,1978). Osmotic pressure is a property closely related to freezing point lowering. For completeness, we briefly derive here the formula for its computation from G(S,t,p), following LANDAU and

The GIBBS thermodynamic potential of seawater

27 9

LIFSCHITZ(1966). If pure water under pressure p is in equilibrium with seawater under pressure (p+Tr), both separated by a membrane not permitting salt to pass through, 7r(S,t,p) is called the osmotic pressure of seawater with salinity S. Equilibrium requires equal chemical potentials of water on both sides together with equal temperatures: ~tw(0,t,p) = law(S,t,p+r0

(5.19)

Using the relation ~,v = G - S G/S, and expanding into powers of x, we obtain (all functions taken at t,p) G(0) = G(S) - S ~t(S) + x V(S) (1+8S)

(5.20)

where the haline contraction coefficient B accounts only for less than 3% of the effect. Solving equation (5.20) for g(S,t,p) yields the relation we look for. This formula for the osmotic pressure, (5.21)

Ir = -(G(S) - G(0) - S Gs)/(G v - S Gsp), reduces in the ideal solution case (S small) to V~a~'Y H O ~ ' s law = Ns

kT/V

(5.22)

A table of osmotic pressures, obtained by Eq. (5.21), is given in the appendix, table A.22. The values at one atmosphere are in good agreement with those computed by MILLEROand LEUNG (1976). The weak dependence of osmotic pressure on pressure and temperature and its almost linear increase with salinity are clearly visible in Figs 5.4a-b. Osuotlc

Pressure

II a t

t

=

"C

4"

3 .....................................................................................

1o

2O $~llnltM

3O

in

!J/~

"~iL ...........

4O

lqlU

Fig. 5.4. Osmotic pressures of seawater calculated after Eq. (5.21), a) (above) at tempemtme t---0°C, b) (overleaf) at salinity S=35 PSU, show only weak variations with depth. JPO 36:4-B

2 80

R. FEISTELand E. HAGEN

Osnotlc

Pressure

n at

$ = 35 P.,~,Li

41

100 -~ 0

MPa MPa

c

! 2 ..................................................................................................................................................

o E

J. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. ............................

O0

10

20 Telmlpe~atu~

30 in

40

eC

6. DILUTIONHEAT When seawater with different salinities is mixed, it may warm up or cool down. This heat production depends firstly on ion-ion interaction forces, secondly, with opposite sign and similar magnitude, on the polarization ofwaterin the electrostatic ion field, and f'mally (< 10% of the effec0 on isobaric haline contraction (FALKENI-IAGEN,1971, see also chapter 4). For ideal solutions, there is no dilution heat at all. In practical oceanography, the sign and amount of dilution heat depends strongly on salinity, temperature, pressure, and mixing ratio, however, for the majority of cases there is a weak cooling during the mixing process. It can hardly exceed 200 J per kg of seawater (or, roughly, 0.05°C), which can appear when, e.g., rain falls on salty hot (Red Sea) or cold (Arctic) surface water (see below for some estimates). These small direct effects are not the main reason for their thermodynamic treatment; much more important is their contribution to the general quantitative knowledge of seawater entropy and enthalpy for the study of slow water aging processes in the deep ocean. We consider an experiment, where at constant pressure and constant temperature a mass m 1 of seawater with salinity S 1 is mixed with a mass m e of seawater with salinity S 2 to form a mass m with salinity S. Conservation of mass and salt leads to m = m I + m2

(6.1)

m S = m 1 S ! + m 2S 2

(6.2)

and,

Let H(S) be the specific enthalpy of a sample with salinity S. Then, the released heat Q divided by the total amount of salt is dilution enthalpy L = Q/(m S):

The GIBBS thermodynamic potential of seawater

m I H(SI) + m 2 H(S2) = m H(S) + Q,

281

(6.3)

or, with the relative mass fractions w 1 = ml/m, w 2 = metm, L = (w 1 H(S1) + w 2 H(S2) - H(S))/S

(6.4)

If one of the initial samples is pure water, $2=0, we have m S = m I S v or, S = w I S l

(6.5)

for the amount of salt in the process, and L can be expressed as L = (H(S1)-H(0))/S 1 - (H(S)-H(0))/S

(6.6)

= ~L(S1) - ~L(S)

(6.7)

Here, the relative apparent specific enthalpy~L(S) (LEWIS and RANDALL, 1961; BROMLEY, 1968; MILLERO and LEUNG, 1976) is defined as • L(S) = (H(S)-H(0)-S 0H(0)/OS)/S

(6.8)

Note that q~L and L are invariant with respect to linear ,,gauge transformations" of the form (Eq. 4.20), H'(S) = H(S) + A + BS, i.e., they do not depend on the reference state used for H. We further hint on the fact that L after Eq.(6.6) is a difference of differences of H(S), and, if S 1 is close to S, it expresses the nonlinearity (curvature) of H and behaves similar to a numerical second derivative, such that small noise in H may become amplified to serious errors in L. Vice versa, H itself is not very sensitive to experimental or other noise in dilution heats. Measurements of dilution heats for seawater have been reported by BROMLEY (1968), B ROMLEY et al. (1970), CONNORS (1970) and Mn.LERO, HANSm,~and HOFF (1973). BROMLEY et al. extrapolated the measurements of BROMLEY (1968) tO other temperatures than 25°C using extended measurements of heat capacities, so that for our purpose - we use IOT-4 heat capacities, chapter 2 - there is no relevant new information compared to his earlier paper. CONNORSreported mixing heats, but one of his samples was always clearly above 40 PSU, where our model is no longer valid. His computed enthalpies (J/g, c in g/kg), H(c,t) = .001 (4204.4-0.57 t-c (6990-34.3 t)) t - c 2 (464-19.6 t+0.3 t2)

(6.9)

(S= 10-40, t = 0-30°C, claimed error 5%), however, are not sufficiently precise to retrieve dilution heats for our application (Eq. 6.6). Relative apparent enthalpies (Eq.6.8) cannot be derived from his data because the limit S=0 is outside range; we will use them only for rough additional comparison. The extended measurements of MILLERO, HANSEN and HOFF (103 points from 0 to 30°C) have almost completely been used for our fit (except for 8 samples with rather high deviations), after

282

R. FFas'n~and E. HAGEN

converting them from cal/eq to J/kgPSU multiplying by 4.184 / 57.754 as proposed in the paper (remember that I kgPSU is 1.00488 gram of salt, Eq. 4.4). All experimental temperatures were considered as IPTS-68 values. From BROMLEY's 1968 data we used 24, which fell into the range 0-40 PSU. Salinity in % was converted to PSU dividing by. 100488. For comparison, we have additionally fitted the computed ~L data at 25°C of his paper to the polynomial (r.m.s.= .01 J/kgPSU, ~L in J/kgPSU, S in PSU):

~L = 5.364 ~/S + S (-.95803 + .072311 ~/S + S (-.0052206 + 2.3606E-04 ~/S))

(6.10)

Dilution heat (Eq. 6.2) depends on the limiting law coefficients g300 and g310 as well as on 6 free parameters gi00 and gi 10, i=4,5,6, however, because of H = G - T G t, they appear as pairs and only 3 of them are independent unknowns. To determine them all, freezing points have to be considered simultaneously. The resulting minimization problem of type (3.1) consists of three different data types, which have to be weighted by their expected accuracy (~k: (i) Dilution heats after BROMLEY(1968): Eq. (6.3) was adjusted at 24 measured points. BROMLEYestimated a precision of 1 cal in measured heats, hence a limit for Q of Gk=~---4J was assumed, with a resulting r.m.s, of 2.4 J of the fit. (ii) Dilution heats after MELERO, HANSEN and HOFF (1973): Eq. (6.6) was adjusted at 95 measured points. As discussed in their paper, we estimated a precision of 5 cal/eq in dilution heat, corresponding to a limit for L of o k = 8L -- 0.4 J/kgPSU, with a resulting r.m.s, of 0.36 J/kgPSU of the fit. (iii) Freezing points after DOHERTY and KESTER(1974): Eq. (5.2) was adjusted at 32 measured points. Assuming a precision of 8 0 ~- 2mK of freezing temperatures, a weight for G of (see Eq. 5.16) t~k = 8G = Q/T °. 8 0 = 2 J/kg was applied, with a resulting r.m.s, of 2.0 J/kg of the fit ( 1.6 mK in temperatures). Six coefficients, g400-g600, g410-g610, have been determined as reported in table A. 1 of the appendix. The relative apparent enthalpy obtained by this fit is shown in Figs 6.1 a-b, and values are given in table A.21 in the appendix.

The GIBBS thermodynamic potential of seawater

A p p a r e n t S p e c i f i c gatlmlla3 I~ a t p = O b a r

llelatlee 20

283

.-r

~ 3 5 o C

~ £0'

3

0

o

C

~'i;.c

m

20eC

.%

15eC 0

¸

10oC

1

]

3e¢

-10

OoC

-20

I

.

.

.

.

,

0

. . . .

,

.

.

.

.

,

.

.

.

.

,

.

.

.

.

,

. . . .

,

5

.t

Salinl

tM

ROot

Relative l~plmrent £ n t h a l p g el- a t

= 35 PSU

20-

i 0

NPa

e

;

:,o

/

I

--20

.

0

.

.

.

,

£0

-

•

;

-

,

.

.

.

.

,

zO TOI~POi~atUI~O

30 iN

.

.

.

.

,

40

eC

Fig. 6.1. Relative Apparent Enthalpy after Eq.(6.8) in J/kgPSU, a) at aunospheric lXeSsureover the square root of salinity, b) at salinity S=35 PSU over temperature up to high pressures.

284

R. F~STELand E. HAGEN

Table 6.1. Fit of dilution heats Q after BROMLEY(1968) at 25°C, "B68", compared with this paper, "-Q now", after Eq. (6.3). Heat values in J/kgo

S1/PSU

ml/kg

33.238 33.238 33.238 33.238 33.238 33.238 33.238 33.238 33.238 33.238 33.238 33.238 33.238 33.238 33.238 33.238 33.238 33.238 33.238 33.238

0.2715 0.2698 0.2938 0.2804 0.2710 0.2920 0.2734 0.2943 0.2455 0.2512 0.2560 0.2610 0.2756 0.2850 0.2790 0.2706 0.2772 0.2836 0.2832 0.2640

2.985 10.250 23.784 0.000

0.2530 0.3010 0.2400 0.2776

S2/PSU

m2/kg

B68

-Q now

Diff

0.000 2.110 4.090 6.080 7.862 11.066 13.783 15.037 0.000 1.921 0.000 1.990 3.851 5.682 7.483 9.135 10.618 12.051 13.584 14.897

3.9892 3.9807 4.0022 4.0071 3.9677 4.0170 4.0112 4.0335 4.0027 4.0055 3.9890 4.0163 4.0163 4.0279 4.0226 4.0321 4.0001 4.0145 4.0275 4.0181

-22.57 1.63 7.79 14.57 10.30 11.93 8.08 8.12 -25.96 3.64 -24.58 3.81 10.38 13.69 13.65 17.79 13.48 13.10 9.13 10.68

-24.66 2.75 12.00 14.65 15.00 15.22 12.36 12.14 -24.16 0.99 -24.35 1.69 10.46 14.48 15.37 14.91 14.71 14.15 12.96 11.06

-2.09 1.11 4.21 0.08 4.70 3.28 4.28 4.02 1.80 -2,65 0.23 -2,12 0.08 0.78 1.72 -2.88 1.23 1.05 3.83 0.38

0.000 0.000 0.000 33.238

4.0987 4.0095 4.0244 4.1126

-6.07 -18.34 -27.76 31.69

-3.95 -17.66 -26.49 32.29

2.12 0.68 1.27 0.60

The GIBBS thermodynamic potential of seawater

285

Table 6.2. Fit o f dilution heats L after MILLERO et al., (1973) and BROMLEY (1968). t: temperature in °C; S 1, $2: initial and f'mal salinity in PSU; M H H 7 3 : Measurements o f MILLERO, HANSEN and HOFF (1973); L now: this paper, in J/g, Eq.(6.6); C70: Enthalpy (6.9, 6.6) after CONNORS (1970); B68: Relative enthalpy (6.10, 6.7) after BROMLEY (1968). t

S1

S2

MHH73

L now

C70

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

39.267 39.267 39.267 39.267 39.267 39.267 39.267 39.267 39.267 39.267 35.000

33.190 29.364 25.094 19.031 14.931 14.922 10.496 7.734 6.249 3.158 3.390

2.926 4.865 6.091 9.176 10.459 9.486 12.023 12.902 12.895 14.113 11.619

2.533 4.115 5.875 8.374 10.060 10.064 11.858 12.929 13.467 14.373 12.541

2.833 4.617 6.608 9.435 11.347 11.351 13.415 14.703 15.395 16.836 14.739

9.998 9.998 9.998 9.998 9.998 9.998 9.998 9.998 9.998 9.998 9.998 9.998

41.466 41.466 41.466 41.466 41.466 41.466 41.466 41.466 41.466 9.370 9.370 9.370

30.125 26.083 19.031 13.175 12.430 10.401 10.064 8.714 6.876 4.152 3.253 1.935

3.265 4.498 6.437 7.806 8.253 8.275 8.758 8.621 8.195 0.101 -0.454 -0.887

3.222 4.285 6.012 7.256 7.394 7.733 7.783 7.962 8.128 0.210 0.061 -0.416

3.396 4.607 6.719 8.473 8.696 9.304 9.404 9.809 10.359 1.563 1.832 2.227

14.996 14.996 14.996 14.996 14.996 14.996 14.996 14.996 14.996 14.996 14.996 14.996 14.996 14.996 14.996 14.996 14.996 14.996 14.996 14.996 14.996 14.996 14.996

35.000 35.000 34.762 34.762 29.769 29.478 29.478 25.319 25.319 23.627 23.627 19.684 19.684 19.246 16.481 15.301 15.301 10.349 10.216 10.216 5.110 4.479 4.479

16.966 11.350 11.154 6.542 10.556 8.981 5.894 7.432 6.316 7.223 4.597 5.663 3.776 7.783 3.068 4.852 2.838 4.532 3.121 2.009 1.600 0.738 0.508

3.604 3.972 3.871 4.137 2.934 2.645 2.573 1.924 2.141 1.694 0.872 0.901 0.548 0.692 -0.469 0.252 -0.872 -0.555 -1.060 -1.225 -1.651 -2.220 -2.804

3.389 3.990 3.949 3.954 2.908 2.899 2.752 2.059 1.981 1.737 1.398 0.924 0.503 1.023 -0.238 0.165 -0.498 -0.346 -0.813 -1.454 -1.569 -2.575 -3.070

4.305 5.645 5.635 6.736 4.586 4.893 5.629 4.270 4.536 3.916 4.542 3.347 3.797 2.736 3.202 2.494 2.975 1.389 1.694 1.959 0.838 0.893 0.948

B68

•.. continued

28 6

R. Pssar_~ and E. HAGEN

t

Sl

S2

MHH73

L now

C70

19.995 19.995 19.995 19.995 19.995 19.995 19.995 19.995 19.995 19.995 19.995 19.995 19.995 19.995 19.995 19.995

41.466 41.466 41.466 41.466 41.466 41.466 41.466 41.466 41.466 41.466 9.370 9.370 9.370 9.370 9.370 9.370

34.292 25.616 21.763 18.524 13.981 11.253 11.083 8.260 8.085 6.967 4.526 2.417 2.267 1.860 0.953 0.694

1.088 2.429 2.768 3.063 3.373 3.402 3.323 3.244 3.186 3.135 -1.254 -2.040 -2.487 -2.811 -4.289 -4.959

1.331 2.640 3.093 3.394 3.642 3.648 3.643 3.446 3.425 3.256 -0.950 -2.123 -2.248 -2.633 -3.877 -4.413

1.384 3.059 3.802 4.427 5.304 5.830 5.863 6.408 6.442 6.657 0.935 1.342 1.371 1.449 1.624 1.674

24.994 24.994 24.994 24.994 24.994 24.994 24.994 24.994 24.994

41A66 41.466 41.466 41.466 41.466 41.466 9.370 9.370 9.370

34.742 31.294 25.419 10.724 10.202 4.714 1.987 0.876 0.757

0.714 1.074 1.420 1.514 1.492 0.303 -3.049 -4.858 -4.606

0.951 1.353 1.887 1.869 1.801 0.298 -3.423 -5.203 -5.486

1.091 1.651 2.605 4.990 5.075 5.965 1.198 1.379 1.398

29.993 29.993 29.993 29.993 29.993 29.993 29.993 29.993 29.993 29.993 29.993 29.993 29.993 29.993 29.993 29.993 29.993 29.993 29.993 29.993 29.993 29.993 29.993 29.993

39.267 39.267 39.267 39.267 39.267 39.267 35.000 35.000 34.762 34.762 34.762 34.762 29.769 29.769 29.478 29.478 25.320 25.320 23.627 23.627 19.684 19.684 19.246 19.246

34.079 29.873 24.786 22.028 18.072 14.353 11.478 8.318 11.536 11.039 8.057 6.976 9.504 6.691 9.719 6.108 8.515 4.731 8.133 4.808 6.421 4.097 5.989 3.612

0.360 0.512 0.714 0.742 0.591 0.303 -0.368 -0.959 -0.339 -0.411 -1.175 -1.355 -1.038 -1.802 -1.067 -1.780 -1.297 -2.854 -1.521 -2.883 -1.845 -2.898 -2.033 -3.186

0.476 0.751 0.930 0.944 0.832 0.538 -0.274 -1.015 -0.283 -0.375 -1.115 -1.491 -1.045 -1.937 -1.008 -2.201 -1.474 -3.068 -1.613 -3.046 -2.191 -3.459 -2.367 -3.806

0.761 1.378 2.125 2.529 3.110 3.655 3.451 3.915 3.408 3.481 3.918 4.077 2.973 3.386 2.899 3.429 2.466 3.021 2.273 2.761 1.946 2.287 1.945 2.294

B68

0.758 1.105 1.594 1.559 1.490 0.005 -3.281 -4.885 -5.135

The GIBBS thermodynamicpotential of seawater

287

Let us finally briefly stress the question again, what maximum heating effects can be expected for ocean water mixing? More mathematical details about the irreversible heat production of mixing can be found in FOFONOFF (1992) and MAMAYEV (1975). Equation (6.3) can be written as

Q = - H()

(6.11)

where the brackets <> stand for an average, weighted by mass fractions. JENSEN's inequality results in :_>f() if f(x) is a convex function, Q is always positive (warming) where H(S) is a convex function, and always negative (cooling), wherever H(S) is concave. Especially if the mixing waters have only slightly different salinities, we may write S = + 6S, <6S> = 0

(6.12)

and expand (6.11) into a power series of 6S Q = - H() = ½ Hss <6S2> +...

(6.13)

Obviously, heat is released if the second derivative of H is positive, i.e. H is a convex function of S, as it is shown in Figs 6.2 a-b. However, because <8S2> is small, only small temperature changes 5t = Q/CP can be expected here. If waters with strong differences in salinity mix (rain, river discharge, or salt water inflow into fresh or brackish water seas, or mixing through pronounced haloclines as in cases of double diffusion), stronger effects may happen. A study of equation (6.11) shows maximum warming of about 30 J/kg (8 mK) at t=40°C, if75% fresh water mixes with 25% ocean water of salinity 35 PSU, and a maximum cooling below- 160 J/kg (-40 mK) at t below 0°C, when waters with 0 and 40 PSU mix fifty-fifty (compare Fig. 6. l-a). Under pressure, these effects become generally smaller (Fig. 6-1b).

JPO 36:4-C

288

R. FEISTELand E. HAGEN

Entl~Ipy

£-

C o n v e x i t y d Z H ( E , t , p ) / d S z a t p = O MPa

.5.

-.5

-2 0

I

2

3 Salinit9

Enthall~

Convexity

4 Root

dZH(S,t,p)/dS

5 45

z

at

p

= 40

MPa

.5

\ g N 6~

40oC 35oC 30oC

|~:~

-.5

0

£

Z

3

$alinit9

4 Root

5

6

45

Fig. 6.2. Second salinity derivative of cnthalpy, which determines the sign of mixing heat (T~]. 6.13), a) at aunosph~ic pressure, b) at 4000 m depth. Especially at low temperatures there is a visible pressure effect.

The GIBBS thermodynamicpotential of seawater

2 89

7. ENTHALPY AS POTENTIALFUNCTION For many oceanographic applications, water motion can be considered as conserving salt (salinity S) and heat (entropy ~), and it is reasonable to use these two quantities as independent variables. Then, enthalpy H(S,~,p) = G + Tt~ is the responsible potential function. If it is known, we compute first ~(S,t,p) with in-situ values S(r), t(r), p(r) at a given location r, and can then obtain: BERNOULLI'Sfunction (v advection speed, • geopotential),

-

B(r) = H(S,t~,p) + v2/2 + ~(r),

(7.1)

potential temperature0(S,t~(S,t,p),p) at reference pressure Pr (index of H means partial derivative keeping both others fixed), -

T°+0 = H o

at P=Pr'

(7.2)

(which becomes temperature t(S,t~,p) = 0(S,o(S,t,p),p) if pr=p, or in other words, t and t~ are inverse functions to each other.) potential density I/V at reference pressure Pr,

-

V = Hp

at P=Pr'

(7,3)

adiabatic compressibility K' and sound speed U,

-

K' = -Hpp/V = V / U 2, -

adiabatic lapse rate F and the adiabatic thermal expansion coefficient ct', F = Hpo = Vot',

-

(7.5)

heat capacity CP, CP = qTHoo,

-

(7.4)

(7.6)

chemical potential, t.t = H s,

(7.7)

- adiabatic haline contraction coefficient, /5' =-Hps/Vo

(7.8)

If density is written asp(S,o,p), its total differential can be expressed in terms of these adiabatic coefficients (Eqs. 7.3, 7.4, 7.5 and 7.8),

290

R. ~EISTELand E. HAGEN

dp = (~p/OS)o, p dS + (~p/~O)s. p d o + (3p/3p)s. o dp = p (B'dS - ~ ' d o + K'dp).

(7.9)

Values for 8', a ' , K' are listed in tables A.15, A.17 and A.19 of the appendix. To determine H(S,o,p) numerically we have expressed it in almost polynomial form with dimensionless variables x,u,z by S = 40 P S U . x 2, a = 625 J/kgK o u, p = 100 M P a . z as H = 1J/kg • h(x,u,z),

(7.10)

where h(x,u,z) is the dimensionless polynomial-like sum h(x,u,z) = ZjX {hojk +

hlj k x21n(x) + Y'i>l hiik xi}

uJ Zk

The coefficients h,,~ have been determined in 5 successive fits of derivatives of H(S,a,p) to the equivalent thermodyni,~'unic quantifies expressed in G(S,t,p). With t = 40°C y , o = -G t, we have used (i)

(iii)(ii) (iv) (v)

H = G - Gto2/Gtt HOp po = G p - Gtp2/Gtt nppPP = G~p - Gtp2/Gtt H = Gv H p =GP-(T°+t) G t

at at at at at

x=0-1, x=0, x=0-1, x=0-1, x=0-1,

y=0-1, y=0-1, y=0-1, y=0-1, y=0-1,

z=0, z=0-1, z=0-1, z=0-1, z=0-1

(7.11)

The resulting root mean square differences for some quantities, derived from G on one hand and from H on the other hand, taken over the full ranges of salinity, temperature and pressure, are: sound speed : r.m.s. spec. volume : r.m.s. temperature : r.m.s. spec. heat : r.m.s. enthalpy : r.m.s.

= = = = =

1.6E-02 1.8E-10 1.7E-04 2.7E-01 3.7E-03

m/s mS/kg °C J/kgK J/kg

(7.12)

This precision should suffice for most practical purposes, although one can make H more accurate if more free coefficients are involved. The resulting coefficients are given in the following table A.2 of the appendix. Details of the differences (7.12) derived from both potential functions G and H are magnified in Figs 7.1-7.4.

The GIBBS thermodynamic potential of seawater Error of EntbalptJ H ( S , e , p )

-

(G+Ts) at

S

291

: 35 PSU

ii1................................................................................................................. ............................ ............ ' ........................................ iiill ~:

100

--.

02

-.

03

HPa

.......................................................................................................................................

•

0

10

ZO

40

30

Teml=o~atul~e

in

oC

Fig. 7.1. Enthalpy Has thermodynam~potentialfuncfioncomparedwithenthalpycomputedfrom GIBBS' l ~ n f i ~ G at s~ini~ S=35 PSU. The v~ue of H is tyl~cally between 0 and 250 kJ/kg. Error o f P o t . Temp. f r o m H ( S , r , p ) a t S = 35

O MPa

.4 ......................... .3"

. 2 - i ..........................

.~

I

i

~

£

~........................

i ..................

i

i

i

i

/iao JJ

i

i

i

"'//i 40 HPa

i .............................

i ............................

i ........................... ...//~6o...P

i

i

i

0

0

.................../i

...........................

[//~so

.e~

................ m P. MPu

.1"

~.

o

-w - . 1

¸

S, Q

-.

............................

0

i

i

i

i

i .............................

i ............................

i .............................

i ............................

10

20 Te~pe~atuPe

in

30 e¢

40

Fig. 7.2. Po~nfi~ ~mperature at salinity S=35 PSU compu~d from enth~py H as thermodynamic po~nfiN afar F-xl.(7.2)compared with po~nfi~ ~mperamre from GIBBS' potenfiN G.

292

R. FEISTELand E. HAGEN

Error

of

l)ensit

U from

H(S,f,p)

at

S

= 35

PSU

x

i ! tO0

i

0 ~

'

i

i

"

i

NP~

oo .r.

"

"

"'"'"'"'"

i 4 0 " 'f'f~P~ . . . . . . . . . . . . . . . • 20

0

I,,L1Pa

MIPa

-.5.

0

~0

30

20

in

TeMz~e~atuz, e

40

°C

Fig. 7.3. Density at salinity S=35 PSU computed from enthalpy H as thermodynamic potential after Eq.(7.3) compared with density from GIBBS' potential G. Error of ~mnd Speed

.05

..........................

from H(3,w,p)

: ....................

:

..................

Y .............

S................

at S =

35 PSU

!. . . . . . . . . .

! ....................

C

~

-.

05

-

...........

i .........

80 100

MPa

MPa

-.1 0

10

20 Te~pe~atur~

30 in

40

°C

Fig. 7.4. Sound speed at salinity S=35 PSU computed from enthalpy H as thermodynamic potential after F_N.(7.4) coml~.Xl with sound speed from GIBBS' potential G.

The GIBBS thermodynamicpotentialof seawater

293

8. DISCUSSION Two objectives have led us to the present recomputation of the thermodynamic potential of seawater, namely, to make mechanical P-V-T properties compatible with CALDWELL's maximum density temperatures and with DEL GROSSO's sound speed formula, and, to have colligative properties computed from as many data as possible on mixing heat and freezing point measurements, using well defined stoichiometry of seasalt and ice properties, thus to improve the reliability of entropies, enthalpies and chemical potentials. The results of these refinements of the G93 formula are altogether within the anticipated absolute error limits: density for example has changed by less than 10 g/m 3for "Neptunian waters", and freezing points still agree quite well (about 2 mK) with the formulas recommended by UNESCO, even under pressure. In our opinion, further progress can only be made through new measurements, particularly for the following characteristics: (i) Sound speed: Since DEL GROSSO'S polynomial applies only to natural combinations of S,t,p, therefore we had to use IOT-4 sound speed (with smaller weight) to fill the remaining gaps in S-t-p space. Both formulas, however, deviate systematically from each other, so it would be a much better idea to determine a systematic correction to IOT-4 sound speed (based on the physical or technical reason for the deviation), such that both coincide for Neptunian waters. Then, the regression algorithm for G(S,t,p) would no longer need to find a compromise between them, and a better reproduction of sound speed figures could result even for Neptunian waters. (ii) Density: There is a special oceanographic interest in having very precise knowledge of densities at low temperatures, which determine the intensity of winter convection and the rate of deep water formation near the poles. IOT-4 densities are defined down to -2°C, but not based on any measurements < 0°C (UNESCO, 1981). The extrapolation of a polynomial beyond this limit may cause rapidly increasing errors (compare Figs 3. la,c) of the order of several g/m 3 (which is, however, still about the standard deviation of the data used). At, say, 4000 m depth the freezing point is clearly below -2°C; is the computation of high-pressure density from one-atmosphere density pressure still justified, if at that temperature liquid water no longer exists at the surface? Here we have tried to improve the situation by involving maximum density temperatures from CALDWELL'Sexperiments. But even if the thermal expansion coefficient vanishes at the correct temperatures, this does not imply that the densities themselves are correct as well. Therefore, density measurements below 0°C for salinities 0 - 40 PSU are needed for an improved description of the low-temperature range. They might also resolve "CALDWELL'Sparadox" (compare table 5.3), where he observed, liquid saline water apparently not supercooled below the usual freezing point temperature. There is a similar extrapolation problem for salinities >42 PSU, such as are found in Red Sea waters (UNESCO, 1991). (iii) Heat capacity: Specific heat at constant pressure, CP (as a function of salinity, as given by IOT-4) originally was fitted to a polynomial with only three powers of salinity, S°, S 1 and S m (UNESCO, 1981). Correspondingly, in G(S,t,p) only three coefficients g020, g220 and g320 appear. Looking at the

294

R. F~aS~Land E. HACEN

table of coefficients in the introduction, the terms around g420 (a term S2in CP) seem to be missing, especially when the determination of dilution heats is considered. In other words, new precision measurements of heat capacity at atmospheric pressure may reveal corrections to its present salinity dependence, thus leading, in turn, to modified dilution heat treatments. (iv) Dilution heat: Knowledge of heat of mixing (or dilution) is needed to calculate entropy, enthalpy, and chemical potential of seawater. Data scatter by about 10% of their values (compare table 6.2), and there are only few measurements under conditions of the most pronounced heating or cooling effects (see Fig. 6. la) - transitions from high to low salinities at either very high or very low temperatures° (v) Ice properties: As the review of YEN etal. (1991) demonstrates that our knowledge of the properties of ice, even for pure water as required here, is based on only few experiments, many conducted in the first half of this century. Some reported quantities vary considerably from author to author. The high precision of freezing point temperatures can only be transferred to coefficients of the thermodynamic potential if ice properties, especially melting heat, are known to at least the same accuracy. The GIBBS potential of ice derived here (Eq. 5.13) may serve as a first attempt for a consistent description of ice properties close to the melting point. (vi) Chemical composition: A standard stoichiometry of seasalt is desired as a reference. The chemical composition enters directly into the theoretical limiting laws expressed by the coefficients gO, g 1, g300 and g310 (eqs. 4.13, 4.16), and strongly influences the treatment of freezing point and mixing heat data this way. All the thermodynamic properties defined here are only for standard seawater, which has a certain relation between practical salinity, absolute salinity, and stoichiometry. Waters, like for example those from the Baltic, where river discharge of fertilizers and other polluting substances modifies the natural salt mixture, possess, by definition, a practical salinity as measured by conductivity. But it is unsolved still, exactly how all the thermodynamic functions depend on such a salinity, in which there are variations in the seasalt composition. Another topic that may need to be addressed is the definition of the thermodynamic reference state. The GIBBS potential of seawater depends on four arbitrary constants (Eq.4.18), which may be interpreted as the absolute entropy and absolute energy (or enthalpy) both of water and of salt. Formalisms working only with relative apparent thermodynamic quantities avoid such a lack of uniqueness completely (LEWISand RANDALL,1961; MILLERO, 1982). We believe, however, that the compactness and mathematical clarity of using a thermodynamic potential function G(S,t,p) has more benefits than inconvenience because reference states can be chosen at will. The shape of surfaces of constant entropy or enthalpy in the ocean is, in general, not independent of these free constants (eqs. 4.18 - 4.21). The state we propose here (vanishing entropy and enthalpy at t=0°C, p=0 MPa, S=0 and 35 PSU, see section 4) differs from the one used before (FEISTEL,1993). It was the outcome of a longer discussion about its various "pros and cons"; the future will decide whether this is a good choice or not.

The GIBBS thermodynamic potential of seawater

29 5

9. ACKNOWLEDGEMENT For substantial hints, discussions, or other kind of helpful support the authors like to thank J.Ahlheit, T.McDougall, H.Eicken, N.P.Fofonoff, E.Nordmeyer, S.Weinreben, Y.-C.Yen., J.P~Zarling, and last not least, the IOW library staff. We thank as well the editor's referees for their careful criticism, thus improving various details of the manuscript. In this paper, a number of coefficients of several polynomials have been derived. It is no problem to handle them on a PC, but boring and insecure to type them in again. An ASCII file copy of appendix coefficient tables can be obtained by interested readers on request from the author via intemet: rfeistel @physik.io-wamemuende.d400.de.

10. REFERENCES ABRAMOWlTZ,M. and I.A. STEGUN(1968), editors, Handbook of Mathematical Functions. Dover Publications, New York. ALEKIN, O.A. and YU.I. LYAKHIN(1984) Ocean Chemistry (in Russian). Leningrad, Gidrometeoizdat. BARLOW, A.J and E. YAZGAN (1967) Pressure dependence of the velocity of sound in water as a function of temperature. British Journal of Applied Physics, 18, 645-651. BETTIN, H. and F. SPIEWECK(1990) Die Dichte des Wassers als Funktion der Temperatur nach Einfilhrung der Intemationalen Temperaturskala yon 1990. PTB-Mitteilungen 100, 195-196. BIGG, P.H. (1967) Density of water in SI units over the range 0-40°C, British Journal of Applied Physics. 18, 521-525. BIGNELL,N. (1983) The effect of dissolved air on the density of water. Metrologia, 19, 57-59. BLANKE, W. (1989) Eine neue Temperaturskala. Die Internationale Temperaturskala von 1990 (ITS-90). PTBMitteilungen, 99, 411-418. BOLZ, E.R. and G.L. TUVE (1985), editors, CRC Handbook of Tables for Applied Engineering Science. CRC Press,

Boca Raton. BREWER,P.G. and A. BRADSHAW(1975) The effect of the non-ideal composition of sea water on salinity and density, Journal of Marine Research. 33, 157-175. BROMLEY,L.A. (1968) Relative enthalpies of sea salt solutions at 25°C, Journal of Chemical and Engineering Data. 13, 399-402. BROMLEY,L.A., A.E. DIAMOND,E. SALAMIand D.G. WILKINS(1970) Heat capacities and enthalpies of sea salt solutions to 200°C, Journal of Chemical and Engineering Data. 15, 246-253. CALDWELL,D.R. (1978) The maximum density points of pure and saline water. Deep-Sea Research. 25, 175-181. CHEN,C.-T. and F.J. MILLERO(1976) The specific volume of seawater at high pressures. Deep-Sea Research, 23, 595 -612. CHEN, C.-T. and F.J. MILLERO(1977) Speed of sound in seawater at high pressures. The Journal of the Acoustical Society of America, 62, 1129-1135o CHEN, C.-T. and F.J. MILLERO(1978) The equation of state of seawater determined from sound speeds. Journal of Marine Research, 36, 657-691. CONNORS, D.N. (1970) On the enthalpy of seawater. Limnology and Oceanography, 15, 587-594. DEL GROSSO, V.A. (1974) New equation for the speed of sound in natural waters (with comparisons to other equations). The Journal of the Acoustical Society of America, 56, 1084-1091. DOHERTY, B.T. and D.R. KESTER(1974) Freezing point of seawater. Journal of Marine Research, 32, 285-300. DORON1N, YU.P. and D.E. KHEISlN (1975) Sea Ice (in Russian). Gidrometeoizdat, Leningrad. DORONIN, YU.P. (1978), editor, Ocean Physics (in Russian). Gidrometeoizdat, Leningrad. DORSEY, N.E. (1940) Properties of Ordinary Water-Substance. Reinhold, New York. DUSftAW, B.D., P.F. WORCESTERand B.D. CORNUELLE(1993) Equations for the speed of sound in seawater. The Journal of the Acoustical Society of America, 93, 255-275. FALKENHAGEN,H. (1971) Theorie der Elektrolyte. S.Hirzel, Leipzig. FEISTEL, R. (1993) Equilibrium thermodynamics of seawater revisited. Progress in Oceanography, 31, 101-179. FEISTEL,R. and E. HAGEN(1994) Thermodynamic quantities in oceanography. In: The Oceans: Physical-Chemical Dynamics and Human Impact. S.K. MAJUMDARet aL, editors, The Pennsylvania Academy of Science, Easton, 1-16.

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R. FEISTELand E. HAGEN

FOFONOFF,N.P. (1962) Physical properties of sea-water. In: The Sea. M.N.HtLL,editor,Wiley,New York, 3-30. FOFONOFF,N.P. (1985) Physical properties of seawater: A new salinity scale and equation of state for seawater. Journal of Geophysical Research 90, 3332-3342.

FOFONOFF,N.P. (1992) Lecture Notes EPP-226. Harvard Umversity. FOFONOFF,N.P. and R.C. MILLARD(1983) Algorithms for computation of fundamental properties of seawater. UNESCO Technical Papers in Marine Science, 44, UNESCO. FUJINO,R., E.L. LEWISand R.G. PERKIN(1974) The freezing point of seawater at pressures up to 100 bars. Journal of Geophysical Research, 79, 1792-1797. HAAR, L., J.S. GALLAGHERand G.S. KELL (1984) NBS/NRC Steam Tables. Hemisphere Publishing Corp., Washington New York London. HAAR,L., J.S. GALEAGHERand G.S. KEEL(1988) NBS/NRC Wasserdampftafeln. Springer-Verlag, Berlin. HOBBS, P.V. (1974) Ice Physics. Clarendon Press, Oxford. ]-[OLDEN, N.E. (1980) IUPAC Commission on atomic weights and isotopic abundances, atomic weights of the elements 1979. Pure & Applied Chemistry, 52, 2349-2384. KEEL,G.S. (1975) Density, thermal expansivity, and compressibility of liquid water from 0° to 150°C: Correlations and tables for atmospheric pressure and saturation reviewed and expressed on 1968 temperature scale. Journal of Chemical and Engineering Data, 20, 97-105. KELE, G.S. and E. WHALLEY(1975) Reanalysis of the density of liquid water in the range 0-150°C and 0-1 kbar. The Journal of Chemical Physics, 62, 3496-3503. KENNISrt,M.J. (1994) Practical Handbook of Marine Science. CRC Press, Boca Raton. KESTER,D.R. (1974) Comparison of recent seawater freezing point data. Journal of Geophysical Research, 79, 4555-4556. LANDAU,L.D. and I.M. LIFSrlITZ(I 966) Statistische Physik. Lelarbuchder Theoretischen Physik Bd.V. AkademieVerlag, Berlin. LECHNER,M.D. (1992), editor, D'Ans-Lax Taschenbuchfl2r Chemiker und Physiker. Springer, Berlin, Heidelberg. LEWIS, E.L. and R.G. PERrdN (1981) The practical salinity scale 1978: Conversion of existing data. Deep-Sea Research, 28A, 307-328. LEWIS,G.N. and M. RANDALL(1961) Thermodynamics. McGraw-Hill, New York, Toronto, London. LINKE,F. and F. BAUR(1970), editors, Meteorologisches Taschenbuch. Geest & Portig, Leipzig. MAMAYEV, O.I. (1975) Temperature-salinity analysis of world ocean waters. Elsevier Scientific Publishing Company, Amsterdam. MAMAYEV,O.I. (1976) On the thermohaline analysis of ocean waters (in Russian). Doklady Akademii Nauk, 229, 195-198. MAMAYEV,O.I., H. DOOLEY,B. MILLARDand K. TAIRA( 1991), editors, Processing of oceanographic station data. UNESCO. MILLERO,F.J. (1974) Seawater as a multicomponent electrolyte solution. In: The Sea, Vol.$, Marine Chemistry, E.W.GOLDBERG,editor, Wiley Interscience, New York, London, Sydney, Toronto, p.3-80. MILLERO, F.J. (1978) Freezing point of seawater. In: 8th Report of JPOTS. Unesco technical papers in marine science, 28, UNESCO. MILLERO, F.J. (1982) The thermodynamics of seawater. Part I. The PVT properties. Ocean Science and Engineering, 7, 403-460. MILLERO,F.J. (1983) The thermodynamics of seawater. Part II. Thermochemical properties. Ocean Science and Engineering, 8, 1-40. MIEEERO,F.J. and W.H. LEUNG(1976) The thermodynamics of seawater at one atmosphere. American Journal of Science, 276, 1035-1077. MIELERO, F.J. and M.L. StUN (1992) Chemical Oceanography. CRC Press, Boca Raton. MILLERO, FJ., L.D. HANSENand E.V. HOFF (1973) The enthalpy of seawater from 0 to 30°C and from 0 to 40 salinity. Journal of Marine Research, 31, 21-39. MILLERO,F.J., G. PERRONand J.F. DESNOYERS(1973) Heat capacity of seawater solutions from 5°C to 35°C and 0.05 to 22 chlorinity. Journal of Geophysical Research, 78, 4499-4506. OWEN, B.B., R.C. MILLER,C.E. MILNERand H.L.COGAN(1961) The dielectric constant of water as a function of temperature and pressure. Journal of Physical Chemistry, 65, 2065-2070. PRESTON-THOMAS,H. (1990) The International Temperature Scale of 1990 (ITS-90). Metrologia, 27, 3-10. POUNDER,E.R. (1965) Physics oflce. Pergamon Press, Oxford. SAUNDERS,P. (1990) The International Temperature Scale of 1990, ITS-90. WOCE Newsletter 1O.

The GIBBS thermodynamic potential of seawater

29 7

SEYFRIED,P. (1989) Die Avogadro-Konstante and das Kilogramm - Stand und Aussichten. PTP-Mitteilungen, 99, 336-342.

SOMMERFELD,A. (1962) Thermodynamik und Statistik. Akademische Verlagsgesellsehaft Geest & Portig, Leipzig. SIEDLER, G. and H. PETERS (1986) Properties of sea water: Physical properties. In J.SONDERMANN, editor, Oceanography, Landolt-BOrnstein V/3a, Springer Berlin Heidelberg. UNESCO (1976) Seventh report of the joint panel on oceanographic tables and standards, UNESCO Technical Papers in Marine Science, 24, UNESCO. UNESCO (1978) see MILLERO, 1978. UNESCO (1981) Background papers and supporting data on the International Equation of State of Seawater 1980, UNESCO Technical Papers in Marine Science, 38, UNESCO. UNESCO (1983) see FOFONOFFand MILLaP,9, 1983. UNESCO (1987) International oceanographic tables, UNESCO Technical Papers in Marine Science, 40, UNESCO. UNESCO (1991) see MAMAYEVet al., 1991. WAGENBRETH,n. and W. BL~KE (1971) Die Dichte des Wassers im Internationalen Einheitensystem und in der Intemationalen Praktischen Temperamrskala von 1968. PTB-Mitteilungen, 81, 412-415. WHITE, H.J. (1977) Release on static dielectric constant of water substance. International Association for the Properties of Steam (lAPS), National Bureau of Standards, Washington D.C. WEICHART, G. (1986) Chemical Properties of Sea Water. In: Landolt-BOrnstein, Oceanography, Vol.3a, J. SUNDERMANN,editor, Springer Berlin. WILLE,P. (1986) Properties of sea water:, acoustical properties of the ocean. In J.SONDERMANN,editor, Oceanography, Landolt-B6rnstein V/3a, Springer Berlin Heidelberg. WILSON,W.D. (1959) Speed of sound in distilled water as a function of temperature and pressure. The Journal of the Acoustical Society of America, 31, 1067-1072. YEN, Y.C. (1981) Review of thermal properties of snow, ice and sea ice. USA Cold Regions Research and Engineering Laboratory, CRREL Report 81-10, Hanover, New Hampshire. YEN, Y.C., K.C. CHENGand S. FUKUSAKO(1991) Review of intrinsic thermophysical properties of snow, ice, sea ice, and frost. In: J.P.ZARLINGand S.L.FAUSSETr,editors, Proceedings 3rd International Symposium on Cold Regions Heat Transfer, Fairbanks, Alaska, 187-218.

The GIBBS thermodynamic potential of seawater

299

11. APPENDIX T a b l e A.1. Coefficients go, g~ and gijk of the t h e r m o d y n a m i c potential (Specific Free E n t h a l p y ) G(S,t,p) to be c o m p u t e d as G(S,t,p) = 1 J/kg (g0+gloy) x21n(x) + E gi-k xi YJ zk S=40PSU-x 2,t=40 C.y,p=100Ml3a .z logarithm terms go = 5813.3468, gl = 851.3047 X0

X2

X3

X4

X5

X6

Zo-

~) yl y2 y3 y4 y5 y6

0.0 0.0 -12351.7944 747.2334 -161.5446 58.2481 - 11.8396

1531.9856 160.9033 895.0020 -209.2699 55.5839 -3.1543 0.0

-2435.7962 -469.9126 -130.7027 44.7568 -11.0422 0.0 0.0

2007.7988 617.8957 0.0 0.0 0.0 0.0 0.0

-996.8895 -348.4918 0.0 0.0 0.0 0.0 0.0

253.0520 104.5879 0.0 0.0 0.0 0.0 0.0

100015.6317 -268.6529 1431.9976 -571.4559 199.1250 -10.4327 - 11.6532

-3299.5893 712.1452 -683.0344 303.8209 -64.4393 0.0 0.0

141.4839 -146.2849 209.6177 -70.8685 0.0 0.0 0.0

28.9105 -38.2516 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0 0.0

-2544.4971 769.0566 -711.2225 382.1835 - 152.3299 27.5081

385.5770 -340.9772 311.9769 -151.6012 44.5991 0.0

-52.9754 83.2066 -58.9981 0.0 0.0 0.0

-4.2198 2.6286 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0

278.1432 - 160.7702 176.5732 -88.8296 21.9035

-63.0288 55.2513 5.2864 -87.6880 27.5419

16.3245 5.0646 -49.3330 43.6584 0.0

-5.9393 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0

-24.7502 4.6854 - 14.1270 4.3018

4.2007 - 13.5434 36.4541 7.4147

0.8337 0.0 0.0 0.0

0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0

0.4941 2.8582 0.0

-0.2284 4.1059 - 16.5790

0.0 0.0 0.0

0.0 0.0 0.0

0.0 0.0 0.0

0.0 0.0 0.0

Zl:

yO yl y2 y3 y6 Z2:

yO y~ y2 y3 y~ y5 Z3~

yO yl ),2 y3 y4 z4: yO y~ ),2 y3 Z57

yO yl y2

300

R. FEISTELand E. HAGEN

T a b l e A . 2 . C o e f f i c i e n t s h~jk o f the t h e r m o d y n a m i c p o t e n t i a l ( S p e c i f i c E n t h a l p y ) H(S,(~,p) to be computed as H ( S , o , p ) = 1 J / k g Z {ha.. + h... x21n(x) + Z h.~ x ~} u i z k, UjK

IJK

U~

S = 4 0 ° C • x 2, o = 6 2 5 J / k g K • u, p = 100 M P a o z x°

x21n(x)

X2

X3

X4

Xs

X6

Z o.

u° u1 u2 u3 u4 u5 u6 u7

0.0 170718.7111 12650.5879 770.9610 -52.5565 2.7434 23.5198 -10.0052

5813.2879 868.8495 63.7675 2.7770 19.4851 -27.8257 10.6018 0.0

1531.4884 183.9358 893.1168 -9.9761 -22.0764 17.7715 0.0 0.0

-2423.8311 -548.5834 -92.8484 -45.6886 23.1436 0.0 0.0 0.0

1991.8523 666.0947 46.6475 9.6517 0.0 0.0 0.0 0.0

-1001.4788 -349.8016 -6.6096 0.0 0.0 0.0 0.0 0.0

262.2127 118.4626 0.0 0.0 0.0 0.0 0.0 0.0

100015.6367 -272.0487 1443.0006 -324.2998 32.2248 76,6009 - 15.4605 - 10.0865

-9.6457 112.6834 -94.6191 108.6468 -53.0497 -16.1723 20.8072 0.0

-3300.1123 730.0348 -467.3002 104.3974 56.3781 -24.5821 0.0 0.0

134.1105 -178.2801 175.0458 -5.0520 -38.4563 0.0 0.0 0.0

20.2894 43.9690 -34.9825 5.3975 0.0 0.0 0.0 0.0

20.9564 -45.0215 9.1301 0.0 0.0 0.0 0.0 0.0

-3.7740 8.3775 0.0 0.0 0.0 0.0 0.0 0.0

-2543.0410 747.1691 -477.6693 113.5262 16.1375 - 15.3624 -3.1380

22.2954 -16.6872 13.2905 -70.0238 89.9813 -36.5855 0.0

374.3880 -156.0386 -25.3956 69.5202 -5.8132 0.0 0.0

-44.3462 -28.8369 129.6701 -82.1771 0.0 0.0 0.0

8.7814 9.2822 -21.0847 0.0 0.0 0.0 0.0

-10.3273 3.0472 0.0 0.0 0.0 0.0 0.0

1.4069 0.0 0.0 0.0 0.0 0.0 0.0

269.8109 -60.7570 -24.3631 109.6875 -89.0391 26.0113 -0.0122

3.7661 -66.5397 189.9213 -233.9952 150.8011 -42.8463 0.0

-26.9173 -168.2888 406.3395 -379.2325 101.3579 0.0 0.0

-7.8205 192.2959 -342.1119 168.4422 0.0 0.0 0.0

-4.3206 -38.2767 33.4068 0.0 0.0 0.0 0.0

0.2006 2.0491 0.0 0.0 0.0 0.0 0.0

0.1850 0.0 0.0 0.0 0.0 0.0 0.0

-11.4577 -49.7267 70.9687 -63.6210 25.2964 - 1.7472

-0.8015 19.7618 -60.8276 65.5233 -24.6301 0.0

-7.0025 46.8885 -53.9862 46.1055 0.0 0.0

2.3893 -39.1219 43.7600 0.0 0.0 0.0

1.5958 5.8612 0.0 0.0 0.0 0.0

0.2417 0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0 0.0

-3.7162 14.7436 -14.0497 7.1615 - 1.1527

-1.9476 5.3568 -0.6755 -4.2221 0.0

-2.4319 10.2056 -16.4782 0.0 0.0

4.8140 -4.3602 0.0 0.0 0.0

-1.5296 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0 0.0

0.4155 -0.4779 0.0382 0.2801

0.4610 -2.1414 2.4078 0.0

0.6909 -2.4435 0.0 0.0

-0.4619 0.0 0.0 0.0

0.0 0.0 0.0 0.0

0.0 0.0 0.0 0.0

Z 1:

u° uI u2 u3 u4 u5 u6 u7 Z 2-

u° u1 u2 u3 u4 u5 u6 Z3:

u° uI u2 us u4 u5 u6 z4: u° uI u2 u3 u4 u5 Z5.-

u° uI u2 u3 u4 Z6:

u° uI u2 u3

0.0 0.0 0.0 0.0

The GIBBS thermodynamic potential of seawater

3 01

T a b l e A.3. S p e c i f i c F r e e E n t h a l p y G(S,t,p) in kJ/kg, the potential f u n c t i o n itself. Its d e p e n d e n c e on salinity, t e m p e r a t u r e and pressure is s h o w n in F i g s A. 1 a-c, o v e r l e a f 0PSU 0oc 5oC 10°C 15°C 20°C 25°C 30°C 35°C 40°C

5PSU

0.000 -0.645 -0.192 -0.848 -0.761 -1.426 -1.700 -2.371 -3.003 -3.676 -4.662 -5.336 -6.672 -7.343 -9.1)26 -9.694 -11.720 -12.380

0oc 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

9.976 9.784 9.218 8.285 6.992 5.345 3.349 1.012 -1.663

9.291 9.089 8.515 7.577 6.281 4.634 2.641 0.307 -2.361

0°C 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

19.904 19.713 19.151 18.226 16.943 15.308 13.327 11.006 8.350

19.179 18.978 18.410 17.480 16.195 14.560 12.582 10.265 7.616

0oC 5oC 10°C 15°C 20°C 25°C 30°C 35°C 40°C

39.616 39.432 38.884 37.976 36.715 35.107 33.157 30.870 28.251

38.816 38.623 38.069 37.157 35.895 34.288 32.340 30.058 27.446

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35oC 40°C

97.725 97.590 97.102 96.266 95.087 93.569 91.718 89.539 87.036

96.713 96.572 96.081 95.242 94.063 92.547 90.699 88.524 86.028

10PSU

15PSU 20PSU 25PSU 30PSU 35PSU 0 MPa = 0 dBar -0.831 -0.844 -0.745 -0.562 -0.310 -0.000 -1.036 -1.049 -0.947 -0.758 -0.499 -0.181 -1.613 -1.622 -1.514 -1.318 -1.049 -0.721 -2.555 -2.558 -2.441 -2.234 -I.954 -1.612 -3.855 -3.849 -3.721 -3.501 -3.207 -2.850 -5.507 -5.490 -5.348 -5.114 -4.803 -4.428 -7.504 -7.474 -7.317 -7.065 -6.736 -6.342 -9.842 -9.796 -9.622 -9.351 -9.001 -8.585 -12.515 -12.451 -12.257 -11.965 -11.593 -11.153 10 MPa -- 1000 dBar 9.066 9.013 9.074 9.219 9.432 9.704 8.862 8.811 8.875 9.026 9.247 9.528 8.289 8.243 8.313 8.473 8.704 8.996 7.355 7.315 7.395 7.565 7.808 8.114 6.065 6.034 6.125 6.309 6.567 6.888 4.426 4.407 4.511 4.710 4.984 5.323 2.443 2.437 2.557 2.773 3.066 3.425 0.122 0.132 0.270 0.505 0.819 1.199 -2.532 -2.505 -2.347 -2.091 -1.754 -1.350 20 MPa = 2000 dBar 18.915 18.824 18.847 18.954 19.130 19.365 18.714 18.625 18.652 18.766 18.950 19.194 18.147 18.063 18.097 18.220 18.414 18.670 1 7 . 2 2 1 17.144 17.188 17.322 17.529 17.799 15.942 15.875 15.930 16.077 16.299 16.585 14.316 14.260 14.329 14.492 1 4 . 7 3 1 15.035 12.348 12.306 1 2 . 3 9 1 1 2 . 5 7 1 12.829 13.153 10.044 10.018 10.120 10.320 10.598 10.944 7.409 7.400 7.522 7.743 8.044 8.414 40 MPa = 4000 dBar 38.477 38.312 38.260 38.294 38.398 38.560 38.285 38.123 38.077 38.119 38.232 38.404 37.733 37.577 37.539 37.590 37.714 37.899 36.826 36.678 36.650 36.714 36.851 37.050 35.571 35.433 35.417 35.495 35.647 35.863 33.972 33.846 33.845 33.938 34.107 34.342 32.035 31.923 31.938 32.049 32.237 32.492 29.766 29.670 29.702 29.832 30.042 30.319 27.168 27.090 27.142 27.294 27.526 27.827 100 MPa = 10000 dBar 96.166 95.793 95.535 95.365 95.265 95.224 96.027 95.661 95.411 95.250 95.161 95.133 95.540 95.181 94.941 94.791 94.715 94.701 94.708 94.358 94.130 93.994 93.932 93.934 93.537 93.198 92.983 92.861 92.816 92.835 92.030 91.704 91.504 91.399 91.371 91.409 90.193 89.882 89.698 89.611 89.603 89.661 88.032 87.736 87.570 87.503 87.516 87.597 85.550 85.272 85.126 85.080 85.115 85.220

40PSU 0.360 0.188 -0.339 -!.216 -2.437 -3.997 -5.889 -8.110 -10.653 10.027 9.860 9.341 8.474 7.265 5.719 3.842 1.639 -0.886 19.651 19.490 18.979 18.123 16.927 15.395 13.534 11.349 8.843 38.774 38.629 38.138 37.305 36.135 34.634 32.805 30.655 28.187 95.237 95.158 94.742 93.991 92.911 91.505 89.779 87.737 85.385

302

R. FEZSTeLand E. HA6EN

Specific

Free

Enthalpg

G(S,t,p)

at

p = 0 bar

2

i o

................

i i .............................

i ..................................................................

~

~

: ~

:OeC .

~

e

........

• _~,39~c

~

......

.

................................................................. !................................ ~................ i................ ! . . . . . . . .

-e

-10

.................................................................

:. . . . . . . . . . . . . . . .

; .................................................................

40oC

-14

. . . .

,

. . . .

,

. . . .

,

20

0

. . . .

i

15

. . . .

$

Enthal M

. . . .

,

30

25

Salinity

Free

i

20

G at

in

t

. . . .

,

. . . .

,

35

40

PSU

= 8 *C

100. 100

80

%.

"......................................................................................

............................................................

60

.........................

40

...........................

20

........................

HlPa

BO

'. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

MPa

~ ~ ' rHp

~ ...............

c .w

:. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

'

......................................................

~"

"40"

"MPsk

................

~..,.~ , 2 0 .

MP.a

...............

dl

1

..........................................................................................

b,

o ........................

i .............................

~ .....

:

i

'. . . . . . . . . . . . . . . . . . . . . . .

i 9..NP~

............

i

-20

. 0

.

.

.

~ JLO

.

.

.

.

,

.

.

.

.

.

20

. 30

Salintt~

in

.

. 40

PSU

Fig. A.i. Free enthalpy G, a) at atmospheric pressture, b) at temperature t=0°C, c) (right) at salinity S=35 PSU

The GIBBS ~ermodynamic po~ntial of seawater

Free E n t h a l p g G a t S = 3 5 PSU

100.

80

303

................................................................................................................................................

~ 60

8

0

N

P

a

...........................................................................................................................

60NPa 40

..........................................................

~ ...................................................................................

~ 20

4

0

M

...................................................................................................................

P

a i

............................ 20HPa

O~ -20"

0

10

O

N 20

P

a 30

TempePatuPe in °C

40

304

R. FEISTELand E. HAGEN

Table A.4. Specific Enthalpy H(S,t,p) in kJ/kg, computed from G(S,t,p) as: H(S,t,p) = G(S,t,p) - (T°+t) (0G/Ot)s. p , T ° = 273.15 K. Its dependence on salinity, temperature and pressure is shown in Figs A.2 a,b. 0PSU

5PSU

10PSU

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

0.000 21.046 42.029 62.971 83.888 104.790 125.684 146.574 167.464

0.061 20.931 41.749 62.536 83.305 104.065 124.821 145.576 166.332

0.103 20.806 41.466 62.104 82.730 103.353 123.976 144.601 165.228

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

10.108 30.936 51.726 72.495 93.256 114.013 134.772 155.533 176.301

10.075 30.739 51.374 71.996 92.615 113.237 133.863 154.493 175.130

10.029 30.535 51.021 71.501 91.984 112.474 132.971 153.476 173.987

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

20.069 40.699 61.312 81.924 102.540 123.164 143.797 164.438 185.090

19.949 40.425 60.892 81.364 101.845 122.339 142.843 163.357 183.883

19.819 40.147 60.474 80.810 101.161 121.527 141.907 162.299 182.702

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

39.579 59.868 80.176 100.510 120.871 141.257 161.664 182.089 202.532

39.300 59.453 79.630 99.837 120.075 140.340 160.626 180.931 201.253

39.019 59.040 79.089 99.173 119.290 139.436 159.606 179.794 200.000

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

95.354 114.963 134.639 154.384 174.191 194.049 213.946 233.874 253.831

94.713 114.212 133.783 153.427 173.135 192.895 212.693 232.516 252.361

94.081 113.470 132.938 152.482 172.094 191.757 211.456 231.177 250.914

15PSU 20PSU 25PSU 0 MPa = 0 dBar 0.124 0.124 0.103 20.664 20.506 20.330 41.172 40.864 40.542 61.664 61.214 60.753 8 2 . 1 5 1 8 1 . 5 6 6 80.971 102.641 101.924 101.201 123.134 122.290 121.442 143.631 142.662 141.691 164.132 163.039 161.946 10 MPa = 1000 dBar 9.965 9.883 9.783 3 0 . 3 1 8 3 0 . 0 8 7 29.840 5 0 . 6 5 9 50.285 4 9 . 8 9 9 7 1 . 0 0 0 70.491 6 9 . 9 7 2 9 1 . 3 5 1 90.711 9 0 . 0 6 3 111.712 110.945 110.173 132.083 131.194 130.301 152.464 151.453 150.440 172.853 171.721 170.591 20 MPa = 2000 dBar 1 9 . 6 7 5 19.515 19.340 3 9 . 8 5 8 39.557 39.243 6 0 . 0 4 7 59.611 59.163 8 0 . 2 5 1 7 9 . 6 8 4 79.109 100.474 9 9 . 7 8 2 9 9 . 0 8 3 120.716 119.902 119.083 140.974 140.041 139.104 161.246 160.194 159.142 181.530 180.361 179.194 40 MPa = 4000 dBar 38.729 38.428 38.116 58.619 58.190 57.752 7 8 . 5 4 3 77.990 7 7 . 4 3 0 98.505 97.833 97.153 118.504 117.715 116.920 138.535 137.631 136.723 158.590 157.573 156.555 178.664 177.536 176.408 198.757 197.518 196.281 100 MPa = 10000 dBar 9 3 . 4 4 7 92.811 9 2 . 1 7 0 112.728 111.984 111.236 132.093 131.247 130.399 151.540 150.596 149.651 171.055 170.016 168.976 190.623 189.489 188.354 210.224 208.994 207.763 229.844 228.514 227.184 249.474 248.039 246.606

30PSU

35PSU

40PSU

0.062 20.137 40.205 60.279 80.367 100.471 120.589 140.717 160.851

0.000 19.926 39.853 59.793 79.752 99.731 119.727 139.736 159.752

-0.084 19.697 39.485 59.292 79.124 98.981 118.857 138.748 158.647

9.665 29.578 49.500 69.441 89.406 109.394 129.402 149.424 169.458

9.528 29.299 49.087 68.899 88.740 108.607 128.497 148.403 168.322

9.372 29.004 48.658 68.343 88.061 107.810 127.583 147.375 167.180

19.150 38.915 58.704 78.524 98.376 118.258 138.163 158.086 178.025

18.943 38.573 58.233 77.929 97.660 117.424 137.215 157.026 176.853

18.718 38.216 57.748 77.321 96.933 116.582 136.259 155.959 175.676

37.793 57.303 76.860 96.466 116.119 135.811 155.532 175.278 195.043

37.456 56.842 76.279 95.770 115.310 134.891 154.505 174.143 193.803

37.106 56.369 75.688 95.064 114.492 133.964 153.471 173.004 192.560

91.524 110.484 129.547 148.702 167.933 187.217 206.531 225.854 245.173

90.873 109.727 128.690 147.749 166.886 186.077 205.296 224.522 243.739

90.214 108.964 127.827 146.790 165.834 184.932 204.057 223.187 242.303

305

The GIBBS tbermod ¢namic potential of seawater

Enthall~J

H at

t

= 0 °C

300

;ZOO

.................................................................................................................

I00

.................................................................................................................................................. 100 NPa

£

¢

0

. 0

.

.

.

.

. 10

.

.

, 20

Np a

: 60

N]Pa

: 40

M]Pa

i 20

I~Pa

:O

.

Salinity

180

30 in

N]Pa

40

PSU

EnChalp9 H a t S = 35 PSU 300-

100

200

MPa

80

NPa

60

MPa

40

N]Pa

20

MPa

J<

0

¢

e 4* ¢

kl

100

O~ O

10

20 Tempe~at~

30 in

40

eC

Fig. A.2. Enthalpy H, a) ~ ~mperature t--0°C, b) at salmi~ S=35 PSU

MPa

30 6

R. FEISTELand E. HAGEN

Table A.5. Specific Internal Energy E(S,t,p) in kJ/kg, computed from G(S,t,p) as E(S,t,p) = G(S,t,p) - (TO+t) (OG/Ot)s. p - (P°+p) (3G/~P)s, t, T ° = 273.15 K, po = 0.101325 MP a Its dependence on salinity, temperature and pressure is shown in Figs A.3 a,b. 0PSU

5PSU

10PSU

0oc 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

-0.101 20.945 41.928 62.870 83.786 104.688 125.582 146.472 167.362

-0.040 20.830 41.648 62.435 83.204 103.963 124.719 145.474 166.230

0.002 20.705 41.366 62.003 82.629 103.252 123.875 144.499 165.126

0oC 5oc 10°C 15°C 20°C 25°C 30°C 35oc 40°C

0.056 20.883 41.669 62.431 83.182 103.927 124.671 145.416 166.164

0.063 20.726 41.356 61.970 82.579 103.188 123.799 144.413 165.031

0.056 20.561 41.041 61.513 81.986 102.463 122.945 143.432 163.925

0oc 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

0.163 20.788 41.392 61.987 82.582 103.180 123.783 144.391 165.005

0.120 20.591 41.047 61.501 81.961 102.428 122.902 143.383 163.870

0.068 20.389 40.703 61.021 81.349 101.689 122.039 142.396 162.761

0oc 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

0.237 20.507 40.785 61.080 81.393 101.724 122.068 142.423 162.791

0.107 20.239 40.384 60.550 80.739 100.948 121.171 141.406 161.653

-0.026 19.970 39.986 60.028 80.095 100.185 120.291 140.409 160.539

0oc 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

-0.407 -0.714 -1.016 19.092 18.672 18.258 38.643 38.116 37.597 58.250 57.620 5 7 . 0 0 0 77.906 77.177 76.458 97.602 96.773 9 5 . 9 5 7 117.325 116.397 115.482 137.068 136.035 135.018 156.828 155.685 154.560

15PSU 20PSU 25PSU 0 MPa = 0 dBar 0.024 0.024 0.004 20.564 20.406 20.231 4 1 . 0 7 1 40.764 40.442 6 1 . 5 6 3 6 1 . 1 1 4 60.653 82.051 81.466 80.872 102.540 101.824 101.102 123.033 122.190 121.342 143.530 142.562 141.591 164.031 162.938 161.845 10 MPa = 1000 dBar 0.031 -0.012 -0.074 20.382 20.188 19.979 4 0 . 7 1 7 40.380 40.032 61.050 60.577 6 0 . 0 9 5 8 1 . 3 8 9 8 0 . 7 8 6 80.175 101.737 101.007 100.271 122.093 121.240 120.383 142.457 141.482 140.506 162.827 161.731 160.637 20 MPa = 2000 dBar -0.000 -0.084 -0.184 2 0 . 1 7 5 19.948 19.708 40.350 39.987 39.612 6 0 . 5 3 5 60.041 59.538 8 0 . 7 3 5 80.115 79.488 100.950 100.208 99.460 121.177 120.315 119.449 141.415 14(-).434 139.452 161.661 160.563 159.466 40 MPa = 4000 dBar -0.170 -0.326 -0.494 1 9 . 6 9 4 19.408 19.112 3 9 . 5 8 3 39.171 38.751 5 9 . 5 0 2 58°969 5 8 . 4 2 9 7 9 . 4 5 0 78.799 78.143 99.422 98.657 97.888 119.414 118.535 117.654 139.418 138.428 137.437 159.435 158.333 157.233 100 MPa = 10000 dBar -1.320 -I.630 -1.946 1 7 . 8 4 2 17.422 16.998 3 7 . 0 7 6 36.553 3 6 . 0 2 5 5 6 . 3 7 9 55.756 5 5 . 1 3 0 7 5 . 7 4 0 75.021 7 4 . 2 9 8 9 5 . 1 4 3 94.328 9 3 . 5 1 0 114.570 113.658 112.744 134.005 132.993 131.981 153.440 152.323 151.207

30PSU

35PSU

40PSU

-0.037 20.039 40.106 60.180 80.268 100.371 120.489 140.617 160.751

-0.099 19.828 39.754 59.694 79.653 99.632 119.628 139.637 159.652

-0.182 19.598 39.386 59.194 79.026 98.882 118.759 138.649 158.547

-0.154 19.755 39.669 59.601 79.554 99.529 119.520 139.525 159.540

-0.253 19.514 39.293 59.095 78.924 98.777 118.651 138.540 158.439

-0.371 19.256 38.902 58.576 78.282 98.016 117.773 137.547 157.333

-0.300 19.454 39.226 59.025 78.852 98.705 118.578 138.467 158.368

-0.433 19.184 38.827 58.501 78.207 97.942 117.701 137.477 157.266

-0.584 18.900 38.414 57.964 77.551 97.170 116.815 136.480 156.159

-0.674 18.804 38.321 57.881 77.480 97.112 116.769 136.443 156.132

-0.868 18.484 37.880 57.323 76.808 96.329 115.877 135.445 155.028

-1.077 18.151 37.428 56.754 76.127 95.538 114.979 134.441 153.920

-2.267 16.568 35.492 54.499 73.572 92.690 111.829 130.968 150.091

-2.596 -2.934 16.131 15.687 3 4 . 9 5 4 34.408 53.863 53.221 7 2 . 8 4 2 72.105 9 1 . 8 6 6 91.036 110.910 109.986 129.951 128.931 148.973 147.852

The GIBBS thermodynamic potential of seawater

I n t e r n a l Ener~j E X-

o

-1

at

307

t = 0 "C

i

t~

'

..............

-3

.

.

i

"

.

~

.

0

,

.

o,~........... "

.

.

.

10

i

"40" "HPi

.

.

.

.

i

20

.

.

.

.

30

$alinitg

in

...............

i

40

1~3U

I n t e r n a l Enes'g~l E a t S : 3'5 P3U 180-

0

HPa

155

130-

iiiiii iii

\ ,X

105.

c :m 0 c bl

C f,, ##

80'

55

C 30

5

-20

.

0

.

.

.

i

10

.

.

.

.

i

.

.

.

.

i

20 Teml~l,

30 r i t ¢ ~

In

.

.

.

.

i

40

eC

Fig. A.3. Internal energy E, a) at temperature t--0°C, b) at salinity S=35 PSU

30 8

R. FE1STELand E. HA~EN

Table A.6. Specific Free Energy F(S,t,p) in kJ/kg, computed from G(S,t,p) as F(S,t,p) = G(S,t,p) (pO+p) (~G/~P)s.t ' po = 0.101325 MPa. Its dependence on salinity, temperature and pressure is shown in Figs A.4 a,b. 0PSU

5PSU

10PSU

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

-0.101 -0.746 -0.931 -0.293 - 0 . 9 4 9 - 1 . 1 3 7 -0.862 - 1 . 5 2 7 -1.714 -1.802 -2.472 -2.656 -3.105 - 3 . 7 7 7 -3.956 -4.764 -5.437 -5.608 -6.774 -7.445 -7.605 -9.128 - 9 . 7 9 5 - 9 . 9 4 3 -11.822 -12.482 -12.616

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

-0.076 - 0 . 7 2 1 -0.907 -0.268 -0.925 -1.113 -0.838 - 1 . 5 0 3 - 1 . 6 9 1 -1.778 -2.449 -2.633 -3.082 -3.755 -3.934 -4.741 -5.415 -5.586 -6.751 - 7 . 4 2 3 -7.584 -9.106 - 9 . 7 7 3 -9.922 -11.800 -12.460 -12.594

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

-0.003 -0.650 -0.836 -0.198 - 0 . 8 5 5 -1.044 -0.770 - 1 . 4 3 5 -1.624 -1.711 -2.382 -2.568 -3.015 - 3 . 6 8 9 - 3 . 8 6 9 -4.676 -5.350 -5.522 -6.686 - 7 . 3 5 9 -7.520 -9.042 - 9 . 7 0 9 - 9 . 8 5 9 -11.735 -12.397 -12.532

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

0.274 -0.377 -0.568 0.071 - 0 . 5 9 1 -0.784 -0.508 - 1 . 1 7 8 -1.370 -1.454 -2.129 -2.319 -2.763 -3.440 -3.624 -4.426 -5.104 -5.279 -6.439 -7.115 -7.280 -8.796 -9.467 -9.619 -11.490 -12.155 -12.293

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

1.964 1.286 1.719 1.032 1.106 0.414 0.132 -0.564 -1.198 -1.896 -2.878 - 3 . 5 7 5 -4.903 - 5 . 5 9 7 -7.267 -7.957 -9.966 -10.649

1.069 0.816 0.199 -0.774 -2.099 -3.770 -5.781 -8.127 -10.805

15PSU 20PSU 25PSU 0 MPa = 0 dBar -0.944 -0.845 -0.661 -1.149 -1.046 -0.857 -1.723 -1.614 -1.417 - 2 . 6 5 8 - 2 . 5 4 1 -2.334 -3.949 -3.821 -3.601 -5.590 - 5 . 4 4 9 -5.214 -7.574 - 7 . 4 1 8 -7.165 - 9 . 8 9 7 -9.722 -9.451 -12.552 -12.358 -12.066 10 MPa = 1000 dBar -0.920 - 0 . 8 2 1 -0.638 -1.126 -1.023 -0.835 -1.700 -1.591 -1.395 -2.636 - 2 . 5 1 9 -2.312 - 3 . 9 2 7 -30799 -3.580 - 5 . 5 6 8 - 5 . 4 2 7 -5.192 - 7 . 5 5 3 -7.396 -7.144 - 9 . 8 7 5 - 9 . 7 0 1 -9.430 -12.531 -12.337 -12.045 20 MPa = 2000 dBar -0.851 -0.753 -0.571 - 1 . 0 5 8 - 0 . 9 5 7 -0.769 -1.634 -1.527 -1.331 - 2 . 5 7 1 - 2 . 4 5 5 -2.249 -3.864 -3.737 -3.518 -5.506 -5.365 -5.131 - 7 . 4 9 1 - 7 ° 3 3 5 -7.084 - 9 . 8 1 3 -9.640 -9.370 -12.469 -12.276 -11.984 40 MPa = 4000 dBar - 0 . 5 8 8 -0.494 -0.316 -0.802 -0.705 -0.521 -1.384 -1.28 0 -1.088 -2.326 - 2 . 2 1 3 -2.010 -3.622 - 3 . 4 9 8 -3.282 -5.266 -5.129 -4.898 - 7 . 2 5 3 -7.100 -6.852 - 9 . 5 7 7 -9.406 -9.139 -12.233 -12.042 -11.754 100 MPa = 10000 dBar 1.025 1.094 1.249 0.775 0.850 1.011 0.164 0.246 0.417 -0.802 -0.710 -0.527 -2.117 -2.013 -1.816 -3.775 -3.657 -3.444 -5.772 - 5 . 6 3 8 -5.407 - 8 . 1 0 3 -7.950 -7.700 -10.762 -10.590 -10.318

30PSU

35PSU

40PSU

-0.409 - 0 . 0 9 9 0.262 - 0 . 5 9 8 -0.280 0.090 - 1 . 1 4 8 - 0 . 8 1 9 -0.437 - 2 . 0 5 3 - 1 . 7 1 1 -1.315 - 3 . 3 0 7 - 2 . 9 4 9 -2.536 - 4 . 9 0 3 - 4 . 5 2 7 -4.095 -6.836 - 6 . 4 4 1 -5.988 - 9 . 1 0 1 -8.684 -8.209 -11.693 -11.253 -10.753 -0.386 -0.076 0.284 -0.576 - 0 . 2 5 8 0.112 - 1 . 1 2 7 - 0 . 7 9 8 -0.416 -2.032 -1.690 -1.294 -3.285 - 2 . 9 2 8 -2.515 -4.882 - 4 . 5 0 7 4.075 - 6 . 8 1 5 -6.420 -5.968 - 9 . 0 8 1 -8.664 -8.189 -11.673 -11.232 -10.732 -0.320 - 0 . 0 1 1 0.348 -0.512 -0.195 0.174 -1.064 -0.736 -0.355 -1.970 -1.629 -1.234 -3.224 - 2 . 8 6 8 -2.456 - 4 . 8 2 1 - 4 . 4 4 7 -4.016 - 6 . 7 5 5 - 6 . 3 6 1 -5.909 - 9 . 0 2 1 - 8 . 6 0 5 -8.130 -11.613 -11.173 -10.674 - 0 . 0 6 9 0.236 0.591 - 0 . 2 6 7 0.046 0.411 -0.824 -0.500 -0.123 - 1 . 7 3 5 - 1 . 3 9 7 -1.005 -2.992 - 2 . 6 3 9 -2.230 - 4 . 5 9 1 -4.220 -3.792 -6.526 - 6 . 1 3 5 -5.686 - 8 . 7 9 3 -8.380 -7.908 -11.385 -10.948 -10.452 1.473 1.244 0.661 -0.271 -1.544 -3.156 -5.099 -7.371 -9.967

1.755 1.536 0.965 0.048 -1.209 -2.802 -4.725 6.974 -9.546

2.089 1.881 1.323 0.421 -0.818 -2.391 -4.292 -6.519 -9.066

The GIBBS thermodynamic potential of seawater

3 09

F r e e E n e r ~ j F a t t = 0 *C 3 :

~ ~ ~

100 : 80 60

MPa MPa MPa .

-3

-6

t~ k~

.

.

.....................................................................................................................................

...............................................................................................................................................

-9

.................................................................................................................................................

-12

..............................................................................................................................................

-15 O

10

20 Salinity

30 in

40

PSU

Fr'~e Enes-gy F a t S = 35 ~b-~lJ

-3 \ ,.T e. -£. e. r. /, t..

100 MPa 80 MPa

-12

-15

-I" 0

10

20 Telepel-atuz~

30 in

40

oC

Fig. A.4. Free energy F, a) at temperature t=0°C, b) at salinity S=35 PSU

.

.

.

.

.

.

.

.

.

.

.

.

3 10

R. FEISTELand E. HAGEN

T a b l e A.7. R e l a t i v e C h e m i c a l P o t e n t i a l ~t(S,t,p) in J / k g P S U , c o m p u t e d f r o m G ( S , t , p ) as la(S,t,p) = (0G/0S)t. p. Its d e p e n d e n c e o n s a l i n i t y , t e m p e r a t u r e a n d p r e s s u r e is s h o w n i n F i g s A . 5 a,b. 5PSU

10PSU

15PSU

20PSU

25PSU

30PSU

35PSU

40PSU

56.46 57.95 59.98 62.49 65.47 68.88 72.70 76.93 81.54

67.20 68.96 71.25 74.01 77.23 80.88 84.93 89.38 94.22

76.69 78.70 81.23 84.23 87.67 91.54 95.81 100.47 105.50

48.85 50.47 52.60 55.19 58.23 61.68 65.54 69.79 74.41

59.64 61.53 63.91 66.75 70.02 73.72 77.80 82.28 87.12

69.19 71.31 73.93 77.00 80.50 84.40 88.71 93.39 98.43

41.39 43.13 45.34 48.01 51.10 54.60 58.49 62.75 67.38

52.24 54.23 56.69 59.60 62.93 66.66 70.78 75.27 80.12

61.83 64.06 66.75 69.88 73.43 77.38 81.71 86.40 91.45

26.90 28.82 31.19 33.98 37.17 40.73 44.67 48.96 53.60

37.83 40.00 42.61 45.64 49.06 52.85 57.02 61.53 66.39

47.51 49.91 52.74 55.98 59.61 63.62 67.99 72.71 77.77

- 13.74 -11.42 -8.73 -5.70 -2.34 1.35 5.35 9.66 14.30

-2.60 -0.06 2.85 6.11 9.69 13.59 17.81 22.34 27.19

7.26 10.01 13.13 16.59 20.37 24.48 28.89 33.62 38.67

0 MPa = 0 dBar

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

-62.34 -63.36 -63.79 -63.66 -63.02 -61.90 -60.32 -58.30 -55.85

-16.95 -17.07 -16.62 -15.63 -14.15 -12.19 -9.79 -6.96 -3.72

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

-70.30 -71.15 -71.45 -71.22 -70.50 -69.33 -67.70 -65.66 -63.20

-24.81 -24.78 -24.20 -23.12 -21.56 - 19.55 -17.11 -14.26 -11.00

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

-78.08 -78.79 -78.96 -78.65 -77.86 -76.63 -74.98 -72.91 -70.43

-32.51 -32.34 -31.65 -30.48 -28.86 -26.80 -24.32 -21.45 - 18.18

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

-93.18 -93.62 -93.59 -93.12 -92.22 -90.90 -89.19 -87.09 -84.61

-47.45 -47.03 -46.15 ,-44.83 -43.09 -40.95 -38.42 -35.51 -32.23

0°C 5°C lO°C 15°C 20°C 25°C 30°C 35°C 40°C

- 135.15 -135.10 -134.67 -133.88 -132.75 -131.29 -129.49 -127.37 -124.90

-89.06 -88.17 -86.91 -85.29 -83.33 -81.04 -78.43 -75.49 -72.21

9.76 28.92 43.97 10.20 29.77 45.17 11.19 31.18 46.91 12.71 33.09 49.14 14.71 35.48 51.84 17.17 38.31 54.99 20.07 41.58 58.56 23.39 45.26 62.53 27.11 49.34 66.89 10 MPa = 1000 dBar 1.98 21.20 36.31 2.56 22.19 37.64 3.67 23.70 39.48 5.28 25.70 41.80 7.35 28.16 44.57 9.86 31.05 47.76 12.80 34.35 51.36 16.14 38.05 55.36 19.88 42.14 59.74 20 MPa = 2000 dBar -5.65 13.63 28.80 -4.94 14.75 30.25 -3.72 16.36 32.19 -2.03 18.44 34.58 0.10 20.96 37.40 2.67 23.89 40.64 5.64 27.23 44.27 9.00 30.95 48.29 12.75 35.05 52.68 40 MPa = 4000 dBar -20.46 -1.06 14,21 -19.52 0.27 15.86 - 18.12 2.06 17.96 -16.29 4.27 20.48 -14.04 6.89 23.41 - 11.40 9.90 26.72 -8.37 13.29 30.40 -4.98 17.05 34.45 -1.21 21.16 38.85 100 MPa = 1000(3 dBar -61.78 -42.14 -26.64 -60.39 -40.37 -24.57 -58.62 -38.24 -22.14 -56.50 -35.75 -19.36 -54.05 -32.93 -16.24 -51.26 -29.78 -12.81 -48.16 -26.32 -9.05 -44.73 -22.53 -4.98 -40.97 -18.41 -0.58

The GIBBS thermodynamic potential of seawater

Chenieal P o t e n t i a l

31 1

p a t t = 0 "C

100-~

0

MPa

20

NPa

= 50-

n. \ al ,,~ \

•4 0 " M P a

C

60

MPa

80

MPa

100

...........

MPa

o

~ U

-50

-

~

0

0

~

0

10

20

30

Salinit9

Chemical P o t e n t i a l

in

P at S

= 35

100

PSU

i

~

50

40

PSU

...........................

:.

.

.

.

.

.

.

.

.

.

: 0

2

.

.

MPa

0

.

.

.

MPa

.

.

.

.

.

.

.

0

-50

.................................................................................................................................

-I0010

20 TeMPet-atttx~e

30 in

40

eC

Fig. A.5. Chemical potential Ix, a) at temperature t=0°C, b) at salinity S=35 PSU

3 12

R. FEISTELand E. HAGEN

Table A.8. Entropy o(S,t,p) in J/kgK, computed from G(S,t,p) as a(S,t,p) = -(OG/Ot)s. p Its dependence on salinity, temperature and pressure is shown in Figs A.6 a,b. 0PSU

5PSU

10PSU

O°C 5oC IO°C 15°C 20°C 25°C 30°C 35°C 40°C

0.000 76.353 151.120 224.436 296.404 367.104 436.601 504.949 572.197

2.584 78.301 152.482 225.253 296.711 366.931 435.970 503.875 570.693

3.417 78.526 152.144 224.393 295.361 365.118 433.714 501.193 567.594

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

0.483 76.044 150.123 222.834 294.264 364.476 433.523 501.450 568.302

2.872 77.838 151.365 223.561 294.504 364.256 432.861 500.360 566.794

3.525 77.920 15.).916 222.615 293.090 362.396 430.574 497.659 563.688

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

0.606 75.448 148.900 221.058 291.991 361.752 430.380 497.914 564.396

2.818 77.103 150.035 221.703 292.172 361.490 429.691 496.810 562.884

3.308 77.056 149.485 220.679 290.699 359.588 427.375 494.092 559.773

0oc 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

-0.136 73.471 145.833 217.019 287.075 356.030 423.907 490.732 556.542

1 . 7 7 3 1.984 74.887 74.617 146.782 146.058 217.525 216.368 287.155 285.584 355.700 353.728 423.177 420.817 489.608 486.868 555.029 551.914

0oc 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

-8.679 62.460 132.570 201.693 269.840 337.009 403.193 468.393 532.635

-7.320 63.416 133.153 201.924 269.733 336.571 402.421 467.279 531.163

-7.632 62.708 132.076 200.500 267.976 334.486 400.009 464.532 528.067

15PSU 20PSU 25PSU 0 MPa = 0 dBar 3.543 3.182 2.434 78.063 77.126 75.817 151A35 149.666 147.835 222.875 222).908 218.590 293.366 290.932 288.156 362.671 359.794 356.583 430.836 427.536 423.908 497.898 494.189 490.159 563.891 559.783 555.360 10 MPa = 1000 dr3ar 3.483 2.962 2.065 7 7 . 3 2 3 7 6 . 2 5 9 74.830 149.802 148.232 146.306 221.015 218.970 216.577 291.032 288.539 285.704 359.903 356.982 353.727 427.663 424.332 420.674 494.344 490.616 486.567 559.978 555.862 551.433 20 MPa = 2000 dBar 3.113 2.449 1.415 7 6 . 3 3 7 75.158 73.620 148.274 146.613 144.600 219o005 216.889 214.428 288.586 286.038 283.151 357.054 354.093 350.800 424.437 421.078 417.394 490.761 487.017 482.953 556.058 551.937 547.503 40 MPa = 4000 dBar 1.528 0.616 -0.651 7 3 . 6 8 8 72.310 7 0 . 5 8 3 144.681 142.863 140.700 214.566 212.328 209.751 283.375 280.735 277.761 351.127 348.100 344.744 417.834 414.433 410.708 483.514 479.748 475.664 548.195 544.071 539.636 100 MPa = 10000 dBar -8.587 -9.975 -11.696 61.360 5 9 . 5 8 2 57.474 130.363 128.223 125.755 198.443 195.961 193.153 265.588 262.778 259.643 331.774 328.642 325.188 396.972 393.519 389.747 461.165 457.385 453.289 524.358 520.240 515.809

30PSU

35PSU

40PSU

1.361 74.193 145.699 215.976 285.091 353.091 420.007 485.861 550.675

0.000 72.291 143.295 213.101 281.773 349.352 415.864 481.328 545.760

-1.625 70.136 140.645 209.989 278.224 345.388 411.501 476.580 540.636

0.850 73.092 144.080 213.892 282.585 350.194 416.743 482.252 546.742

-0.646 71.083 141.590 210.950 279.215 346.415 412.572 477.702 541.822

-2.398 68.826 138.859 207.773 275.616 342.414 408.184 472.939 536.694

0.071 71.779 142.292 211.679 279.982 347.230 413.438 478.623 542.809

-1.546 69.672 139.724 208.675 276.565 343.416 409.243 474.060 537.887

-3.413 67.322 136.920 205.441 272.921 339.381 404.832 469.285 532.758

-2.216 68.564 138.250 206.890 274.508 341.114 406.713 471.315 534.942

-4.042 66.288 135.547 203.781 271.011 337.244 402.483 466.735 530.023

-6.105 63.779 132.616 200.447 267.292 333.155 398.038 461.946 524.899

-13.694 55.091 123.015 190.074 256.240 321.468 385.710 448.931 511.121

-15.932 52.470 120.038 186.761 252.604 317.516 381.444 444.347 506.209

-18.388 49.634 116.847 183.236 248.757 313.355 376.970 439.558 501.095

The GIBBS thermodynamic potential of seawater

Entropy f at t

:

313

I] "C

ZO-

10 X \ m \ ,~

0

~

-10

....... ~

~

60 HPa

-20

0

10

20

30

S~linit~

Entropy

•

at

in

40

PSU

S = 35

600

:

~

0 40

HPa MPa

500'

40O

~ 200

.......i!i! ...i....'..... . . . . . .i'i.'i.'i'i.i.i.'.i'i." i ................

100

0

0

10

20 Teptpe~attu~

30 in

40

eC

Fig. A.6. Entropy ~, a) at temperature t=0°C, b) at salinity S=35 PSU

3 14

R. FEISTELand E. HAGEN

Table A.9. Density A n o m a l y x in kg/m 3, computed from G(S,t,p) as x(S,t,p) = 9 - 1000 kg]m 3, V = 1/p = (OG/OP)s. , 0PSU

5PSU

10PSU

Ooc 5oc IO°C 15°C 20°C 25°C 30°C 35°C 40oc

-0.1563 3.9164 7.9606 -0.0336 3.9482 7.9071 -0.2980 3.6099 7.4989 -0.8987 2.9495 6.7812 -1.7949 2.0053 5.7901 -2.9540 0.8076 4 . 5 5 3 9 -4.3511 -0.6203 3.0943 -5.9666 -2.2605 1.4284 -7.7845 -4.0976 -0.4293

Ooc 5oc IO°C 15°C 20°C 25°C 30°C 35°C 40°C

4.8739 4.8300 4.4307 3.7212 2.7381 1.5100 0.0595 -1.5961 -3.4420

8.8967 8.7682 8.3001 7.5348 6.5067 5.2425 3.7628 2.0840 0.2195

12.8936 12.6856 12.1521 11.3332 10.2609 8.9604 7.4510 5.7477 3.8636

Ooc 5oc IO°C 15°C 20°C 25°C 30°C 35°C 40°C

9.7907 9.5875 9.0589 8.2446 7.1771 5.8818 4.3787 2.6835 0.8091

13.7655 13.4838 12.8911 12.0248 10.9152 9.5862 8.0559 6.3389 4.4471

17.7168 17.3612 16.7074 15.7908 14.6399 13.2769 11.7188 9.9789 8.0688

0oc 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

19.2925 18.7935 18.0233 17.0121 15.7847 14.3609 12.7564 10.9835 9.0526

23.1768 22.6110 21.7854 20.7289 19.4649 18.0118 16.3840 14.5926 12.6468

27.0420 26.4127 25.5341 24.4335 23.1333 21.6506 19.9987 18.1878 16.2262

0oC 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

45.3194 44.1194 42.7664 41.2704 39.6416 37.8887 36.0187 34.0370 31.9468

48.9797 47.7426 46.353 44.8274 43.1730 41.4003 39.5160 37.5238 35.4250

52.6292 51.3544 49.9310 48.373 46.6935 44.9009 43.0014 40.9971 38.8869

15PSU 20PSU 25PSU 0 MPa = 0 dBar 11.9937 16.0218 20.0481 11.8589 15.8088 19.7600 11.3835 15.2686 19.1573 10.6102 14.4415 18.2779 9.5732 13.3596 17.1521 8.2989 12.0477 15.8032 6.8077 10.5250 14.2492 5 . 1 1 5 7 8.8068 12.5051 3 . 2 3 7 0 6.9070 10.5843 10 MPa = 1000 dBar 16.8808 20°8642 24.8467 16.5970 20.5074 24.4199 16.0006 19.8504 23.7042 15.1296 18.9288 22.7334 14.0141 17.7710 21.5343 12.6777 16.3991 20.1275 11.1384 14.8301 18.5290 9.4104 13.0775 16.7519 7.5063 11.1534 14.8081 20 MPa = 2000 dBar 21.6599 25.6001 29.5406 21.2336 25.1060 28.9811 20.5210 24.3366 28.1568 19.5556 23.3237 27.0978 18.3642 22.0925 25.8277 16.9674 20.6626 24.3650 15.3814 19.0487 22.7237 13.6185 17.2628 20.9150 11.6897 15.3155 18.9493 40 MPa = 4000 dBar 30.9019 34.7612 38.6226 30.2117 34.0123 37.8170 29.2817 33.0325 36.7891 28.1382 31.8472 35.5630 26.8022 30.4762 34.1578 25.2901 28.9351 32.5881 23.6142 27.2352 30.8646 21.7835 25.3849 28.9950 19.8060 23.3915 26.9859 I00 MPa = 10000 dBar 56.2798 59.9359 63.6000 54.9675 58.5862 62.2132 53.5093 57.0937 60.6870 51.9213 55.4762 59.0405 50.2166 53.7472 57.2880 48.4045 51.9159 55.4382 46.4897 49.9860 53.4935 44.4726 47.9559 51.4502 42.3494 45.8187 49.2984

30PSU

35PSU

40PSU

24.0750 23.7145 23.0513 22.1212 20.9526 19.5673 17.9825 16.2126 14.2709

28.1040 27.6737 26.9521 25.9728 24.7624 23.3415 21.7264 19.9310 17.9688

32.1362 31.6388 30.8608 29.8338 28.5827 27.1269 25.4821 23.6616 21.6793

28.8305 28.3362 27.5639 26.5454 25.3058 23.8648 22.2373 20.4360 18.4727

32.8171 32.2578 31.4307 30.3659 29.0870 27.6123 25.9563 24.1312 22.1488

36.8077 36.1857 35.3058 34.1961 32.8788 31.3713 29.6873 27.8388 25.8378

33.4832 32.8608 31.9834 30.8795 29.5714 28.0766 26.4082 24.5771 22.5934

37.4293 36.7463 35.816 34.6701 33.3250 31.7987 30.1038 28.2507 26.2495

41.3801 40.6387 39.6604 38.4707 37.0896 35.5325 33.8116 31.9369 29.9188

42.4879 41.6277 40.5531 39.2873 37.8487 36.2511 34.5043 32.6158 30.5914

46.3584 45.4455 44.3257 43.0214 41.5503 39.9253 38.1559 36.2487 34.2096

50.2352 49.2714 48.1079 46.7663 45.2635 43.6119 41.8203 39.8952 37.8418

67.2736 65.8502 64.2910 62.6163 60.8410 58.9734 57.0143 54.9578 52.7909

70.9581 69.4986 67.9069 66.2049 64.4077 62.5229 60.5501 58.4806 56.2981

74.6544 73.1593 71.5360 69.8075 67.9893 66.0881 64.1020 62.0197 59.8215

The GIBBS thermodynamic potential of seawater

3 15

Table A. 10. Specific Heat at Constant Pressure CP(S,t,p) in J/kgK, computed from G(S,t,p) as CP(S,t,p) = (T°+t) (~(~/~t)s,p = -(T°+t) (~2G/~tE)s,p 0PSU

5PSU

10PSU

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

4217.37 4181.14 4202.02 4168.08 4 1 9 1 . 8 7 4159.90 4 1 8 5 . 4 7 4155.17 4181.62 4152.66 4 17 9 . 4 1 4151.48 4178.24 4151.02 4 1 7 7 . 8 5 4151.02 4 1 7 8 . 3 9 4151.63

4146.55 4135.60 4129.25 4126.04 4124.78 4124.54 4124.73 4125.09 4125.76

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

4170.84 4137.09 4161.02 4129.19 4155.45 4125.27 4 15 2 . 7 1 4123.91 4 1 5 1 . 6 8 4123.98 4151.56 4124.69 4151.92 4125.60 4 1 5 2 . 7 3 4126.68 4 1 5 4 . 3 8 4128.35

4104.72 4098.60 4096.21 4096.13 4097.23 4098.71 4100.15 4101.52 4103.23

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

4128.82 4097.29 4123.76 4093.85 4122.15 4093.61 4 1 2 2 . 5 9 4095.17 4 1 2 4 . 0 1 4097.47 4125.70 4099.80 4 1 2 7 . 3 8 4101.88 4129.22 4103.89 4 1 3 1 . 8 7 4106.48

4066.91 4064.97 4066.00 4068.63 4071.76 4074.71 4077.21 4079.42 4082.03

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

4 0 5 6 . 9 1 4029.09 4059.34 4032.68 4 0 6 4 . 0 3 4038.31 4069.54 4044.55 4 0 7 4 . 8 5 4050.39 4079.42 4055.28 4083.20 4059.18 4 0 8 6 . 6 8 4062.60 4090.96 4066.63

4002.02 4006.70 4013.22 4020.18 4026.56 4031.81 4035.91 4039.36 4043.28

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

3915.80 3892.81 3 9 2 8 . 3 0 3906.78 3 9 4 2 . 1 3 3921.68 3 9 5 5 . 4 2 3935.62 3 9 6 6 . 8 9 3947.33 3 9 7 5 . 9 7 3956.20 3 9 8 2 . 8 0 3962.39 3 9 8 8 . 3 4 3966.86 3 9 9 4 . 4 3 3971.42

3870.19 3885.57 3901.50 3916.08 3928.02 3936.74 3942.36 3945.85 3949.04

15PSU 20PSU 25PSU 0 MPa = 0 dBar 4113.02 4080.35 4048.42 4104.07 4073.31 4043.22 4099.44 4070.33 4041.82 4097.69 4069.96 4042.76 4097.60 4070.98 4044.85 4098.25 4072.48 4047.16 4099.05 4073.86 4049.10 4099.75 4074.88 4050.42 4100.48 4075.67 4051.27 10 MPa = 1000 dBar 4073.24 4042.50 4012.38 4068.81 4039.68 4011.11 4067.88 4040.13 4012.90 4069.01 4042.43 4016.31 4071.09 4045.45 4020.23 4073.32 4048.39 4023.88 4075.27 4050.85 4026.82 4076.92 4052.79 4029.05 4078.70 4054.64 4031.01 20 MPa = 2000 dBar 4037.28 4008.25 3979.74 4036.75 4009.09 3981.89 4039.01 4012.51 3986.44 4042.64 4017.12 3992.00 4046.59 4021.85 3997.49 4050.15 4026.02 4002.26 4053.07 4029.36 4006.03 4055.52 4032.06 4009.00 4058.18 4034.82 4011.88 40 MPa = 4000 dBar 3975.43 3949.23 3923.36 3981.15 3955.97 3931.09 3988.56 3964.22 3940.18 3996.22 3972.58 3949.23 4003.14 3980.05 3957.26 4008.78 3986.11 3963.73 4013.12 3990.72 3968.65 4016.67 3994.43 3972.56 4020.57 3998.37 3976.61 100 MPa --- 10000 dBar 3847.80 3825.61 3803.59 3864.56 3843.70 3822.99 3881.49 3861.62 3841.87 3896.70 3877.45 3858.31 3908.89 3889.90 3871.03 3917.47 3898.37 3879.41 3922.58 3903.00 3883.59 3925.17 3904.73 3884.51 3927.06 3905.40 3884.02

30PSU

35PSU 40PSU

4017.16 4013.72 4013.84 4016.05 4019.16 4022.25 4024.72 4026.33 4027.24

3986.50 3984.76 3986.34 3989.78 3993.87 3997.70 41300.68 4002.57 4003.53

3956.41 3956.31 3959.29 3963.90 3968.94 3973.49 3976.96 3979.12 3980.14

3982.82 3983.05 3986.12 3990.61 3995.41 3999.72 4003.14 4005.67 4007.74

3953.77 3955.44 3959.75 3965.28 3970.93 3975.90 3979.79 3982.61 3984.81

3925.19 3928.25 3933.76 3940.31 3946.77 3952.38 3956.74 3959.86 3962.19

3951.69 3955.11 3960.75 3967.22 3973.47 3978.82 3983.03 3986.29 3989.32

3924.06 3928.71 3935.41 3942.77 3949.75 3955.69 3960.33 3963.90 3967.11

3896.83 3902.66 3910.39 3918.62 3926.32 3932.83 3937.91 3941.80 3945.21

3897.80 3906.49 3916.39 3926.13 3934.72 3941.64 3946.89 3951.04 3955.25

3872.52 3882.13 3892.85 3903.26 3912.41 3919.79 3925.41 3929.84 3934.25

3847.49 3858.01 3869.52 3880.61 3890.32 3898.17 3904.18 3908.92 3913.58

3781.71 3802.40 3822.23 3839.28 3852.27 3860.57 3864.33 3864.49 3862.89

3759.97 3781.92 3802.68 3820.34 3833.60 3841.84 3845.21 3844.65 3841.99

3738.36 3761.54 3783.23 3801.49 3815.02 3823.22 3826.22 3824.97 3821.30

316

R. FEISXELand E. HAGEN

Table A. 11. Sound Speed U(S,t,p) in m/s, computed from G(S,t,p) as U(S,t,p) = { (0P/0P)s.a }1r2 V {-(02G/Ot2)s. p / D e t } 1/2, V = (OG/OP)s. ', Det = (O2G/0t2)s.p (02G/OP2)s., - (O2G/O~p)s2 0PSU

5PSU

10PSU

15PSU

20PSU 25PSU dBar 1422.53 1429.13 1435.75 1445.34 1451.65 1457.96 1465.60 1471.63 1477.67 1483.48 1489.26 1495.04 1499.17 1504.71 1510.28 1512.87 1518.21 1523.57 1524.78 1529.94 1535.10 1535.08 1540.07 1545.04 1543.88 1548.71 1553.51 10 MPa = 1000 dBax 1438.53 1445.20 1451.87 1461.50 1467.85 1474.20 1481.91 1487.97 1494.03 1499.92 1505.72 1511.51 1515.72 1521.26 1526.81 1529.50 1534.80 1540.09 1541.46 1546.50 1551.54 1551.74 1556.55 1561.33 1560.50 1565.07 1569.60 20 MPa = 2000 dBar 1454.98 1461.70 1468A1 1477.97 1484.36 1490.73 1498.44 1504.53 1510.60 1516.52 1522.33 1528.14 1532.40 1537.94 1543.48 1546.25 1551.52 1556.78 1558.24 1563.23 1568.20 1568.55 1573.26 1577.93 1577.34 1581.78 1586.16 40 MPa = 4000 dBar 1489.09 1495.86 1502.58 1511~76 1518.17 1524.54 1532.05 1538.16 1544.24 1550.08 1555.92 1561.74 1565.98 1571.55 1577.09 1579.89 1585.18 1590A3 1591.97 1596.96 1601o91 1602.40 1607.11 1611.77 1611.40 1615.90 1620.34 100 MPa = 10000 dBar 1597.12 1603.56 1609.90 1616.74 1622.88 1628.96 1634.84 1640.74 1646.60 1651.26 1656.91 1662.51 1665.91 1671.24 1676.50 1678.80 1683.76 1688.64 1690.10 1694.73 1699.27 1700.22 1704.68 1709.09 1709.83 1714.49 1719.20

30PSU

35PSU

40PSU

1442.39 1464.29 1483.72 1500.85 1515.86 1528.95 1540.28 1550.02 1558.28

1449.06 1470.64 1489.79 1506.68 1521.47 1534.35 1545.48 1555.00 1563.02

1455.77 1477.03 1495.90 1512.54 1527.12 1539.79 1550.70 1559.98 1567.74

1458.54 1480.55 1500.10 1517.32 1532.36 1545.39 1556.58 1566.08 1574.07

1465.22 1486.91 1506.18 1523.14 1537.93 1550.70 1561.61 1570.83 1578.51

1471.93 1493.29 1512.27 1528.97 1543.52 1556.03 1566.65 1575.55 1582.90

1475.10 1497.09 1516.68 1533.95 1549.02 1562.03 1573.15 1582.56 1590.49

1481.78 1503.45 1522.75 1539.75 1554.55 1567.27 1578.08 1587.16 1594.77

1488.45 1509.80 1528.82 1545o56 1560.09 1572.51 1582.99 1591.73 1598.99

1509.25 1530.87 1550.30 1567.53 1582.61 1595.65 1606.82 1616.38 1624.73

1515.88 1537.18 1556.33 1573.30 1588.10 1600.83 1611.68 1620.94 1629.05

1522.45 1543.44 1562.33 1579.04 1593.57 1605.98 1616A9 1625.43 1633.31

1616.13 1634.98 1652.42 1668.04 1681.67 1693A1 1703.70 1713.42 1723.93

1622.24 1640.92 1658.15 1673.49 1686.74 1698.05 1708.00 1717.66 1728.64

1628.20 1646.75 1663.79 1678.83 1691.68 1702.55 1712.15 1721.77 1733.32

0 MPa = 0

0°C 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

1402.42 1426.09 1447.23 1465.93 1482.36 1496.71 1509.16 1519.86 1528.92

1409.26 1432.65 1453.47 1471.88 1488.04 1502.16 1514.43 1525.02 1534.04

1415.92 1439.02 1459.56 1477.70 1493.62 1507.53 1519.62 1530.07 1539.00

0°C 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

1418.10 1442.06 1463.40 1482.29 1498.91 1513.46 1526.11 1537.02 1546.29

1425.06 1448.67 1469.69 1488.26 1504.59 1518.87 1531.29 1542.01 1551.15

1431.83 1455.11 1475.82 1494.11 1510.17 1524.20 1536.39 1546.90 1555.87

0°C 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

1434.32 1458.40 1479.85 1498.83 1515.56 1530.24 1543.03 1554.10 1563.54

1441.37 1465.06 1486.15 1504.81 1521.24 1535.64 1548.17 1559.00 1568.25

1448.21 1471.55 1492.32 1510.69 1526.84 1540.96 1553.22 1563.80 1572.83

0oc 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

1468.18 1492.13 1513.44 1532.36 1549.09 1563.82 1576.74 1587.97 1597.62

1475.31 1498.79 1519.73 1538.32 1554.76 1569.22 1581.86 1592.83 1602.26

1482.25 1505.31 1525.91 1544.22 1560.39 1574.57 1586.93 1597.64 1606.85

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

1577.43 1598.40 1617.39 1634.55 1650.02 1663.88 1676.22 1687.16 1696.93

1584.02 1604.44 1623.09 1640.00 1655.22 1668.77 1680.75 1691.35 1700.89

1590.60 1610.59 1628.94 1645.61 1660.56 1673.78 1685.42 1695.76 1705.27

The GIBBS thermodynamic potential of seawater

3 17

Table A. 12. Potential Temperature 0(S,t,p) in °C, computed from G(S,t,p) solving the equation 6(S,t,p) = 6(S,0,0), 6(S,t,p) = -(cgG/Ot)s,p, numerically by NEWTONiteration. 0PSU Ooc 5oc I O°C 15°C 20°C 25°C 30°C 35°C 40°C

0.0000 5.0000 10.0000 15.0000 20.0000 25.0000 30.0000 35.0000 40.0000

0oc 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

0.0313 4.9795 9.9326 14.8897 19.8500 24.8126 29.7768 34.7421 39.7083

0°C 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

0.(1393 4.9401 9.8501 14.7675 19.6908 24.6184 29.5490 34.4816 39.4159

0oc 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

-0.0088 4.8093 9.6431 14.4898 19.3467 24.2110 29.0804 33.9532 38.8289

0oc 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

-0.5614 4.0822 8.7501 13.4388 18.1439 22.8610 27.5859 32.3156 37.0488

5PSU

10PSU

15PSU 20PSU 25PSU 30PSU 35PSU 40PSU 0 MPa -- 0 dBar 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 5.0000 5 . 0 0 0 0 5 . 0 0 0 0 5 . 0 0 0 0 5 . 0 0 0 0 5 . 0 0 0 0 5 . 0 0 0 0 5.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 15.0000 15.0000 15.0000 15.0000 15.0000 15.0000 15.0000 15.0000 20.0000 20.0000 20.0000 20.0000 20.0000 20.0000 20.0000 20.0000 25.0000 25.0000 25.0000 25.0000 25.0000 25.0000 25.0000 25.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 30.0000 35.0000 35.0000 35.0000 35.0000 35.0000 35.0000 35.0000 35.0000 40.0000 40.0000 40.0000 40.0000 40.0000 40.0000 40.0000 40.0000 10 MPa = 1000 dBar 0.0188 0.0071 -0.0040 -0.0147 -0.0249 -0.0348 -0.0443 -0.0534 4.9691 4 . 9 5 9 3 4 . 9 4 9 9 4 . 9 4 0 8 4.9321 4 . 9 2 3 8 4 . 9 1 5 7 4.9079 9.9240 9.9158 9 . 9 0 7 9 9 . 9 0 0 3 9 . 8 9 2 9 9 . 8 8 5 8 9 . 8 7 8 9 9.8723 14.8827 14.8759 14.8692 14.8628 14.8566 14.8505 14.8447 14.8390 19.8443 19.8386 19.8331 19.8277 19.8224 19.8172 19.8122 19.8074 24.8079 24.8033 24.7987 24.7941 24.7897 24.7853 24.7811 24.7769 29.7731 29.7693 29.7654 29.7617 29.7579 29.7543 29.7507 29.7472 34.7391 34.7361 34.7330 34.7299 34.7269 34.7239 34.7210 34.7182 39.7061 39.7037 39.7013 39.6989 39.6966 39.6943 39.6921 39.6901 20 MPa = 2000 dBar 0.0153 -0.0072 -0.0286 -0.0491 -0.0688 -0.0877 -0.1059 -0.1234 4.9201 4.9012 4 . 8 8 3 1 4 . 8 6 5 7 4 . 8 4 8 9 4 . 8 3 2 8 4 . 8 1 7 2 4.8023 9.8335 9 . 8 1 7 7 9 . 8 0 2 4 9 . 7 8 7 7 9 . 7 7 3 5 9 . 7 5 9 8 9 . 7 4 6 5 9.7337 14.7539 14.7408 14.7280 14.7156 14.7035 14.6918 14.6805 14.6695 19.6798 19.6689 19.6582 19.6477 19.6375 19.6276 19.6179 19.6085 24.6095 24.6005 24.5916 24.5829 24.5742 24.5658 24.5576 24.5496 29.5418 29.5345 29.5271 29.5198 29.5127 29.5056 29.4987 29.4920 34.4759 34.4700 34.4640 34.4581 34.4523 34.4466 34.4410 34.4356 39.4115 39.4069 39.4023 39.3977 39.3933 39.3889 39.3848 39.3808 40 MPa = 4000 dBar -0.0529 -0.0943 -0.1338 -0.1717 -0.2081 -0.2431 -0.2768 -0.3092 4.7723 4 . 7 3 7 3 4 . 7 0 3 7 4.6714 4.6402 4.6102 4.5813 4.5534 9.6124 9 . 5 8 3 0 9 . 5 5 4 6 9 . 5 2 7 2 9 . 5 0 0 6 9 . 4 7 5 0 9 . 4 5 0 2 9.4263 14.4646 14.4401 14.4163 14.3931 14.3706 14.3488 14.3276 14.3072 19.3262 19.3060 19.2861 19.2666 19.2476 19.2291 19.2110 19.1936 24.1945 24.1778 24.1613 24.1450 24.1291 24.1135 24.0983 24.0835 29.0672 29.0536 29.0400 29.0265 29.0133 29.0003 28.9877 28.9754 33.9427 33.9318 33.9208 33.9099 33.8991 33.8887 33.8786 33.8688 38.8207 38.8121 38.8035 38.7951 38.7869 38.7790 38.7714 38.7643 100 MPa = 10000 dBar -0.6461 -0.7267 -0.8042 -0.8791 -0.9515 -1.0216 -1.0893 -1.1548 4.0087 3 . 9 3 8 4 3 . 8 7 0 5 3 . 8 0 4 7 3.7411 3 . 6 7 9 5 3 . 6 1 9 9 3.5623 8.6877 8 . 6 2 7 5 8 . 5 6 9 0 8 . 5 1 2 4 8 . 4 5 7 4 8 . 4 0 4 2 8 . 3 5 2 7 8.3029 13.3869 13.3364 13.2871 13.2391 13.1925 13.1473 13.1036 13.0613 18.1019 18.0602 18.0194 17.9795 17.9406 17.9029 17.8663 17.8310 22.8276 22.7939 22.7606 22.7278 22.6958 22.6647 22.6346 22.6055 27.5598 27.5328 27.5058 27.4791 27.4529 27.4273 27.4026 27.3787 32.2952 32.2734 32.2513 32.2292 32.2075 32.1863 32.1657 32.1459 37.0323 37.0140 36.9952 36.9762 36.9575 36.9392 36.9214 36.9043

3 18

R. FEISTELand E. HAGEN

Table A. 13. Potential Density Anomaly x' (S,t,p) in kg/m 3, computed from G(S,t,p) as x' (S,t,p) = I/V' - 1000 kg/m 3, V' = (OG(S,0,0)/Op)s o, 0 from solving 6(S,t,p) = 6(S,0,0), 6(S,t,p) = -(3G/ t)s,p 0PSU 0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

-0.1563 -0.0336 -0.2980 -0.8987 -1.7949 -2.9540 -4.3511 -5.9666 -7.7845

0oc 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

-0.1542 -0.0332 -0.2921 -0.8822 -1.7640 -2.9061 -4.2839 -5.8782 -7.6732

0oc 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

-0.1537 -0.0326 -0.2849 -0.8640 -1.7315 -2.8569 -4.2159 -5.7895 -7.5622

0°C 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

-0.1569 -0.0308 -0.2674 -0.8233 -1.6622 -2.7547 -4.0772 -5.6112 -7.3414

0oc 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40oc

-0.1968 -0.0254 -0.1985 -0.6777 -1.4300 -2.4277 -3.6482 -5.0732 -6.6879

5PSU

10PSU

15PSU 20PSU 25PSU 0 MPa = 0 dBar 3.9164 7.9606 11o9937 16.0218 20.0481 3.9482 7.9071 11.8589 15.8088 19.7600 3.6099 7.4989 11.3835 15.2686 19.1573 2.9495 6.7812 10.6102 14.4415 18.2779 2.0053 5 . 7 9 0 1 9.5732 13.3596 17.1521 0.8076 4.5539 8.2989 12.0477 15.8032 -0.6203 3.0943 6.8077 10.5250 14.2492 -2.2605 1.4284 5.1157 8.8068 12.5051 -4.0976 -0.4293 3.2370 6.9070 10.5843 10 MPa = 1000 dBar 3.9173 7.9608 11.9937 16.0219 20.0487 3.9492 7.9091 11.8620 15.8133 19.7661 3.6176 7.5085 11.3950 15.2823 19.1732 2.9684 6.8024 10.6339 14.4678 18.3067 2.0387 5.8260 9.6118 13.4008 17.1960 0.8580 4.6068 8.3545 12.1060 15.8641 -0.5508 3.1663 6.8822 10.6021 14.3290 -2.1699 1 . 5 2 1 3 5.2109 8.9044 12.6050 -3.9844 -0.3141 3.3542 7.0263 10.7056 20 MPa = 2000 dBar 3.9171 7.9604 11.9934 16.0221 20.0497 3.9507 7.9118 11.8661 15o8190 19.7735 3.6266 7.5195 11.4082 15.2976 19.1908 2.9889 6.8254 10.6593 14.4957 18.3372 2.0737 5.8636 9.6519 13.4436 17.2415 0.9097 4.6611 8.4113 12.1654 15.9262 -0.4804 3.2391 6,9575 10.6799 14.4093 -2.0791 1 . 6 1 4 3 5.3062 9.0021 12.7050 -3.8716 -0.1994 3.4709 7.1450 10.8263 40 MPa = 4000 dBar 3.9139 7.9579 11.9922 16.0227 20.0526 3.9552 7.9192 11.8768 15.8333 19.7917 3.6482 7.5454 11o4385 15.3326 19.2305 3.0344 6.8757 10.7147 14.5561 18.4027 2.1479 5.9429 9.7363 13.5332 17.3363 1.0167 4.7730 8.5283 12.2874 16.0533 -0.3373 3.3869 7.1100 10.8372 14.5714 -1.8967 1.8009 5.4972 9.1974 12.9047 -3.6473 0.0287 3.7028 7.3807 11.0658 I(X}MPa = 10000 dBar 3.8824 7.9357 11.9803 16.0219 20.0636 3.9730 7.9498 11.9208 15.8912 19.8639 3.7317 7.6435 11.5516 15.4610 19.3743 3.1952 7.0519 10.9064 14.7634 18.6257 2.3951 6.2051 10.0137 13.8258 17.6440 1 . 3 5 7 5 5.1280 8.8975 12.6710 16.4513 0.1041 3.8413 7.5776 11.3180 15.0655 -1.3478 2.3615 6.0698 9.7823 13.5018 -2.9842 0.7024 4.3876 8.0769 11.7733

30PSU

35PSU

40PSU

24.0750 23.7145 23.0513 22.1212 20.9526 19.5673 17.9825 16.2126 14.2709

28.1040 27.6737 26.9521 25.9728 24.7624 23.3415 21.7264 19.9310 17.9688

32.1362 31.6388 30.8608 29.8338 28.5827 27.1269 25.4821 23.6616 21.6793

24.0763 23.7223 23.0695 22.1526 20.9992 19.6310 18.0649 16.3149 14.3943

28.1063 32.1399 27.6835 31.6506 26.9727 30.8839 26.0069 29.8705 24.8118 28.6348 23.4079 27.1960 21.8113 25.5695 20.0357 23.7686 18.0942 21.8066

24.0784 23.7316 23.0895 22.1857 21.0473 19.6957 18.1477 16.4172 14.5170

28.1096 27.6948 26.9951 26.0426 24.8626 23.4753 21.8967 20.1403 18.2188

32.1446 31.6640 30.9087 29.9088 28.6883 27.2660 25.6574 23.8754 21.9331

24.0841 23.7540 23.1340 22.2564 21.1474 19.8279 18.3146 16.6213 14.7602

28.1185 27.7214 27.0446 26.1185 24.9679 23.6125 22.0684 20.3487 18.4656

32.1569 31.6951 30.9632 29.9900 28.7988 27.4084 25.8339 24.0881 22.1834

24.1076 23.8408 23.2935 22.4950 21.4703 20.2401 18.8219 17.2306 15.4792

28.1551 27.8232 27.2198 26.3728 25.3058 24.0388 22.5888 20.9701 19.1959

32.2073 31.8121 31.1542 30.2598 29.1516 27.8486 26.3671 24.7215 22.9249

The GIBBS thermodynamic potential of seawater

3 19

Table A. 14. Th er m a l Expansion C o e f f ic ie n t tx(S,t,p) in ppm/K, computed from G(S,t,p) as o~(S,t,p) = ( l / V ) (~V/0t)s, = (~2G/~t0P)s / (~G/~P)s. t 0PSU

5PSU

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

-67.15 16.02 88.07 151.04 206.85 257.11 303.16 345.91 385.82

-46.86 32.28 100.94 161.12 214.63 263.03 307.54 348.99 387.76

-28.04 47.46 113.10 170.79 222.26 268.98 312.09 352.32 389.96

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

-30.06 45.74 111.73 169.72 221.36 268.11 311.11 351.17 388.67

- 11.42 60.69 123.58 179.00 228.53 273.56 315.15 354.03 390.51

5.87 74.68 134.80 187.93 235.58 279.05 319.34 357.10 392.56

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

5.01 73.93 134.27 187.60 235.34 278.78 318.90 356.42 391.63

22.06 87.64 145.15 196.12 241.91 283.75 322.56 358.99 393.29

37.90 100.49 155.50 204.36 248.40 288.78 326.37 361.76 395.13

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

69.05 125.65 175.89 220.85 261.57 298.99 333.85 366.69 397.72

83.10 137.09 185.04 228.02 267.05 303.04 336.74 368.64 398.94

96.23 147.89 193.80 235.00 272.49 307.18 339.79 370.78 400.33

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

214.34 244.84 273.55 300.40 325.54 349.26 371.92 393.87 415.44

220.68 250.98 278.99 304.82 328.78 351.33 372.99 394.30 415.71

227.13 257.11 284.36 309.16 331.99 353.47 374.30 395.16 416.68

JPO 36:4-D

10PSU

15PSU 20PSU 25PSU 30PSU 35PSU 40PSU 0 MPa -- 0 dBar -10.26 6.64 22.75 38.13 5 2 . 8 1 66.82 61.87 75.61 88.74 101.29 113.30 124.77 1 2 4 . 7 2 135.85 146.53 1 5 6 . 7 7 166.59 176.00 1 8 0 . 1 2 189.11 197.78 2 0 6 . 1 4 2 1 4 . 1 8 221.92 2 2 9 . 7 1 2 3 6 . 9 5 2 4 3 . 9 7 2 5 0 . 7 8 2 5 7 . 3 6 263.72 2 7 4 . 8 7 2 8 0 . 6 5 2 8 6 . 3 0 2 9 1 . 8 0 2 9 7 . 1 5 302.33 3 1 6 . 6 5 321.17 325.63 3 2 9 . 9 8 3 3 4 . 2 3 338.35 3 5 5 . 7 0 359.08 3 6 2 . 4 2 3 6 5 . 6 9 3 6 8 . 8 8 371.96 3 9 2 . 2 1 3 9 4 . 4 7 3 9 6 . 6 9 3 9 8 . 8 4 400.90 402.87 10 MPa = 1000 dBar 22.22 37.75 52.55 66.67 8 0 . 1 3 92.97 87.96 100.63 112.74 124.32 135.39 145.96 1 4 5 . 5 4 155.84 165.72 1 7 5 . 2 0 184.28 192.99 1 9 6 . 5 6 204.89 212.93 2 2 0 . 6 8 2 2 8 . 1 4 235.30 2 4 2 . 4 7 2 4 9 . 1 8 2 5 5 . 6 9 262.00 2 6 8 . 1 0 273.98 2 8 4 . 4 9 2 8 9 . 8 4 2 9 5 . 0 6 3 0 0 . 1 4 3 0 5 . 0 7 309.84 3 2 3 . 5 5 3 2 7 . 7 2 331.81 3 3 5 . 8 0 3 3 9 . 6 9 343.44 3 6 0 . 2 2 3 6 3 . 3 2 3 6 6 . 3 6 3 6 9 . 3 2 3 7 2 . 1 9 374.94 3 9 4 . 6 4 396.69 398.69 4 0 0 . 5 9 402.40 404.08 20 MPa = 2000 dBar 52.88 67.13 80.70 93.64 105.98 117.74 1 1 2 . 7 2 124.40 135.55 1 4 6 . 2 2 156.42 166.16 165.41 1 7 4 . 9 2 1 8 4 . 0 4 192.81 201.21 209.25 2 1 2 . 3 2 2 2 0 . 0 2 2 2 7 . 4 5 234.61 2 4 1 . 5 0 248.12 2 5 4 . 7 4 2 6 0 . 9 2 2 6 6 . 9 2 2 7 2 . 7 3 2 7 8 . 3 4 283.74 2 9 3 . 7 6 2 9 8 . 6 5 3 0 3 . 4 2 3 0 8 . 0 7 3 1 2 . 5 6 316.90 3 3 0 . 1 9 333.97 337.67 3 4 1 . 2 7 3 4 4 . 7 6 348.12 3 6 4 . 5 6 3 6 7 . 3 3 3 7 0 . 0 4 3 7 2 . 6 6 375.18 377.58 3 9 6 . 9 8 3 9 8 . 8 0 4 0 0 . 5 3 4 0 2 . 1 7 403.70 405.09 40 MPa = 4000 dBar 1 0 8 . 7 0 120.57 131.91 1 4 2 . 7 3 1 5 3 . 0 6 162.92 1 5 8 . 2 0 168.07 177.53 1 8 6 . 5 7 195.23 203.51 2 0 2 . 2 2 2 1 0 . 3 2 2 1 8 . 1 0 2 2 5 . 5 8 2 3 2 . 7 5 239.62 2 4 1 . 7 6 248.31 2 5 4 . 6 3 2 6 0 . 7 2 2 6 6 . 5 8 272.19 2 7 7 . 8 3 2 8 3 . 0 2 2 8 8 . 0 6 2 9 2 . 9 2 297.61 302.11 3 1 1 . 2 9 315.31 3 1 9 . 2 2 323.01 3 2 6 . 6 6 330.16 3 4 2 . 8 4 345.85 348.78 351.61 3 5 4 . 3 3 356.93 3 7 2 . 9 4 375.08 3 7 7 . 1 4 3 7 9 . 1 3 3 8 1 . 0 0 382.76 4 0 1 . 7 5 403.12 404.42 4 0 5 . 6 3 4 0 6 . 7 3 407.70 100 MPa = 10000 dBar 233.57 239.96 2 4 6 . 2 7 2 5 2 . 4 9 258.59 264.58 2 6 3 . 1 6 2 6 9 . 0 9 2 7 4 . 9 0 2 8 0 . 5 6 2 8 6 . 0 6 291.41 2 8 9 . 6 1 294.71 2 9 9 . 6 5 3 0 4 . 4 2 309.01 313.41 3 1 3 . 3 7 317.43 321.23 3 2 5 . 0 3 3 2 8 . 5 5 331.88 3 3 5 . 1 0 338.07 3 4 0 . 8 9 3 4 3 . 5 5 3 4 6 . 0 3 348.33 3 5 5 . 5 7 357.59 3 5 9 . 5 0 3 6 1 . 2 8 3 6 2 . 9 3 364.42 3 7 5 . 6 6 377.01 378.33 3 7 9 . 5 9 3 8 0 . 7 6 381.84 396.21 3 9 7 . 3 7 398.58 3 9 9 . 8 3 4 0 1 . 0 7 402.30 4 1 8 . 0 1 419.60 4 2 1 . 3 7 4 2 3 . 2 8 425.30 427.39

320

R. I~ISTELand E. HAGEN

Table A. 15. Adiabatic Expansion Coefficient c~'(S,t,p) in mK/kJ, computed from G(S,t,p) as (x' (S,t,p) = ( I / V ) (bV/~C~)s,p = F / V = - (~R3/~tOp) s / { (bZG/bt2)s, p (3G/3P)s,t} 0PSU

5PSU

10PSU

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

-4.349 1.061 5.949 10.399 14.501 18.342 21.996 25.513 28.915

-3.061 2.154 6.870 11.173 15.152 18.890 22.460 25.907 29.248

-1.847 3.192 7.755 11.928 15.796 19.444 22.937 26.319 29.598

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

-1.968 3.058 7.613 11.776 15.630 19.254 22,715 26.058 29.297

-0.754 4.088 8.482 12.507 16.245 19.774 23.157 26.436 29.622

0.391 5.068 9.318 13.221 16.856 20.299 23.611 26.830 29.959

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

0.332 4.986 9.223 13.112 16.729 20.146 23.423 26.598 29.681

1.471 5.954 10.040 13.800 17.308 20.635 23.839 26.956 29°992

2.546 6.876 10.829 14.473 17.884 21.130 24.267 27.327 30.312

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

4.649 8.610 12.255 15.638 18.818 21.852 24.786 27.650 30.444

5.634 9.455 12.974 16.245 19.328 22.280 25.149 27.962 30.720

6.568 10.267 13.673 16.844 19.839 22.716 25.522 28.285 31.006

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

14.951 17.336 19.648 21.884 24.057 26.190 28.308 30.432 32.569

15.485 17.869 20.143 22.317 24.417 26.477 28.537 30.630 32.779

16.031 18.405 20.637 22.748 24,777 26,770 28.782 30.860 33.042

15PSU 20PSU 25PSU 30PSU 0 MPa = 0 dBar - 0 . 6 8 2 0.444 1.535 2.593 4.193 5.163 6.105 7.020 8.614 9.450 10.265 1 1 . 0 5 9 1 2 , 6 6 6 13.389 14.097 1 4 . 7 9 0 1 6 , 4 3 4 17.062 17.682 18.291 1 9 , 9 9 7 2 0 . 5 4 6 21.091 2 1 . 6 3 0 2 3 . 4 1 8 2 3 . 9 0 0 24.379 2 4 . 8 5 5 2 6 . 7 3 6 2 7 . 1 5 5 27.573 2 7 . 9 8 8 2 9 . 9 5 3 3 0 . 3 0 9 30.663 3 1 . 0 1 3 10 MPa = 1000 dBar 1.490 2.551 3.578 4.572 6.013 6.929 7.818 8.682 1 0 . 1 3 1 1 0 . 9 2 2 11.693 1 2 . 4 4 5 1 3 . 9 2 0 14,605 15.277 1 5 . 9 3 5 1 7 . 4 6 0 18.056 18.645 1 9 . 2 2 3 2 0 . 8 2 4 21.345 2 1 . 8 6 2 2 2 . 3 7 3 2 4 . 0 6 8 24.525 2 4 . 9 8 0 2 5 . 4 3 0 2 7 . 2 2 7 2 7 . 6 2 4 2 8 . 0 2 0 28.411 3 0 . 2 9 9 30,638 3 0 . 9 7 2 31.301 20 MPa = 2000 dBar 3.578 4.574 5.539 6.472 7.767 8.631 9.469 10.283 1 1 . 5 9 6 12o343 13.072 1 3 . 7 8 4 1 5 . 1 3 4 15.782 16.418 1 7 . 0 4 0 1 8 . 4 5 5 19.018 19.574 2 0 . 1 2 1 21.625 22.117 22.604 23.085 2 4 . 6 9 7 2 5 . 1 2 6 25.553 2 5 . 9 7 4 2 7 . 7 0 0 28.073 2 8 . 4 4 3 2 8 . 8 0 8 3 0 . 6 3 3 30.951 3 1 . 2 6 4 3 1 . 5 6 9 40 MPa = 41300 dBar 7.469 8.340 9.184 10.002 1 1 . 0 5 3 11.818 12.561 1 3 . 2 8 5 1 4 . 3 5 5 1 5 . 0 2 2 15.673 1 6 . 3 0 9 1 7 . 4 3 2 18.011 1 8 . 5 7 9 19B135 20,345 20.846 21.339 21.824 23.152 23.584 24.012 24.433 25.898 26,272 26.642 27.006 2 8 . 6 1 1 28.935 2 9 . 2 5 5 2 9 . 5 6 9 3 1 . 2 9 1 3 1 . 5 7 2 31.847 3 2 . 1 1 5 100 MPa = 10000 dBar 1 6 . 5 8 1 17.133 17.686 18.237 18,941 19.473 20.001 2 0 . 5 2 3 2 1 . 1 2 7 2 1 . 6 0 9 22.085 22.551 2 3 . 1 7 3 2 3 . 5 9 0 23.997 2 4 . 3 9 4 25.131 25.477 25.815 26.143 27.062 27.349 27.629 27.902 2 9 . 0 3 2 29,283 2 9 . 5 3 2 2 9 . 7 7 8 3 1 . 1 0 5 31.359 31.619 3 1 . 8 8 2 3 3 . 3 3 3 33,645 33.973 3 4 . 3 1 4

35PSU

40PSU

3.618 4.613 7.909 8.772 11.833 12.587 15.469 16.132 18.891 19.478 2 2 . 1 6 2 22.686 2 5 . 3 2 6 25.792 2 8 . 3 9 9 28.805 3 1 . 3 5 8 31.697 5.536 9.521 13.178 16.578 19.792 22.877 25.875 28.797 31.623

6,470 10.335 13.891 17.208 20.350 23.373 26.313 29.177 31.936

7.377 11.074 14.477 17.650 20.658 23.559 26.390 29.166 31.867

8.253 11.842 15.152 18.245 21.185 24.025 26.799 29.517 32.154

10.796 13.988 16.929 19.679 22.300 24.847 27.364 29.876 32.374

11.566 14.672 17.534 20.211 22.765 25.252 27.715 30.174 32.623

1 8 . 7 8 6 19.332 2 1 . 0 3 9 21.549 2 3 . 0 0 9 23.456 24.781 25.156 26.461 26.766 2 8 . 1 6 5 28.419 3 0 . 0 1 9 30.253 3 2 . 1 4 6 32.410 3 4 . 6 6 5 35.024

The GIBBS thermodynamic potential of seawater

3 21

Table A. 16. Isothermal Compressibility K(S,t,p) in ppm/MPa, computed from G(S,t,p) as K(S,t,p) = - ( l / V ) (OV/0P)s, ~= -(~2G/0p2)s. , / (OG/OP)s. t 0PSU

5PSU

10PSU

0oc 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

508.82 491.74 478.11 467.33 458.91 452.45 447.68 444.38 442.39

501.70 485.37 472.34 462.03 453.96 447.77 443.20 440.02 438.08

494.91 479.27 466.79 456.90 449.16 443.23 438.84 435.80 433.93

0oc 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

494.91 478.71 465.74 455.43 447.32 441.07 436.41 433.14 431.10

488.08 472.60 460.20 450.33 442.57 436.59 432.14 429.02 427.09

481.57 466.74 454.86 445.39 437.95 432.23 427.99 425.04 423.23

0oc 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

481.37 466.06 453.76 443.94 436.17 430.14 425.61 422.39 420.34

474.84 460.21 448.44 439.03 431.59 425.83 421.51 418.47 416.55

468.59 454.59 443.31 434.28 427.15 421.64 417.53 414.67 412.88

0oC 5oC 10°C 15°C 20°C 25°C 30°C 35°C 40°C

455.45 441.92 430.97 422.14 415.09 409.56 405.34 402.29 400.28

449.49 436.59 426.10 417.62 410.85 405.55 401.52 398.64 396.76

443.78 431.44 421.37 413.22 406.71 401.63 397.80 395.08 393.33

0oC 5oc 10°C 15°C 20°C 25°C 30°C 35oC 40°C

387.53 378.93 371.75 365.76 360.83 356.83 353.70 351.33 349.64

383.19 375.04 368.15 362.36 357.59 353.75 350.77 348.57 346.99

378.95 371.18 364.53 358.94 354.33 350.66 347.85 345.77 344.27

15PSU 20PSU 25PSU 0 MPa ---0 dBar 488.32 481.90 4 7 5 . 6 1 473.34 467.55 461.86 4 6 1 . 3 7 4 5 6 . 0 7 450.85 4 5 1 . 8 9 446.96 442.11 444.46 439.83 4 3 5 . 2 7 4 3 8 . 7 7 4 3 4 . 3 8 430.04 434.57 430.37 426.22 4 3 1 . 6 7 4 2 7 . 6 1 423.60 429.90 425.94 4 2 2 . 0 5 10 MPa = 1000 dBar 475.25 469.10 463.08 46 1.05 4 5 5 . 4 8 4 5 0 . 0 3 449.64 444.54 439.52 440.56 435.82 4 3 1 . 1 5 4 3 3 . 4 3 4 2 8 . 9 8 424.60 427.96 423.76 419.61 423.93 419.93 415.99 421.14 417.32 4 1 3 . 5 5 419.46 415.77 412.15 20 MPa = 2000 dBar 462.55 4 5 6 . 6 5 450.90 449.13 443.79 438.56 438.30 433.39 428.57 4 2 9 . 6 3 425.06 4 2 0 . 5 6 4 2 2 . 7 9 418.50 4 1 4 . 2 8 4 1 7 . 5 2 413.48 409.50 4 1 3 . 6 3 409.80 4 0 6 . 0 3 4 1 0 . 9 5 407.30 403.71 409.31 405.81 402.37 40 MPa = 4000 dBar 4 3 8 . 2 5 4 3 2 . 8 6 427.61 426.42 421.52 416.72 416.74 412.21 4 0 7 . 7 6 408.90 404.66 4 0 0 . 4 9 402.64 398.65 394.72 3 9 7 . 7 8 3 9 4 . 0 0 390.29 394.15 390.56 387.04 3 9 1 . 6 0 3 8 8 . 1 7 384.81 3 8 9 . 9 7 3 8 6 . 6 6 383.41 100 MPa = 10000 dBar 374.81 370.78 366.86 3 6 7 . 3 7 363.63 3 5 9 . 9 6 3 6 0 . 9 6 357.43 3 5 3 . 9 6 3 5 5 . 5 5 3 5 2 . 2 0 348.91 3 5 1 . 1 2 347.95 3 4 4 . 8 4 3 4 7 . 6 1 3 4 4 . 6 2 341.68 344.96 342.12 339.34 3 4 3 . 0 0 340.27 3 3 7 . 5 8 3 4 1 . 5 2 338.79 336.08

30PSU

35PSU 40PSU

469.46 4 6 3 . 4 1 457.47 4 5 6 . 2 8 4 5 0 . 7 9 445.38 4 4 5 . 7 1 440.65 435.66 437.32 4 3 2 . 5 9 427.92 4 3 0 . 7 5 4 2 6 . 2 9 421.88 425.76 421.51 417.31 4 2 2 . 1 1 4 1 8 . 0 5 414.03 4 1 9 . 6 5 4 1 5 . 7 5 411.89 4 1 8 . 2 2 4 1 4 . 4 5 410.73 457.20 444.68 434.59 426.55 420.27 415.52 412.10 409.84 408.59

451.42 439.42 429.73 422.01 416.00 411.47 408.26 406.19 405.09

445.75 434.24 424.94 417.53 411.77 407.47 404.46 402.57 401.64

445.28 433.43 423.83 416.14 410.12 405.57 402.31 400.18 398.99

439.77 428.40 419.16 411.78 406.02 401.70 398.65 396.70 395.67

434.37 423.45 414.58 407.48 401.97 397.88 395.04 393.28 392.40

422.49 412.02 403.39 396.39 390.85 386.63 383.58 381.51 380.22

4 1 7 . 4 8 412.59 407.42 402.91 399.11 394.89 3 9 2 . 3 6 388.40 3 8 7 . 0 5 383.31 3 8 3 . 0 4 379.51 380.18 376.83 3 7 8 . 2 7 375.08 377.08 374.00

363.05 356.38 350.56 345.69 341.79 338.82 336.63 334.96 333.41

359.35 352.88 347.24 342.53 338.82 336.03 333.99 332.40 330.77

355.77 349.47 343.99 339.45 335.92 333.32 331.43 329.90 328.18

322

R. b-lnSTELand E. HAGEN

Table A. 17. Adiabatic Compressibility K ' (S,t,p) in ppm/MPa, computed from G(S,t,p) as K'(S,t,p) = - ( l / V ) (OV/OP)s~ = V / U s =

( ~ 2 G / ~ P ) s 2 - (~2G/~t2)s, e (~G/~p2)s.t (~2G/~t2)s. p (~G/~P)s. t

0PSU

5PSU

10PSU

15PSU

20PSU

25PSU

30PSU

35PSU

40PSU

0 MPa = 0 dBar

0°C 5oC 10°C 15°C 20°C 25°C 30°C 35°C 40°C

508.53 491.72 477.59 465.76

501.56 485.30 471.65 460.23

494.85 479.12 465.92 454.88

488.31 473.09 460.31 449.63

481.89 467.16 454.80 444.46

475.58 461.33 449.37 439.37

469.36 455.59 444.02 434.34

463.22 457.17 4 4 9 . 9 2 444.32 438.73 433.51 4 2 9 . 3 6 424.44

455.90

450.71

445.67

440.72

435.84

431.02

426.26

421.55

447.72 440.99 435.50 431.15

442.81 436.28 430.96 426.69

438.02 431.71 426.54 422.39

0oc 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

494.85 478.57 464.89 453.44 443.87 435.92 429.34 423.97 419.68

488.07 472.35 459.16 448.11 438.88 431.21 424.87 419.68 415.52

481.56 466.37 453.61 442.94 434.02 426.62 420.51 415.51 411.51

0oc 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

481.37 465.69 452.53 441.50 432.26 424.56 418.17 412.93 408.72

474.80 459.70 447.00 436.36 427.45 420.03 413.88 408.85 404.80

468.50 453.92 44 1.65 431.37 422.77 415.61 409.71 404.88 401.00

0°C 5°C 10°C 20°C 25°C 30°C 35°C 40°C

455.14 440.86 428.85 418.75 410.25 403.12 397.17 392.26 388.28

449.04 435.32 423.75 413.99 405.79 398.91 393.19 388.48 384.66

443.17 429.96 418.78 409.35 401.42 394.80 389.30 384.78 381.12

0oC 5oC 10°C 15°C 20°C 25°C 30°C 35°C 40°C

384.46 374.87 366.59 359.45 353.29 348.02 343.54 339.74 336.52

379.94 370.76 362.78 355.85 349.89 344.82 340.54 336.93 333.83

375.49 366.68 358.94 352.23 346.48 341.61 337.52 334.06 331.01

433.32 428.68 424.10 4 2 7 . 2 1 4 2 2 . 7 7 418.39 422.20 417.94 413.73 4 1 8 . 1 9 414.06 410.01 10 MPa = 1000 dBar 475.22 469.00 462.90 4 6 0 . 5 3 454.80 4 4 9 . 1 7 448.19 442.87 437.63 4 3 7 . 8 7 432.88 4 2 7 . 9 7 429.26 424.56 4 1 9 . 9 3 422.11 417.67 413.29 4 1 6 . 2 2 412.01 4 0 7 . 8 5 411A3 407.41 403.46 4 0 7 . 5 9 4 0 3 . 7 5 399.98 20 MPa = 2000 dBar 462.36 456.35 450.47 4 4 8 . 2 7 4 4 2 . 7 4 437.31 436.42 431.28 4 2 6 . 2 3 4 2 6 . 4 7 421.66 4 1 6 . 9 3 418.17 413.65 409.19 4 1 1 . 2 8 407.01 402.80 405.60 401.57 3 9 7 . 5 9 400.98 397.16 393.40 3 9 7 . 2 9 393.65 390.08 40 MPa = 4000 dBar 437.46 4 3 1 . 8 9 426.45 424.72 419.60 4 1 4 . 5 7 413.92 409.15 404.46 404.80 400.32 3 9 5 . 9 2 3 9 7 . 1 4 3 9 2 . 9 2 388.77 3 9 0 . 7 5 386.77 3 8 2 . 8 6 3 8 5 . 4 7 381.72 378.03 3 8 1 . 1 5 377.59 374.09 3 7 7 . 6 4 374.22 3 7 0 . 8 7 100 MPa = 10000 dBar 371.15 366.90 362.76 3 6 2 . 6 4 358.68 3 5 4 . 7 9 355.15 351.40 347.72 3 4 8 . 6 5 345.11 341.63 3 4 3 . 1 0 3 3 9 . 7 7 336.51 3 3 8 . 4 3 3 3 5 . 3 2 332.27 3 3 4 . 5 3 3 3 1 . 6 0 328.73 3 3 1 . 2 0 328.38 3 2 5 . 6 0 328.16 325.29 322.44

15°C

416.88

419.57 415.08 414.06 409.77 4 0 9 . 5 8 405.48 4 0 6 . 0 3 402.10

410.63 405.52 401.42 398.23

456.90 443.63 432A7 423.12 415.36 408.96 403.74 399.56 396.28

450.99 438.17 427.37 418.34 410.84 404.68 399.69 395.72 392.64

445.17 432.79 422.35 413.62 406.38 400.45 395.69 391.93 389.06

444.69 431.98 421.25 412.26 404.79 398.65 393.68 389.70 386.58

439.01 426.73 416.35 407.66 400A5 394.56 389.82 386.06 383.14

433.43 421.56 411.53 403.12 396.17 390.52 386.01 382.48 379.76

421.12 409.65 399.86 391.59 384.70 379.02 374.40 370.66 367.58

415.90 404.81 395.33 387.33 380.68 375.24 370.84 367.29 364.35

410.80 400.07 390.89 383.14 376.73 371.52 367.34 363.98 361.19

358.73 350.98 344.11 338.23 333.32 329.30 325.94 322.88 319.61

354.81 347.25 340.58 334.90 330.21 326A1 323.22 320.22 316.81

351.01 343.62 337.13 331.65 327.19 323.60 320.58 317.63 314.06

The GIBBS thermodynamic potential of seawater

32 3

Table A. 18. Haline Contraction Coefficient B(S,t,p) in ppm/PSU, computed from G(S,t,p) as l~(S,t,p) = -(I/V) (0V/0S)t. p = -(~G/0~p) s / (0G/OP)s,t 0PSU

5PSU

10PSU

15PSU

20PSU

25PSU

30PSU

35PSU

40PSU

786.60 773.00 761.88 752.80 745.37 739.33 734.47 730.70 728.02

784.05 771.06 760.42 751.70 744.54 738.69 733.97 730.31 727.72

781.73 769.32 759.15 750.79 743.90 738.25 733.68 730.13 727.65

0 MPa = 0 dBar

O°C 5oC IO°C 20°C 25°C 30°C 35°C 40°C

824.77 805.14 789.70 777.76 768.70 761.98 757.17 753.89 751.88

807.48 790.02 776.02 764.90 756.15 749.33 744.12 740.25 737.56

801.08 784.62 771.34 760.69 752.21 745.50 740.27 736.31 733.51

OoC 5oC IO°C 15°C 20°C 25°C 30°C 35°C 40°C

810.02 792.01 777.86 766.92 758.62 752.49 748.08 745.06 743.17

793.93 777.86 764.96 754.70 746.63 740.34 735.52 731.95 729.45

788.00 772.84 760.59 750.75 742.90 736.70 731.87 728.22 725.63

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

795.91 779.48 766.53 756.53 748.97 743.40 739.43 736.73 735.04

780.93 766.20 754.33 744.89 737.46 731.69 727.31 724.07 721.82

775.46 761.54 750.25 741.17 733.94 728.24 723.84 720.54 718.21

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

769.65 756.15 745.43 737.11 730.85 726.31 723.19 721.19 720.03

756.60 744.33 734.34 726.36 720.10 715.31 711.76 709.22 707.51

751.96 740.32 730.77 723.06 716.95 712.20 708.62 706.02 704.25

7 9 6 . 4 3 792.65 7 8 9 . 4 3 780.78 777.74 775.19 7 6 8 . 1 1 765.61 7 6 3 . 5 9 7 5 7 . 8 8 7 5 5 . 7 8 754.13 7 4 9 . 6 6 747.83 7 4 6 . 4 4 7 4 3 . 0 9 7 4 1 . 4 2 740.21 7 3 7 . 9 2 736.33 7 3 5 . 2 2 733.95 732.40 731.37 7 3 1 . 1 2 729.60 728.61 10 MPa = 1000 dBar 783.70 780.22 777.27 7 6 9 . 3 0 766.49 7 6 4 . 1 6 7 5 7 . 5 9 755.30 7 5 3 . 4 4 7 4 8 . 1 3 7 4 6 . 1 9 744.68 740.52 738.82 737.55 734.44 732.89 731.78 7 2 9 . 6 6 728.19 7 2 7 . 1 9 726.01 724.59 723.67 7 2 3 . 4 2 7 2 2 . 0 4 721.18 20 MPa = 2000 dBar 771.50 768.32 7 6 5 . 6 3 7 5 8 . 2 7 755.71 7 5 3 . 5 8 747.48 745.37 743.69 738.74 736.96 735.59 7 3 1 . 7 2 730.15 729.01 7 2 6 . 1 3 724.71 7 2 3 . 7 2 7 2 1 . 7 7 720.43 7 1 9 . 5 4 7 1 8 . 4 8 7 1 7 . 1 9 716.38 716.17 714.93 714.20 40 MPa = 4000 dBar 748.67 746.06 743.89 737.56 735.45 733.75 728.41 726.67 725.32 7 2 0 . 9 7 7 1 9 . 5 0 718.42 715.03 713.74 712.85 7 1 0 . 3 8 709.21 7 0 8 . 4 6 706.84 705.75 705.10 704.25 703.22 702.65 7 0 2 . 4 8 7 0 1 . 5 0 701.01

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

706.08 700.00 694.29 689.42 685.69 683.24 682.03 681.89 682.44

696.16 689.82 683.99 679.07 675.24 672.55 670.89 669.95 669.29

693.33 687.01 681.28 676.46 672.74 670.10 668.39 667.28 666.25

691.65 685.40 679.79 675.12 671.53 669.00 667.31 666.12 664.88

15°C

774.67 772.34 770.22 7 6 2 . 1 6 760.40 758.82 7 5 1 . 8 9 7 5 0 . 5 7 749.43 7 4 3 . 4 7 7 4 2 . 4 9 741.69 7 3 6 . 5 9 7 3 5 . 8 6 735.31 7 3 1 . 0 0 7 3 0 . 4 5 730.09 7 2 6 . 5 3 7 2 6 . 1 2 725.91 7 2 3 . 1 0 7 2 2 . 8 0 722.71 7 2 0 . 7 1 720.51 720.55 763.27 751.77 742.30 734.51 728.16 723.03 718.98 715.92 713.84

761.17 750.19 741.12 733.66 727.54 722.59 718.67 715.73 713.76

759.26 748.79 740.13 732.97 727.09 722.32 718.55 715.73 713.90

742.02 732.32 724.25 717.62 712.25 708.01 704.77 702.42 700.88

740.37 731.11 723.39 717.02 711.86 707.79 704.68 702.44 701.02

738.90 730.07 722.68 716.59 711.65 707.74 704.77 702.67 701.37

689.40 683.49 678.34 674.18 671.10 668.98 667.55 666.36 664.82

689.15 689.07 6 8 3 . 3 8 683.44 678.40 678.64 674.44 674.87 671.54 672.16 6 6 9 . 5 8 670.37 6 6 8 . 2 6 669.19 6 6 7 . 1 3 668.12 6 6 5 . 5 5 666.52

100 MPa = 10000 dBar

690.56 684.41 678.94 674.43 671.00 668.58 666.96 665.74 664.36

689.84 683.81 678.49 674.16 670.89 668.62 667.09 665.87 664.39

324

R. FEXSTELand E. HAGEN

Table A. 19. Adiabatic Contraction Coefficient l]'(S,t,p) in ppm/PSU, computed from G(S,t,p) as 13'(S,t,p) = -(l/V) (0V/0S)ov (0:G/OS0t)p (02G/Otc3p)s - (O2G/Ot2)s,p (O2G/OS0P)t (~TG/Ot2)s,p (OG/OP)s,t 0PSU

5PSU

10PSU

20°C 25°C 30°C 35°C 40°C

825.21 804.89 787.57 772.85 760.31 749.53 740.13 731.78 724.23

806.67 790.32 776.22 764.03 753.46 744.21 736.01 728.66 722.01

800.92 784.51 770.21 757.73 746.76 737.02 728.27 720.32 713.09

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

810.30 791.20 774.94 761.14 749.37 739.24 730.37 722.42 715.16

793.75 778.32 765.01 753.51 743.53 734.79 727.02 720.02 713.65

788.02 772.53 759.04 747.25 736.88 727.66 719.36 711.81 704.91

0oc 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

795.85 777.99 762.79 749.88 738.87 729.38 721.06 713.57 706.67

781.22 766.71 754.18 743.36 733.96 725.73 718.42 711.83 705.80

775.50 760.94 748.23 737.12 727.34 718.66 710.84 703.73 697.22

0oc 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

768.51 753.15 739.98 728.73 719.12 710.84 703.58 697.05 690.98

757.38 744.71 733.68 724.11 715.81 708.56 702.16 696.40 691.09

751.73 738.98 727.76 717.90 709.24 701.58 694.73 688.50 682.75

0°C 5°C 10°C

700.68 692.32 684.09 676.50 669.83 664.20 659.53 655.57 651.88

696.64 688.94 681.52 674.76 668.90 663.93 659.71 655.89 651.93

691.13 683.03 675.30 668.31 662.21 656.99 652.42 648.09 643.43

0oc 5oc 10°C

15°C

15°C 20°C 25°C 30°C 35°C 40°C

15PSU 20PSU 25PSU 30PSU 0 MPa -- 0 dBar 796.44 7 9 2 . 6 0 789.15 785.97 7 8 0 . 1 8 776.57 7 7 3 . 3 9 7 7 0 . 5 1 765.94 762.47 759.50 756.85 7 5 3 . 4 1 750.01 7 4 7 . 1 6 7 4 4 . 6 7 742.31 738.90 736.09 733.68 732.37 728.86 726.04 723.67 723.35 719.72 716.84 714.45 7 1 5 . 1 2 711.33 7 0 8 . 3 7 705.96 707.61 703.67 700.65 698.22 10 MPa = 1000 dBar 7 8 3 . 6 1 779.86 776.50 773.43 768.27 764.75 761.67 758.89 7 5 4 . 8 3 751.45 7 4 8 . 5 7 7 4 6 . 0 1 7 4 3 . 0 0 739.67 7 3 6 . 9 0 7 3 4 . 4 9 732.50 729.16 726.42 724.10 7 2 3 . 0 8 7 1 9 . 6 6 716.91 7 1 4 . 6 2 714.54 710.99 708.20 705.90 7 0 6 . 7 2 703.05 7 0 0 . 2 0 6 9 7 . 8 9 699.58 695.79 692.90 690.60 20 MPa = 2000 dBar 7 7 1 . 1 8 767.54 764.30 761.35 7 5 6 . 7 6 753.34 750.37 747.70 7 4 4 . 1 0 740.81 7 3 8 . 0 2 735.56 7 3 2 . 9 4 729°70 727.01 724.70 7 2 3 . 0 3 7 1 9 . 7 7 717.13 7 1 4 . 8 9 7 1 4 . 1 6 7 1 0 . 8 2 708.17 705.97 7 0 6 . 1 2 702.67 699.98 697.79 6 9 8 . 7 7 695.21 6 9 2 . 4 7 6 9 0 . 2 8 6 9 2 . 0 4 688.38 685.62 683.46 40 MPa = 4000 dBar 7 4 7 . 6 1 7 4 4 . 2 3 7 4 1 . 2 8 738.62 7 3 4 . 9 7 7 3 1 . 7 6 729.02 7 2 6 . 6 1 7 2 3 . 7 6 7 2 0 . 6 6 718.07 7 1 5 . 8 3 713.85 710.79 708.29 706.18 7 0 5 . 0 7 701.99 6 9 9 . 5 4 697.50 6 9 7 . 2 5 6 9 4 . 1 0 691.65 689.66 6 9 0 . 1 9 686.97 6 8 4 . 5 0 6 8 2 . 5 3 6 8 3 . 7 6 6 8 0 . 4 4 6 7 7 . 9 4 675.99 6 7 7 . 8 1 674.41 6 7 1 . 9 0 6 6 9 . 9 8 100 MPa = 10000 dBar 687.69 685.19 683.24 681.66 6 7 9 . 4 0 6 7 6 . 8 0 674.80 673.20 6 7 1 . 5 6 668.93 6 6 6 . 9 6 665.42 6 6 4 . 5 0 6 6 1 . 9 0 6 6 0 . 0 0 658.57 6 5 8 . 3 6 655.79 653.98 652.67 6 5 3 . 0 6 650.51 6 4 8 . 7 6 647.54 6 4 8 . 3 5 6 4 5 . 7 4 643.98 642.79 6 4 3 . 7 5 6 4 0 . 9 7 6 3 9 . 1 0 637.82 6 3 8 . 6 4 6 3 5 . 5 2 6 3 3 . 3 7 631.85

35PSU

40PSU

782.97 780.11 7 6 7 . 8 4 765.33 7 5 4 . 4 3 752.19 7 4 2 . 4 3 740.38 7 3 1 . 5 6 729.65 721.61 719.78 712.41 710.63 703.93 702.19 6 9 6 . 2 2 694.54 770.54 756.33 743.68 732.34 722.06 712.64 703.96 695.97 688.72

767.79 753.91 741.53 730.38 720.22 710.89 702.27 694.33 687.16

758.59 755.98 7 4 5 . 2 5 742.95 7 3 3 . 3 4 731.29 7 2 2 . 6 4 720.77 7 1 2 . 9 4 711.20 704.08 702.43 695.94 694.35 6 8 8 . 4 7 686.95 681.71 680.28 736.17 724.41 713.84 704.33 695.76 688.00 680.92 674.43 668.49

733.87 722.38 712.02 702.68 694.23 686.57 679.57 673.16 667.31

680.32 679.17 6 7 1 . 8 9 670.78 6 6 4 . 2 0 663.20 6 5 7 . 4 7 656.63 6 5 1 . 7 2 651.03 646.71 646.16 642.01 641.53 6 3 6 . 9 8 636.45 6 3 0 . 7 8 630.04

The GIBBS thermodynamic potential of seawater

325

Table A.20. Adiabatic Lapse Rate F(S,t,p) in m K / M P a , computed from G(S,t,p) as F(S,t,p)= (~t/3p)o, s = - (~R3/~tc3p) s / (~3/~t2)s.p 0PSU

5PSU

10PSU

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

-4.350 1.061 5.950 10.408 14.527 18.396 22.092 25.667 29.142

-3.049 2.145 6.846 11.140 15.121 18.875 22.474 25.966 29.369

-1.833 3.167 7.698 11.847 15.706 19.356 22.866 26.281 29.611

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

-1.959 3.043 7.580 11.733 15.588 19.225 22.714 26.100 29.398

-0.747 4.053 8.412 12.414 16.140 19.671 23.070 26.381 29.615

0.386 5.004 9.206 13.072 16.685 20.119 23.437 26.676 29.844

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

0.328 4.939 9.140 13.005 16.610 20.028 23.321 26.527 29.657

1.451 5.875 9.912 13.636 17.121 20.439 23.648 26.786 29.859

2.501 6.759 10.651 14.248 17.626 20.853 23.986 27.057 30.070

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

4.561 8.451 12.038 15.376 18.525 21.542 24.474 27.349 30.171

5.506 9.246 12.698 15.915 18.959 21.886 24.743 27.559 30.337

6.395 10.003 13.333 16.442 19.390 22.235 25.022 27.780 30.511

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

14.303 16.604 18.842 21.016 23.140 25.234 27.324 29.430 31.561

14.762 17.055 19.251 21.360 23.406 25.425 27.452 29.522 31.657

15.229 17.506 19.656 21.699 23.671 25.620 27.595 29.644 31.805

15PSU 20PSU 25PSU 0 MPa = 0 dBar -0.674 0.437 1.505 4.144 5.083 5.987 8.517 9.308 10.072 1 2 . 5 3 3 13.198 1 3 . 8 4 4 1 6 . 2 7 8 16.837 17.384 1 9 . 8 3 2 2 0 . 3 0 2 20.763 2 3 . 2 6 0 23.651 2 4 . 0 3 7 26.600 26.917 27.232 2 9 . 8 5 6 30.101 30.341 10 MPa = 1000 dBar 1.465 2.499 3.491 5.915 6.790 7.632 9.971 10.709 1 1 . 4 2 2 1 3 . 7 1 2 14.334 14.937 1 7 . 2 1 9 17.741 18.251 20.563 21.001 21.431 2 3 . 8 0 3 24.167 2 4 . 5 2 5 26.973 27.268 27.558 30.073 30.300 30.520 20 MPa = 2000 dBar 3.502 4.460 5.380 7.606 8.419 9.202 11.362 12.050 12.714 1 4 . 8 4 3 15.422 15.984 1 8 . 1 2 2 18.607 19.081 21.264 21.669 22.066 24.323 24.657 24.985 2 7 . 3 2 8 27.597 2 7 . 8 6 0 30.279 30.484 30.682 40 MPa = 4000 dBar 7.245 8.059 8.842 1 0 . 7 2 9 11.429 12.103 1 3 . 9 4 7 14.542 15.117 1 6 . 9 5 5 17.455 17.941 1 9 . 8 1 4 20.229 2 0 . 6 3 4 2 2 . 5 8 1 22.921 2 3 . 2 5 4 25.300 25.575 25.844 2 8 . 0 0 1 2 8 . 2 1 9 28.431 3 0 . 6 8 3 30.851 31.011 100 MPa = 10000 dBar 1 5 . 6 9 8 1 6 . 1 6 5 16.628 1 7 . 9 5 4 18.395 18.829 2 0 . 0 5 4 2 0 . 4 4 2 20.821 22.029 22.350 22.659 2 3 . 9 2 9 24.178 2 4 . 4 1 7 2 5 . 8 1 2 2 5 . 9 9 9 26.178 2 7 . 7 4 2 2 7 . 8 8 9 28.033 29.780 29.924 30.072 3 1 . 9 7 9 32.171 3 2 . 3 7 7

30PSU

35PSU

40PSU

2.532 3 . 5 1 9 4.470 6.857 7.696 8.503 1 0 . 8 1 0 11.523 12.210 1 4 . 4 7 0 1 5 . 0 7 7 15.664 17.916 18.434 18.937 2 1 . 2 1 5 21.656 22.087 2 4 . 4 1 6 2 4 . 7 8 8 25.151 27.541 2 7 . 8 4 4 28.139 3 0 . 5 7 7 3 0 . 8 0 4 31.024 4.444 5.360 8.443 9.223 12.111 1 2 . 7 7 6 15.523 16.090 18.749 19.232 21.852 22.262 2 4 . 8 7 7 25.220 27.842 28.119 30.733 30.938

6.240 9.974 13.417 16.639 19.702 22.662 25.555 28.387 31.132

6.263 9.956 13.356 16.530 19.543 22.454 25.306 28.117 30.872

7.111 10.682 13.976 17.058 19.992 22.833 25.619 28.365 31.051

7.925 11.380 14.574 17.569 20.427 23.200 25.922 28.604 31.220

9.595 12.754 15.673 18.412 21.028 23.578 26.106 28.635 31.162

10.318 13.380 16.211 18.868 21.410 23.893 26.359 28.831 31.303

11.013 13.983 16.729 19.308 21.780 24.197 26.602 29.017 31.433

17.087 19.255 21.189 22.957 24.644 26.348 28.172 30.221 32.593

17.541 19.672 21.546 23.242 24.859 26.508 28.305 30.370 32.817

17.989 20.080 21.890 23.515 25.062 26.657 28.431 30.517 33.047

326

R. PEISTELand E. HAGEN

Table A.21. Relative Apparent Specific Enthalpy ~L(S,t,p) in J/kgPSU, computed from G(S,t,p) as ~L(S,t,p) = {H(S,t,p)-H(0,t,p)}/S - (0H/OS)t.p ~ s=o, H(S,t,p) = G(S,t,p) - (T°+t) (OG/0t)s, p 5PSU

10PSU

15PSU

0°C 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

1.270 3.137 4,810 6,314 7.674 8.919 10.078 11.183 12.266

-0.558 2.083 4.449 6.576 8.499 10.260 11.899 13.461 14.993

-2.581 0.654 3.552 6.156 8.512 10.668 12.676 14.589 16.466

0oc 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

2.313 3.891 5.310 6.598 7.780 8.887 9.950 11.001 12.076

0.970 3.201 5.209 7.029 8.701 10.267 11.770 13.257 14.777

-0.660 2.073 4.532 6.761 8.809 10.727 12.568 14.389 16.251

0°C 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

3.243 4.554 5.740 6.831 7.853 8.839 9.822 10.834 11.913

2.338 4.191 5.869 7.411 8.857 10.252 11.641 13.073 14.598

1.064 3.334 5.389 7.277 9.049 10.757 12.458 14.212 16.080

0oc 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

4.764 5.619 6.411 7.168 7.923 8.709 9.560 10.514 11.608

4.588 5.798 6.918 7.989 9.057 10.168 11.371 12.720 14.268

3.915 5.397 6.768 8.080 9.388 10.748 12.223 13.875 15.770

0oC 5oc 10oC 15°C 20°C 25°C 30°C 35°C 40°C

6.524 6.927 7.268 7.576 7.883 8.224 8.633 9.148 9.806

7.352 7.922 8.404 8.839 9.274 9.756 10.335 11.063 11.994

7.558 8.256 8.846 9.380 9.912 10.502 11.211 12.102 13.244

20PSU 25PSU 30PSU 35PSU 0 MPa = 0 dBar - 4 . 6 3 8 - 6 . 6 9 8 -8.760 -10.830 - 0 . 9 0 3 - 2 . 5 2 3 -4.186 -5.890 2.443 1,219 - 0 . 0 8 7 - 1 . 4 6 3 5.450 4,581 3.596 2.515 8.171 7.622 6.927 6.114 1 0 . 6 6 0 1 0 . 4 0 6 9.977 9.407 1 2 . 9 7 8 12.997 12.816 1 2 . 4 7 4 1 5 . 1 8 8 15.468 15.522 15.397 1 7 . 3 5 5 17.891 1 8 . 1 7 6 18.263 10 MPa = 1000 dBar -2.372 - 4 . 1 1 8 -5.886 -7.680 0.784 - 0 . 5 8 9 -2.020 -3.504 3.623 2.585 1.457 0.252 6.197 5.464 4.610 3.657 8.562 8.107 7.506 6.785 1 0 . 7 7 6 10.583 1 0 . 2 1 8 9.714 1 2 . 9 0 2 12.959 12.821 1 2 . 5 2 6 1 5 . 0 0 5 15.311 15.397 15.308 1 7 . 1 5 5 1 7 . 7 1 4 18.029 1 8 . 1 5 2 20 MPa = 2000 dBar -0.334 -1.792 -3.292 -4.832 2.287 1.138 -0.082 -1.365 4.660 3.792 2.824 1.775 6.840 6.229 5.495 4.659 8.886 8.516 8.000 7.365 1 0 . 8 5 8 10.721 1 0 . 4 1 5 9.974 1 2 . 8 2 3 12.917 12.821 1 2 . 5 7 3 1 4 . 8 4 8 15.181 15.301 15.251 1 7 . 0 0 5 17.593 17.944 18.105 40 MPa = 4000 dBar 3.050 2.082 1.042 -0.061 4.762 3.996 3.138 2.203 6.345 5.766 5.077 4.297 7.860 7.460 6.932 6.301 9.370 9.148 8.782 8.298 10.941 10.904 10.706 10.377 1 2 . 6 4 3 12.808 12.791 1 2 . 6 2 9 1 4 . 5 5 0 14.940 15.127 1 5 . 1 5 2 1 6 . 7 3 9 17.387 17.807 1 8 . 0 4 7 100 MPa = 10000 dBar 7.508 7.314 7.018 6.636 8,314 8.215 8.005 7.703 8.995 8.976 8.839 8.603 9.611 9.665 9.593 9.418 1 0 . 2 2 6 10.353 1 0 . 3 4 6 10.232 10.908 11.114 11.181 1 1 . 1 3 3 1 1 . 7 2 6 12.029 12.183 1 2 . 2 1 6 1 2 . 7 5 5 13.180 13.444 1 3 . 5 7 7 14.073 14.653 15.058 15.321

40PSU -12.920 -7.639 -2.906 1.347 5.194 8.715 11.993 15.118 18.182 -9.505 -5.042 -1.027 2.614 5.958 9.089 12.096 15.070 18.110 -6.416 -2.709 0.647 3.731 6.623 9.413 12.191 15.054 18.105 -1.226 1.194 3.433 5.575 7.710 9.932 12.340 15.037 18.132 6.176 7.315 8.278 9.149 10.019 10.983 12.140 13.596 15.459

The GIBBS thermodynamic potential of seawater

32 7

Table A.22. Osmotic Pressure g(S,t,p) in MPa, computed from G(S,t,p) solving equation (5.19) ISw(0,t, p) = law(S,t,p+n), law = G - S (~gG/OS)t~ in linear approximation; G(S,t,p) - G(0,t,p) - S (igG/OS),.p, ~(S,t,p) =

(~G/~P)s. t - S (~2G/~S~p)t 5PSU

10PSU

15PSU

0oc 5oc 10°C 15°C 20°C 25°C 30°C 35°C 40°C

0.3336 0.3398 0.3459 0.3517 0.3574 0.3629 0.3682 0.3734 0.3785

0.6611 0.9906 0.6738 1.0103 0.6861 1 . 0 2 9 2 0.6980 1.0474 0.7094 1.0649 0.7205 1.0818 0.7313 1.0982 0.7417 1.1140 0.7518 1.1293

Ooc 5°C IO°C 15°C 20°C 25°C 30°C 35°C 40°C

0.3356 0.3418 0.3478 0.3536 0.3592 0.3647 0.3701 0.3753 0.3804

0.6654 0.6780 0.6902 0.7020 0.7134 0.7245 0.7352 0.7457 0.7558

0.9976 1.0170 1.0357 1.0537 1.0711 1.0880 1.1044 1.1203 1.1356

Ooc 5oC IO°C 15°C 20°C 25°C 30°C 35°C 40°C

0.3376 0.3437 0.3496 0.3554 0.3611 0.3666 0.3720 0.3772 0.3823

0.6696 0.6821 0.6942 0.7059 0.7173 0.7283 0.7391 0.7496 0.7598

1.0043 1.0234 1.0419 1.0598 1.0772 1.0941 1.1105 1.1264 1.1418

0°C 5°C 10°C 15°C 20°C 25°C 30°C 35°C 40°C

0.3413 0.3474 0.3532 0.3590 0.3646 0.3701 0.3755 0.3808 0.3860

0.6777 0.6899 0.7018 0.7134 0.7247 0.7358 0.7466 0.7572 0.7674

1.0171 1.0358 1.0539 1.0717 1.0889 1.1058 1.1222 1.1381 1.1536

0oc 5oc 10°C i 5°C 20oc 25°C 30°C 35°C 40°C

0.3515 0.3574 0.3631 0.3688 0.3745 0.3800 0.3855 0.3909 0.3962

0.6992 1.0510 0.7110 1.0690 0.7226 1.0866 0.7341 1.1039 0.7453 1.1210 0.7564 1.1378 0.7674 1.1543 0.7781 1.1706 0.7886 1.1865

20PSU 25PSU 30PSU 35PSU 40PSU 0 MPa = 0 dBar 1 . 3 2 3 8 1.6615 2.0044 2.3534 2.7096 1 . 3 5 0 9 1.6962 2.0473 2 . 4 0 5 0 2.7706 1 . 3 7 6 7 1.7295 2 . 0 8 8 3 2 . 4 5 4 3 2.8287 1 . 4 0 1 6 1.7613 2.1276 2 . 5 0 1 5 2.8843 1 . 4 2 5 4 1.7919 2.1653 2 . 5 4 6 8 2.9376 1 . 4 4 8 5 1.8214 2 . 2 0 1 6 2.5902 2.9888 1 . 4 7 0 7 1.8498 2 . 2 3 6 5 2 . 6 3 2 1 3.0381 1 . 4 9 2 2 1.8772 2.2701 2 . 6 7 2 4 3.0855 1 . 5 1 2 9 1.9036 2 . 3 0 2 5 2 . 7 1 1 2 3.1311 10 MPa = 1000 dBar 1 . 3 3 3 6 1.6743 2.0204 2.3729 2.7328 1 . 3 6 0 2 1.7084 2 . 0 6 2 5 2 . 4 2 3 5 2.7924 1 . 3 8 5 7 1.7412 2 . 1 0 2 9 2.4720 2.8496 1 . 4 1 0 3 1.7727 2 . 1 4 1 8 2.5186 2.9045 1.4341 1.8031 2 . 1 7 9 2 2 . 5 6 3 5 2.9573 1.4571 1.8325 2 . 2 1 5 3 2 . 6 0 6 7 3.0082 1.4793 1.8609 2 . 2 5 0 2 2 . 6 4 8 5 3.0574 1 . 5 0 0 8 1.8883 2 . 2 8 3 8 2 . 6 8 8 8 3.1047 1 . 5 2 1 6 1.9147 2.3163 2 . 7 2 7 6 3.1503 20 MPa = 2000 dBar 1 . 3 4 3 1 1.6867 2.0359 2 . 3 9 1 8 2.7552 1 . 3 6 9 2 1.7202 2.0773 2 . 4 4 1 3 2.8136 1 . 3 9 4 4 1.7525 2.1171 2.4891 2.8698 1 . 4 1 8 9 1.7838 2 . 1 5 5 5 2.5352 2.9241 1 . 4 4 2 5 1.8140 2.1927 2 . 5 7 9 7 2.9765 1 . 4 6 5 4 1.8433 2 . 2 2 8 7 2 . 6 2 2 8 3.0271 1 . 4 8 7 7 1.8717 2.2635 2 . 6 6 4 5 3.0761 1 . 5 0 9 2 1.8991 2 . 2 9 7 2 2 . 7 0 4 7 3.1234 1 . 5 3 0 0 1.9256 2 . 3 2 9 7 2 . 7 4 3 6 3.1690 40 MPa = 4000 dBar 1 . 3 6 1 0 1.7102 2.0654 2.4275 2.7975 1 . 3 8 6 5 1.7427 2 . 1 0 5 4 2 . 4 7 5 3 2.8538 1 . 4 1 1 2 1.7743 2 . 1 4 4 2 2 . 5 2 1 8 2.9084 1 . 4 3 5 3 1.8051 2 . 1 8 2 0 2.5670 2.9615 1 . 4 5 8 7 1.8350 2 . 2 1 8 7 2.6110 3.0131 1 . 4 8 1 6 1.8641 2 . 2 5 4 4 2 . 6 5 3 7 3.0633 1 . 5 0 3 8 1.8924 2.2891 2 . 6 9 5 2 3.1120 1 . 5 2 5 4 1.9199 2.3228 2 . 7 3 5 4 3.1592 1 . 5 4 6 3 1.9465 2 . 3 5 5 4 2 . 7 7 4 2 3.2047 100 MPa = 10000 dBar 1 . 4 0 8 3 1.7716 2 . 1 4 1 8 2.5196 2.9063 1 . 4 3 2 5 1.8024 2 . 1 7 9 5 2 . 5 6 4 6 2.9589 1 . 4 5 6 4 1.8327 2.2165 2 . 6 0 8 7 3.0104 1 . 4 7 9 8 1.8625 2 . 2 5 2 9 2 . 6 5 2 0 3.0611 1 . 5 0 2 9 1.8918 2 . 2 8 8 7 2 . 6 9 4 6 3.1109 1 . 5 2 5 6 1.9206 2 . 3 2 3 8 2 . 7 3 6 5 3.1599 1 . 5 4 7 9 1.9489 2 . 3 5 8 4 2 . 7 7 7 6 3.2080 1 . 5 6 9 8 1.9767 2.3923 2 . 8 1 7 9 3.2551 1 . 5 9 1 2 2 . 0 0 3 9 2 . 4 2 5 4 2 . 8 5 7 3 3.3012