Properties of electrolyte solutions at high pressures and temperatures

Properties of electrolyte solutions at high pressures and temperatures SEFTON D. HAMANN Commonwealth Scientific and Industrial Research Organization, G.P.O. Box 4331, Melbourne, Victoria 3001, Australia Abstract--This article surveys some of the experimental physico-chemicalproperties of water and aqueous solutions to the highest temperatures and pressures at which they are known, and discusses theories that have been advanced to describe and predict these properties. Attention is drawn, particularly, to uncertainties in the P-V-T equation of state for water above 10 kbar, and to conflicting evidence about its fluidity above 25 kbar at high temperatures.

SCOPE OF THE ARTICLE In this article, the writer has set out to review some of the experimentally known physico-chemical properties of solutions to the highest pressures and temperatures at which they have been measured, and to discuss a few of the theories that have been advanced to describe and predict those properties. The paper is concerned only with water and aqueous solutions, since these are the systems of most interest to this meeting (in a sense pure water is a solution--though normally a very dilute o n e - - o f hydrogen and hydroxyl ions). The physical properties reviewed include P - V - T and thermodynamic properties, phase behaviour, transport phenomena, and electrical and optical properties. The chemical effects discussed are mainly those arising from changes in ionic dissociation. M a n y of these properties have been well reviewed in earlier articles by Franck (1961, 1973, 1977, 1978), by T6dheide (1972) and by Helgeson and K i r k h a m (1974), and in such cases it has been deemed sufficient here to summarize the conclusions of those reviews and bring them up to date. For the sake of numerical convenience, as well as familiarity, the pressures have been expressed in the units: bar, kilobar (kbar) and megabar (Mbar), which are respectively 105, 108 and 1011 pascal. Temperatures are quoted in both degrees Celsius (°C) and kelvin (K).

EQUATION OF STATE Water: experimental results The experimental P-V-T(pressure-volume-temperature) behaviour of water has been reviewed and summarized in articles by Kennedy and Holser (1966), Jfiza (1966), K6ster and Franck (1969), Burnham et al. (1969), T6dheide (1972) and Helgeson and K i r k h a m (1974), and in the bibliography of Hawkins (1975). In summary, the behaviour is known accurately to 100°C and 1 kbar, rather less accurately to 1000°C and 10 kbar, and becomes progressively more uncertain beyond those limits. The highest temperature reached in static measurements is 1200°C (at 4 kbar (Goranson, 1938)) and the highest pressure is 3 4 k b a r (at 175°C (Bridgman, 1942)). Beyond those limits, our information comes from shock-wave experiments, which have the serious shortcoming that, although the pressure and volume can be measured in the shocked state, it is very difficult to measure the shock temperature directly, and this is nearly always estimated by making assumptions about the caloric equation of state. The shock-wave P - V - T data are therefore not really experimental, and the main purpose of this section is to consider how reliable they may be. 89

90

SEFTON D. HAMANN 120

p=l.Tgcm -3 100

/

/ / 1,5

/ / /

80

ice VII

/ /

/

.,Q

60 gh

fluid z.O

_

20

0 ~ 0

-r 200

400

I'"

10

0.8 0.7 0.6

~

0.5 600

800

1000

Temperature/°C Fig. 5.1. Isochores for water at high temperatures and pressures. The number on each line denotes the density p in g cm-3.

The most generally cited shock-wave P - V - T data for water are those in the range to 450kbar and circa 3100°C, which Rice and Walsh (1957) derived from their P - V measurements by making the assumptions that the derivative (aH/OV)p = Cp/(OV/OT)p is independent of the temperature (H is the specific enthalpy and Cp is the specific heat at constant pressure) and that Cp is independent of P above 25 kbar. Their results, which they listed in the form of isotherms, are plotted in Fig. 5.1 as isochores. This diagram is an extension to much higher pressures of Fig. 4 of Helgeson and Kirkham (1974)--in fact the isochores below 10 kbar are taken from their compilation. Figure 5.2 shows a plot of Rice and Walsh's estimated shock temperatures against their experimental pressures. It now appears that their assumptions may have led Rice and Walsh to overestimate the shock temperatures, and hence the temperatures of their derived isotherms. The evidence for this is as follows. (1) Jfiza (1966) found that his empirical equation of state for water, without the assumptions made by Rice and Walsh, predicted shock temperatures lower than theirs, and he suggested that his were "probably closer to reality". (2) Cowperthwaite and Shaw (1970) have replaced the assumptions of Rice and Walsh by assumptions that (OP/OT)v and Cv are constant and, on that basis, have derived the middle curve in Fig. 5.2. (3) Bakanova et al. (1975) have made some valuable new measurements of the shock compression of loosely-packed ice; of the double shock compression of water, and of the speed of sound in shock-compressed water, and combined their results with simpler single-shock compression data to derive an equation of state which gives the bottom curve in Fig. 5.2. This curve is probably most reliable in the region of the majority of their experiments--150-450 kbar--where their estimated temperatures are several hundred degrees lower than Rice and Walsh's: the curve is clearly

Properties of electrolyte solutions at high pressures and temperatures

91

4000 Rice and Wolsh (1957)

3000 -

~

o

;:~

"

NOVO

2000

-

J

E

J

J.

J

J~,,s"" J J

S

.

e t a!

~-

(1975)

~3owperthwoiteone Show (1970)

.e

1000

o

0

I

100

I

200

I

300

I

400

500

Pressure P/kbar Fig. 5.2. The curves are calculated P - T relationships for water under shock-wave compression from atmospheric pressure and 20°C. The dots are experimental determinations by Kormer (1968).

wrong at low shock pressures, where Rice and Walsh's and Cowperthwaite and Shaw's curves have the correct limiting slope of 1.44°C kbar-1, corresponding to isentropic compression. (4) The only directly measured temperatures of shock-compressed water are some approximate ones which Kormer (1968) made by a photometric method and which are shown as dots in Fig. 5.2. They lie considerably below Rice and Walsh's curve. On this evidence, the best guess we can make at present is that Rice and Walsh's temperatures are probably too high and that Cowperthwaite and Shaw's may be nearer the truth. The difference is about 120°C at a shock pressure of 100 kbar, rising to 350°C at 300 kbar. If we were to use Cowperthwaite and Shaw's temperatures, it would change the isochores in Fig. 5.1 in the manner illustrated by the dashed curve A for the p = 1.5 g c m - 3 isochore. This uncertainty is clearly unsatisfactory and it is to be hoped that it may soon be resolved--perhaps by the use of platinum resistance thermometry in shock-wave experiments or by extension of the P - T range of static measurements. There is another aspect altogether of the shock-wave results for water which should cause us to question some of the published P - V relationships (as well as the P - T relationships) along the shock-compression curve. It will be shown later in this paper that there is now good evidence that, between shock densities of 1.6 and 1.9 g cm -3, water transforms from a virtually un-ionized state into one in which it is virtually fully dissociated into hydrogen and hydroxyl ions, and we should certainly expect this transition to be reflected in the P - V behaviour. Although Rice and Walsh's (1957) P - V curve was smooth throughout and showed no abnormality in that density region, Al'tshuler et al. (1958) later found that their own experimental P - V curves for shock waves in water formed two fairly distinct sections separated by a transition zone centred near p = 1.75 g cm 3. And van Thiel (1967), in summarizing his compilation of shock-wave data for water, displayed them in the form of a plot of shock velocity Us against particle velocity Up which consisted of two straight lines of different slopes below and above a shock density of about 1.6-1.8 g c m - 3. Plots of Us versus Up are known to be particularly sensitive to transitions. A large number of additional Us, Up data have been published since van Thiel's work. Some of these were included in a plot by H a m a n n and Linton (1969a) but more have come to light since: there are now over 80 Us, Up points, determined by 14 different sets of authors. Apart from the data cited by van Thiel and by H a m a n n and Linton, there are further measurements by Deal (1958), Ahrens and Ruderman (1966), Lysne (1970),

92

SEFTON D. HAMANN Density p / g crn-3 1.0

1.L

1.6

1.8

I

I

I

2.0

2.2

2.L

26

2.8

30

I

I

I

I

I

I

1312-I•

/

1200 1000

800 11 -

B 600

10--

g=_ E E 8

~-~

--

LO0

.13

- 300

>"

7

oj

-- 2 0 0

QO >

2. i1/

6

(3.

o

5

--

~00

t/3

¢ I'

3

-

50

--

20

i/ -0 1

0

m

I 1

I 2

I 3

I &

I 5

I 6

P a r t i c l e v e l o c i t y Up/ram

I 7

I 8

I 9

10

-1 14s

Fig. 5.3. The experimental relationship between the shock velocity and particle velocity for water under shock-wave compression from atmospheric pressure and circa 20'C.

Podurets et al. (1972), Bakanova et al. (1975), Mitchell and Nellis (1979) and Baconin (1979). Figure 5.3 shows a plot of all the measured U~, Up points. In spite of the scatter, which understandably increases with increasing shock velocity and shock pressure, the points seem to lie fairly uniformly about the two straight lines A and B, defined by the formulae: A: U~ = 1.50 + 1.82 Up

(1)

B: Us = 3.15 + 1,17 Up

(2)

where the U are in mm #s-1. If these relationships are translated into P, V (or P, p) points by the Stokes-Rankine-Hugoniot formulae

p/pO= Vo/V = UJ(U~- Up) p _ po = 10 x U s U p p °

(3) (4)

(where the U are in mm/~s- 1, the p are in g cm- 3, the P are in kbar and the superscript o denotes values for the initial uncompressed state) then we arrive at the values of pressure and density that are plotted in Fig. 5.4 and indicated on the top and right-hand axes of Fig. 5.3. It is apparent that, if the lines A and B of Fig. 5.3 are to be trusted, the corresponding curves A and B in Fig. 5.4 are linked by a region of abnormally high compressibility where the ionization transition is thought to occur. If this abnormality is real, it means

Properties of electrolytesolutions at high pressures and temperatures

93

1000 900 800 700 #.

600 500

ul

/

400

O_

300 s

200

s s S s

100 0 10

~ ~ l k 1.2

l z,

it I 1.8

1.6

I 2.0

I 2.2

I 2./,

I 2.6

I 2.8

30

Density £ / g c m -3

Fig. 5.4. The P-p relationship for water under shock-wavecompression from atmospheric pressure and circa 20°C, derived from the lines A and B of Fig. 5.3. that there must be errors in any equations of state that have assumed that the P - V (or P p ) behaviour along the shock-compression curve is smooth and without inflections (e.g. the equations of state of Rice and Walsh (1957), Cowperthwaite and Shaw (1970), and Bakanova et al. (1975)). In particular, it is hardly to be expected that the P - T curve for shock compression will be neatly linear, as it is shown in Fig. 5.2. Until these questions are properly resolved, it would be best to consider the "experimental" equation of state for water to be rather uncertain above 30kbar and quite uncertain above 100 kbar.

Electrolyte solutions: experimental results The effects of a dissolved electrolyte, 2, on the P - V - T properties of water, 1, are conveniently represented by the partial molar volume V2 of the electrolyte, which is related to the total volume Vof the solution by

I/2 = (?V fi~n2)l,.r.,~

(5)

where the n denote the number of moles of the components. V2 is strongly dependent on the concentration n2/(nl + n2) of the electrolyte and upon temperature and pressure. N o t many measurements have been made of V2 as a function of temperature and pressure, but Franck (1978) has recently published the results of some valuable measurements by Hilbert (1979) on aqueous solutions of NaC1 to 4 kbar and 400°C. These results are shown in Fig. 5.5. F r o m this diagram, and from the pioneering work of Adams (1931), the behaviour of NaC1 can be summarized as follows: (1) V2 increases with increasing pressure at constant temperature. A corollary of this is that dissolved NaC1 reduces the isothermal compressibility of water. (2) At higher temperatures, 1/2 decreases with increasing temperature at constant pressure, although at low temperatures the trend is in the opposite direction. (3) V2 is relatively insensitive to the temperature at a constant density of 1 g c m - 3. (4) V2 increases with increasing concentration of NaC1 at all pressures (Adams, 1931). These trends for NaC1 seem to be typical of electrolytes in general, although the temperature dependence (2) of V2 is more marked for multiply charged ions (Ellis and McFadden, 1972).

94

SEFTOND. HAMANN

20

10 '7_ O E mE -.2

o

-10

-20 0

100

200

300

Z,O0

Temperature/°C

Fig. 5.5. The experimental partial molar volume at infinite dilution V2 of NaC1 in water (after Franck 1978).The number on each curve denotes the pressure in kbar. The dots are the results of Adams (1931) and are rather lower than those of Franck.

Water: theoretical approaches The severe experimental difficulties involved in measuring the P - V - T properties of water at extremely high temperatures and pressures provide a strong incentive to develop theories to predict its behaviour in these conditions. The last 10 years have brought enormous progress in this direction, with the development of powerful computing methods and the derivation of reliable potential energy functions for describing the interaction of water molecules with each other. At normal and high temperatures, but relatively low densities, it is possible to use the virial equation of state

P V / R T = 1 + B(T)/V + C(T)/V z + ...

(6)

where the coefficients B(T), C(T)... are functions of the temperature and depend on the forces between the water molecules. Johnson and Spurling (1971) have applied this equation to water, calculating B(T) and C(T) on the assumption of an additive intermolecular potential which included angle-dependent multipolar electrical interactions. The precise form of the potential was determined from experimental data for the viscosity of water vapour at low pressures and for the polarizability of water molecules, and a theoretical value for the quadrupole moment of water molecules. They limited their calculations to B(T) and C(T) because the integrations needed for D(T) and higher terms are very laborious. Even with that truncated form of the equation, they were able to derive the following theoretical critical constants for water: T~(K) Theoretical values Experimental values

536 647

Pc(bar) V~(cm3mol-l) 426 218

52 45

Properties of electrolyte solutions at high pressures and temperatures

95

The agreement with experiment would almost certainly be improved by the inclusion of higher virial coefficients, but it is already surprisingly good when it is considered that the theoretical values were derived entirely from low-pressure data. The virial equation is unsuitable at densities as high as that of normal liquid water. However, there are now alternative computer methods for directly simulating the behaviour of assemblies of the order of 10 z 103 molecules present in dense phases. They are of two distinct types. The Monte Carlo (MC) method generates a large number of configurations of the molecules, subject to their interaction by an appropriate intermolecular potential, by trial r a n d o m displacements at a fixed temperature and density. Quantities like the energy, pressure and radial distribution function are then derived as averages over all configurations in the sequence. In the Molecular Dynamics (MD) technique, the classical equations of motion of the molecules are solved by step-by-step methods at a fixed total energy and density, and the other thermodynamic properties are then estimated as averages over time rather than configurations. The first computer simulations of water were carried out by Barker and Watts (1969) using the M C method, and there have been subsequent MC calculations by several groups of workers, using both empirical and theoretical molecular interaction potentials. They have generally yielded fairly good agreement with the experimental properties of water at normal temperature and pressure. For instance, the calculations of Lie et al. (1976), based on an ab initio theoretical potential function, gave the values listed in Table 5.1 for the properties of water at 25°C and a density of 0.998 g c m -3. Unfortunately for our present purposes, no M C calculations have yet been made at higher densities or high temperatures. The M D approach has been applied to water in a series of remarkably illuminating papers by R a h m a n and Stillinger (1971 and later), who have considered at least a few states of moderately high density and temperature. It is particularly encouraging that their results at low pressures show a temperature of m a x i m u m density and a temperature of minimum compressibility--both of which are found experimentally, and are generally considered to be anomalous. Some of their high-pressure results are shown in Table 5.2 (Stillinger and Rahman, 1974b), where they are compared with low pressure results for the same temperature (97°C) (found here by interpolation in the earlier work of Stillinger and R a h m a n (1974a)) and with the experimental results. The quantities are of the right order and show the right kind of density-dependence; there is little doubt that the numerical agreement could be improved by allowing for quantum effects and for the Table 5.l. Computed and experimental thermodynamic properties of water at 25 C and a density of 0.998 gcm 3 Internal energy (kcalmol 1)

Specific heat at constant volume (calmol-lK-1)

Isothermal compressibility (bar 1)

-6.8 -8.1

18 17.9

5.3 x l0 5 4.6 x 10- 5

Theoretical (MC) Experimental

Table 5.2. Computed and experimental thermodynamic properties of water at 97 C Density (g cm 3) 1.000 1.346

Property Pressure (kbar) Internal energy (kcal tool - 1) Specific heat at constant volume (cal tool 1K- ~) Isothermal compressibility(bar -~)

Theoretical (MD) Experimental Theoretical (MD) Experimental Theoretical(MD) Experimental Theoretical (MD) Experimental

1.0 ll 1.0 22 -6.85 -7.20 - 6.7 25 17 16 7.4 x 10-5 1.5 x 10 ~ 4.9 x 10 5 0.75 x 10 5

96

SEFTON D. HAMANN

non-additivity of the intermolecular potentials, and by improving the forms of the potentials. In this last connection, Stillinger and David (1978) have proposed a "polarization" model for water molecules, which realistically allows them to vibrate internally and to ionize into hydrated H ÷ and O H - ions. It would be extremely interesting to see whether simulations using this model would give rise to the transition behaviour shown in Figs 5.3 and 5.4 at high temperatures and pressures. Electrolyte solutions: theoretical considerations

Here we can consider two distinct types of theories of electrolyte solutions. The first treats the solvent (water) as a continuum and attempts to describe its interaction with dissolved ions in terms of its bulk properties. The second takes account of the discrete molecular structure of the solvent and tries to derive the effects of solvent-solute interactions from the molecular properties by methods of computer simulation. The simplest "continuum" model for ionic solutions is that of Born (1920), which gives the electrostatic free energy of hydration of an ion of radius r and charge ze in the form 2r

1-

(7)

where N denotes Avogadro's constant and E is the dielectric constant of water. By differentiating eqn. (7) with respect to pressure, we obtain the following expression for the electrostatically-induced volume change associated with the hydration of the two ions of a 1 : 1 electrolyte: AVh _ Ne2 0(1/~) r t~P

(~)0(1/r)

Ne 2 1-

(8)

OP

where r is roughly equal to the mean ionic radius of the two ions. If this equation is applied to a solution of NaC1 in water at 25°C, it yields the results shown in Table 5.3 (Hamann, 1957). The bottom two rows of the Table compare the calculated increases of AVh with the increases that Adams (1931) found experimentally for the partial molar volume V2 of NaC1 (see the points on the left of Fig. 5.5). It is apparent that almost the whole of the observed change of V2 with increasing pressure can be attributed to hydration effects. Moreover, the continuum formula (8) gives a good qualitative description of the temperature dependence of V2, since the more important derivative 3(1/e)t3P, which is negative, becomes increasingly so as the temperature is raised (e.g. Helgeson and Kirkham, 1974, 1976). The concentration dependence of V2 is, of course, described exactly at low concentrations of solute by the D e b y e - H i c k e l theory. "Molecular" theories of ionic solutions show that the ionic charges attract the surrounding polar water molecules and compress them locally to a higher density than n o r m a l - - i n this way contributing a substantial negative volume term (electrostriction) to 1/2. Watts and his co-workers (e.g. Watts, 1976) have carried out MC simulations for a number of aqueous solutions, and Heinzinger and his colleagues (e.g. Heinzinger, et al., 1978) have made similar calculations by the M D method. Although neither group has explicitly derived or discussed the extent of electrostriction of water around the ions, Table 5.3. The partial molar volume of NaCI in water at 25°C and at infinite dilution. All the volumes are in cm3 mol- 1 Pressure (kbar) AVh calc. AVh - AV~ calc.* V2 - V~ exp.*

0

0.5

1.0

-22.2 0 0

-20.6 +1.6 +2.1

-19.6 +2.6 +3.2

* The superscript o indicates the value at 0 kbar.

1.5

2.0

3.0

-18.7" +3.5 +4.2

-18.0 +4.2 +4.9

-17.1 +5.1 +6.0

Properties of electrolyte solutions at high pressures and temperatures

s.o O ; . , n ' ~ ' ' " 4.5 R(Li-F)=3.5A. T=500K f~/I

/

4.0 3.5

97

"

I ~'':

-~

'

-"

~. ~_.:.~.-.

:

/ ~

.

.

.

.

"

.

.

.

2.0

.

', i

' ~\

1.0 #"

0.5

, ;II

\ •

5.ol

4.5 4.0 / 3.5 :" 3.0' "J /' 2.0

~/

,

/ '

f/

,

i

~

~L

•

-4,0

. /. . .

~',,

"X~',

~;

\

:~

~,,,~ "\

FU ~ I -2.0

,

~;,:,,1~

. . . . Hydrogen ~.~-' R(Li-F)=3:5,~,T=5OOK ~-~b) ~ ) ' ' " ,,

//- "~-~-~'\ \

i~,~ ~I ' ,~,I -6.0

.

' / ~ : - _' - ~ , I '

~ ~\\ ~/~

'

o.~

,~ "

/'

~

0.0

,: ~,~ 2.0

~

'~L~,\~ T~ 4.0

'

,

\ ~¢'-,,

6.0

Fig. 5.6. Calculated (MC) densities of O and H atoms of H~O molecules in the neighborhood of an Li+F - ion pair (solvent separated) at 500 K and an average water density of 0,125 g cm -3 (from Watts et al., 1974). The n u m b e r s on the axes indicate distances in ~ from the midpoint between the ions.

some information on this is contained in their calculated plots of the average number of water molecules within particular distances from the centre of an ion. Both the MC and M D results show that, for the alkali halide ions at normal temperature and pressure, the first hydration shell (as defined by Heinzinger and Vogel, 1976) contains 1-2 more water molecules than it would if it had the normal density of water, although this rather large contraction is partly counterbalanced by the lower density of the next shell. No calculations have yet been made for high pressures and the only ones that have been made for a relatively high temperature are a few by Watts et al. (1974) for LiF ion-pairs at 500 K, but at a low average water density of 0.125 g cm-3, corresponding to a pressure of about 200 bar for supersaturated water vapour. The results, based on the use of Hartree-Fock interaction potentials, are shown in Fig. 5.6 in the form of plots of the density of oxygen and hydrogen atoms around the ions. It will be seen that, in spite of the fairly high temperature and low pressure, a substantial amount of electrostrictive "condensation" of the water has occurred around the ions. There are about 10 water molecules within a radius of 5A from the centre of each ion, and this "condensation" corresponds to a contraction of the water at constant pressure by an amount AVh ~ -- 1500 cm 3 per mole of LiF. It explains the fact that contractions of that order occur when salts dissolve in supercritical water at fairly low pressures (Benson et al., 1953).

PHASE BEHAVIOUR

At room temperature, water freezes to ice-VI at about 10kbar and transforms to ice-VII at about 23 kbar: ice-VII appears to have no further transformation below at least 150kbar, where its density reaches about 2.05 g c m -3. At higher temperatures, ice-VII melts according to the phase line shown in Fig. 5.1, taken from the measurements of Mishima and Endo (1978). The high-pressure phases VI and VII have interesting crystal structures, each consisting of two interpenetrating but fully hydrogen-bonded lattices. One framework has cavities into which the molecules of the other fit, and the frameworks are interpenetrating but not interconnected. It has been suggested (Vereshchagin et al., 1975; Kawai et al., 1977) that water undergoes a transition into a metallic state at a pressure of the order of 1 Mbar at normal temperature, but the evidence is not convincing. It was based on an observed sharp drop of electrical resistance which could well have been caused by a short circuit between the

98

SEFTON D. HAMANN

anvils of the apparatus. Recent, better designed, experiments by Nelson and Ruoff (1979) show that ice remains an insulator to at least 0.72 Mbar at 78 K. On the theoretical side, it should be possible, at least in principle, to use computer simulation methods to predict the phase diagram of water under extreme conditions, and indeed this has been done for simpler substances. But the calculations so far made on water have not considered any P T states lying well within the experimental phase boundaries of the solid phases. It is to be hoped that they will do so in the future. The phase behaviour of aqueous solutions is extremely complex. For instance, the solubilities of salts may either increase or decrease with increasing temperature and pressure and the trends often invert. In many cases, the solubilities are governed more by chemical than by physical factors, and we are certainly very far from having any adequate theory of the behaviour of the multicomponent systems that are important geologically. We must rely, at this stage, on experimental studies such as the one that Gehrig et al. (1979) made recently of the HEO-CO2-NaC1 system. One interesting experimental fact is that, although an increase of pressure at constant temperature normally favours the formation of hydrated forms of salts, it can cause their dehydration at very high pressures (e.g. above 30 kbar at 1000°C in the case of Mg(OH)2 (Yamaoka et al., 1970)) where the partial molar volume of water in the fluid phase becomes less than that in the crystal (about 13 cm 3 mol ~ at normal temperature and pressure, compared with 18 cm a tool-1 for liquid water).

FLUIDITY Experimental results

The term fluidity is used here to embrace features both of viscous flow and of diffusional processes. At moderate temperatures and pressures, it is well established experimentally that the fluidity of water as measured by the reciprocal of its viscosity increases with increasing temperature at constant pressure and decreases with increasing pressure at constant temperature (apart from a small initial increase at temperatures below 30°C). And similar behaviour is shown by the coefficient of self-diffusion and by related properties such as the electrolytic mobilities of dissolved ions. There have been a number of reviews of these properties at normal and high temperatures and at pressures up to l0 kbar (e.g. T6dheide, 1972). The agreement between the results of different workers is generally good and there is no need to review the work again here. Instead, attention will be given to an apparent contradiction concerning the fluidity of water at much higher pressures--the highest at which attempts have been made to measure it. These attempts have used shock-wave techniques, which automatically generate high temperatures as well as high pressures (see Fig. 5.2). It might be expected that in these circumstances the two trends mentioned in the last paragraph would be to some degree self-compensatory, and that there might therefore be no enormous change of fluidity in the shocked state. One group of experiments tends to support that view, whereas another group suggests that the fluidity drops very steeply with increasing shock pressure, reaching a more or less constant low value above 80 kbar. In the second group, Mincer and Zaidel' (1968) and Mineev and Savinov (1975) have described experiments designed to measure the viscosity of shocked water by following the progressive change of shape of shock-wave fronts that are, initially, corrugated. They consider that the rate of this change is related in a simple way to the viscosity, rl, of water behind the shock fronts, and they have concluded that q rises very steeply from its normal value of 10 -2 poise until it levels off above 80kbar at a value near 104 poise. Their results are shown as crosses in Fig. 5.7. Recently, Al'tshuler et al. (1977) have applied a more direct method based on the acceleration of thin transverse wires by the flow of water behind plane shock-wave fronts. Their results are shown as circles in

99

Properties of electrolyte solutions at high pressures and temperatures

Temperature/*C 20

250 I

105

500 I

1000 I

1500 I

2000 I

(*)

104 O O

103

(*) o

o

20

102

Temperature/*C 1.0 60

10-1.6

I

I

o

/ P\~ ,/"

10-1.8

10

t f

o

s

c"

f

t/1

>, J

~ 10-2 .

.%

10-2.2

10-1

T 10-2.4

10-2

0 A

i~ 3

I 0

I 100

I 5 P/kbor

I

I 200

"~.. I 10

I 300

Pressure P / k b o r Fig. 5.7. The viscosity r/of water under shock-wave compression from atmospheric pressure and circa 20°C. The crosses are the results of Mineev and co-workers (1968, 1975) and the circles are those of Al'tshuler et al. (1977). The crosses in brackets represent upper limits for r/. The curves A, P and T are discussed on this page.

Fig. 5.7 and generally substantiate those of Mineev, although they are lower by about an order of magnitude in the neighbourhood of 75 kbar. Here, it should be noted that it is possible to calculate the viscosity of shock-compressed water quite precisely at pressures up to 10kbar by combining the static P - V - T - E data for water (e.g. Helgeson and Kirkham, 1974) with the Stokes-RankineHugoniot relationships and then extracting values of r/for the resulting P - T shock loci from the static data of Bett and Cappi (1965). The results, obtained here, are shown as the curves A in Fig. 5.7. The curve T represents the change of 1/ with temperature at constant pressure (atmospheric) and P represents the change of t/ with pressure at constant temperature (20°C). It will be seen that, as was suggested above, these two trends are largely self-compensatory for shock-compressed water below 10 kbar. If the results of Al'tshuler et al. (1977) are correct, this compensation must break down and there must be an enormous and unheralded increase in r/ between 10 and 29 kbar, at 56 and 140°C respectively. In complete contrast to the findings of Mineev et al. and of Al'tshuler et al., a number of experiments in the first group, though not concerned directly with viscosity, have shown that the rates of migration of dissolved species in shocked water remain normal, implying that the fluidity also does so. Hamann and Linton (1969a, b) measured the electrical conductivities of aqueous solutions of a number of simple salts at shock pressures in the range 45-135kbar and found that they were close to the corresponding conductivities at normal temperature and pressure. It follows from Walden's rule that the viscosity is probably also normal. Moreover, a number of workers have observed precipi-

100

SEFTOND. HAMANN

tation reactions (brought about by the enhanced ionization of water--see p. 104) occurring in shocked aqueous solutions in times less than a microsecond. For example, Yakusheva et al. (1972) observed the H+-induced precipitation of sulphur from thiosulphate solutions by the reaction

820 ~- "+ H + ---, HSO3 + SJ.

(9)

showing, unambiguously, that the sulphur atoms can diffuse together and coagulate within a microsecond. The authors concluded that this sets an absolute upper limit to the viscosity of water of 30 poise at 150kbar and 1000°C--which is much lower than the viscosities shown in Fig. 5.7. Unless there is some unexpected and very drastic breakdown of the usual relationship between viscosity and diffusion, the cause of the above contradiction must lie in the experiments or their interpretation. Concerning first the viscosity measurements, the method of Mineev and co-workers (1968, 1975) involves conditions of unsteady flow for which a complete analysis is extremely difficult: dissipative effects other than simple viscous flow may be important. In some of the experiments the shock-front corrugations actually inverted to greater than their original amplitude, which is hardly a dissipative effect at all and suggests that other shock-wave processes such as Mach interactions may have occurred. It is highly suspicious that their method led them to the improbable conclusion that "the obtained values of ~/for water and mercury are the same as those of shock-compressed aluminium and lead". The method of Al'tshuler et al. (1977) is more direct, but, again, the results are very difficult to interpret rigorously. Experimentally, the authors found that the drag coefficients were about three times larger than normal and, by assuming steady-state laminar flow, they concluded from this (relatively small) change that q must be greater than normal by a factor of about 105. However, in shock-wave experiments it is improbable that steady flow would be reached in the experimental period of 3 ~ts, and the drag coefficient is known to be considerably greater than normal in accelerated and in suddenly-started (impulsive) flow. In addition there are inertial drag effects, and there could be complicating factors due to the high temperatures and gradients of temperature near the wires, since the temperature boundary layers and velocity boundary layers would interact in a complex way. Concerning the mobility measurements, these are straightforward in showing that the mobilities of ions and molecules are normal in shock-compressed water. However Mineev and Savinov (1975) have suggested that Walden's relationship may not apply under shock conditions and that the high mobilities need not imply a low viscosity. To summarize: although there is good evidence that the fluidity of water, as measured by the mobilities of dissolved species, is normal in shock-compressed water at P > 80kbar, T > 700K, there is also, apparently contradictory, evidence that the macroscopic viscosity is high. Maybe the question can only be resolved by hydrostatic viscosity measurements, since all attempts to use shock-wave methods for viscosity will probably be bedevilled by the complexities of unsteady hydrodynamic flow. Theoretical approaches

Dudziak and Franck (1966) found that Enskog's theory of the viscosity of dense fluids, using a simple hard-sphere molecular model, gave quite a good description of their experimental results for water at hydrostatic pressures up to 3.5 kbar and temperatures to 560°C, and Hamann and Linton (1969b) later extended the calculations, to estimate under the conditions of strong shock-wave compression. For what the results are worth, they indicate that r/ should be of the order of 10 - 2 poise in the regions where the experimental work of Mineev and co-workers and Al'tshuler et al. (see pp. 98, 99) led them to conclude that it is 103-104 poise. An alternative and very promising theoretical approach is by way of computer simulations using the molecular dynamics (MD) method, which, unlike the Monte Carlo (MC) method, provides valuable information about the evolution of molecular motions with

Properties of electrolyte solutions at high pressures and temperatures

101

time. It has already given remarkably encouraging results for water and aqueous solutions. For instance, Stillinger and Rahman (1972), using a realistic intermolecular potential, derived a theoretical self-diffusion coefficient ~ = 4.3 × 10- 5 cm 2 s- 1 for water at 53~C and p = 1 g c m -3, compared with the experimental value ~ = 4.0 z 10 -5 cm 2 s x At 350°C and p = 1 g cm -a, corresponding to an experimental pressure of 6.4 kbar, they derived a value ~ = 24 x 10 -s c m 2 s - l , compared with the value ~ = 27 × 10 s c m 2 s - I given by applying the Stokes-Einstein relation to Dudziak and Franck's (1966/ experimentally determined viscosities. Also, Stillinger and Rahman (1974b1 calculated that at 9TC, 2 should drop from 7.2 x 1 0 - 5 c m 2 s i at p = l g c m 3 to 6.5 x 10 5 cm 2 s-1 at p = 1.346 g cm -3, corresponding to theoretical pressures of 1 and 11 kbar respectively. This is a remarkably small decrease for such a large pressure increase, and it accounts for the relatively small decrease that is found experimentally for water (Kisel'n i k e t al., 1973) in comparison with, say, simple organic liquids (e.g. Wade and Waugh, 1965: McCool and Woolf 1972). The normal decrease in fluidity brought about by the more dense molecular packing is compensated by the tendency of pressure to break down the hydrogen-bonded network of water. It would be particularly interesting to have some M D results for the region of supposed high viscosity shown in Fig. 5.3. Heinzinger et al. (1978) have used the M D method in preliminary estimates of the self-diffusion coefficient of water in concentrated aqueous solutions of the alkali halides. The results are of the right order, but longer simulation times will be needed to give numerically significant results.

DIELECTRIC AND OPTICAL EFFECTS Experimental results Static measurements of the dielectric constant • of water at high temperatures and pressures have been thoroughly reviewed by T6dheide (1972) and Helgeson and Kirkham (1974). The data extend to 550°C and 5 kbar and show a general trend for • to decrease with increasing temperature and increase with increasing pressure. Measurements of • at high shock-wave pressures and temperatures are feasible (e.g. Yakushev et al., 1975) but have not so far been made for water. T/Sdheide (1972) also reviewed the static data for the refractive index n of water, which at that time were restricted to normal temperatures and a maximum pressure of 1.5 kbar. They have now been taken to 11 kbar at 25°C (Vedam and Limsuwan, 1978). Also, there are some shock-wave measurements of n that extend the range to about 900°C and 150 kbar. The results of these are plotted against the shock density p in Fig. 5.8, where the circles are values measured by Zel'dovich et al. (1961); the triangles were measured by Ahrens and Ruderman (1966), and the squares by Yadav et al. (1973). The curves in this diagram will be discussed in the next section. Franck and his colleagues have made a series of very interesting and informative studies of the spectra of water and aqueous solutions at high static temperatures and pressures, in the i.r., visible and u.v. regions. The results have been reviewed by Franck (1973) and show, inter alia, that "free" water molecules as distinct from hydrogen-bonded molecules are not present in water vapour at high temperatures, at densities greater than 0.1 g c m -3. Under more extreme shock-wave conditions, Ewald (see Hamann, 1966) succeeded in photographing the visible electronic spectra of a wide range of aqueous solutions of salts of the rare earth and transition metals, to about 100 kbar. The results showed shifts of the absorption bands that generally paralleled shifts that Drickamer and his colleagues had found earlier for salts in the solid state under static compression. However, there were sometimes significant differences, notably a greater band-broadening, that could be attributed to the high shock temperatures (circa 600°C). Solutions of uranyl ions UO22÷ showed enormous and quite untypical increases of absorption over the whole spectrum.

102

SEFTON D. HAMANN

1.6

s. A"

j J

L-L / I

0~

Z

x 1.5

-8C

,~ 1.l, rY

1.~ 1.0

I 1.2

I 1l.

I 1.6

18

-3 Density p / g cm Fig. 5.8. The refractive index n of water under shock compression from atmospheric pressure and circa 20°C. The circles are the results of Zel'dovich et al. (19611, the triangles are those of Ahrens and Ruderman (19661 and the squares are those of Yadav et al. (19731. The curves are discussed on pp. 102, 105.

T h e o r e t i c a l approaches

T6dheide (1972) has reviewed the application of Onsager and Kirkwood's treatment of the dielectric constant E of water at high temperatures and pressures and concluded that, at moderate and high densities, considerable local ordering of the molecules persists above the critical temperature. In a more refined analysis, Jansoone and Franck (1972) found that, with the addition of polarization effects, Wertheim's treatment of E for fluids of hard spherical molecules with embedded point dipoles gives a good description of e for water at high temperatures and pressures, but overestimates E below 300°C. In that connection, Rushbrook (1979) has recently argued that "E for water can never be accounted for by treating the molecules as (polarizable) dipolar hard spheres: charge separation must play a dominant role in determining E". For more realistic molecules with charge separation, Rahman and Stillinger (1971) calculated Kirkwood's orientational correlation factor gK for water by M D simulations, and obtained a value close to previous estimates. They have not, as yet, derived ~ a priori, nor have they considered high-pressure states; but they have concluded that there is a substantial temperature-induced decrease of 9K in going from - 1 3 to +118°C at p = 1 g c m - 3, accompanied by a slight increase in the dipole moments of the molecules (Stillinger and Rahman 1974a). There is no adequate t h e o r y - - o r even an approximate o n e - - o f the refractive index n of dense water. T6dheide (1972) referred to "the approximate validity" of the LorenzLorentz formula (n 2 - 1)/p(n 2 + 2) =constant (C)

(10)

or

n = [-(1 + 2pC)/(1 - pC)] ~

(11)

but, in fact, the recent results of Vedam and Limsuwan (1978) show that this formula overestimates the increase of n between 0 and l l . 3 k b a r at 25°C by 12%. At higher pressures, it gives the relationship between n and p that has been plotted in Fig. 5.8 as the dashed curve L-L, which is clearly much too high. The solid curve Z is an empirical one which Zel'dovich et al. (1961) obtained from the known influences of temperature and density on n under normal conditions, and is defined by n = 1.334 + 0.334(p - 1) - (1.90 x 1 0 - s p t ) where p is in g c m - 3 and t is in °C.

(12)

Properties of electrolyte solutions at high pressures and temperatures

103

ELECTRICAL CONDUCTIVITY

The influence of high temperatures and pressures on the electrical conductivity of water and aqueous solutions has been adequately surveyed by Marshall (1968), T6dheide (1972), Brummer and Gancy (1972), H a m a n n (1974) and Franck (1977, 1978). There is no need to go over the same ground in detail here. It will be sufficient to summarize the results by saying, first, that the conductivity of pure water rises fairly dramatically with increasing temperature at constant density and with increasing density at constant temperature. The effect undoubtedly arises from enhancement of the self-ionization of water into H + and O H - ions (see p. 104). On the other hand, the conductances of dissolved strong electrolytes generally follow the fluidity of water, and ultimately tend to decrease with increasing density at constant temperature. The conductances of weak or associated electrolytes (particularly at high concentrations) reflect, of course, the influence of pressure and temperature on their ionic dissociation.

IONIC DISSOCIATION EQUILIBRIA

Weak electrolytes One of the most striking and general of the effects of pressure on aqueous systems is the remarkable increase that it causes in the degree of ionization of dissolved weak electrolytes--and also of water itself. A few experimental results at moderate pressures, typical of many others, are shown in Fig. 5.9 (Hamann, 1974), where the logarithm of the molal ionization constant K is plotted against the pressure. Figure 5.10 shows the behaviour of aqueous ammonia and of water over a rather wider pressure range (cf. El'yanov and Hamann, 1975). It will be seen that the curves rise continuously, and that a pressure of 12 kbar increases the ionization constant of ammonia at 45°C by a factor of about 500. The changes stem from the fact that compression enhances the stability of the free ions with respect to their parent molecules. It does so because it increases the magnitude of the free energy of hydration of the ions ]AGhl, both by increasing the dielectric constant E of water and by reducing the ionic radius r. Indeed, if these changes are allowed for in the Born formula (7), they yield the theoretical curve A in Fig. 5.10 for a typical pair of small ions in water at 45°C (Hamann, 1957). It gives at least a qualitative description of the behaviour of real electrolytes. The effect of temperature upon K is more complex than the effect of pressure. An increase of temperature at constant pressure reduces e and increases r and so lowers tAGh! t

u

125

piperidine

20

~

15

in

100

10

\amrnonic

o

0.50

1

2

Pressure P / k b o r

Fig. 5.9 The effect of pressure on the ionization constants of some acids and bases in water at 25°C ( H a m a n n 1974). Kp is the value of the molal ionization constant at the pressure P, and K1 is the value at atmospheric pressure.

104

SEFTON D. HAMANN

1000

3.0

5OO

2.5 200 100

2.0

50 ~

(3++ o

1.5

/

~

~

~vater

20

o~

o_

10

1.0

5 0.5 2 0

I

I

I

2

0

I

I

4

I

I

6

I

I

8

I

I

10

I

12

Pressure P/kbor

Fig. 5.10. T h e effect of pressure on the i o n i z a t i o n c o n s t a n t of a m m o n i a in water at 45°C and on the self-ionization constant of water at 25"C ( E F y a n o v and H a m a n n 19751. Kp and K i are defined as for Fig. 5.9. Curve A is discussed on p. 103.

and tends to cause a reduction of K. However this effect of reduced hydration may be counterbalanced by the existence of a large positive "internal" enthalpy change for the ionization reaction, as it is in the case of water itself. The greatest extremes of pressure and temperature have been reached in conductimetric studies of the self-ionization of water by both static (Holzapfel and Franck, 1966) and shock-wave techniques (David and Hamann, 1959; Hamann and Linton, 1966, 1969a). The two methods agree in showing that pure water becomes quite highly ionized at pressures near 100kbar and temperatures near 1000°C. The shock-wave results are shown in Fig. 5.11, where K'w denotes the molal ionization product, uncorrected by activity coefficients. The curve A is a precise one calculated for weak shock waves of up to 8 kbar from static P - V - T - E data (e.g. Helgeson and Kirkham, 1974) and static measurements of Kw (Linov and Kryukov, 1972). The line D indicates the value that Kw Pressure P/kbar

10

20

50

100

i

i

i

i

150 200 I

I

\

102

D

,,f

1

c\/¢111°°o °c z ~ ~°°c

10-2 10-4

5OO°C o

E

10-6 10-8

J

10-10

/

d' of

10-12

10-+;4 10-16 10

I

I

12

I

I

1.&

I

I

1.6

I

I

1.8

I

2.0

Density p/g cm-3

Fig. 5.11. T h e m o l a l i o n i z a t i o n c o n s t a n t of water under s h o c k - w a v e c o m p r e s s i o n from a t m o s pheric pressure and circa 20°C. T h e curve A is discussed o n this page.

Properties of electrolyte solutions at high pressures and temperatures

105

would have if the water were completely ionized to H3 O+ and O H - ions (mH3o÷ = mon- = 27.8 tool k - l ) : complete ionization to H90~- and O H - ions would occur at a lower value than D by a factor of 6.25. It will be seen that K'w increases by a factor of more than 1013 between atmospheric pressure and a shock pressure of 133 kbar, and that the trend is such that conversion to a completely ionized state might be expected to occur in shock-wave compression to P = 150-250 kbar, p = 1.8-2.0 g cm-3, t > 1000°C. In this connection, however, it must be mentioned that the scales of P, p and t shown in Fig. 5.11 are based on the tables of Rice and Walsh (1957), which are now suspect and could well require amendment for the reasons given on p. 91. If, for instance, the densities p were taken from the relationships A in Figs 5.3 and 5.4, rather than from Rice and Walsh's values, the ionization curve would steepen from B to C in Fig. 5.11. Also, it appears that Rice and Walsh may have overestimated the temperature of shocked water. For both these reasons, the transition to a fully ionized state may occur under rather milder conditions than those indicated above. Whether complete ionization does actually occur, is difficult to establish by conductance measurements alone, because the very high conductivity of shocked water beyond the highest experimental point in Fig. 5.11 makes it impossible to carry out an independent measurement of the mobilities of the hydrogen and hydroxyl ions--which are needed in order to derive the concentrations of the ions from the measured conductivity (Hamann, 1974). However, the transition should be reflected in other properties than conductivity--e.g, in the thermodynamic and optical properties of shock-compressed water and there are some indications that it is. The possible existence of a transition region in the P - p plot has already been discussed on p. 91. Also, the refractive index results in Fig. 5.8 appear to be abnormally low in the region near p = 1.8 g cm- 3, which is consistent with the fact that the ionic salt ammonium fluoride NH~-F- (which is iso-electronic with H 3 0 + O H -) has a lower refractive index than molecular water and than ice, if the refractive index of ice is adjusted by the Gladstone-Dale formula to the density of ammonium fluoride (Hamann and Linton, 1969a). Another significant point is that the two coefficients of Up in eqs. (1) and (2) are, respectively, close to the average values for molecular materials such as organic compounds (1.78) on the one hand, and for ionic materials such as the alkali halides (1.34) on the other hand (taken from Van Thiel, 1967). In short, the indications are that water becomes fully ionized to Ha O+ and O H ions at circa 150-200kbar, 1.7-1.9 g cm -3 and 1000°C. It is then essentially a molten salt, iso-electronic with N H 4 F and N a O H . At higher pressures it may further ionize to H + and O H - ions or H + and O 2- ions, although these transitions require much higher energies. In this connection, it is perhaps significant that a single pressure-density measurement for water at the extremely high shock pressure of 14 Mbar (Podurets et al., 1972) lies close to extrapolations of the curves B in Figs 5.3 and 5.4. On the theoretical side, it would be very interesting to see whether a computer simulation of water based on Stillinger and David's (1978) "polarization" molecular model, which is capable of ionizing, would predict a high degree of ionization in the P V - T region where it appears to occur experimentally.

Associated ions and ion complexes Many salts form ion-pairs and complex ions in aqueous solutions under normal conditions, and even more do so at high temperatures and low pressures, where the density and dielectric constant E of water drop steeply. On the other hand, isothermal compression raises E and favours the dissociation of associated ions, as it does that of weak electrolytes (Hamann, 1974). There have been quite extensive studies of these equilibria under conditions of combined high temperatures and high pressures, notably by Franck and his associates and by

106

SEFTOND. HAMANN

10

i ~ .. ~. ~. .,. ... .-","~e

10-2

LiCI

.,0 .3 J

.

..;i/

//.AY

~

.

~

ti° J

.- . - ~ ~ / ' f / /

1'°-' 10-3

10-7 10-B

105 I 0.3

I O.L

I 0.5

I 0.6

I 07

10`6

-3 Density 9/g crn

Fig. 5.12. The molar dissociation constants of ion pairs of Li+CI- (Franck 1961) and Na+C1 (Quist and Marshall, 1968) at high temperatures and pressures. The dashed curves are given by formula (13). Marshall and Quist. The results have been reviewed by Franck (1973, 1978) and by Brummer and Gancy (1972). As in the case of weak electrolytes, although it is difficult to predict the influence of temperature at constant density, it is possible to make a fair estimate of the influence of density (or pressure) at constant temperature. H a m a n n et al. (1964) found that this could be done for ion-pairs of lanthanum ferricyanide in water at 25°C by applying Fuoss's continuum theory of ion-pair formation, which gives the molar dissociation constant in the form K = ( 3 0 0 0 / 4 ~ N a 3) e x p ( z l z 2 e 2 / a e k T )

(13)

where N denotes Avogadro's constant, k is Boltzmann's constant, z l e and z2e are the charges on the two ions, and a is the "contact distance" between their centres in the paired state, which is here assumed to be independent of the pressure. There is no reliable a priori way of estimating the parameter a, but if it is derived by applying eq. (13) to the measured value of K for a particular temperature and pressure (say, atmospheric pressure), the eq. (13) can be used to predict the effect of compression through its influence on • ( H a m a n n et al., 1964). If it is applied in this way to some of the high-temperature, high-pressure results, it yields the dashed curves in Fig. 5.12, where the upper set of solid curves represents Franck's (1961) experimental results for the dissociation of Li+C1 - and the lower set are for Quist and Marshall's (1968) results for Na+C1 -. The values of a have been found by fitting eq. (13) to the experimental results for K at p = 0.3 g c m -3. There is fair agreement between the predicted and experimental values at higher densities. Marshall and Quist (1967) have proposed an alternative predictive method, but its validity is very dubious (Hamann, 1974). Ultimately, no doubt, the best predictions will be made by computer simulation. REFERENCES

ADAMSL. H. (1931) Equilibrium in binary systems under pressure. I. An experimental and thermodynamic investigation of the system NaC1-H20 at 25°C. J. Amer. Chem. Soc. 53, 3769-3813. Ar{gENST. J, and RUDERMANM. H. (1966) Immersed-foilmethod for measuring shock wave profiles in solids. J'. Appl. Phys. 37, 4758-4765.

Properties of electrolyte solutions at high pressures and temperatures

107

AL'TSHULER L. V., BAKANOVAA. A. and TRUNIN R. F. (1958) Phase transformations of water compressed by strong shock waves. Soy. Phys. Doklady 3, 761-763. AL'TSHULER L. V., KANEL' G. I. and CHEKIN B. S. (1977) New measurements of the viscosity of water behind a shock wave front. Sot'. Phys. J E T P 45, 348-350. BACONIN J. (1979) Personal communication to the author. BAKANOVAA. A., ZUBAREVV. N., SUTULOV YU, N. and TRUNIN R. F. (1975) Thermodynamic properties of water at high temperatures and pressures. Soy. Phys. J E T P 41,544-548. BARKER J. A. and WATTS R. O. (1969) Structure of water: A Monte Carlo calculation. Chem. Phys. Lett. 3, 144- 145. BENSON S. W., COPELAND C. S. and PEARSON D. (1953) Molar volumes and compressibilities of the system NaC1 H20 above the critical temperature of water. J. Chem. Phys. 21, 2208-2212. BErT K. E. and CAPPI J. B. (1965) Effect of pressure on the viscosity of water. Nature 207, 620-621. BORN M. (1920) Volumes and heats of hydration of ions. Z. Physik 1, 45~48. BRIDGMAN P. W. (1942) Freezing parameters and compressions of twenty one substances to 50,000kg/cm 2. Proc. Amer. Acad. Arts Sci. 74, 399M24. BRUMMER S. B. and GANCV A. B. (1972) Aqueous solutions under extreme conditions. In Water and Aqueous Solutions (HORNE R. A., ed.) Chap. 19, pp. 745-803. Wiley-Interscience. BURNHAM C. W., HOLLOWAVJ. R. and DAVIS N. F. (1969) The specific volume of water in the range 1000 to 8900 bars, 20 to 900C. Amer. J. Sci. 267-A, 70-95. COWPERTHWAITE M. and SHAW R. (1970) Cv(T) equation of state for liquids. Calculation of the shock temperature of carbon tetrachloride, nitromethane and water in the 100-kbar region, J. Chem. Phys. 53, 555-560. DAVID H. G. and HAMANN S. D. (1959) The electrical conductivity of water at high shock pressures. Trans. Faraday Soc. 55, 72 78. DEAr, W. E. (1958) Measurement of the reflected shock Hugoniot and isentrope for explosive reaction products. Phys. Fluids 1, 523 527. DUDZIAK K. H. and FRANCK E. U. (1966) Messungen der ViscositS.t des Wassers bis 560°C und 3500 bar. Ber. Busenqes. 70, 1120-1128. ELLIS A. J. and MCFADDEN I. M. (1972) Partial molar volumes of ions in hydrothermal solutions. Geochim. Cosmochim. Acta 36, 413-426. EL'VANOV B. S. and HAMAr,,I~ S. D. (1975I Some quantitative relationships for ionization reactions at high pressure. AuNt. J. Chem. 28, 945 954. FRANCK E. U. (196l) [Jberkritisches Wasser als elektrolytisches L6sungsmittel. Angew. Chem. 73, 309-322. FRANCK E. U. (1973) Concentrated electrolyte solutions at high temperature and pressure. J. Sol. Chem. 2, 339 356. FRAYCK E. U. (1977) Equilibria in aqueous electrolyte systems at high temperatures and pressures. In Phase Equilibria and Fluid Properties in the Chemical Industry (SToRVlCK T. S. and SANDLER S. I., eds), Chap. 5, pp. 99-117. Amer. Chem. Soc. FRANCK E. U. (1978) Experimental investigations of fluids at high pressures and elevated temperatures. In High Pressure Chemistry (KELM H., ed.), pp. 221 225. Reidel. GEHRIG M., LZNTZ H. and FRANCK E. U. (1979) Thermodynamic properties of water-carbon dioxide-sodium chloride mixtures at high temperatures and pressures. In High Pressure Science and Technology (TIMMERHAUS K. D. and BARBER M. S., eds) Vol. 1, pp. 539-542. See also GEHRIG M. (1979) Dissertation, Institute of Physical Chemistry, University of Karlsruhe. GORANSON R. W. (1938) Silicate-water systems--Phase equilibria in the NaAISi3Os H20 and KA1Si3Os-H2 O systems at high temperatures and pressures. Amer. J. Sci. 35A, 71-91. HAMANN S. D. (1957) Physico-Chemical Effects of Pressure. Butterworths. HAMANN S. D. (1966) Effects of intense shock waves. In Advance in High Pressure Research (BRADLEY R. S., ed.), Vol. 1, Chap. 2, pp. 85 141. Academic. HAMANN S. D. (1974) Electrolyte solutions at high pressure. In Modern Aspects of Electrochemistry (CONWAV B. E. and BOCKRISJ. O'M., eds), Vol. 9, Chap. 2, pp. 47-158. Plenum. HAMANN S. D. and LINTON M. (1966) Electrical conductivity of water in shock compression. Trans. Faraday Soc. 62, 2234~2241. HAMANN S. D. and LINTON M. (1969a) Electrical conductivities of aqueous solutions of KC1, KOH and HC1 and the ionization of water at high shock pressures. Trans. Faraday Soc. 65, 2186-2196. HAMANN S. D. and LINTON M. (1969b) The viscosity of water under shock compression. J. Appl. Phys. 40, 913 914. HAMANN S. D., PEARCE P. J. and STRAUSSW. (1964) The effect of pressure on the dissociation of lanthanum ferricyanide ion pairs in water. J. Phys. Chem. 68, 375-380. HAWKINS D. T. (1975) A bibliography on the physical and chemical properties of water: 1969 1974. J. Solution Chem. 4, 621 744. HEINZINGER K., RIEDE W. O., SCHAEFERL. and SzAsz GY. I. (1978) Molecular dynamics simulations of liquids with ionic interactions. ACS Symposium Series 86, 1-28. HEINZINGER K. and VOGEL P. C. (1976) A molecular dynamics study of aqueous solutions. III. A comparison of selected alkali halides. Z. Naturforsch. 31a, 463 475. HELGESON H. C. and KIRKHAMD. H. (1974) Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures: I. Summary of the thermodynamic/electrostatic properties of the solvent. Amer. J. Sci. 274, 108%1198. HELGESON H. C. and KIRKHAMD. H. (1976) Theoretical prediction of the thermodynamic properties of aqueous electrolytes at high pressures and temperatures. III. Equation of state for aqueous species at infinite dilution. Amer. J. Sci. 276, 97-240. HILBERT R. (1979) PVT-Daten yon Wasser und yon w~isserigen NaCl-L6sungen bis 873 K, 4000bar und 25 Gewichtsprozent NaCl. Thesis, Institute of Physical Chemistry, Karlsruhe University (cited by FRANCK, 1978).

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HOLZAPFELVON W. and FRANCKE. U. (1966) Leitf~ihigskeit und Ionendissoziation des Wassers bis 1000°C und 100 kbar. Ber. Bunsenges. 70, 1105-1112. JANSOONE V. M. and FRANCK E. U. (1972) The dielectric constant of water at high temperatures and pressures from the mean spherical model in the Wertheim solution. Ber. Bunsenyes. 76, 943-946. JOHNSON C. H. J. and SPURL1NGT. H. (1971) Pairwise additive third virial coefficients for multipolar molecules: application to water. Aust. J. Chem. 24, 1567-1580. JI~ZA J. (1966) An equation of state for water and steam--Steam tables in the critical region and in the range from 1000 to 10,0000 bars. Naklad. Cesk. Akad. V~d, Praffue. KAWAI N., TOGAYA M. and MISHIMA O. (1977) Metallic transition of oxides, H20 and hydrogen. In High Pressure Research--Applications to Geophysics (MANGHANI M. H. and AKIMOTOS., eds), pp. 267--279. Academic Press. KENNEDY G. C. and HOLSER W. T. (1966) Pressure-volume-temperature and phase relations of water and carbon dioxide. In Handbook of Physical Constants (CLARKS. P. JR, ed.), Section 16, pp. 371 383. Geological Society of America. KISEL'N1K V. V., MALYUKN. G., TORYANIK A. N. and TORYANIKV. M. (1973) Effect of pressure and temperature on the self-diffusion of water. J. Struct. Chem. 14, 911-914. K~)STER H. and FRANCKE. U. (1969) Das spezifische Volumen des Wassers bei hohen Driicken bis 600C und l0 kbar. Ber. Bunsenges. 73, 716-722. KORMER S. B. (1968) Optical study of the characteristics of shock-compressed condensed dielectrics. Sot. Phys. Uspekhi II, 229-254. LIE G. C., CLEMENTI E. and YOSHIMINE M. (1976) Study of the structure of molecular complexes. XIII. Monte Carlo simulation of liquid water with a configuration interaction pair potential. J. Chem. Phys. 64, 2314 2323. LINOV E. D. and KRVUKOVP. A, (1972)The ionization of water at pressures between 1 and 8000 kg/cm 2 and at temperatures 18, 25, 50 and 75~C. Izv. Sib. Otd, Akad. Nauk SSSR Ser. Khim. Nauk 4, 10-13. LVSNE P. C. (1970) A comparison of calculated and measured low-stress Hugoniots and release adiabats of dry and water-saturated tuff. J. Geophys. Res. 75, 4375-4386. MARSHALL W. L. (1968) Conductances and equilibria of aqueous electrolytes over extreme ranges of temperature and pressure. Pure Appl. Chem. 18, 167 186. MARSHALLW. L. and QU1STA. S. (1967) A representation of isothermal ion-ion-paj,r-solvent equilibria independent of changes in dielectric constant. Proc. Nat. Acad. Sci. 58, 901 906. McCooL M. A. and WOOLE L. A. (1972) Pressure and temperature dependence of the self-diffusion of carbon tetrachloride. JCS Faraday Trans. 1 68, 1971-1981. MINEEV V. N. and SAVINOVE. V. (1975) Relationship between the viscosity and possible phase transformations in shock-compressed water. Soy. Phys. J E T P 41,656-657. M1NEEV V. N. and ZAIDEL' R. M. (1968) The viscosity of water and mercury under shock loading. Soy. Phys. J E T P 27, 874 878. MISHIMA O. and ENDO S. (1978) Melting curve of ice VII. J. Chem. Phys. 68, 4417-4418. M1TCHELL A. C. and NELLtS W. J. (1979) Water Hugoniot measurements in the range 30 to 220 GPa. In High Pressure Science and Technolo~ty (T1MMERHAUSK. D. and BARBERM. S., eds), Vol. l, pp. 428 434. Plenum. NELSON D. A. and RUOEF A. L. (1979) Metallic xenon at static pressures. Phys. Rer. Lett. 42, 383 386. PODURETS M. A., SIMAKOVG. V., TRUNIN R. F., POPOVL. V. and MOIS~EVB N. (1972) Compression of water by strong shock waves. Sov, Phys. J E T P 35, 375-376. QulsT A. S. and MARSHALLW. L. (1968) Electrical conductances of aqueous sodium chloride solutions from 0 to 800C and at pressures to 4000 bars. J. Phys. Chem. 72, 684-703. RAHMAN A. and ST|LLINGER F. H. (1971) Molecular dynamics study of liquid water. J. Chem. Phys. 53, 3336-3359. Rice M. H. and WALSH J. M. (1957) Equation of state of water to 250 kilobars. J. Chem. Phys. 26, 824-830. RUSHBROOKEG. S. (1979) On the dielectric constant of dipolar hard spheres. Mol. Phys. 37, 761-778. SILLINGER F. H. and DAVID C. W. (1978) Polarization model for water and its ionic dissociation products. J. Chem. Phys. 69, 1473 1484. STILLINGER F. H. and RAHMANA. (1972) Molecular dynamics study of temperature effects on water structure and kinetics. J. Chem. Phys. 57, 1281-1292. ST1LL1NGER F. H. and RAHMAN A. (1974a) Improved simulation of liquid water by molecular dynamics. J. Chem. Phys. 60, 1545 1557. STIEL1NGER F. H. and RAHMANA. (1974b) Molecular dynamics study of liquid water under high compression. J. Chem. Phys. 61, 2973-2980. TODHEIDE K. (1972l Water at high temperatures and pressures. In Water--A Comprehensive Treatise (FRANKS F,, ed.}, Vol. 1, Chap. 13, pp. 463 514. Plenum. VAN THIEL M. (1967) Compendium of Shock Wave Data UCRL Rep. 50108, Lawrence Radiation Laboratory. VEDAM K. and LIMSUWAN P. (1978) Piezo- and elasto-optic properties of liquids under high pressure. I. Refractive index vs. pressure and strain. J. Chem. Phys. 69, 4762-4771. VERESHCHAGIN L. F., YAKOVLEVE. N. and TIMOFEEVYU. A. (1975) Transition of H20 into the conducting state at static pressures P ~ 1 Mbar. J E T P Lett. 21,304-305. WADE C. G. and WAUGHJ. S. (1965) Temperature and pressure dependence of self-diffusion in liquid ethane. J. Chem. Phys. 43, 3555-3557. WATTS R. O. (1976) The effect of ion pairs on water structure. Mol. Phys. 32, 659~568. WATTS R. O., CLEMENT!E. and FROMMJ. (1974) A theoretical study of the lithium fluoride molecule in water. J. Chem. Phys. 61, 2550-2555. YADAV H. S., MUETY D. S., SINHA K. H. C., GUPTA B. M. and DAL CHAND (1973) Measurement of refractive index of water under high dynamic pressures. J. App. Phys. 44, 2197-2198. YAKUSHEV V. V,, NABATOV S. S. and DREMIN A. N. (1975) Measurement of dielectric permittivity at high dynamic pressures. Soviet Phys. J E T P 42, 646 647.

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YAKUSHEVAO. B., YAKUSHEVV. V. and DREMINA. N. (1972) Formation of particles of sulphur in solutions of sodium thiosulphate behind fronts of shock waves. In Combustion and Explosion, pp. 544-548. Izdat. 'Nauk'. YAMAOKA S., FUKUNAGAO. and SAITOS. (1970) Phase equilibrium in the system MgO-H20 at high temperatures and very high pressures. J. Amer. Ceram. Soc. 53, 179-181. ZEL'DOVICHYA. B., KORMERS. B., SINITSYNM. V. and YUSHKOK. B. (1961) A study of the optical properties of transparent materials under high pressure. Soy. Phys. Doklady 6, 494-496. DISCUSSION HELGESON: I have a c o m m e n t regarding the kinetics of the reactions you discussed. We have been investigating the kinetics of silicate reactions with water (hydrolysis reactions) for m a n y years, and we now believe that we have a viable model based on transition state theory. It appears thus that the overall rates of silicate hydrolysis in geochemical processes are controlled in acid solutions by the hydrogen ion concentration or, in other words, the hydronium ion. This fits in, in a much more general sense, with your remarks concerning carbonate reactions. There seems to be an activated complex that forms on the surface of alumino-silicates which would, by analogy with the carbonates, have the configuration which you propose. I therefore think the significance of your observations can be much broader than their application to carbonates would imply. HAMANN: Yes, I would agree with you. In fact, not only the rate of isotope exchange but also the rate of recrystallization increases very much, which would fit your model. ROWLINSON: May I ask a very simple-minded question about the interpretation of the shock measurements of Rice and Walsh and Cowperthwaite and Shaw and so on that you mentioned earlier'? These are obviously based on alternative assumptions as to which heat capacities or which compressibilities or which coefficients of thermal expansion were unchanged with pressure or unchanged with density. My simple question is, if you use these to deduce the equation of state of water, and if you have more than one starting point, then presumably you can calculate post facto whether the assumptions were correct, and, indeed, iterate to improve them. Why cannot you check which of these assumptions, if any, is correct after you have done the calculations? HAMANN: This is essentially what B a k a n o v a did. By using different starting conditions and having double compression, she achieved different final states from different initial states, and this bypassed some of the hypotheses. ROWLINSON: Can one then not discriminate between them? HAMANN: Well, it's not so very simple, it's quite complex. Jfiza, I should mention, produced an empirical equation of state which, as you probably know, also predicts temperatures lower than Rice and Walsh's and which doesn't make any of these assumptions at all. It's a van der Waals type of equation of state which has adjustable a's and b's. It seems to work rather nicely under static conditions, and it predicts lower shock wave temperatures than Rice and Walsh did. ROWLINSON: A van der Waals equation, of course, normally commits you to Cv being a function of density but not of temperature. HAMANN: Well, I think in the case of Jfiza's equation, they're probably functions of both temperature and pressure. It's a considerably modified van der Waals type of equation. FRANCK : You expressed doubts as to the possibility to calculate the ion product of water beyond a limit of specific conductivity of, say, 1 o h m - 1 c m - 1 because you said that there is a lack of mobility data which would be needed. Would you not think that, within the range of reasonable accuracy that one needs there, we could take conductivity data of

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fused sodium hydroxide and extrapolate those at constant density to 1000°C? Since the sodium ion in fused sodium hydroxide is very like the H3 O+ ion, it is probably rather similar in size. So, in the range you pointed out, we could perhaps consider water as very similar to fused sodium hydroxide. Sodium hydroxide has, in the fused state, a density of about 2.1 g c m - 3 and a molecular weight of 40. Two water molecules have 36 and the comparable density would be about 1.8. I think that this extrapolation is uncertain by 50 or even 100~o. However, in discussing the Kw-value one is dealing with a quantity which at unusual conditions has not to be known very precisely. In other words, it need not be known as accurately as the mobility data. One main difficulty is that the ion product concept is certainly not the appropriate concept to deal with a fused salt. HAMANN: Yes, I agree, this is probably a fairly good approximation. We tried, of course, to keep to experimental data as far as we could and not make any assumptions. I think we've done that, apart from the assumption about the activity coefficients which we can do nothing about. But I think there is a question of what is the conductivity of sodium hydroxide under pressure. What do you think pressure would do, a pressure of, say, 15 kbar? FRANCK: Oh, I would look at the conductivity of this fused sodium hydroxide at its melting temperature (318°C) and then try to extrapolate this to high temperature at constant density. HAMANN: Oh, I see, you think density is the important thing'? FRANCK: Yes. PITZER: How far can you go by using sodium hydroxide aqueous and some acid aqueous? What is the limit? HAMANN: Only to the stage where water is about 5 or 10~ ionized. Beyond that it masks the contribution of the solute. So, that's where our curve ends, at about 5~o ionization. PITZER: I understand, but it gets you quite a way in that, even beyond there, the conductance should be the sum of the two. I'm quite convinced that you're correct in saying that this is partially approaching a fused salt, but, in contrast to fused N a O H , for example, the H 2 0 molecule is still a fairly low energy, reasonably abundant constituent. So, I suspect that the H 2 0 molecule does not disappear completely in the population distribution in this region. Going to the other extreme, what's the chance that there's s o m e H402+ or 0 2 - in the population too? In other words, it would seem to me that maybe one has a distribution of all these species, in which the middle three--H3 O+, O H - and H z O - - a r e fairly comparable in population. HAMANN: Of course, yes, there must be a mixture of all these things. I oversimplified it, I'm afraid, by suggesting it was only H 3 0 + and O H - . But H402+?? PITZER: Oh, I agree that would be a minor species, but maybe not totally negligible. It might be appreciable. HAMANN: That would form at a lower level even than the H 3 0 + . . . . PITZER : Yes and it'd be less populous, but it might not be negligible. HELGESON: Well, notwithstanding these ambiguities, it seems to me that if we base the extrapolation on evidence that you now have, and consider H 2 0 to be a hydronium hydroxide fused salt at these high pressures and temperatures it becomes rather intriguing from a

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geophysical point of view. An oceanic or continental geotherm should pass through the curves in the region where H 2 0 would correspond to hydronium hydroxide. Which bring up the interesting possibility that, if there is indeed H 2 0 in the lower mantle of the earth, it may be there in a form analogous to fused sodium hydroxide, as Professor Franck suggests. However, it also seems likely that the Clapeyron slope of the melting curve of a solid polymorph of liquid hydronium hydroxide would be small. In that case there might well be an ice polymorph at higher pressures than we know ice-VII to melt at, which might be intersected by the geotherms. It follows that the possibility of a solid ice polymorph in the lower mantle cannot be dismissed. I wonder if you have any comments about the possibility of a melting curve of this kind existing at high pressures? HAMANN: I must confess that I have not thought about this possibility. It is interesting, but l have not considered it. FRANCK: I'd like to make one comment to Professor Pitzer's remark. I suppose that the chances for bivalent ions become lower the higher the temperature is here. We have tried very crude estimates, not calculations, to investigate the possibilities of higher charged species at high temperature. They are certainly there, but reduced. The dielectric constant is low. Even if you apply just a simple Born equation, you see that the chances of the higher charged ions are lower. In another example, Dr. T6dheide measured the conductivity of bismuth chloride at high supercritical temperatures and high density. Here too, there is in principle the chance of having neutral, monovalent and bivalent ions, and you certainly get all of these just by reasonably using the existing thermochemical data. But it turns out, that the bivalent ions are less by one or two orders of magnitude. So that, within the crude approximation which one can only make here, I think it's not too bad a position if one just neglects everything else except single valent species. KESTIN : I'd like to reinforce your doubts about the measurements of viscosity, on two grounds. Firstly, from what we know about shock experiments in gases, the flow is absolutely certain to be turbulent. There is no possibility of its being laminar. Secondly, it seems to me from the very brief description of the measurement, that probably pressure drag is much more important than viscous drag in an experiment like this. And, out of those conditions, you can make errors of orders of magnitude. The second remark concerns the experimental technique. You seem to have gone to a lot of trouble to have a condition which occurs after two shock waves have collided head on. In the gas experiments, we usually avoided this kind of thing by making one shock wave reflect from a rigid wall so that behind the reflected shock a static mass of substence is created in which measurements are made very comfortably. HAMANN: Yes, we have used reflected methods also, but the pressures that these give were not as high as in the direct collision of two shocks. They're never as high in liquids. WHALLEY: I'd like to make a comment about the simple sphere model of the ions that you mentioned. It really has no business to work at all, because the electric field near a sphere the size of an ion is enormous, and the model assumes that the dielectric is incompressible, whereas it is actually very compressible. It seems likely that there are two things happening. The high field an ion does both things: it increases the density which tends to increase the Gibbs function change, and it decreases the dielectric constant by inducing partial saturation. And the only sense one can make of the fact that you mentioned, as other people have done, for high temperature solutions that are highly compressible is that there is a compensation of two effects which work in opposite directions: the saturation and the electrostriction. It is very simple to calculate both of them for a dilute gas. They're essentially of the same magnitude but opposite in sign. HAMANN: Yes, I'd agree with these comments. This is why I prefer simulations on computers.