Thermodynamic representation of the state of a saturated fluid

Volume 130, number 4,5

PHYSICS LETTERS A

11 July 1988

THERMODYNAMIC REPRESENTATION OF THE STATE OF A SATURATED FLUID Albrecht ELSNER Max-Planck-Institutfuir Phasmaphysik, Garching, FRG Received 17 April 1988; revised manuscript received 2 March 1988; accepted for publication 9 May 1988 Communicated by J.P. Vigier

The Massieu function of a saturated fluid (composed of one particle species, two phases, and one thermodynamic degree of freedom) is given as a function of the temperature, vapor pressure, surface tension, and chemical potential, the latter being expressed in terms of the bulk densities. It is thus possible to give a complete thermodynamic description of the state of this system that includes the absolute values of the entropy and internal energy.

The aim of this Letter is to give the complete set of the equations of state of a saturated single-cornponent fluid as a function of the measured temperature, vapor pressure, surface tension, and the bulk densities. The theoretical framework in which the fluid can be subjected to a thermodynamic treatment was elaborated by Callen [1] in particular and is generally known. This framework not only clearly defines the approach but also gurantees primarily the uniqueness and completeness of the solution of the problem. The entropy representation is chosen to describe the system. The treatment enlists the known solutions of the ideal gas in order, on the one hand, to illustrate the difference from the real fluid and, on the other, to

equilibrium state is described by the entropy function, which takes the form [1]

determine the entropy constant of the real fluid by comparison with the ideal gas laws. The equations of state of the fluid are thus obtained phenomenologically here. The homogeneity of the fluid allows its macroscopic properties to be described by extensive and intensive parameters. The extensive parameters are additive system quantities; there are the entropy S, the internal energy U, the volume V, in the case of two fluid phases of different densities the free interface area A, and the particle number N. The intensive parameters characterize the thermodynamic (i.e. thermal, mechanical and diffusive) system equilibrium; these are the temperature T, the pressure p, the surface tension y, and the chemical potential jt. The

v= v~ + v2,

S~U

+ V~—A

—

N ~.

(1)

Eq. (1) contains all conceivable thermodynamic information on the single-component fluid. Below its critical temperature the fluid decomposes into the two bulk phases vapor (index v) and liquid (index 2), which are spatially separated from one another by an interface (index i). The separation of the extensive parameters leads to the relations [2—5] S Sv + S5 + Si,

U Uv + U5 + U~, N= N~+ N5,

(2)

and the equilibrium relation (1) is split into the equations 1 p ~ Uv

—

~

—

Sv = =

+ Nv

—

+N

—

T

T

T’

1 U1

—5, =A ~

(3)

As discussed at length in ref. [2], the saturated single-component fluid has exactly one thermodynamic degree offreedom, i.e. for a temperature value Tthere

0375-9601/88/s 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

225

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PHYSICS LETTERS A

exists exactly one value ofp (saturation vapor pressure), y, ~i, VV/NV v~,V5/N5 v~(reciprocal bulk densities). It can also be shown that the following thermodynamic relations between particle densities (v~,v5) and energy densities (U~/N~u.,,, U5/N5 u9) are valid (see ref. [2] and appendix): _____ —

—

d(p/T)

— —

d (1 / T)

v 5

u~v~—u5v~ ——

d(u/T) d(l/T)’

v~/v5—l—ln(v~/v5) u + v~/v5—1— (v~/v~) ln(v~/v5)~

(4)

The first of these equations represents the temperature dependence of the vapor pressure and is called the Carnot—Clapeyron—Clausius equation. The second equation is obtained from the Gibbs—Duhem equation and represents the temperature dependence of the chemical potential. The third equation represents the correspondence principle and states that the ratio of the energy densities of the two bulk phases is an analytical function ofthe corresponding particle densities. The thermodynamic theory shows that the state of a(in system can be uniquely and completely the entropy representation) either byformulated means of the entropy S as a function ofthe extensive variables U, V, A, N (e.g. fundamental equation (7)) or, equivalently,by a function M obtained from S( U, ~ A, N) by Legendre transformation. Such a function M that has at least one intensive parameter as independent variable is called a Massieu function. In the function M considered below all variables are intensive parameters. According to eq. (1) this function vanishes identically [1]: M M(l

p y T’ V ~

=

S

U~

V~ +A

+N

0.

(5)

The intensive variables of the Massieu function of the ideal gas are presented as an example and with a view to their later use. The quantum mechanical calculation of the state of the model system of identical, structureless particles without spin and inter226

action leads in the region of high temperatures to the following relations [5]: 1 ~ N T= ~

~—k~ ~=—kln(—~—~), T J7~ T \NVq I —

3 Vq=

(2itmkT)312 h

(6)

(m particle mass, h Planck constant, k Boltzmann constant). These equations are the long-known laws of Clausius (U/N), Boyle, Mariotte, Gay-Lussac (VI N), and Sackur, Tetrode (i.e. according to eqs. (1)

‘

—

Vv —V~

11 July 1988

and (6) S/N=(U/N)(l/T)+(V/N)(p/T)—~/T =k[~+ln( V/Nv0)]). As the ideal gas does not condense, one has Ay/ T~0. The Massieu function (eq. (5)) then becomes M” 1 ~ ~T’ T’ T

~

~kN—kN—kNln

(_~_)=o. \NV0

Substituting the parameters U for the parameter Tin one now obtains S as a function of the extensive variables, i.e. instead of eqs. (6) the following fundamental equation: Vq,

S( U, V ,N) = k

[~

~ 4itm~\

)

1n(e3~ 312 V U + ln( N5t2 N (ideal gas) (7) With the absolute values U, V, Nthe value Sis thus also known absolutely. Eqs. (6) and (7) are equivalent to one another and each contains the complete information on the ideal gas. The real fluid is distinguished from the ideal gas by the effect of the interaction forces between the particles. In the case of the fluid condensation occurs. Taking the interaction potential into account makes it difficult to treat the fluid of high particle .

density. For and example, calculation of the entropy energythe of numerical a liquid from a thermodynamic potential which is described by a modeled pair-potential function or by experimentally determined interaction potentials is very extensive and can only be performed on large computers. Here the model calculations for comparison purposes are on an equal footing with calculations in which experimental data are used [3]. Such computer simulation calculations are not confined to investigations of a

Volume 130, number 4,5

PHYSICS LETTERS A

11 July 1988

fluid below the critical temperature (saturated fluid). They are also applied to fluids in the region of supercritical temperatures and high pressures where, among other things, continuous transitions between

density of the gas is now set equal to that of the real vapor, i.e. the temperature-dependent value v~,. is chosen for V/N, the condition for neglecting the interaction forces v0/v~<< 1 ought to be all the better

otherwise clearly separate states of matter (insulating—ionic, ionic—metallic, metallic—insulating) were identified, in agreement with measurements [6]. The here discussed method of describing the fluid state uses, instead of specified interaction potentials (i.e. Gibbs phase space distribution functions for the proposed microscopic structures), the potentials p, y, ~u.Of these three quantities p and y can immediately be determined by measurement. The value of /1, on the other hand, can only be determined according to eqs. (4) without the integration constant: r r d(u/T)=J vd(p/T)

satisfied the further below the critical temperature T~the reference temperature T is chosen. The evaluation of 20 substances showed without exception that the values of (S/N)gas = k[ ~+ ln (v~/v0)]and s.. with a large temperature difference to the critical point (T< 0.8 T~)are in good agreement (in particular, their relative deviation at the triple point is about 1%), if one of the two numerically equivalent expressions for (ui T) is used: r / kT = k’ 2 + lnl —s--

j

with

—

=— k

—

v~v.,. + v5

—

n

VV V5

The theory requires, however, that the complete chemical potential / ~ (p/T)c =T T v d (piT)

(\T/C ~)

—

J

p/T

be known. The index c here denotes the corresponding state value at the critical point. Fortunately, however, there is the possibility of determining the integration constant (j~/T)~empirically and thus describing the state of the fluid uniquely and completely. For this purpose the data of the chemical potential or, equivalently, of the entropy of the ideal gas have to be compared with the corresponding data of the fluid. The entropy values of the ideal gas according to eq. (6) are /5 \ I—I \J~JideaIgas

rs2 /\NVqJ] v ~I =kl—+ln(——I L

and those of the real vapor according to eq. (3) are ~sv

~~PIL

(for u.,, see eq. (12)). As is known, the ideal gas laws are very well satisfied for a real gas with low particle density, i.e. under the condition NVq/ V.cz< 1. If the

L

T~

kTc

‘\.VqcPc

[~~

+ ln(-~-)]<0.

+

(8)

Equating the two expressions in eq. (8) yields the numerical value 0.3034 for the compressibility factor of the fluid at the critical point, v~p~/kT~. As the chemical potential is known, all intensive variables of the Massieu function ofthe fluid can be specified. The relations analogous to eqs. (6) are -~-

T

(measured), ~ (measured), T /

=

-~

(measured)

T

~

l~-~~)j —

v=— v,,, + v5

v d (piT)

piT

C

VV — V5

—

(9)

ln(v~/v5) With eqs. (8) and (9) it is possible to determine the state of a single-component fluid completely. The corresponding Massieu is ~ —S function V~°A ~T’ T’ T’ T) T T ‘~‘-~-

—

— “

(p/T)c

+N[(~)~

—

S

vd(P/T)]=0.

(10)

According to eq. (10) the absolute entropy value S can be specified, although not as an explicit function of the extensive variables U, V, A, N as in the case 227

Volume 130, number 4,5

PHYSICS LETTERS A

11 July 1988

(~5

absolutely determinable quantities U, V, A, N, T, p. of the ideal gas (eq. (7)) but as a function of the y, v~,v5 and l~u/T)C. As a selection of possible entropy representations one finds

The energy can then also be written in the form

~

U=(— V Aa+Nv)~” d(!/T)~

g=g v~ ——

v~

(fig.!).

(13)

T~ —

— —

V~-~ —A dT

—

u,,, + v~ p sv =

N

—

dT

U5 + vs p —

—

T

S5 =

Consequently, the energy value, which is associated with the interaction potential and which is a

U+ Vp —Ay— N,i T

u

(11)

T

From the representation of the specific bulk entropies S~/N~ s~and S5/N5 s5 one obtains the equations of Carnot, Clapeyron and Gaussius: (s~—s5)T= (zs~—U5)

+

(v~—v5)p

function of V, A, N, T, p, y, v~,v 5, can obviously be obtained from the difference LxU Usaturated fluid —

Uideal gas, if Usaturated fluid = U describes the energy of the fluid system with particle interaction and Uideal gas ~NkT that whithout particle interaction. Then

~ U= V (—f—- — l’~d (p/T) + U, V/N )d(l/T)

—

~NkT< 0.

(14)

The interaction potentials in the bulk phases, that are ~ and z~U5/N5=—i~u5, can be deduced from ~ as E~u~=u~—~kT 3 and and i~u~ versus v~/3presents the interaction potential as a Au5=u5— ~kT. Thus a plot ofi~u5versus vU

and dp

s~—s~ = (v~—v5) _~

The absolute value of the energy U follows from

function of the particle distance (where L~u 5~

eqs. (2)—(4) [2,3]:

~

A few results are discussed: (a) From eqs. (12) it follows that the internal energy of the particles in the volume V~exceeds the

d ( y/ T) U U~+ U5 + U~ NvUv +N5u5 +A d(!/T) —

—

~

—

UV

=

/ i.’5 —

v5 v.. —! \ d (p/T) ln(vs/v~))Vvd(l/T) ~0, _________

\V~

v,,,/v5 U5=

limiting value 0 (= u~)and that of the particles in the volume V5 is below 0. Particles with positive energy are called freeparticles (vapor), those with negative energy bound particles (liquid, solid). Since it holds that

+A d(y/T) +Ndt(~T) d( lIT) d( 1 /T) d( lIT)

—— (Vv

V5

—

_______

1 \

d(p/T)

1n(V~/v5))V5d(1/T)~O.

lim U~=0, (12)

It should be mentioned tht the values of the surface tension y can be calculated if the intrinsic function of the interface is known. It is possible to give an empirical interfacefunction g which (for non-polar substances) has universal particle density dependence (see fig. 1 of the appenddix). If g is known, one obtains

5

(p/T)~ -~

=

a d(p/T),

U( v~)= N u... ( z-’,,

—

v5) + u5 ( i.’~, v~)+ U~ —

vv — V5

vanishes at the critical point; it is negative for temperatures below T~and positive for temperatures above TC, with the fluid in the latter case only existing in a single homogeneous gas phase. At absolute zero it assumes the value of the free interface energy Ay0 and condensation energy —N(u~—u5)o,yielding U

pIT

3, a~g(v~—v5)~ 228

T~TC

the fluid energy

r_.r~N lim —

C°~

limu~=0,

Volume 130, number 4,5

PHYSICS LETTERS A

11 July 1988

lim u5 = — (u~—u5)o U~U5\

T-.O

lim

vv=vqexp( kT limy~y0.

Y~YC=°~

(15)

T-.O

The chemical potential ~ according to eqs. (8) and (9) is a negative, concave temperature function with the maximum value at absolute zero,

kT / p= —exp

c’,

Vq

U.,.

u,,—u5\

kT

—

=kToln( Pr/Tr for T—0 \J1~,/T0)

—~

‘~

(18)

lim j~u~=—(u~—u5)0. (Pr/Tr>Po/To>0). The values T0,p0 (~p(T0))may

(b) In the region of very low temperatures one obtains the temperature scaling ofthe volume V.,. and of the vapor pressure p by combining the relation between the evaporation energy and vapor pressure and the relation between the evaporation energy and chemical potential. With the definition q~(v~—v5)p,limq=klim T T-.O

(16)

T—.O

and the relations

be obtained from

~

k =

~

/ p0/T0

On changing from the vapor to the (real) gas at low temperatures (i.e. v~—~ V/N, u.,,—’ U/N, u5 = 0) one obtains from eqs. (6), (18) 2 \3/2 VT3/2 / eh N c2~mk) —

3/2

dp=(s~—s 5)~dT,(s~—s5)T=u~—u5+q one obtains for the vapor pressure at any temperature exp(

S~

( —J

S~dT)

Tr (‘UV—UQ

q

T

dlnT

J

k,

~f~

JZ.. — NT

Tr

Pr ——exp Tr

I—~-j

7~\eh

~ —5

To/T

k’., (Pri Tr))

p

=

‘\

—

1±=4k

S/N= 4k,

T

in particular it holds that

)~

S

8U

(17)

and The index r here denotes a reference point between the absolute zero and critical point (0< Tr ~ Ta). In the low-temperature region it then holds according to eqs. (16) and (17), on the one hand, that

ca,,

T (0 (~çN)) = 0 for T~0

(c~specific heat at constant x= V or p), whereas in u.. —u 5~ =

—

d ln T

—

——

kTd~~~ —1)5) d ln T

kTd ln V.,. d ln T’

and according to eqs. (9) and (15), on the other, that lim (u.,. —u5)=

—

lim

~t

the case ofthe ideal gas it follows from eqs: (6) that Cv 4k and c~=4k for all temperatures. (c) According to Nerst the entropy of a real system tends to a finite limit at absolute zero: As a result of the complete particle condensation it follows from relations (8), (9), (11), (15) that lim S= lim N5s5 =Nlim v5p/T=0. T-.0 T-.O T-.O

T-.O

= lim IkTlfl(<~l. r-.o[ \p/T)J

These relations yield the scalings

(The vapor entropy [(V~/v~)(2U~+V~p—U5)/T] is calculated as limT.OSV = limT.ONVsV=limT.0 =0 and the interface entropy as limT.oS.,. = A limT.Ody/dT= 0.) The entropy value at the crit229

Volume 130, number 4,5

PHYSICS LETTERS A

ical point is N( v~.p~/ kT.

—

~~/kT~ ) k. This yields the

limiting values S T-. T~N lim

—

[

3

VCPC

)]

+ ln ((2Em ) 3/23PC (kT~)5,’2\ 1 k

=

h

‘

lim (s~T)=(U~—us)o,lims 5=0 .

ters v~,v5, p, y. The entropy constant SC (eq. (19)) required for calculating the entropy, the free energy, and the chemical potential was obtained by comparison with experimental data. For the real gas this constant provides, at the critical point, the numerical relation between thermodynamics quantum statistics, which otherwise is given at and absolute zero according to Nernst (by the constants Sgas4k,

(19)

T-. 0

T—.0

11 July 1988

SCondensate =

For equal particle density, the entropy of the ideal gas at T~exceeds that of the real gas by the amount [4—3vQpC/kTC—ln(kTC/vCpC) ]k=0.3971k. With the critical values S~= NSC and UC = 0 one obtains from eqs. (11) and (12) the relation

0).

The author is grateful to D. Pfirsch for helpful discussions.

Appendix S—SC= J-~dU. 0

The temperature derivative dS/dT (1/ T) dU/dT is the heat capacity of the system in relation to T, C( T) / T> 0, and so it always has a positive value, Hence S~represents the maximum entropy value ~5”, i.e. S—S~<0, and from dU/dT>.0 and .f~’(1 / T) dU< 0 it follows in turn that U< 0. With limT.o S=0 and limT.OU= U0 —Ay0 —N( u.,. — U5 )o one obtains SC = f~(1 / T)dU and hence a relation between system values at theof critical and at absolute zero. In the vicinity T=0, point i.e. 0~T~T*, C( T) can be quantum mechanically calculated and for T~T* be represented in the form C(T)=N,jT” with n~l 5(k4/h3)v 3 (w velocity of sound in (e.g.liquid), i~=~xT*=0.01 K, 5/w the n=3 for 4He [7]). This then yields

$ o

$ 0

~dU=c~T)

110

+

U(v~) U(V)

N

UV(VCV5)+UQ(VVVC) V~—vs

~

dT U(vC)~UC=~lim

(u~+u 5)

T-.T~

0

j

~ dU

—

U(T)

This study shows that it is possible to represent any thermodynamic function of a saturated fluid as a temperature function of the experimental parame230

as

lim

U(T)

and with U( T*) = U0+ C( T*) T* it finally follows that C

—

This yields

~dU

s = U( T*) —Ay0 + N( U-,, — Ug ),~+

Energy: The expressions for u.,. and u5 (see eqs. (4) and (12)) can be given if the temperature behavior of the energy at the critical point and absolute zero is known and suitable ansatzes for separately representing u.,. and u5 can be found. First it is shown that the energy functions u.. and u5 in the critical region have the same temperature scaling: The bulk energy of the system with the critical volume V/N= v~is calculated with allowance for 2 d (pIT) u.~—vU5 = T v,,. 5 dT

i.e. in the vicinity ofthe critical point the system energy is composed of vapor energy and liquid energy in equal to parts. scaling u5 conform that The of thetemperature order parameter v.,. —ofV5 u~. (with the critical-point exponent fi) since —

Volume 130, number 4,5

ln( U., hm T-.T,,

PHYSICS LETTERS A

u5)

—

ln ( TC — T) ln(v., —v5) +ln [T ln( TC — T)

T-.T~

ln ( v.~ v5) T-’ T~ln (TC T) —

= hm

—

—

p

—

This yields the required result: lim

T.

lnJuC—uVI

ln ( TC

—

T)

ergy density and particle density in the coexisting bulk phases — if it can be found may take the place of an equation for the specific As theV.,local teraction potential in the partialheat. volumes and inV 5 are unique functions ofthe local particle density and determine the values of U., and u5, respectively, it should be possible by means of a density coefficient and a temperature fuction to find suitable ansatzes —

2 d(p/T)/dT] =hm

11 July 1988

= lim

T. ~,

for the functions u., and u5 depending on the density and temperature. As density coefficient, a function

lfl(uC—U5)

ln ( TC

—

T) = p.

of the density parameter z is now chosen, and as temperature function the evaporation energy u., U5, and U., =p(l /z) (u., u5) is written for the energy in the vapor phase. The energy in the condensed phase must then be u5= —p(z)(u.,—u5) since the transfer ofaparticlefrom V.,to V5causesthevalueofu~to change to u5, the density parameter from l/z to z and the phase transition energy from u., u5 to— (u., u5), while the form of p is preserved because the functional density dependence of the microscopic interaction forces is ofcourse phase invariant. This yields for the density coefficient the equation

p

—

It is now shown that the energy vanishises at the critical point. It is found that the expression T2d(~/T) — u.~.v5—u5v.,— v5(u.,—u2) dT — V., V~ V., ~‘s T) = v5 T2d + (u., — u5) U., —

~,

—

—U5

—

—

is positive for every temperature T~7’,, since the term U., is always outweighed by the two positive terms preceding it. If z~ v~/Vg>~ 1 is introduced, then the relation U., — zu5>0 is valid for every z> 1, i.e. for T< T~.On the other hand, for T< TC the relation U.,— i~5>0 is also valid. Both relations can only be satisfied for u5 <0, and the inequalities U5 < u.,,/z< U., can only be satisfied for u.,,> 0. This yields u5 <0< u,,, for z>l or T
U.,

.

—

—

—

p(l [Z) +p (z) = and the boundary conditions 0~p( liz) p(z) which are satisfied by 1--0~p(l/z)=—’--—-lnz z—l

du.,,/dT

T-.T~U hm—= hm =—l. 5 T—.T~dUQ/dT

The energy value at absolute zero for the vapor (in agreement with (U/N)idealga. according to eq. (6)) is given by limu.,=0,

— ~

-‘

1

— —

—

+

lflz

z z—

~ 1

1

The ratio u.,,/u5 is then generally a pure function of v.,/v5 (=z):

T—. 0

and for the condensate by lim u5 =

—

lim

[(u.,,—u5)—u~]

—1~

=

—

u5

—

z—l—lnz z—l—zlnz

~ 0,

(u.,, —u5)0,

T-.0

where (u.,—u5)o is the condensation energy at absolute zero. In order to obtain the temperature dependence of the system energy, it is customary to determine the heat capacity of the system. In the case ofa saturated fluid, however, the unique relation between the en—

U., —

yielding a precise formulation of the correspondence principle (see eq. (4)). Substitutingthe equation of Carnot, Clapeyron, Clausius, viz.’ 7-2 d(p/T) U., — U5 = (v., v5) dT —

in the ansatzes for u., and u5, one obtains the energy 231

Volume 130, number 4,5

PHYSICS LETTERS A

expressions (12), which are symmetric when the phase indices are changed. Interface function: As both the surface tension and its temperature derivative vanish at the critical point, the following identity is obtained: T,

$

T~

y=— J~dT=T~+ T

T~-~dT

7-

11 July 1988

the difference g(x) —g0 (x) being the greater the higher the molecular moment of the polar substance is; in the critical region (i.e. x< 10) and with decreasing x the values g(x) and g0(x) get closer and closer. If the experimental data in g are plotted versus ln x (see fig. 1), then ln g approaches the asymptote lng0+A0lnx for x—’c~ or T-60 and the asymptote ln g,, + ~,,inx for x—#0 or T—~T~.Here g0 and g,, are constants, and )~o=2/3; the value yields the scaling of g in the critical region; it has to be > 1/3 (to conform to the identity) and, according to the measurements, it is ~ 1/3 (thus dy/dT scales as (v.,—v 3~”). At absolute zero the free inter‘~

or —

2 d(y/T) dT =

T

5

TJ.?’~dT

—

.

T

(As T2 d ( y/ T) /dT is the interface energy per unit surface area, Ui/A, the identity can also be repre-

5)”

sented in the form U~=—f~’dU1.)To treaty further, an ansatz suggested by the known relations

face energy Ay and the internal interface energy U, are equal to one another (since lim 7-..’Ay= limT~.O[Ay—Td(Ay)/dT] =limT.OUI), and it holds that

T2dC~~~’T) U%,U5

limy=yo=g0

—

dT

V.,—V5

and 2 dCu/T) T

—

—

(u~—Us)o 2/3 V50

u.,v 5

dT

T-.0

—u5v.,

-~

o

..,l....l...,l

C)

v.,,—v5

0

is made for the identity: T2d~~/T)=gIv.,_vsI1~~3Us~_U5 dT

tz:

v.,—v5~

The new function g introduced here is called inter-

~

non~~~Ier

_

face function because its value is obtained from experimental data determined at the interface: >

1 g=

Iv.,—vsI”

Because of the relation between p and dence of g on it can be written as g

(Vv+

Ii

d(y/T) 3d(p/T)’

(20) ~t

C I

.9

the depen‘~

- -

v..—v

Vs_lfl(V/v5))(Vv5)I/3d(p/T). 5 \ 1 d(y/T)

-~ .9

—

—2—1

If the dimensionless quantity x~Iv.,/v5—v5/v.,I is now chosen as argument of g, the latter is found to be a function growing monotonically with x which appears to have the particular property of being universally valid for non-polar substances, i.e. for the same argument x of two substances the same value g is obtained. It is also found that the function value for a polar substance, g0(x), is smaller than g(x), 232

0

1

2

3

density parameter V.,/Vj—Vi/v,

4

5

loglO scale

Fig. 1. Interface functions (solid lines) of water and non-polar substances (gained from data on H2, D2, Ne, N2, F2, Ar, 02, CH4, Xe, CCIF3, CO2. C2H6, SF6, CBrF3, C3H8 to C6H,4, Cd4). The scaling in the low-temperature region is given by the slope 2/3. In the really critical region the scaling cannot be given exactly since no measurements were made. The interface function is therefore continued by a dashed line, and the dotted lines merely show candidates for scaling asymptotes with slopes 1/3 and 0.

Volume 130, number 4,5

PHYSICS LETTERS A

with

11 July 1988

since according to eq. (12) for T-+ T~:

dy lim v5 = V50, lim = 0. T—.OdT

du.,~ ln

—

—

dT

ln

d[(v.,—V5)/ln(v.,/v5)—v5]I dT

(~7i.~)

For the interface entropy one obtains

=ln I dv5 I =ln

Ui_AY=s~>o

=ln(v.,—v5)—ln(T~—T).

(i.e.alsody/dT<0),limT~S~=0andlimT.T,S~ =0 (i.e. dy/dT= 0). For 0 ~ T~T~the above stated identity then reads

hence 2v=l+(l/3+)~~)P=5/4with fl=1/(7/3+ 2~)= 3/8 for 2~= 1/3.

Ay=—TS1+U1

(withA=l),

References

0 (U1

[1]H.B. Callen, Thermodynamics (Wiley, New York, 1960) pp. 31—36,47—53, 90—102, 154—157. [2] F. Garcia-Moliner, Surface thermodynamics in surface sci-

whence U, —Si T A =

(

_5. fl \ OA

)NTV

= (8 (U—ST) OA JN,T,V

The critical-point exponent of y is 2 v [5], where v is the critical-point exponent of the density fluctuations, andone these scale as the energy fluctuations 2>. Then obtains <ü ln<ü2> v = lim T-.T~ln(T~—T) —

in I du.,/dTp T—.T~ln(T~—T)

=lim

ence, Vol. 1 (IAEA, Vienna, 1975) pp. 3—12. [3] C. Croxton, Introduction to liquid state physics (Wiley, New York, 1975) pp. 111—124,223—236; Statistical mechanics of the liquid surface (Wiley, New York, 1980) pp. 6—16. [41J.G. Dash, Films on solid surface (Academic Press, New York, 1975) pp. 62,63. [5] L.D. Landau, and L.P. Pitaevskii, physics, Vol. 5,E.M. part Lifshitz 1 (Pergamon Press, Oxford, Statistical 1980) pp. 108, 122, 133, 134, 163, 164, 518—520. [6]E.U.Franck,PhysicaB+C 139/140(1986) 21. [7] J. Wilks, Liquid and solid helium (Clarendon, Oxford, 1967) p.114.

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