NOTE ON THERMODYNAMIC EQUILIBRIUM I N THE GRAVITATIONAL FIELD F. 0. KOENIG Department of Chemistry, Stanford University, California Received September 18, 1936 I. THE FUNDAMENTAL OIBBSIAN EQUATIONS FOR THE CHARACTERISTIC FUNCTIONS OF A SUBPHASE
In his excellent treatise, E. A. Guggenheim (1) has recently given an exact (non-relativistic) account of thermodynamic equilibrium in gravitational fields, starting from differential equations of Gibbsian type for the characteristic functions of an infinitely thin homogeneous layer of matter of given gravitational potential. Thus for the energy, Ea, of such a layer, CY, he writes dE" = T"dS"
- PadV" + C (14 + S i p a ) dn: t
where Ta,Sa, Pa,Va, and p a denote respectively temperature, entropy, pressure, volume, and gravitational potential of the layer, n': the number of moles of the ithcomponent in the layer, the molecular weight of the ithcomponent, and p: a function which is assumed to be completely determined by Pa, To, n:. It seems worth pointing out, for reasons stated below, that equation 1 can be generalized so as to take account of the fact that the energy of the layer in question can be varied without absorption of heat (TadSa), performance of volume work (-PedVa), or changes of composition ((,: + &ipe)diL:), merely by a change of gravitational potential pa, as for instance through a shift of level in the earth's field. The contribution to Ea of an infinitesimal change in p a a t constant Xe,Va, n; is evidently Madpa where M a is the mass of the layer. Equation 1 is accordingly replaced by d E = TdS - P d V + Mdp
+ C 7,dn, z
(2.1)
where for simplicity the superscript a is omitted and the sum fi$ + Sip denoted by 7i. Of the variables E, S, V , ni, p all but one are independent, so that if the number of components is k, the number of independent differentials in equation 2.1 is k + 3. The corresponding equations for 373
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the other characteristic functions, i.e., the heat content H, the Gibbs free energy F , and the Helmholtz free energy A , are of course dH =
VdP
+ TdX + Mdp + T,d%., + VdP + Mdp + C T,dfii - SdT + Mdp + r,dn,
(2.2)
I
dF = -XdT
(2.3)
%
dA = -PdV
z
(2.4)
For convenience the thin layer to which these equations apply may be called a “subphase.” The new equations (2) are of interest because (i) they further illustrate the point clearly made by Guggenheim (ref. I, p. 154) that “the gravitational potential difference pa between two [sub-]phases, in contrast to the electric potential difference 9“ is thermodynamically determinate, owing to the fact that its value is independent of the presence and nature of the phase there;” (ii) they show that the statement (ref. 1, p. 154) that “in all thermodynamic formulae the quantity pa occurs only in combinations of the form (fi: + Mzpa)”is generally true only for changes a t constant pa; (iii) they are of the same form as the general thermodynamic equations for external electric and magnetic fields (2) and thus serve t o emphasize certain important analogies. Equations 2 furthermore yield deductions of the laws of hydrostatic equilibrium and of sedimentation equilibrium which to the author seem more roncise than previous ones and are accordingly given below. These deductions fall naturally into the stages indicated by the titles of the following paragraphs. 11. THERMAL AIiD CHEMICAL EQUILIBRIUM BETWEEN SUBPHASES
The general criteria for thermodynamic equilibrium in conjunction with the equations 2 lead in the usual manner to the familiar conditions
Tu = TB
(3)
r: = r t
(4)
for thermal and chemical equilibrium, respectively, between any two subphases a and 6. 111. THE GIBBS-DUHEM EQUATION FOR A SUBPHASE
The energy of a subphase like that of a bulk phase (in the absence of gravity) is clearly homogeneous of the first order in the capacity factors S, V , 12%. By Euler’s theorem equation 2.1 therefore yields
E = TS - P V
+
r,n,
(5)
THERMODYNAMIC EQUILIBRIUM IN GRAVITATIONAL FIELD
375
which, on differentiation and comparison with equation 2.1 in the usual manner, gives SdT
- VdP - Mdp +
fiidTi = 0
(6)
the analogue of the Gibbs-Duhem equation for a bulk phase. For the treatment of hydrostatic and sedimentation equilibrium it is convenient to write equation 6 in the form d P = sdT
- pdp + C cidri
(7)
%
where s is the entropy per unit volume, ci the volume concentration of the ithcomponent, and p the density of the subphase. Of the k 3 intensive variables T, P, p, ri in equations 6 and 7 , any k + 2 are independent.
+
IV. HYDROSTATIC EQUILIBRIUM
If T, p, ri are taken as the independent variables, it follows from equation 7 that for any subphase
Now by equations 3 and 4 d l the (infinitely numerous) subphases of a fluid in thermal and chemical equilibrium have the same values of T and of the T ~ . But since any two adjacent subphases whatever differ infinitesimally in p, it follows from equation 8 that any fluid in thermal and chemical equilibrium in a gravitational field is also in hydrostatic equilibrium, Le., that *dP = -pdp (9) where the operator d refers to the difference between two adjacent subphases. Any equation such as equation 9, in which d has this significance, is conveniently prefixed by an asterisk, to distinguish it from equations such as 1, 2, 6, 7 in which d refers to an infinitesimal change within a single subphase. V. ACTIVITY COEFFICIENTS I N A SUBPHASE
For compactness in the expression for sedimentation equilibrium to be derived below it is expedient to introduce activity coefficients defined in a manner analogous to that for a bulk phase (ref. 1, p. 115), namely fii
= fi:(T, P )
+ R T log Nifi
(10)
where fi; is a function only of T and P, N; is the mole fraction of the ith component in the subphase, and f+is its activity coefficient, that is, a
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F. 0. KOENIG
function of T , P, ni so chosen as to be unity a t infinite dilution. This definition evidently implies that pi is completely determined independently of p by T, P, ni alone, a fact readily proved from the equations 2, which herein reveal a decided advantage over the equations of type 1. From equation 2.3 it follows that
where the subscript n indicates the constancy of ail the ni except a particular one with respect to which the differentiation is carried out. The introduction of the relations
into equation 11 gives
VI. PARTIAL MOLAR VOLUMES AND
PARTIAL MOLAR
EXTROPIES IN A
SUBPHASE
Besides the activity coefficients fi it is expedient to introduce the partial molar volumes 7; and the partial molar entropies 9,. That these quantities as well as V and S themselves are functions of T, P, ni alone follows from equation 2.3 by proofs similar to the one given above for pi. Particularly useful for compact derivation of the law of sedimentation equilibrium are two equations readily deduced from equations 2.3, 12, and 10, namely
in which
7;and 8: denote the values of Pi and 8i a t infinite dilution. VII. SEDIMENTATION EQUILIBRIUM
From equation 12 it follows that for any component of a subphase, say the jthl dpi + a j d p
- dri
=
0
(18)
THERMODYNAMIC EQUILIBRIUM IN GRAVITATIONAL FIELD
377
Differentiation of equation 10 and substitution of the result into equation 18 gives dT + - dP (?i)T
(%)P
+ RTd log N, f, + R log N , f, d T + B J d 9 - d7, = 0
(19)
which, on introduction of equations 16 and 17 and elimination of d P by means of equation 7, becomes RTd log N j f i
- (pyq - Rj) dp + ($7; - 8; + R log N j f j )d T + Pq Cid7i - drj = 0
(20)
i
If T, ‘p, 7;are taken to be the independent variables it follows that in any subphase
for each component. This equation in conjunction with equations 3 and 4 shows that in any solution in thermal and chemical equilibrium in a gravitational Jield *RTd log Nj f j =
(pBg
- Rj)d9
(22)
for each component. Equation 22 is the exact law of sedimentation equilibrium in a form more compact than any hitherto given. It is readily shown to be identical with the expression given by Guggenheim (ref. 1, p. 157). The integration of the sedimentation equation, neglecting compressibility, has been carried out by Guggenheim (ref. 1, pp. 157-8) for the binary ideal solution of any concentration and for the “extremely dilute” ideal solution of any number of components. The result for a third simple case of physical interest, that of the “extremely dilute” non-ideal solution of any number of components, seems worth giving; it is
where
po
is the density of the pure solvent. VIII. SUMMARY
The fundamental equations for the thermodynamics of the gravitational field are written in a form more general than hitherto by the inclusion of
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F. 0. KOENIG
the gravitational potential (p as an independent variable. The ,theoretical interest of the resulting new equations is briefly pointed out, and their practical usefulness shown by deducing from them the known laws of hydrostatic equilibrium and sedimentation equilibrium. The latter law is expressed in a new and compact form involving the activity coefficient. REFERENCES (1) GUGGENHEIM, E. A.: Modern Thermodynamics by the Methods of Willard Gibbs, pp. 153-9. Methuen and Go., London (1933). (2) KOENIG,F. 0.: Paper read before the Division of Physical and Inorganic Chemistry of the American Chemical Society, San Francisco, August, 1935.
