P. 1. Robinson
University of Manchester Institute of Science and Technology Manchester I, England
Chemical Potentials and the Gibbs-Duhem Equation
The first difficulty concerns the integration of eqn. (I), which refers to constant temperature and pressure, and in which the symbols have their usual meaning, to give eqn. (2).
The result. can, of course, be justified by the usual physical argument ("build up the mixture from the components keeping the composition constant"), but this can be a disturbing argument for the student and has even aroused vigorous discussion in THIS JOURNAL.' The use of Euler's Theorem2 is mathematically too esoteric for most chemistry undergraduates. The difficulty is easily overcome when one realizes that it really lies in the unfamiliar integration of an expression involving several terms which have to be integrated with respect to different variables. This suggests an alternative derivation in which the equation is transformed into an integration with respect to a single variable, as follows. Let the mole fractions of the components be XI, xt, . . . , and let the total number of moles of mixture be n = nl n2 . . . , so that n, = xln, n2 = x,n, etc. Consider the change in G when dn moles of the mixture are added
Roams, D. W., J. CHEM. EDUC., 39,527 (1962), and ensuing correspondence, J. &EM. Eouc., 40, 225-8 (1963). See, for example, MILLER, D. G., J. CHEM.EDUC.,40, 648 (1963).
= (am ~ 2 ~ . .2 . )dn (3) The term in parentheses is a constant for the mixture (it is, in fact, the mean molar free energy), and eqn. (3) is integrated without any qualms to give eqn. (2)
A n y teacher of chemical thermodynamics a t a moderately advanced level must have encountered the great difficulties which exist in finding a convincing presentation of some of the mathematics involved in derivations involving the Gibhs-Duhem equation. These difficulties appear to be centered on two main points; the integration of the equation dG = pldnl p2dnp . . . to give G = plnl p2n2 . . . , and the subsequent complete differentiation of this equation to produce the Gibhs-Duhem equation. Methods are presented below by which each of these difficulties can be avoided.
+
+
+
+
Integration of dG to give G
+ +
+
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Journal of Chemiml Education
+
G
+ rrm + . . .In
=
Grxl
=
alnl
+
pxnn
(2)
+ . ..
Experience shows that this dcrivatiou is accept,ed wit,hout difficulty. The apparently awkward step has been t,ransformed into the reasonable application of eqn. (1) t o the addition of mixture rather than of individual components, and this does not even arousc commcnt.
constant temperature and pressure, and the qualifying subscripts T and P are omitted for clarity. &I , = (L) hs , an, ", = (A), hl , (E) anl , = an, ,,. (5)
(&)
.("1
.
(*)
(since the order of differentiation is irrelevant). Next we consider the differentials with respect to X I and x2 rather than 721 and nl. Since xz = ne/(nl n J , it follows that (?IX~/?I?~).~ = nl(nl n2)%,and t,hus
+
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Derivation of the Gibbs-Duhem Equation
The Gibbs-Duhem equation is usually derived by differentiating eqn. (2) completely, allowinp the composition to change as well as the amount of mixture (but still a t constant temperaturc and pre'isure) dG = (wdnl n l d d (rczdnz nrdrrr) ... and subtracting eqn. (1) to give
+
+
+
( ~ J I ~ / ~ P Z I T .= P -(ndn11
+
(4 I
for a two-component mixture. Although this procedure can be thoroughly justified' there is no doubt that it creates grave difficulties for the student. A geometrical proofa involves techniques which are too unfamiliar to be generally useful. An alternative derivation, given below, again avoids the steps which give rise to the difficulty. We consider a two-component mixture and start from the definitions
and
We evaluate first the changes in pi when the othev component is added a t constant n,. The whole process is a t BESTE, L.F.,J. CHIN EDUC., 38,509 (1861).
and similarly
Equating eqns. ( G ) and (7) in view of eqn. ( 5 ) ; wc have
Now since the chemical p~t~entials are funct,ionsof composition alone (at constant temperature and pressure), piwill vary with xi a t exactly the same rate irrespective of whether nl and n2vary or are constant, i.e., the qualifying subscripts in eqn. (8) are now redundant and should be omitted. Since in addition XI x2 = 1, and therefore 6x1 = -62% under all conditions, we thus have, with the reintroduction of the subscripts for constant T and P
+
I t will be seen that this derivation involves only the mauipulation of part,ial differential equations, with no debatable physical arguments, and is therefore basically more acceptable to the studcnt. Unfortunat,ely, the algebraic manipulations are themselves not as simple as might be desired and may well present some difficulty. Xevertheless, it is a worthwhile observation t,hat such a derivation a t lcast, exists and can be followed in outline, cven if not in detail.
Volume
47, Number 2, February 1970
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161
