Page 270 - Advanced Thermodynamics for Engineers, Second Edition
P. 270
12.6 VARIATION OF GIBBS ENERGY WITH COMPOSITION 259
This can be written, by substituting for m from Eqns (12.49) and (12.50),as
0 0 0
a b c
G ¼ð1 εÞ m þ<Tlnpr a þð1 εÞ m þ<Tlnpr b þ 2ε m þ<Tlnpr c
(12.66)
h i h i
0
0
¼ ð1 εÞm þð1 εÞm þ 2εm 0
a b c þ<T ð1 εÞlnpr a þð1 εÞlnpr b þ 2εlnpr c
The partial pressures are defined by Eqn (12.60) as p i ¼ x i p and the mole fractions of the con-
stituents are
1 ε 1 ε 2ε
; ; ¼ ε: (12.67)
2 2 2
x a ¼ x b ¼ x c ¼
Substituting these terms in Eqn (12.66) gives
0 0 0 1 ε 1 ε
G ¼ ð1 εÞm þð1 εÞm þ 2εm c þ<T ð1 εÞln p r þð1 εÞln p r þ 2εlnεp r
a
b
2 2
0 0 0
¼ ð1 εÞm þð1 εÞm þ 2εm
a b c þ<T½ð1 εÞþð1 εÞþ 2εlnp r
1 ε 1 ε
þ<T ð1 εÞln þð1 εÞln þ 2εlnε
2 2
(12.68)
Equation (12.68) can be rearranged to show the variation of the Gibbs energy of the mixture as the
0
reaction progresses from the reactants A and B to the product C by subtracting 2m from the left-hand
c
side, giving
0
0
0
G 2m ¼ð1 εÞ m þ m 2m 0 c þ 2<Tlnp r þ 2<T ð1 εÞln 1 ε þ εlnε (12.69)
b
a
c
2
Equation (12.69) consists of three terms; the second one simply shows the effect of pressure and
will be neglected in the following discussion. The first term is the difference between the standard
0
0
chemical potentials of the separate components (m þ m ) before any reaction has occurred and the
a b
0
standard chemical potential of the mixture (2m ) after the reaction is complete. Since the standard
c
chemical potentials are constant throughout this isothermal process, this term varies linearly with the
fraction of reaction, ε. The third term defines the change in chemical potential due to mixing and is a
function of the way in which the entropy of the mixture (not the specific entropy) varies as the reaction
progresses. The manner in which the first and third terms might vary is shown in Fig. 12.2 and the sum
of the terms is also shown. It can be seen that for this example, the equilibrium composition is at
ε ¼ 0.78. This figure illustrates that the Gibbs energy of the mixture initially reduces as the compo-
sition of the mixture goes from A þ B to C. The standard chemical potential of compound C is less than
the sum of the standard chemical potentials of A and B, and hence the reaction will tend to go in the
direction shown in Eqn (12.63). If the standard chemical potentials were the only parameters of
importance in the reaction then the reactants A and B would be completely transformed to the product,
C. However, as the reaction progresses the term based on the mole fractions varies non-monotonically,
1 ε
as shown by the line labelled <T 1 ε ln þ 2εlnε , and this affects the composition of the
2
mixture which obeys the law of mass action. If the two terms are added together then the variation of