Page 170 - Modern physical chemistry

162 Relationships among Reactants

whence
x = 0.0945 bar
and
P = 0.0945 + 0.5945 bar = 0.689 bar.

Z 15 Variation of Equilibrium Constants with Temperature

The equilibrium constant K for a reaction is related to the temperature T by equation
(7.58). So if we want to learn how In K varies, we need to investigate the behavior of !:lGo IT.
Let us first consider properties of a single reactant or product. Its Gibbs energy G is
defined by equation (5.79). Dividing this by temperature T,

G H
-=--s [7.84]
T T '
then differentiating, gives

[a~~)p =-:Z+~[~~)p -[:~)p' [7.85]

But at constant pressure, we have

dB= dqp = dH [7.86]
T T
and
[ as) 1 [aH) [7.87]
aT p = T aT p'

So equation (7.85) reduces to
[7.88]

This is called the Gibbs - Helm1wltz equation.
Since formula (7.88) applies to each reactant and product separately, one can con-
struct the difference for the standard states:

[7.89]

Introducing formula (7.62) converts (7.89) to the van't Hoff equation

[7.90]

Generally, the standard enthalpy change t:J{J varies slowly with temperature T. In the
approximation that it is constant, integration of equation (7.90) yields
InKp =----+C. [7.91 ]
mO 1
R T