Physical Chemistry , 1st ed.

Since (^) rxnG°RTln K, we can substitute for ( (^) rxnG°/T) and get





T

(Rln K)p^ r

T

xn

2

H°

Ris a constant, and the two negative signs cancel. This equation rearranges to
yield the van’t Hoff equation:





ln

T

K^

R

rx

T

nH

2

° (5.18)

A qualitative description of the changes in Kdepends on the sign of the en-

thalpy of reaction. If (^) rxnHis positive, then Kincreases with increasing Tand
decreases with decreasing T. Endothermic reactions therefore shift towards
products with increasing temperatures. If (^) rxnHis negative, increasing tem-
peratures decrease the value ofK, and vice versa. Exothermic reactions there-
fore shift toward reactants with increasing temperatures. Both qualitative
trends are consistent with Le Chatelier’s principle, the idea that equilibria that
are stressed will shift in the direction that minimizes the stress.
A mathematically equivalent form of the van’t Hoff equation is





(1

ln

/T

K

)



(^) rx
R
nH° (5.19)
This is useful because it implies that a plot of ln Kversus 1/Thas a slope of
( (^) rxnH°)/R. Values of (^) rxnHcan be determined graphically by measuring
equilibrium constants versus temperature. (Compare this with the analogous
plot of the Gibbs-Helmholtz equation. What differences and similarities are
there in the two plots?) Figure 5.4 shows an example of such a plot.
A more predictive form of the van’t Hoff equation can be found by moving
the temperature variables to one side of equation 5.18 and integrating both
sides:
dln K
(^) r
R
x
T
nH
2

°

dT



K 1

K 2

dln K

T 1

T 2



R

rx

T

nH

2

°

dT

5.5 Changes in Equilibrium Constants 133

0

 3

1.2

ln

K rxnH°

Slope   R

T^1 (^10 ^3 K^1 )

 2

 1

0.7 0.8 0.9 1.0 1.1

Figure 5.4 Plot of the van’t Hoff equation as given in equation 5.19. Plots like this are one

graphical way of determining (^) rxnH.