Chemistry 360Spring 2012Dr. Jean M. StandardMarch 28, 2012

The van't Hoff Equation

The van't Hoff equation provides information about the temperature dependence of the equilibrium constant. Thevan't Hoff equation may be derived from the Gibbs-Helmholtz equation, which gives the temperature dependence ofthe Gibbs free energy. For the standard Gibbs free energy, the Gibbs-Helmholtz equation is

G T

1T( )

P

= H . (1)

Substituting the expression

G = RT lnKeq and dividing both sides by R, the first form of the van't Hoffequation is obtained,

lnKeq 1T( )

P

= H

R . (2)

From this form of the van't Hoff equation, we see that at constant pressure, a plot with

lnKeq on the y-axis and

1 T on the x-axis has a slope given by

H R . This is the origin of LeChatelier's Principle for the heat absorbedor evolved during the course of a chemical reaction.

For an endothermic reaction, the slope is negative and so as the temperature increases, the equilibrium constantincreases, as illustrated in Figure 1.

lnKeq

1/T

Endothermic case

Thigh Tlow

slope = -Ho/R < 0

Figure 1. Typical van't Hoff plot for an endothermic reaction.

An example endothermic reaction may be written in the following form,

A + B + heat C . (3)For such a reaction, LeChatelier's Principle predicts that if the temperature increases, the system must shift to theright in order to alleviate the stress of additional heat. A shift to the right leads to an increase in products and adecrease in reactants, which increases the equilibrium constant as predicted by the van't Hoff equation.

2For an exothermic reaction, the slope is positive and so as temperature increases, the equilibrium constant decreases,as illustrated in Figure 2.

lnKeq

1/T

Exothermic case

Thigh Tlow

slope = -Ho/R > 0

Figure 2. Typical van't Hoff plot for an exothermic reaction.

An example exothermic reaction may be written in the following form,

A + B C + heat . (4)For this type of reaction, LeChatelier's Principle predicts that if the temperature increases, the system must shift tothe left in order to alleviate the stress of additional heat. A shift to the left leads to a decrease in products and anincrease in reactants, which decreases the equilibrium constant as predicted by the van't Hoff equation.

An integrated form of the van't Hoff equation may be obtained at constant pressure by multiplying Eq. (2) by

d 1 T( ) and integrating,

d lnKeq = H R d 1 T( )

d lnKeq1

2

= H

R d 1 T( )1

2

lnK2 lnK1 = H R

1T2

1T1

.

The integral on the right side was obtained assuming that the standard molar enthalpy of reaction is constant over thetemperature range. Combining the natural logs, the integrated form of the van't Hoff equation is

ln K2K1

= H

R1T2

1T1

. (5)

Finally, an additional linear form of the van't Hoff equation may be obtained starting from the relation

G = RT lnKeq . (6)

Solving for the natural log,

lnKeq = G RT .

(7)

3At any given temperature, the change in Gibbs free energy may be written from the definition of the Gibbs freeenergy as

G = H TS . (8)

Substituting this relation into Eq. (5), the linear form of the van't Hoff becomes

lnKeq = H RT +

S R .

(9)

This equation indicates that a plot with

lnKeq on the y-axis and

1 T on the x-axis has a slope given by

H R(as we saw before from Eq. (2)) and an intercept given by

S R , as illustrated in Figure 3. Note that in thisfigure, the slope is drawn for an endothermic reaction for example purposes.

lnKeq

1/T

slope = -Ho/R

intercept = So/R

Figure 3. The slope and intercept of a van't Hoff plot.