Physical Chemistry Third Edition

336 7 Chemical Equilibrium

Using Eq. (7.1-20) we can write

dln(K)

dT



∆H◦

RT^2

(7.6-4)

or

dln(K)

d(1/T)

−

∆H◦

R

(7.6-5)

Exercise 7.16

Verify Eqs. (7.6-4) and (7.6-5).

If the value of∆H◦is known as a function of temperature, Eq. (7.6-4) can be

integrated to obtain the value ofKat one temperature from the value at another

temperature:

ln

(

K(T 2 )

K(T 1 )

)



1

R

∫T 2

T 1

∆H◦

T^2

dT (7.6-6)

which is equivalent to

∆G◦(T 2 )

T 2

−

∆G◦(T 1 )

T 1

−

∫T 2

T 1

∆H◦

T^2

dT (7.6-7)

where∆G◦(T 2 ) is the value of∆G◦at temperatureT 2 , and∆G◦(T 1 ) is the value of

∆G◦at temperatureT 1 .If∆H◦is temperature-independent, Eq. (7.6-6) becomes an

equation known as thevan’t Hoff equation.

ln

(

K(T 2 )

K(T 1 )

)

−

∆H◦

R

[

1

T 2

−

1

T 1

]

(7.6-8)

and Eq. (7.6-7) becomes theGibbs–Helmholtz equation:

∆G◦(T 2 )

T 2

−

∆G◦(T 1 )

T 1

∆H◦

[

1

T 2

−

1

T 1

]

(7.6-9)

Exercise 7.17

Carry out the integrations to obtain Eqs. (7.6-8) and (7.6-9).

EXAMPLE7.18

Assuming that∆H◦is temperature-independent, calculate the value ofKand∆G◦at 100◦C

for the reaction

0 2NO 2 (g)−N 2 O 4 (g)