Physical Chemistry Third Edition

5.4 The Gibbs Energy and Phase Transitions 221

We can use this equation to calculate∆Gfor an isothermal process at temperature

T 2 from its value at temperatureT 1. We multiply bydTand integrate fromT 1 toT 2 :

∫T 2

T 1

(

∂(∆G/T)

∂T

)

P,n

dT−

∫T 2

T 1

∆H

T^2

dT (5.4-13)

If∆His temperature-independent,

∆G(T 2 )

T^2

−

∆G(T1)

T 1

∆H

(

1

T 2

−

1

T 1

)

(5.4-14)

EXAMPLE 5.9

The Gibbs energy of formation of monoclinic sulfur at 298.15 K is 0.096 kJ mol−^1 , and the

Gibbs energy of formation of rhombic sulfur is equal to zero by definition. The enthalpy

change of formation of monoclinic sulfur at 298.15 K is 0.33 kJ mol−^1.

a.Assume that∆His temperature-independent and find the Gibbs energy to form monoclinic

sulfur from rhombic sulfur at 500.0K.

b.Can there be a temperature at which the rhombic and monoclinic forms of sulfur can be

at equilibrium at 1 bar pressure?

Solution

a.From Eq. (5.4-14)

∆G◦(500.0K)

500 .0K



96 J mol−^1

298 .15 K

+

330 J mol−^1

8 .3145 J K−^1 mol−^1

(

1

500 .0K

−

1

298 .15 K

)

 0 .322 J K−^1 mol−^1 − 0 .0537 J K−^1 mol−^1  0 .268 J K−^1 mol−^1

∆G◦(500.0K)(0.268 J K−^1 mol−^1 )(500.0K)134 J mol−^1  0 .134 kJ mol−^1

b.Since∆G◦becomes more positive as the temperature increases, no higher temperature

can be found such that the two phases will equilibrate. SinceTcannot be negative, no

lower temperature can be found, because the rate of change of∆Gwith temperature is

too small to reach equilibration at a positive value ofT.

PROBLEMS

Section 5.4: The Gibbs Energy and Phase Transitions

5.31 1 .000 mol of ammonia is vaporized at a constant pressure
of 1.000 atm and a constant temperature of 240 K, the
normal boiling temperature. The enthalpy change of
vaporization at this temperature is 25.11 kJ mol−^1.

a.Find∆G,∆S,w, andq.

b.The gaseous ammonia is expanded from 1.000 atm to

0 .100 atm. Find∆G,∆S,w, andq.

5.32 Construct an accurate graph of the standard-state molar

entropy of water as a function of temperature from

298 .15 K to 398.15 K at a constant pressure of 1.000 atm,

using data in Tables A.7 and A.8 of the appendix. Assume

that the heat capacities are constant.

5.33 Construct an accurate graph of the molar Gibbs energy of

water as a function of pressure from 0.100 atm to

1 .900 atm at a constant temperature of 273.15 K. Since the

zero of the Gibbs energy is arbitrary, let the molar Gibbs

energy of the solid and liquid at 1.000 atm equal zero.