Physical Chemistry Third Edition

3.3 The Calculation of Entropy Changes 123

Entropy Changes for Processes That Begin and End

at the Same Temperature

If a process is not isothermal but has a final temperature that is equal to its initial

temperature, we can calculate∆Sfor the process by integratingdqrev/Ton a reversible

isothermal path. The actual process does not have to be reversible or isothermal, but the

initial and final states must be equilibrium or metastable states at the same temperature.

EXAMPLE 3.4

Calculate the entropy change for the following process: A sample containing 2.000 mol of

helium gas originally at 298.15 K and 1.000 bar is cooled to its normal boiling temperature of

4.00 K, condensed to a liquid, and then cooled further to 2.00 K, where it undergoes another

phase transition to a second liquid form, called liquid helium II. This liquid phase is suddenly

vaporized by a beam of laser light, and the helium is brought to a temperature of 298.15 K

and a pressure of 0.500 bar.

Solution

Since entropy is a state function the entropy change is the same as for an isothermal reversible

expansion from 1.000 bar to 0.500 bar:

∆SnRln

(

V 2

V 1

)

nRln

(

P 1

P 2

)

where we have used Boyle’s law,PVconstant at constant temperature.

∆S(2.000 mol)(8.3145 J K−^1 mol−^1 )ln

(

1 .000 bar

0 .500 bar

)

 11 .5JK−^1

Entropy Changes for Reversible Phase Changes

Two phases of a single substance can be at equilibrium with each other at a fixed

temperature that depends on the pressure. For example, liquid and gaseous water can

be at equilibrium with each other at 100.00◦C if the pressure is 1.000 atm (760.0 torr),

and can be at equilibrium with each other at 25.00◦C if the pressure is 23.756 torr. If an

equilibrium phase change is carried out at constant pressure and temperature Eq. (3.3-2)

applies. Since the pressure is constant,qis equal to∆H, and

∆S

qrev

T



∆H

T

(reversible phase change

at constant pressure)

(3.3-4)

EXAMPLE 3.5

Find the entropy change of the system and of the surroundings if 3.000 mol of water freezes

reversibly at 1.000 atm. The freezing temperature is 0.00◦C at this pressure, and the specific

enthalpy change of fusion is equal to 79.7 cal g−^1 at this temperature.