Physical Chemistry Third Edition

3.2 The Mathematical Statement of the Second Law: Entropy 119

adiabatic process cannot lead to a state that is lower in temperature than the reversible

adiabat.

If state 2 is above the reversible adiabatic curve as in Figure 3.7b, we carry out

a constant-volume reversible step (step 2) from state 2 to state 3, and an adiabatic

reversible step from state 3 to state 1. This time, because state 2 is at a higher temperature

than state 3 and because the capacity of the system must be positive,

qcycleq 2 < 0 (3.2-16)

so that

wsurr−wcycle−∆Ucycle+qcycle−∆Ucycle+q 2 q 2 < 0 (3.2-17)

In this case, heat transferred to the surroundings has been turned completely into work

done on the system. This does not violate the second law of thermodynamics because the

surroundings do not undergo a cyclic process. The final temperature for an irreversible

adiabatic process cannot be lower than for a reversible adiabatic process with the same

final volume, but it can be higher.

Now consider the entropy change for the irreversible adiabatic process that was

depicted in Figure 3.7b. BecauseSis a state function,

∆Scycle∆S 1 +∆S 2 +∆S 3  0 (3.2-18)

Because step 3 is reversible and adiabatic,∆S 3 0, and

∆S 1 −∆S 2 (3.2-19)

Because step 2 is reversible, we can integrate Eq. (3.2-1) for this step:

∆S 2 

∫T 3

T 2

dqrev

T



∫T 3

T 2

CV

T

dT < 0 (3.2-20)

The inequality comes from the fact that the temperature and the heat capacity are

both positive and the fact that the temperature of state 2 must be greater than that of

state 3. Because∆S 1 −∆S 2 ,∆S 1 must be positive. Therefore,

∆Sirrev∆S 1 >0 (irreversible adiabatic process) (3.2-21)

Combining Eqs. (3.2-11) and (3.2-21),

∆S≥0 (any adiabatic process) (3.2-22)

where the equality holds for reversible processes.For any adiabatic process, the entropy

of the system cannot decrease. This is the most important consequence of the second

law of thermodynamics. Because we define the universe to include all objects that

interact with each other, the universe can undergo only adiabatic processes.In any

reversible process, the entropy of the universe remains constant. In any irreversible

process, the entropy of the universe increases.