Physical Chemistry Third Edition

118 3 The Second and Third Laws of Thermodynamics: Entropy

V

1

T

V

T

(a) (b)

1

3

2

1

2

3

Figure 3.7 Reversible and Irreversible Adiabats.(a) Impossible case. An irreversible adiabatic process cannot lead to the low-
temperature side of the reversible adiabat. (b) Possible case. The irreversible adiabatic process can lead to the high-temperature side
of the reversible adiabat.

Figure 3.7 shows schematically two possibilities for an irreversible adiabatic process

of a closed simple system in which the initial state (state 1) and the final state (state 2)

are equilibrium states. During the process, the state of the system is not an equilibrium

state and cannot be represented by a point in theV–Tplane. The broken curve in the

figure indicates that the state point leaves theV–Tplane and then returns to theV–T

plane at the end of the process.

We now show that an irreversible adiabatic process must lead to a higher temperature

than the reversible adiabatic process starting at the same initial state (state 1). The solid

curve in Figure 3.7a represents the reversible adiabat passing through state 1. We first

assume that state 2 lies below this curve (an assertion that we want to disprove). Let

state 3 be the state on the reversible adiabat that has the same volume as state 2. After

the irreversible step 1 has occurred, we carry out a reversible constant-volume step

from state 2 to state 3 (step 2). For a constant-volume process,

q 2 

∫

c

dq

∫

c

CVdT (3.2-12)

It is an experimental fact that the heat capacity of any system is positive. Therefore,

q 2 >0, since the temperature of state 2 is lower than that of state 3. After step 2, we

carry out a reversible adiabatic step from state 3 to state 1 (step 3). Step 1 and step 3

are both adiabatic, so that

qcycleq 2 > 0 (3.2-13)

SinceUis a state function,

∆Ucycle 0 (3.2-14)

The work done on the surroundings in the cycle is

wsurr−wcycle−∆Ucycle+qcycle−∆Ucycle+q 2 q 2 (3.2-15)

Heat transferred to the system in a cyclic process has been completely turned into

work done on the surroundings, which is a violation of the second law. An irreversible