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5.6 THE CLAUSIUS INEQUALITY 97

Figure 5.6b. In this process, which must have the same initial and final states as the
irreversible process, water is slowly and continuously added to the beaker on the
piston to ensure that P = P external . The ideal gas undergoes a reversible isothermal
T
V
transformation described by , T : 1/2V i , . Because ¢U = 0 , q = –w. We cal-
i
i
i
culate ¢S for this process:
1
V
dq reversible q reversible w reversible 2 i
¢S = = =- = nR ln =-nR ln 2 6 0 (5.27)
T T i T i V i
L
For the opposite process, in which the gas spontaneously expands so that it occupies
twice the volume, the reversible model process is an isothermal expansion for which
2V i
¢S = nR ln = nR ln 2 7 0 (5.28)
V i
Again, the process with ¢S 7 0 is the direction of natural change in this isolated
system. The reverse process for which ¢S 6 0 is the unnatural direction of change.
The results obtained for isolated systems are generalized in the following statement:

For any irreversible process in an isolated system, there is a unique direction of
spontaneous change: ¢S 7 0 for the spontaneous process, ¢S 6 0 for the
opposite or nonspontaneous direction of change, and ¢S = 0 only for a
reversible process. In a quasi-static reversible process, there is no direction of
spontaneous change because the system is proceeding along a path, each step of
which corresponds to an equilibrium state.

We cannot emphasize too strongly that ¢S 7 0 is a criterion for spontaneous
change only if the system does not exchange energy in the form of heat or work with its
surroundings. Note that if any process occurs in the isolated system, it is by definition
spontaneous and the entropy increases. Whereas U can neither be created nor destroyed,
S for an isolated system can be created (¢S 7 0) , but not destroyed (¢S 6 0) .

5.6 The Clausius Inequality

In the previous section, it was shown using two examples that ¢S 7 0 provides a crite-
rion to predict the natural direction of change in an isolated system. This result can also
be obtained without considering a specific process. Consider the differential form of
the first law for a process in which only P–V work is possible

dU = dq - P external dV (5.29)
Equation (5.29) is valid for both reversible and irreversible processes. If the process is
reversible, we can write Equation (5.29) in the following form:

dU = dq reversible - PdV = TdS - PdV (5.30)
Because U is a state function, dU is independent of the path, and Equation (5.30) holds
for both reversible and irreversible processes, as long as there are no phase transitions
or chemical reactions, and only P–V work occurs.
To derive the Clausius inequality, we equate the expressions for dU in Equations (5.29)
and (5.30):
dq reversible - dq = (P - P external )dV (5.31)
If P - P external 7 0 , the system will spontaneously expand, and dV > 0. If
P - P external 6 0 , the system will spontaneously contract, and dV < 0. In both possi-
ble cases, (P - P external )dV 7 0 . Therefore, we conclude that

dq reversible - dq = TdS - dq Ú 0 or TdS Ú dq (5.32)