To demonstrate the Clausius inequality, consider two
heat engines that work between the same two reservoirs. Heat
engine A is an reversible heat engine and heat engine B is an irreversible
one. Both of these two engines absorb
heat Q_{H} from a heat source, which has a temperature of T_{H} and
reject heat to a heat sink at temperature T_{L}. Applying the
first law to both of the engines gives,
W_{A,rev} = Q_{H} 
Q_{A,L}
W_{B,irr} = Q_{H}  Q_{B,L}
Since a reversible engine has the highest efficiency of any heat engines
working between the same source and sink, a reversible
heat engine produces more work than an irreversible
heat engine for the
same heat input Q_{H}.
W_{A,rev} = Q_{H} 
Q_{A,L} > W_{B,irr} =
Q_{H}  Q_{B,L}
Q_{A,L} < Q_{B,L}
Since the temperature T_{H} is constant during the heat transfer
Q_{H}, and the temperature T_{L} is constant during the
heat transfer Q_{A,L} and Q_{B,L}, the cyclic integral
of for
both heat engines is
Comparing the above two equations with Q_{A,L} < Q_{B,L} yields,
(1)
A reversible heat engine holds the following relation:
Therefore, the cyclic integral
of for
a reversible engine is zero.
Combining the above equation with equation (1) gives,
To summarize the above analysis, the Clausius inequality states
for
internally reversible cycles
for
irreversible cycles
