4.6 Entropy  (Page 2/5)

As an example, let us determine the net entropy change of a reversible engine while it undergoes a single Carnot cycle. In the adiabatic steps 2 and 4 of the cycle shown in [link] , no heat exchange takes place, so $\text{Δ}{S}_2=\text{Δ}{S}_4={\displaystyle \int d}Q\text{/}T=0.$ In step 1, the engine absorbs heat $Q_{\text{h}}$ at a temperature $T_{\text{h}},$ so its entropy change is $\text{Δ}{S}_1=Q_{\text{h}}\text{/}{T}_{\text{h}}.$ Similarly, in step 3, $\text{Δ}{S}_3=\text{−}{Q}_{\text{c}}\text{/}{T}_{\text{c}}.$ The net entropy change of the engine in one cycle of operation is then

However, we know that for a Carnot engine,

so

There is no net change in the entropy of the Carnot engine over a complete cycle. Although this result was obtained for a particular case, its validity can be shown to be far more general: There is no net change in the entropy of a system undergoing any complete reversible cyclic process. Mathematically, we write this statement as

where ${\displaystyle \oint }$ represents the integral over a closed reversible path .

We can use [link] to show that the entropy change of a system undergoing a reversible process between two given states is path independent. An arbitrary, closed path for a reversible cycle that passes through the states A and B is shown in [link] . From [link] , ${\displaystyle \oint dS=0}$ for this closed path. We may split this integral into two segments, one along I, which leads from A to B , the other along II, which leads from B to A . Then

Since the process is reversible,

Hence, the entropy change in going from A to B is the same for paths I and II. Since paths I and II are arbitrary, reversible paths, the entropy change in a transition between two equilibrium states is the same for all the reversible processes joining these states. Entropy, like internal energy, is therefore a state function.

What happens if the process is irreversible? When the process is irreversible, we expect the entropy of a closed system, or the system and its environment (the universe), to increase. Therefore we can rewrite this expression as

where S is the total entropy of the closed system or the entire universe, and the equal sign is for a reversible process. The fact is the entropy statement of the second law of thermodynamics    :

We can show that this statement is consistent with the Kelvin statement, the Clausius statement, and the Carnot principle.