## Defining Entropy (S) through the Clausius Inequality

Consider two heat engines, one a reversible
(Carnot) engine and the other an irreversible heat engine. For
purposes of developing the Clausius Inequality we assume that
both engines are sized to accept the same amount of heat Q_{H} from the thermal source. Thus since the irreversible
engine must be less efficient than the Carnot engine, it must
reject more heat Q_{L,irrev} to
the thermal sink than that rejected by the Carnot engine Q_{L,rev} , as shown:

Consider first the reversible (Carnot) heat
Engine. We saw in **Chapter 5**
that reversible heat transfer can only occur isothermally, thus
the cyclic integral of the heat transfer divided by the temperature
can be evaluated as follows:

Recall from **Chapter
5** that whenever we considered the efficiency of a reversible
heat engine, we went into "meditation mode", replacing
the ratio of heat flows with the ratio of temperatures:

Notice from the above diagram showing the two
heat engines that for an irreversible engine having the same value
of heat transfer from the thermal source Q_{H} as the reversible engine, the heat transfer to the
thermal sink Q_{L,irrev}> Q_{L,rev}.

Let Q_{diff} = (Q_{L,irrev}- Q_{L,rev}), then the cyclic integral for an irrevesible heat
engine becomes:

Thus finally, for any reversible or irreversible
heat engine we obtain the Clausius Inequality:

### Defining the property Entropy - S

All properties (such as pressure P, volume
V, etc) have a cyclic integral equal to zero.

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Engineering Thermodynamics by Israel Urieli is licensed under a
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