## 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
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