download the script: Clausius Inequality

Since the concept “reversibility” plays a crucial role in the thermodynamics, it is necessary for us to find an approach to determine whether the process reversible or not. And the most well-known Clausius Inequality provides us this possibility.

Image that we have a reversible engine which consists of more than two heat reservoirs. We know from the Carnot Principles that between the same reservoirs (one is a constant temperature source and the other a constant temperature sink), the efficiency of a reversible heat engine is always greater than the efficiency of an irreversible one.(for further explanation about Carnot Principles, please click here:https://thermodynamics-engineer.com/carnot-principals/)

Hence, for the maximum efficiency, Carnot efficiency, all the heat transfer must take place at highest and lowest reservoir temperatures. In other words, heat is absorbed from temperature source with T_{H} and released to temperature sink with T_{L} by heat engine during each sub process.

Now we draw an arbitrary reversible cycle process operated by the engine in p-v diagram (1→2→1).

We can divide this cycle process into infinite small Carnot process (a→b→c→d):

a→b: reversible isothermal expansion at T_{1}; Heat δq_{1} is absorbed.

b→c: reversible adiabatic expansion.

c→d: reversible isothermal compression at T_{2}; Heat δq_{2} is released.

b→c: reversible adiabatic compression.

We calculate the maximum efficiency, the Carnot efficiency:

(Pay attention here, δq_{2}<0)

So we obtain an equation:

As mentioned above, for the whole reversible process 1→2→1, we have:

Now 1→2 is still reversible process, but 2→1 is supposed to be irreversible process. So also according to the Carnot Principles:

For the whole irreversible cycle process:

Therefore, if we combine the both statements for reversible and irreversible processes, we get:

where

- “<” for irreversible process

- “=” for reversible process

#### This inequality is known as Clausius Inequality.

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