Clausius inequality

The Clausius inequality is mathematically expressed as . It shows that the change in entropy of the surroundings, which is in thermal contact with a system where a reversible process is occurring from A to B, is different from the change in entropy of the surroundings, which is in thermal contact with a system where an irreversible process is occurring between the same states.

The motivation behind the derivation of the inequality can be perceived as the search for a less restrictive inequality given by eq130, which states that for any adiabatic process in an isolated system.

As we know, a system and its surroundings forms the universe, with their entropies related by the following equations:

Since , and from eq133  for any process,

Let’s assume that the surroundings is an infinite heat reservoir with uniform heat capacity and temperature. In other words, equilibrium is always attained in the infinite heat reservoir and therefore, any process occuring in the surroundings is reversible. If so, we can write,

Since any transfer of energy to the surroundings  must come from the system, . So,

or

Eq139 is called the Clausius inequality. We have shown in eq119 that  for a reversible process, where . So the component of eq139 must be for a reversible process, which leaves the remaining component for an irreversible process:

We have therefore developed a change in entropy inequality (eq139) for a system that encompasses all processes.

 

Question

If entropy is a state function, why is there a difference in the change of entropy for reversible versus irreversible processes in eq139?

Answer

Entropy is a state function and the change in entropy of a system undergoing a reversible process from state A to B must be the same as that for an irreversible process between the same two states, i.e. .

With reference to eq132 and eq136, . Similarly, from eq131 and eq136, . Since , the change in entropy of the surroundings associated with a reversible process occurring in the system from A to B must be different from the change in entropy of the surroundings associated with an irreversible process occurring in the system between the same states. This is what the Clausius inequality is trying to convey.

 

 

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