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Background: Clausius and Entropy

In the previous lesson, we learned about

, where to find entropy data (NIST WebBook) and

and

In this lesson, we will use the

and the definition of entropy to define

Integral form of the definition of

Path A is irreversible and
path B is internally reversible.

Lesson 7C: page 1 of 9

This lesson is where we begin to apply the 2nd Law to IRREVERSIBLE processes.
The key to the 2nd Law analysis of irreversible processes is the Principle of Increasing Entropy.
Let’s begin by applying the Clausius Inequality to the two-step cycle shown here on a PV Diagram.
The cyclic integral in the Clausius Inequality can be broken into two ordinary integrals: on for each step.
The first integral follows the irreversible path A from state 1 to state 2.
The second integral follows the internally reversible path B from state 2 back to state 1.
The cycle is irreversible because at least one of the steps that make up the cycle is irreversible.
Recall that the equality part of the Clausius Inequality ONLY applies to reversible processes.
The “less than” part of the inequality applies to all irreversible processes.
Therefore, the sum of these two integrals must be less than zero for our irreversible cycle.
Next let’s use the definition of entropy in its integral form.
This is useful because path B is internally reversible.
Notice, in the right-hand sides of the last two equations, that switching the limits on the integral is the same as multiplying by minus 1.
This gets the equation into the same form as the second integral term in the Clausius Inequality, above.
Next, a little algebraic slight-of-hand.