# Background: Clausius and Entropy

In the previous lesson, we learned about

#### entropy

, where to find entropy data (NIST WebBook) and

and

#### HS Phase Diagrams

.
In this lesson, we will use the

#### Clausius Inequality

and the definition of entropy to define

.

#### Clausius Inequality

:
Integral form of the definition of

#### entropy

:
Path A is irreversible and
path B is internally reversible.
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### Ch 7, Lesson C, Page 1 - Background: Clausius and Entropy

• 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.

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