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Chapter 3: Q. 3.27 (page 112)

What partial-derivative relation can you derive from the thermodynamic identity by considering a process that takes place at constant entropy? Does the resulting equation agree with what you already knew? Explain.

Short Answer

Expert verified

The relation that can be derived from the thermodynamic identity is P=-dUdVN,S.

The derived equation agrees with the first law of thermodynamics.

Step by step solution

01

Given Information

The thermodynamic identity is:

dU=TdS-PdV

The process is taking place at constant entropy.

Hence,dS=0

02

Calculation

Since the process is taking place at constant entropy, the thermodynamic identity can be written as:

dU=TdS-PdVdU=-PdVP=-dUdVN,S

This is the derived relation.

From the relation, it can be seen that the volume change is fully responsible for the energy shift. As a result, there is no heat transfer into or out of the system. Heat flows until equilibrium is reached, resulting in a rise in entropy. As a result, the equation follows the first law of thermodynamics.

03

Final answer

The derived relation is P=-dUdVN,S. It agrees with the first law of thermodynamics.

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