Chapter 6.6, Problem 7P

Interpretation Introduction

Interpretation:

Prove that ${\left(\partial \underset{\_}{U}/\partial P\right)}_T$ and ${\left(\partial \underset{\_}{U}/\partial \underset{\_}{V}\right)}_T$ is zero for the ideal gas.

Concept introduction:

The expression fundamental property relation for internal energy as given as follows

$d\underset{\_}{U}=Td\underset{\_}{S}+Pd\underset{\_}{V}$

Here, change in molar internal energy is $d\underset{\_}{U}$, temperature is T, change in molar entropy is $d\underset{\_}{S}$, pressure is $P$, and change in the molar volume is $d\underset{\_}{V}$

Write the total derivative for molar internal energy.

$d\underset{\_}{U}={\left(\frac{\partial \underset{\_}{U}}{\partial \underset{\_}{S}}\right)}_{\underset{\_}{V}}d\underset{\_}{S}+{\left(\frac{\partial \underset{\_}{U}}{\partial \underset{\_}{V}}\right)}_{\underset{\_}{S}}d\underset{\_}{V}$

Here, change in molar entropy and change in molar internal energy at constant molar volume is ${\left(\partial \underset{\_}{S}\right)}_{\underset{\_}{V}}$ and ${\left(\partial \underset{\_}{U}\right)}_{\underset{\_}{V}}$ respectively, and change in molar internal energy and change in molar volume at constant molar entropy is ${\left(\partial \underset{\_}{U}\right)}_{\underset{\_}{S}}$ and ${\left(\partial \underset{\_}{V}\right)}_{\underset{\_}{S}}$ respectively.