Chapter 6.6, Problem 18P

A)

Interpretation Introduction

Interpretation:

Prove that the heat capacity of this gas is independent of pressure.

Concept introduction:

Write the relationship between $C_P$ and P as given as follows

${\left(\frac{\delta C_P}{\delta P}\right)}_T=-T{\left(\frac{{\delta }^2\underset{\_}{V}}{\delta T^2}\right)}_P$

Here, change in constant pressure heat capacity with respect to change in pressure at constant temperature is ${\left(\frac{\delta C_P}{\delta P}\right)}_T$, change in molar volume is $\delta \underset{\_}{V}$, and change in temperature is $\delta T$.

Express the equation of state as given as follows:

$\underset{\_}{V}=\frac{RT}{P}+a{P}^2$

Here, gas constant is R, temperature and pressure is T and P, constant is a, and molar volume is $\underset{\_}{V}$.

B)

Interpretation Introduction

Interpretation:

the expression for ${\left(\frac{\partial \underset{\_}{U}}{\partial P}\right)}_T$ has to be derived.

Concept introduction:

Write the fundamental relationship for molar internal energy.

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

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

Use the expansion rule for ${\left(\frac{\partial \underset{\_}{U}}{\partial P}\right)}_T$.

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

Here, change in molar entropy with respect to change in pressure at constant temperature is ${\left(\frac{\partial \underset{\_}{S}}{\partial P}\right)}_T$ and change in molar volume with respect to change in pressure at constant temperature is ${\left(\frac{\partial \underset{\_}{V}}{\partial P}\right)}_T$.

C)

Interpretation Introduction

Interpretation:

Concept introduction:

Write the total derivative of $d\underset{\_}{U}$ with T and P as intensive variables.

$d\underset{\_}{U}={\left(\frac{\partial \underset{\_}{U}}{\partial T}\right)}_{P}dT+{\left(\frac{\partial \underset{\_}{U}}{\partial P}\right)}_{T}dP$        (6)

Here, change in molar internal energy with respect to change in temperature at constant pressure is ${\left(\frac{\partial \underset{\_}{U}}{\partial T}\right)}_P$ and change in molar internal energy with respect to change in pressure at constant temperature is ${\left(\frac{\partial \underset{\_}{U}}{\partial P}\right)}_T$.

D)

Interpretation Introduction

Interpretation:

the molar change in entropy has to be determined.

Concept introduction:

The change in molar entropy for real gas model is given as follows:

${\underset{\_}{S}}_2-{\underset{\_}{S}}_1={\underset{\_}{S}}_2^R+\left({\underset{\_}{S}}_2^{ig}-{\underset{\_}{S}}_1^{ig}\right)-{\underset{\_}{S}}_1^R$        (11)

Here, residual molar entropy of inlet and exit state is ${\underset{\_}{S}}_1^R$ and ${\underset{\_}{S}}_2^R$, inert gas state molar entropy of inlet and exit state is ${\underset{\_}{S}}_1^{ig}$ and ${\underset{\_}{S}}_2^{ig}$ respectively.