Chapter 6.6, Problem 8P

A)

Interpretation Introduction

Interpretation:

Derive an expression for the following in terms of $P,\underset{\_}{V},,T,C_p,{\text{and C}}_v$.

${\left(\frac{\partial \underset{\_}{H}}{\partial T}\right)}_{\underset{\_}{V}}$

Concept introduction:

Use the fundamental property relationship, total derivatives , and expansion rule to obtain the expression for the given term.

The expression for the fundamental property relation for enthalpy is given as follows:

$d\underset{\_}{H}=Td\underset{\_}{S}+\underset{\_}{V}dP$

Here, change in molar enthalpy is $d\underset{\_}{H}$, temperature is T, change in molar entropy is $d\underset{\_}{S}$, molar volume is $\underset{\_}{V}$, and change in the pressure is dP.

the expression for the total derivative for molar enthalpy is given as follows:

$d\underset{\_}{H}={\left(\frac{\partial \underset{\_}{H}}{\partial \underset{\_}{S}}\right)}_{P}d\underset{\_}{S}+{\left(\frac{\partial \underset{\_}{H}}{\partial P}\right)}_{\underset{\_}{S}}dP$

Here, change in molar entropy and change in molar enthalpy at constant pressure is ${\left(\partial \underset{\_}{S}\right)}_P$ and ${\left(\partial \underset{\_}{H}\right)}_P$, change in molar enthalpy and change in molar entropy at constant molar entropy is ${\left(\partial \underset{\_}{S}\right)}_{\underset{\_}{S}}$ and ${\left(\partial \underset{\_}{H}\right)}_{\underset{\_}{S}}$ respectively.

B)

Interpretation Introduction

Interpretation:

Derive an expression for the following in terms of $P,\underset{\_}{V},,T,C_p,{\text{and C}}_v$.

${\left(\frac{\partial \underset{\_}{H}}{\partial P}\right)}_T$

Concept introduction:

Use the expansion rule and Maxwell’s relation to obtain the expression for the given term.

The expansion rule to ${\left(\frac{\partial \underset{\_}{H}}{\partial P}\right)}_T$, with constant $\underset{\_}{S}$ and P is given as follows:

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

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

C)

Interpretation Introduction

Interpretation:

Derive an expression for the following in terms of $P,\underset{\_}{V},,T,C_p,{\text{and C}}_v$.

${\left(\frac{\partial \underset{\_}{U}}{\partial \underset{\_}{V}}\right)}_P$

Concept introduction:

Use the fundamental property relationship, total derivatives, and expansion rule to obtain the expression for the given term.

The fundamental property relation for molar internal energy.

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

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

The total derivative of molar change in 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 internal energy with respect to change in molar entropy at constant molar volume is ${\left(\frac{\partial \underset{\_}{U}}{\partial \underset{\_}{S}}\right)}_{\underset{\_}{V}}$ and change in molar internal energy with respect to change in molar volume at constant molar entropy is ${\left(\frac{\partial \underset{\_}{U}}{\partial \underset{\_}{V}}\right)}_{\underset{\_}{S}}$.

D)

Interpretation Introduction

Interpretation:

Derive an expression for the following in terms of $P,\underset{\_}{V},,T,C_p,{\text{and C}}_v$.

${\left(\frac{\partial \underset{\_}{U}}{\partial T}\right)}_P$

Concept introduction:

Use the expansion rule to obtain the expression for the given term.

The expansion rule to ${\left(\frac{\partial \underset{\_}{U}}{\partial T}\right)}_P$ with constant $\underset{\_}{S}$ and $\underset{\_}{V}$ is given as follows:

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

E)

Interpretation Introduction

Interpretation:

Derive an expression for the following in terms of $P,\underset{\_}{V},,T,C_p,{\text{and C}}_v$.

${\left(\frac{\partial \underset{\_}{A}}{\partial \underset{\_}{S}}\right)}_P$

Concept introduction:

Use the fundamental property relationship, total derivatives, expansion rule, and triple product rule to obtain the expression for the given term.

F)

Interpretation Introduction

Interpretation:

Derive an expression for the following in terms of $P,\underset{\_}{V},,T,C_p,{\text{and C}}_v$.

${\left(\frac{\partial \underset{\_}{A}}{\partial \underset{\_}{S}}\right)}_P$

Concept introduction:

Use the expansion rule to obtain the expression for the given term.

The expansion rule to ${\left(\frac{\partial \underset{\_}{A}}{\partial \underset{\_}{S}}\right)}_T$ with $\underset{\_}{V}$ and T is given as follows:

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

G)

Interpretation Introduction

Interpretation:

Derive an expression for the following in terms of $P,\underset{\_}{V},,T,C_p,{\text{and C}}_v$.

${\left(\frac{\partial \underset{\_}{G}}{\partial T}\right)}_P$

Concept introduction:

Compare the fundamental property relation and total derivatives of $d\underset{\_}{G}$ to obtain the expression for the given term

Write the fundamental property relation.

$d\underset{\_}{G}=\underset{\_}{V}dP-\underset{\_}{S}dT$

Here, change in pressure is dP, change in temperature is dT, change in molar Gibbs free energy is $d\underset{\_}{G}$, molar volume and entropy is $\underset{\_}{V}$ and $\underset{\_}{S}$ respectively.

Write the total derivative of $d\underset{\_}{G}$.

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

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

H)

Interpretation Introduction

Interpretation:

Derive an expression for the following in terms of $P,\underset{\_}{V},,T,C_p,{\text{and C}}_v$.

${\left(\frac{\partial \underset{\_}{G}}{\partial P}\right)}_S$

Concept introduction:

Use the expansion rule and triple product rule to obtain the expression for ${\left(\frac{\partial \underset{\_}{G}}{\partial P}\right)}_S$, then use the Maxwell’s equation and the $C_P$ identity in the expression for ${\left(\frac{\partial \underset{\_}{G}}{\partial P}\right)}_S$ to obtain the expression for the given term.

The expansion rule with T and P for ${\left(\frac{\partial \underset{\_}{G}}{\partial P}\right)}_S$ is given as follows:

${\left(\frac{\partial \underset{\_}{G}}{\partial P}\right)}_S={\left(\frac{\partial \underset{\_}{G}}{\partial P}\right)}_T{\left(\frac{\partial P}{\partial P}\right)}_{\underset{\_}{S}}+{\left(\frac{\partial \underset{\_}{G}}{\partial T}\right)}_P{\left(\frac{\partial T}{\partial P}\right)}_{\underset{\_}{S}}$

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