Chapter 7.7, Problem 19P

(A)

Interpretation Introduction

Interpretation:

The molar volume $\left(\underset{\_}{V}\right)$ at the critical point

Concept Introduction:

Write the expression for reduced temperature.

$T_r=\frac{T}{T_c}$

Here, critical temperature is $T_c$.

Write the expression for the reduced pressure.

$P_r=\frac{P}{P_c}$

Here, critical temperature is $T_c$.

Write the expression as a function of the reduced temperature.

$\alpha ={\left[1+\kappa \left(1-T_r^{0.5}\right)\right]}^2$

Write the value Peng-Robinson parameter a at the critical point.

$a_c=\frac{0.45724R^2{T}_c^2}{P_c}$

Write the van der Waals parameter $a$.

$a=a_c\alpha$

Write the van der Waals parameter $b$.

$b=\frac{0.07780R{T}_c}{P_c}$

Write the Peng-Robinson equation.

$P=\frac{RT}{\underset{\_}{V}-b}-\frac{a}{\underset{\_}{V}\left(\underset{\_}{V}+b\right)+b\left(\underset{\_}{V}-b\right)}$

Here, molar volume is $\underset{\_}{V}$, parameters of Robinson equation are a, b, gas constant is R, temperature and pressure are T and P respectively.

(B)

Interpretation Introduction

Interpretation:

The molar volume $\left(\underset{\_}{V}\right)$ in liquid phase

Concept Introduction:

Write the expression for reduced temperature.

$T_r=\frac{T}{T_c}$

Here, critical temperature is $T_c$.

Write the expression for the reduced pressure.

$P_r=\frac{P}{P_c}$

Here, critical temperature is $T_c$.

Write the expression as a function of the reduced temperature.

$\alpha ={\left[1+\kappa \left(1-T_r^{0.5}\right)\right]}^2$

Write the value Peng-Robinson parameter a at the critical point.

$a_c=\frac{0.45724R^2{T}_c^2}{P_c}$

Write the van der Waals parameter $a$.

$a=a_c\alpha$

Write the van der Waals parameter $b$.

$b=\frac{0.07780R{T}_c}{P_c}$

Write the Peng-Robinson equation.

$P=\frac{RT}{\underset{\_}{V}-b}-\frac{a}{\underset{\_}{V}\left(\underset{\_}{V}+b\right)+b\left(\underset{\_}{V}-b\right)}$

Here, molar volume is $\underset{\_}{V}$, parameters of Robinson equation are a, b, gas constant is R, temperature and pressure are T and P respectively.

(C)

Interpretation Introduction

Interpretation:

The molar volume in the vapor phase.

Concept Introduction:

Write the expression for reduced temperature.

$T_r=\frac{T}{T_c}$

Here, critical temperature is $T_c$.

Write the expression for the reduced pressure.

$P_r=\frac{P}{P_c}$

Here, critical temperature is $T_c$.

Write the expression as a function of the reduced temperature.

$\alpha ={\left[1+\kappa \left(1-T_r^{0.5}\right)\right]}^2$

Write the value Peng-Robinson parameter a at the critical point.

$a_c=\frac{0.45724R^2{T}_c^2}{P_c}$

Write the van der Waals parameter $a$.

$a=a_c\alpha$

Write the van der Waals parameter $b$.

$b=\frac{0.07780R{T}_c}{P_c}$

Write the Peng-Robinson equation.

$P=\frac{RT}{\underset{\_}{V}-b}-\frac{a}{\underset{\_}{V}\left(\underset{\_}{V}+b\right)+b\left(\underset{\_}{V}-b\right)}$

Here, molar volume is $\underset{\_}{V}$, parameters of Robinson equation are a, b, gas constant is R, temperature and pressure are T and P respectively.

(D)

Interpretation Introduction

Interpretation:

The difference in molar enthalpies

Concept Introduction:

Write the expression for reduced temperature.

$T_r=\frac{T}{T_c}$

Here, critical temperature is $T_c$.

Write the expression for the reduced pressure.

$P_r=\frac{P}{P_c}$

Here, critical temperature is $T_c$.

Write the expression as a function of the reduced temperature.

$\alpha ={\left[1+\kappa \left(1-T_r^{0.5}\right)\right]}^2$

Write the value Peng-Robinson parameter a at the critical point.

$a_c=\frac{0.45724R^2{T}_c^2}{P_c}$

Write the van der Waals parameter $a$.

$a=a_c\alpha$

Write the van der Waals parameter $b$.

$b=\frac{0.07780R{T}_c}{P_c}$

Write the Peng-Robinson equation.

$P=\frac{RT}{\underset{\_}{V}-b}-\frac{a}{\underset{\_}{V}\left(\underset{\_}{V}+b\right)+b\left(\underset{\_}{V}-b\right)}$

Here, molar volume is $\underset{\_}{V}$, parameters of Robinson equation are a, b, gas constant is R, temperature and pressure are T and P respectively.

Write the difference in molar enthalpies.

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

Here, final molar enthalpy is ${\underset{\_}{H}}_2$, initial molar enthalpy is ${\underset{\_}{H}}_1$, final molar enthalpy for an ideal gas state is ${\underset{\_}{H}}_2^{ig}$, and initial molar enthalpy for an ideal gas state is ${\underset{\_}{H}}_1^{ig}$.

Write the difference between molar enthalpy for an ideal gas state $\left(d{\underset{\_}{H}}^{ig}\right)$.

$\begin{array}{l}{\underset{\_}{H}}_2^{ig}-{\underset{\_}{H}}_1^{ig}=C_P^{\ast }dT\\ d{\underset{\_}{H}}^{ig}=C_P^{\ast }dT\end{array}$

Here, constant pressure heat capacity on a molar basis for ideal gas is $C_P^{*}$.