Chapter 7.7, Problem 18P

(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 the 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 $\left(\underset{\_}{V}\right)$ 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 change in molar enthalpy

Concept Introduction:

Write the residual properties of A.

$A=\frac{aP}{R^2{T}^2}$

Write the residual properties of B.

$B=\frac{bP}{RT}$

Write the change in molar enthalpy using residual properties.

$\begin{array}{c}{\underset{\_}{H}}_2-{\underset{\_}{H}}_1=\left({\underset{\_}{H}}_2-{\underset{\_}{H}}_2^{ig}\right)+\left({\underset{\_}{H}}_2^{ig}-{\underset{\_}{H}}_1^{ig}\right)-\left({\underset{\_}{H}}_1-{\underset{\_}{H}}_1^{ig}\right)\\ =\left({\underset{\_}{H}}_2^R\right)+\left({\underset{\_}{H}}_2^{ig}-{\underset{\_}{H}}_1^{ig}\right)-\left({\underset{\_}{H}}_1^R\right)\end{array}$

Here, molar enthalpy at state 1 (liquid) and 2 (vapor) are ${\underset{\_}{H}}_1$ and ${\underset{\_}{H}}_2$, molar enthalpy at state 1 and 2 at inert gas are ${\underset{\_}{H}}_1^{ig}$ and ${\underset{\_}{H}}_2^{ig}$ respectively.

Write the expression for residual molar enthalpy.

$\frac{{\underset{\_}{H}}^R}{RT}=\left(Z-1\right)-\left\{\left(\frac{A}{B\sqrt{8}}\right)\left(1+\frac{\kappa \sqrt{T_r}}{\sqrt{a}}\right)\ln \left[\frac{Z+\left(1+\sqrt{2}\right)B}{Z+\left(1-\sqrt{2}\right)B}\right]\right\}$

Here, compressibility factor is Z, constants of residual properties are A and B.

Write the ideal gas enthalpy change.

${\underset{\_}{H}}_2^{ig}-{\underset{\_}{H}}_1^{ig}={\displaystyle \underset{T_1=300\text{K}}{\overset{T_2=500\text{K}}{\int }}{C}_P^{*}}$

Here, heat capacity at constant pressure for an ideal gas is $C_P^{*}$.