Chapter 7.7, Problem 17P

(A)

Interpretation Introduction

Interpretation:

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

Concept Introduction:

The expression to calculate the molar volume $\left(\underset{\_}{V}\right)$.

$\underset{\_}{V}=\frac{RT}{P}$

Here, pressure is P, gas constant is R, and temperature is T.

(B)

Interpretation Introduction

Interpretation:

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

Concept Introduction:

The expression for van der Waals parameter $a$.

$a=\frac{27R^2{T}_c^2}{64P_c}$

Here, critical temperature is $T_c$, critical pressure is $P_c$, and gas constant is $R$.

Write the expression for van der Waals parameter $b$.

$b=\frac{R{T}_c}{8P_c}$

(C)

Interpretation Introduction

Interpretation:

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

Concept Introduction:

Write the relationship between the parameter, m and Soave’s EOS.

$m=0.480+1.574\omega -0.176{\omega }^2$

Here, the acentric factor is $\omega$.

Write the reduced temperature.

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

Here, critical temperature is $T_c$.

Write the expression $\alpha$ as a function of the reduced temperature.

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

Here, reduced temperature is $T_r$.

Write the reduced pressure $\left(P_r\right)$.

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

Here, pressure is $P$.

Write the expression of a at the critical point.

$a=0.42747R^2\frac{T_c^2}{P_c}\times \alpha$

Write the expression of b as Soave equation.

$b=0.08664R\frac{T_c}{P_c}$

Write the expression to calculate the molar volume using the Soave equation.

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

(D)

Interpretation Introduction

Interpretation:

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

Concept Introduction:

Write the expression to calculate the parameter a for Peng-Robinson EOS.

$\kappa =0.37464+1.54226\omega -0.26993{\omega }^2$

Write the expression to calculate the $\alpha$ expressed as a function of the reduced temperature.

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

Write the expression to calculate the value of parameter a at the critical point.

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

Write the expression to calculate the van der Waals parameter $a$.

$a=a_c\alpha$

Write the expression to calculate the van der Waals parameter $b$.

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

Write the expression to calculate the molar volume $\left(\underset{\_}{V}\right)$ using the Soave equation.

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

(E)

Interpretation Introduction

Interpretation:

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

Concept Introduction:

Write the expression to calculate the virial parameter or coefficient $\left(B^0\right)$.

$B^0=0.083-\frac{0.422}{T_r^{1.6}}$

Write the expression to calculate the virial parameter $\left(B^1\right)$.

$B^1=0.139-\frac{0.172}{T_r^{4.2}}$

Write the expression to calculate parameter as a function of temperature and the acentric factor.

$\frac{B\text{'}{P}_c}{R{T}_c}=B^0+\omega B^1$

Here, function of temperature is $B\text{'}$.

Write the expression to calculate the relative pressure.

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

Write the expression to calculate the virial compressibility factor $\left(Z\right)$.

$Z=1+\left(\frac{B\text{'}{P}_c}{R{T}_c}\right)\frac{P_r}{T_r}$

(F)

Interpretation Introduction

Interpretation:

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

Concept Introduction:

Write the expression to calculate the compressibility factor $\left(Z\right)$.

$Z=Z^0+\omega Z^1$

Here, compressibility of compound with $\omega =0$ is $Z^0$ and difference between $Z^0\text{and}Z$ is $Z^1$.