Chapter 8.6, Problem 18P

(A)

Interpretation Introduction

Interpretation:

The fugacity in the vapor phase.

Concept Introduction:

The expression for the fugacity as a function of pressure and temperature $\left(f\right)$.

$\ln \left(\frac{f}{P}\right)={\displaystyle {\int }_{T=T,P=0}^{T=T,P=P}\left[\frac{Z-1}{P}\right]}dP$

Here, pressure is $P$, temperature is $T$, compressibility factor is $Z$, and change in pressure is $dP$.

The equation of state in terms of compressibility factor.

$\begin{array}{l}\underset{\_}{V}=\frac{RT}{P}+a{P}^2\\ \frac{P}{RT}\left(\underset{\_}{V}\right)=\frac{P}{RT}\frac{RT}{P}+a{P}^2\\ Z=1+\frac{a{P}^3}{RT}\end{array}$

Here, gas constant is $R$ and van der Waals parameter is $a$.

Substitute $1+\frac{a{P}^3}{RT}$ for $Z$ in fugacity equation.

$\begin{array}{l}\ln \left(\frac{f}{P}\right)={\displaystyle {\int }_{T=T,P=0}^{T=T,P=P}\left[\frac{1+\frac{a{P}^3}{RT}-1}{P}\right]}dP\\ ={\displaystyle {\int }_{T=T,P=0}^{T=T,P=P}\left[\frac{a{P}^2}{RT}\right]}dP\\ \ln \left(\frac{f}{P}\right)=\frac{a{P}^3}{3RT}\\ f=P\exp \left(\frac{a{P}^3}{3RT}\right)\end{array}$

(B)

Interpretation Introduction

Interpretation:

The fugacity in the liquid phase.

Concept Introduction:

The fugacity of the compound as a function of pressure and temperature.

$f=P\exp \left(\frac{a{P}^3}{3RT}\right)$

Determine fugacity in liquid phase using the Poynting equation.

$f_L=f_{sat}\exp \left[\frac{\underset{\_}{V}\left(P_2-P_{sat}\right)}{RT}\right]$

Here, saturated pressure is $P_{sat}$, molar volume is $\underset{\_}{V}$, and saturated vapor fugacity is $f_{sat}$.

(C)

Interpretation Introduction

Interpretation:

The fugacity in the vapor phase.

Concept Introduction:

The fugacity of the compound as a function of pressure and temperature.

$f=P\exp \left(\frac{a{P}^3}{3RT}\right)$

(D)

Interpretation Introduction

Interpretation:

The fugacity in the liquid phase.

Concept Introduction:

The fugacity of the compound as a function of pressure and temperature.

$f=P\exp \left(\frac{a{P}^3}{3RT}\right)$

Determine fugacity in liquid phase using the Poynting equation.

$f_L=f_{sat}\exp \left[\frac{\underset{\_}{V}\left(P_2-P_{sat}\right)}{RT}\right]$

Here, saturated pressure is $P_{sat}$, molar volume is $\underset{\_}{V}$, and saturated vapor fugacity is $f_{sat}$.