Chapter 8.6, Problem 24P

(A)

Interpretation Introduction

Interpretation:

The vapour pressure at the temperatures $300\text{K, 500 K, and 700 K}$ using shortcut equation.

Concept Introduction:

The expression of shortcut vapor pressure equation is,

$\begin{array}{l}{\log }_{10}{P}_r^{sat}=\frac{7}{3}\left(\omega +1\right)\left(1-\frac{T_C}{T}\right)\\ P_r^{sat}=e^{\frac{7}{3}\left(\omega +1\right)\left(1-\frac{T_C}{T}\right)}\\ \frac{P}{P_c}=e^{\frac{7}{3}\left(\omega +1\right)\left(1-\frac{T_C}{T}\right)}\\ P=\left(P_c\right)e^{\frac{7}{3}\left(\omega +1\right)\left(1-\frac{T_c}{T}\right)}\end{array}$

Here, reduced temperature is $T_r$, critical temperature is $T_c$, critical pressure is $P_c$, acentric factor is $\omega$, and reduced pressure at saturated point is $P_r^{sat}$.

(B)

Interpretation Introduction

Interpretation:

The Peng-Robinson equation of state.

Concept Introduction:

The Peng-Robinson equation is,

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

Here, gas constant is R, molar volume is $\underset{\_}{V}$, constants are a and b.

The expression to obtain the value of a is,

$\begin{array}{c}a=a_c\alpha \\ =\frac{0.45724R^2{T}_c^2}{P_c}\times \alpha \end{array}$

Here, critical pressure and temperature is $P_c\text{and}{T}_c$ respectively, value of a at critical point is $a_c$, and parameter is $\alpha$.

The expression of parameter, $\alpha$ is,

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

Here, constant is $\kappa$.

The expression of $\kappa$ is,

$\kappa =0.37464+1.54226\omega -0.269932{\omega }^2$

Write the expression to obtain the reduced temperature.

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

Here, critical temperature is $T_c$.

The expression to obtain the value of b is,

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

The expression of fugacity for the Peng-Robinson equation is,

$\ln \left(\frac{f}{P}\right)=\left(Z-1\right)-\ln \left(Z-B\right)-\frac{A}{2B\sqrt{2}}\times \ln \left[\frac{Z+\left(1+\sqrt{2}\right)B}{Z+\left(1-\sqrt{2}\right)B}\right]$

The expression for A and B is,

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

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