Question 9.4.4: Calculation of the Fugacity of a Species in a Mixture by Two...

a. The Lewis-Randall rule is

From Fig. 7.4-1, we have

Thus

and

b. Using the Peng-Robinson equation of state (Eqs. 6.4-2 and 9.4-8 through 9.4-10) and Aspen Plus^R  or the computer programs discussed in Appendix B, we find, for k_{12}=0.0,

P=\frac{R T}{\underline{V}-b}-\frac{a(T)}{\underline{V}(\underline{V}+b)+b(\underline{V}-b)} (6.4-2)

\begin{aligned}a_{\operatorname{mix}} &=\sum_{ i =1}^{ C } \sum_{j=1}^{ C } y_{ i } y_{ j } a_{ ij } \\b_{\operatorname{mix}} &=\sum_{ i =1} y_{ i } b_{ i }\end{aligned} (9.4-8)

a_{ ij }=\sqrt{a_{ ii } a_{ jj }}\left(1-k_{ ij }\right)=a_{ ji } (9.4-9)

\begin{array}{l}\ln \frac{\bar{f}_{ i }^{ V }(T, P, \underline{y})}{y_{ i } P} \\\quad=\ln \bar{\phi}_{ i }^{ V }(T, P, \underline{y}) \\\quad=\frac{b_{ i }}{b_{ mix }}\left(Z_{ mix }^{ V }-1\right)-\ln \left(Z_{ mix }^{ V }-\frac{b_{ mix } P}{R T}\right)\\\quad-\frac{a_{ mix }}{2 \sqrt{2} b_{\text {mix }} R T}\left[\frac{2 \sum_{ j } y_{ j } a_{ ij }}{a_{ mix }}-\frac{b_{ i }}{b_{ mix }}\right] \ln \left[\frac{Z_{ mix }^{ V }+(1+\sqrt{2}) \frac{b_{ mix } P}{R T}}{Z^{ V }+(1-\sqrt{2}) \frac{b_{ mix } P}{R T}}\right]\\\quad =\frac{B_{ i }}{B_{ mix }}\left(Z_{ mix }^{ V }-1\right)-\ln \left(Z_{ mix }^{ V }-B_{ mix }\right)\\\quad -\frac{A_{ mix }}{2 \sqrt{2} B_{ mix }}\left[\frac{2 \sum_{ j } y_{ j } A_{ ij }}{A_{ mix }}-\frac{B_{ i }}{B_{ mix }}\right] \ln \left[\frac{Z_{ mix }^{ V }+(1+\sqrt{2}) B_{ mix }}{Z_{ mix }^{ V }+(1-\sqrt{2}) B_{ mix }}\right]\end{array} (9.4-10)

and using k_{12}=0.09 (from Table 9.4-1),

These last values should be the most accurate.