Chapter 9, Problem 9.80P

In Problem 9.79, the synthesis of methanol from carbon monoxide and hydrogen was described. Further analysis, however, reveals that three reactions can take place:

   $\begin{array}{l}CO+2H_2\rightleftharpoons C{H}_{3}OH\\ C{O}_2+3H_2\rightleftharpoons C{H}_{3}OH+H_{2}O\\ C{O}_2+H_2\rightleftharpoons CO+H_{2}O\end{array}$

(a) Show that only two of these reactions are independent.

(b) The equilibrium constants for the first and third reactions are

   $k_{\left(1\right)}=\frac{y_{C{H}_{3}OH}{y}_{H_2}}{y_{CO}{y}_{H_2}^2{P}^2}=\exp \left(\begin{array}{l}21.225+\frac{9143.6}{T}-7.492InT\\ +4.076\times {10}^{-3}T-7.161\times {10}^{-8}{T}^2\end{array}\right)$

   $\begin{array}{l}CO+2H_2\rightleftharpoons C{H}_{3}OH\\ k_{\left(3\right)}=\frac{y_{CO}{y}_{H_{2}O}}{y_{C{O}_2}{y}_{H_2}P}=\exp \left(\begin{array}{l}13.148+\frac{5639.5}{T}-1.077InT\\ -5.44\times {10}^{-4}T+1.125\times {10}^{-7}{T}^2+\frac{49170}{T^2}\end{array}\right)\end{array}$

As in Problem 9.79, the feed composition is 5.0 mole% methane, 25.0% CO, 5.0% CO₂, and the remainder hydrogen. The temperature and pressure of the equilibrated product stream are 25O°C and 7.5 MPa. Determine the composition (mole fractions) of the product stream and the percentage conversions of CO and H₂.