Chapter 8, Problem 6P

The following chemical reactions take place in a closed system

$\begin{array}{l}2\text{A}+\text{B}\rightleftharpoons \text{C}\\ \text{A}+\text{D}\rightleftharpoons \text{C}\end{array}$

At equilibrium, they can be characterized by

$\begin{array}{l}{K}_1=\frac{c_c}{c_a^2{c}_b}\\ K_2=\frac{c_c}{c_a{c}_d}\end{array}$

where the nomenclature represents the concentration of constituent $i$. $\text{If }{x}_1\text{ and }{x}_2$ are the number of moles of C that are produced due to the first and second reactions, respectively, use an approach similar to that of Prob. 8.5 to reformulate the equilibrium relationships in terms of the initial concentrations of the constituents. Then, use the Newton-Raphson method to solve the pair of simultaneous nonlinear equations for $x_1\text{ and }{x}_2\text{ if }{K}_1=4\times {10}^{-4},K_2=3.7\times {10}^{-2},c_{a,0}=50,c_{b,0}=20,c_{c,0}=5,\text{ and }{c}_{d,0}=10$. Use a graphical approach to develop your initial guesses.