Chapter 8, Problem 5P

A reversible chemical reaction

$2\text{A+B}\rightleftharpoons \text{C}$

Can be characterized by the equilibrium relationship

$K=\frac{c_c}{c_a^2{c}_b}$

where the nomenclature $c_i$ represents the concentration of constituent i. Suppose that we define a variable x as representing the number of moles of C that are produced. Conservation of mass can be used to reformulate the equilibrium relationship as

$K=\frac{\left(c_{c,0}+x\right)}{{\left(c_{a,0}-2x\right)}^2\left(c_{b,0}-x\right)}$

where the subscript 0 designates the initial concentration of each constituent. If $K=0.015,c_{a,0}=42,c_{b,0}=30,\text{ and }{c}_{c,0}=4$, determine the value of x. (a) Obtain the solution graphically. (b) On the basis of (a) solve for the root with initial guesses of $x_l=0\text{ and }{x}_u=20\text{to }{\epsilon }_s=0.5\%$. Choose either bisection or false position to obtain your solution. Justify your choice.