Chapter 8, Problem 3P

In a chemical engineering process, water vapor $\left({\text{H}}_2\text{O}\right)$ is heated to sufficiently high temperatures that a significant portion of the water dissociates, or splits apart, to form oxygen $\left({\text{O}}_2\right)$ and hydrogen $\left({\text{H}}_2\right)$

${\text{H}}_2\text{O}\rightleftharpoons {\text{H}}_2+\frac{1}{2}{\text{O}}_2$

If it is assumed that this is the only reaction involved, the mole fraction $x{\text{ of H}}_2\text{O}$ that dissociates can be represented by

$K=\frac{x}{1-x}\sqrt{\frac{2p_t}{2+x}}\left(\text{P8}\text{.3}\text{.1}\right)$

where $K=$ the reaction equilibrium constant and $p_t=$ the total pressure of the mixture. If $p_t=3\text{ atm and }K=0.05$, determine the value of x that satisfies Eq. $\left(\text{P8}\text{.3}\text{.1}\right)$.