Q. The equilibrium constant for the decomposition of water,
$\left[H_{2} O \left(\right. g \left.\right) \rightleftharpoons H_{2} \left(\right. g \left.\right) + \frac{1}{2} O_{2} \left(\right. g \left.\right)\right]$ is given by

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A

$K=\frac{\left(\alpha \right)^{3} p^{1 / 2}}{\left(\right. 1 - \alpha \left.\right) \left(\right. 2 - \alpha \left(\left.\right)^{1 / 2}}$

B

$K=\frac{\left(\alpha \right)^{3 / 2} p^{1 / 2}}{\left(\right. 1 - \alpha \left.\right) \left(\right. 2 + \alpha \left(\left.\right)^{1 / 2}}$

C

$K=\frac{\alpha ^{3} p^{1 / 2}}{\sqrt{2}}$

D

$K=\frac{\left(\alpha \right)^{3} p^{3 / 2}}{\left(\right. 1 - \alpha \left.\right) \left(\right. 2 + \alpha \left(\left.\right)^{1 / 2}}$

Solution:

$\underset{t_{e q}}{}\underset{1 - \alpha }{H_{2} O \left(\right. g \left.\right)}\rightleftharpoons\underset{\alpha }{H_{2} \left(\right. g \left.\right)}+\underset{\frac{\alpha }{2}}{\frac{1}{2} O_{2} \left(\right. g \left.\right)}$
$\Delta n_{g}=\frac{3}{2}-1=\frac{1}{2}$
$K_{P}=\frac{n_{H_{2}} \times n_{O_{2}}^{1 / 2}}{n_{H_{2} O}}\times \left(\frac{P}{\Sigma n}\right)^{\Delta n_{g}}$
$=\frac{\alpha \times \left(\frac{\alpha }{2}\right)^{1 / 2}}{\left(\right. 1 - \alpha \left.\right)}\times \left(\frac{P}{1 + \alpha / 2}\right)^{1 / 2}$
$=\frac{\left(\alpha \right)^{3 / 2} P^{1 / 2}}{\left(\right. 1 - \alpha \left.\right) \left(\right. 2 + \alpha \left(\left.\right)^{1 / 2}}$

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