The equilibrium constant `(K_(p))` for the decomposition of gaseous `H_(2)O`
`H_(2)O(g)hArr H_(2)(g)+(1)/(2)O_(2)(g)`
is related to the degree of dissociation `alpha` at a total pressure P by

Options

A. `K_(p)=(alpha^(3)p_(2)^(2))/((1+alpha)(2+alpha)^(1//2))`
B. `K_(p)=(alpha^(3)p^(3//2))/((1-alpha)(2+alpha)^(1//2))`
C. `K_(p)=(alpha^(3//2)p^(2))/((1-alpha)(2+alpha)^(1//2))`
D. `K_(p)=(alpha^(3//2)p^(1//2))/((1-alpha)(2+alpha)^(1//2))`

Answer

D

Explanation

`{:(,H_(2)O(g) ,+,H_(2)(g),+,1//2O_(2)(g)),("Initial",1 mol,,0,,0),("At eqm.",1-alpha,,alpha,,alpha//2):}`
Total `=1+alpha//2`
At eqm. `pH_(2)O=(1-alpha)/(1+alpha//2)xxp=(2(1-alpha)p)/(2+alpha)`
`pH_(2)=(alpha)/(1+alpha//2)xxp=(alphap)/(1+alpha//2)=(2alpha p)/(2+alpha)`
`pO_(2)=(alpha//2)/(1+alpha//2)xxp=(alphap)/(2+alpha)`
`:. K_(p)=(pH_(2)xxpO_(2)^(1//2))/(pH_(2)O)`
`=(((2alphap)/(2+alpha))((alphap)/(2+alpha))^(1//2))/((2(1-alpha))/(2+alpha)p)`
`=((alpha)/(1-alpha))((alphap)/(2+alpha))^(1//2)=(alpha^(3//2)p^(1//2))/((1-alpha)(2+alpha)^(1//2))`