The decomposition of `HNO_(3)` represented below attains equilibrium at a given `P_(eq.)` and `T`. If we start: `4HNO_(3(g))hArr4NO_(2(g))+2H_(2)O_((g))+O_(2(g))` with pure `HNO_(3)` having initial pressure `P_(i)`, then at equilibrium. Select the correct statement (`s`).

Options

A. `K_(p)=(1024(P_(O_(2)))^(7))/((P_(eq.)-7P_(O_(2)))^(4))`
B. `K_(p)=(1024(P_(O_(2)))^(7))/((P_(i)-4P_(O_(2)))^(4))`
C. `K_(p)=(1024(P_(O_(2)))^(7))/((P_(eq.)-7P_(O_(2)))^(4))xx((1)/((p^(@))^(3)))`
D. none of these

Explanation

(a,b,c)
`underset(P_(i)-4P)underset(P_(i))(4HNO_(3(g)))hArrunderset(4P)underset(0)(4NO_(2(g)))+underset(2P)underset(0)(2H_(2)O((g)))+underset(P)underset(0)(O_(2(g)))`
Pressure at equilibrium is `P_(eq.)`
`P_(eq.)=P_(i)-4P+4P+2P+P`
`:. P_(i)-4P=P_(eq.)-7P=P_(eq.)-7P_(O_(2))` (`:' P=P_(O_(2))`)
`K_(p)=((P_(NO_(2)))^(4)(P_(H_(2)O))^(2)P_(O_(2)))/((P_(HNO_(3)))^(4))=(1024(P_(O_(2)))^(7))/((P_(eq.)-7P_(O_(2)))^(4))`
`=(1024(P_(O_(2)))^(7))/((P_(i)-4P_(O_(2)))^(4))`
and `K_(p)^(@)=((P_(NO_(2)))^(4)(P_(H_(2)O))^(2)P_(O_(2)))/((P_(HNO_(3)))^(4))xx(1)/((P^(@))^(3))`
`=((4P)^(4)xx(2P)^(2)xxP)/((P_(eq.-7P_(O_(2))))^(4))xx(1)/((P^(@))^(3))`
`=(1024(P_(O_(2)))^(7))/((P_(eq.)-7P_(O_(2)))^(4))xx(1)/((P^(@))^(3))`