Write the two equilibria

$X \rightleftharpoons 2Y$ $Z \rightleftharpoons P+Q$

Given: $K_{P1}:K_{P2}=1:9$

Also, the degree of dissociation of $X$ and $Z$ is same. Let it be $\alpha$.

For reaction $X \rightleftharpoons 2Y$

Start with 1 mole of $X$.

At equilibrium:

• moles of $X = 1-\alpha$
• moles of $Y = 2\alpha$
• total moles $=1+\alpha$

If total pressure is $P_1$, then partial pressures are $p_X=\frac{1-\alpha}{1+\alpha}P_1, \qquad p_Y=\frac{2\alpha}{1+\alpha}P_1$

Hence,

For reaction $Z \rightleftharpoons P+Q$

Start with 1 mole of $Z$.

At equilibrium:

• moles of $Z = 1-\alpha$
• moles of $P = \alpha$
• moles of $Q = \alpha$
• total moles $=1+\alpha$

If total pressure is $P_2$, then

Use the given ratio

$\frac{K_{P1}}{K_{P2}}=\frac{1}{9}$

Substitute expressions: $\frac{\frac{4\alpha^2 P_1}{1-\alpha^2}}{\frac{\alpha^2 P_2}{1-\alpha^2}}=\frac{1}{9}$

$\frac{4P_1}{P_2}=\frac{1}{9}$

$\frac{P_1}{P_2}=\frac{1}{36}$

Therefore, $P_1:P_2=1:36$

Check options

Correct option is A: $1:36$

Compare with stored answer

Stored correct answer = A

This matches our derived answer.