At a given temperature and pressure the equilibrium constant values for the equilibria are given below. The relation and is [2024]

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Solution

Q.
At a given temperature and pressure the equilibrium constant values for the equilibria are given below.

$3 A_{2} + B_{2} ⇌ 2 A_{3} B, K_{1}$

$A_{3} B ⇌ \frac{3}{2} A_{2} + \frac{1}{2} B_{2}, K_{2}$

The relation $K_{1}$ and $K_{2}$ is [2024]

1 $K_{1}^{2} = 2 K_{2}$

2 $K_{2} = K_{1} / 2$

3 $K_{1} = \frac{1}{\sqrt{K_{2}}}$

4 $K_{2} = \frac{1}{\sqrt{K_{1}}}$

Ans.
(4)

$K_{1} = \frac{{[A_{3} B]}^{2}}{{[A_{2}]}^{3} [B_{2}]}$ ...(i)

$K_{2} = \frac{{[A_{2}]}^{3 / 2} {[B_{2}]}^{1 / 2}}{[A_{3} B]}$ ...(ii)

Taking square root on both sides in eq. (i), we get

$\sqrt{K_{1}} = \frac{[A_{3} B]}{{[A_{2}]}^{3 / 2} {[B_{2}]}^{1 / 2}}$

$\Rightarrow [A_{3} B] = \sqrt{K_{1}} {[A_{2}]}^{3 / 2} {[B_{2}]}^{1 / 2}$

Putting the value of $[A_{3} B]$ in eq. (ii), we get

$K_{2} = \frac{1}{\sqrt{K_{1}}}$

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