Question

At a certain temperature \( K_{1} \) and \( K_{2} \) are equilibrium constants for following equations :\[\begin{array}{l} C ( s )+ CO _{2}( g ) \rightleftharpoons 2 CO ( g ) \\2 CO _{2}( g ) \rightleftharpoons 2 CO ( g )+ O _{2}( g )\end{array}\]What will be the equilibrium constant at the same temperature for the given equation\[C ( s )+\frac{1}{2} O _{2}( g ) \rightleftharpoons CO ( g )\]

(1) \( \frac{ K _{1}^{2}}{ K _{2}} \)

(2) \( \frac{ K _{1}}{ K _{2}} \)

(3) \( \frac{ K _{1}}{\sqrt{ K _{2}}} \)

(4) \( K_{1}^{2}-K_{2} \)

Answer 1

Given,

\(C(s) + CO_2 (g) \rightleftharpoons 2CO(g)\)

\(K_1 = \frac{P^2_{Co}}{P_{Co_2}}\)          ........(i)

\(2CO_2 (g) \rightleftharpoons 2CO(g) + O_2(g)\)

\(K_2 = \frac{P^2_{Co} \times P_{O_2}}{P_{Co_2}^2}\)       ......(ii)

\(\therefore C(s) + \frac12 O_2(g) \rightleftharpoons CO(g)\)

\(K_{eq} = \frac{P_{CO}}{P^{\frac12}_{O_2}}\)          ........(iii)

From equation (i) & (ii)

\(\frac{K_1}{\sqrt{K_2}} = \cfrac{\left(\frac{P^2_{CO}}{P_{CO_2}}\right)}{\left(\frac{P_{CO}\times P^{\frac12}_{O_2}}{P_{CO_2}}\right)} = \cfrac{P_{CO}}{P^{\frac12}_{O_2}}\)      ......(iv)

From equation (iii) & (iv)

\(K_{eq} = \frac{K_1}{\sqrt{K_2}}\)

Hence, option (3) is correct.