At $473 \ K$, equilibrium constant $K_{c}$ for decomposition of phosphorus pentachloride, $PCl_{5}$, is $8.3 \times 10^{-3}$. If decomposition is depicted as,
$PCl_{5(g)} \rightleftharpoons PCl_{3(g)} + Cl_{2(g)}, \Delta_{r}H^{\Theta} = 124.0 \ kJ \ mol^{-1}$
$(a)$ Write an expression for $K_{c}$ for the reaction.
$(b)$ What is the value of $K_{c}$ for the reverse reaction at the same temperature?
$(c)$ What would be the effect on $K_{c}$ if $(i)$ more $PCl_{5}$ is added $(ii)$ pressure is increased $(iii)$ the temperature is increased?

$(a)$ The expression for $K_{c}$ is: $K_{c} = \frac{[PCl_{3}][Cl_{2}]}{[PCl_{5}]}$
$(b)$ The equilibrium constant for the reverse reaction $(K_{c}^{\prime})$ is the reciprocal of the forward reaction constant: $K_{c}^{\prime} = \frac{1}{K_{c}} = \frac{1}{8.3 \times 10^{-3}} \approx 120.48$
$(c)$ Effect on $K_{c}$:
$(i)$ No effect, as $K_{c}$ depends only on temperature.
$(ii)$ No effect, as $K_{c}$ is independent of pressure.
$(iii)$ $K_{c}$ increases. Since the reaction is endothermic $(\Delta_{r}H^{\Theta} > 0)$, increasing the temperature shifts the equilibrium to the right, thereby increasing the value of $K_{c}$.