For the reaction $P_{(g)} + 3Q_{(g)} \rightleftharpoons 4R_{(g)}$,the initial concentrations of $P$ and $Q$ are equal. If the equilibrium concentrations of $P$ and $R$ are equal,then the equilibrium constant $K_c$ for the reaction will be .....

Given three reactions and their equilibrium constants:
$N_2 + 3H_2 \rightleftharpoons 2NH_3 ; k_1$
$N_2 + O_2 \rightleftharpoons 2NO ; k_2$
$H_2 + \frac{1}{2}O_2 \rightleftharpoons H_2O ; k_3$
The equilibrium constant for the reaction $2NH_3 + \frac{5}{2}O_2 \rightleftharpoons 2NO + 3H_2O$ in terms of $k_1, k_2,$ and $k_3$ is:

When $20 \ g$ of $CaCO_3$ are subjected to decomposition at $227 \ ^oC$ in a closed container of $10 \ L$ capacity,$50 \%$ of $CaCO_3$ remained unreacted at equilibrium. Calculate $K_P$ for $CaCO_{3(s)} \rightleftharpoons CaO_{(s)} + CO_{2(g)}$ in $atm$.

Consider the following reversible first-order reaction of $X$ at an initial concentration $[X]_{0}$. The values of the rate constants are $K_{f} = 2 \ s^{-1}$ and $K_{b} = 1 \ s^{-1}$.
$X \underset{K_{b}}{\stackrel{K_{f}}{\rightleftharpoons}} Y$
Which of the following plots correctly represents the concentration of $X$ and $Y$ as a function of time?

For the reaction,$0.5 C_{(s)} + 0.5 CO_{2(g)} \rightleftharpoons CO_{(g)}$,the equilibrium pressure is $12 \ atm$. If $CO_2$ conversion is $50 \%$,the value of $K_p$,in $atm$,is:

If $8 \, \text{mol}$ of $PCl_5$ is heated in a closed vessel of $10 \, L$ capacity and $25\%$ of it dissociates into $PCl_3$ and $Cl_2$ at equilibrium,then the value of $K_p$ is .....