Question

Consider the following reaction at \(75^{\circ} \mathrm{C}\) : $$ 3 \mathrm{R}(s)+2 \mathrm{Q}(g) \rightleftharpoons \mathrm{A}(g)+5 \mathrm{~B}(l) \quad K=9.4 $$ A \(10.0-\mathrm{L}\) sample contains \(0.30 \mathrm{~mol}\) of \(\mathrm{R}\) and \(\mathrm{Q}\) and \(0.50 \mathrm{~mol}\) of \(\mathrm{A}\) and \(\mathrm{B}\). In which direction will the reaction proceed?

Short answer

Answer: The reaction will proceed to the left (reactants are favored).

Step-by-step solution

Write down the reaction and given information

Write down the reaction and the given information to have an easy reference during the calculations: $$ 3 \mathrm{R}(s)+2 \mathrm{Q}(g) \rightleftharpoons \mathrm{A}(g)+5\mathrm{~B}(l) \quad K=9.4 $$ Initial molar amounts: $$ [\mathrm R] = 0.3 M $$ $$ [\mathrm Q] = 0.3 M $$ $$ [\mathrm A] = 0.5 M $$ $$ [\mathrm B] = 0.5 M $$ Volume of the reaction: $$ V = 10 L $$

Calculate the initial concentrations

Calculate initial concentrations of the reacting species, using the volume of the reaction: $$ [\mathrm R]_0 = \frac{0.3 \,\text{mol}}{10 \,\text{L}} = 0.03\,\text{M} $$ $$ [\mathrm Q]_0 = \frac{0.3 \,\text{mol}}{10 \,\text{L}} = 0.03\,\text{M} $$ $$ [\mathrm A]_0 = \frac{0.5 \,\text{mol}}{10 \,\text{L}} = 0.05\,\text{M} $$ $$ [\mathrm B]_0 = \frac{0.5 \,\text{mol}}{10 \,\text{L}} = 0.05\,\text{M} $$

Calculate the reaction quotient Qc

Calculate the reaction quotient, Qc, using the initial concentrations: $$ Q_c = \frac{[\mathrm A]_0[\mathrm B]^5_0}{[\mathrm Q]^2_0} = \frac{(0.05\,\text{M})(0.05\,\text{M})^5}{(0.03\,\text{M})^2} ≈ 14.6 $$

Compare Qc to K and determine the direction of the reaction

Compare the calculated Q_c to the equilibrium constant K: $$ \begin{cases} Q_c \gt K \implies \text{Reaction proceeds to the left}, \\ Q_c = K \implies \text{System is already at equilibrium}, \\ Q_c \lt K \implies \text{Reaction proceeds to the right}. \end{cases} $$ Given that \(Q_c ≈ 14.6\) and \(K = 9.4\), we have \(Q_c > K\), which implies that the reaction will proceed to the left (reactants are favored).