Question

Write an equation for an equilibrium system that would lead to the following expressions \((\mathrm{a}-\mathrm{c})\) for \(\mathrm{K}\). (a) \(K=\frac{\left(P_{\mathrm{CO}}\right)^{2}\left(P_{\mathrm{H}_{2}}\right)^{5}}{\left(P_{\mathrm{C}_{2} \mathrm{H}_{6}}\right)\left(P_{\mathrm{H}_{2} \mathrm{O}}\right)^{2}}\) (b) \(K=\frac{\left(P_{\mathrm{NH}}\right)^{4}\left(P_{\mathrm{O}_{2}}\right)^{5}}{\left(P_{\mathrm{NO}}\right)^{4}\left(P_{\mathrm{H}_{2} \mathrm{O}}\right)^{6}}\) (c) \(K=\frac{\left[\mathrm{ClO}_{3}^{-}\right]^{2}\left[\mathrm{Mn}^{2+}\right]^{2}}{\left(P_{\mathrm{Cl}_{2}}\right)\left[\mathrm{MnO}_{4}^{-}\right]^{2}\left[\mathrm{H}^{+}\right]^{4}} ;\) liquid water is a product

Short answer

Question: Write the balanced chemical equations for the equilibrium systems represented by the given K expressions: (a) \(K=\frac{(P_{\mathrm{CO}})^2(P_{\mathrm{H}_{2}})^5}{(P_{\mathrm{C}_{2}\mathrm{H}_{6}})(P_{\mathrm{H}_{2} \mathrm{O}})^2}\) (b) \(K=\frac{(P_{\mathrm{NH}})^4(P_{\mathrm{O}_{2}})^5}{(P_{\mathrm{NO}})^4(P_{\mathrm{H}_{2}\mathrm{O})^6}\) (c) \(K=\frac{([\mathrm{ClO}_{3}^{-}])^2([\mathrm{Mn}^{2+}])^2}{(P_{\mathrm{Cl}_{2}})([\mathrm{MnO}_{4}^{-})^2([\mathrm{H}^{+}])^4}\) Answer: (a) \(2\mathrm{CO} + 5\mathrm{H_2} \rightleftharpoons \mathrm{C_2H_6} + 2\mathrm{H_2O}\) (b) \(4\mathrm{NH} + 5\mathrm{O_2} \rightleftharpoons 4\mathrm{NO} + 6\mathrm{H_2O}\) (c) \(2\mathrm{ClO}_{3}^{-} + 2\mathrm{Mn}^{2+} + 4\mathrm{H}^{+} \rightleftharpoons 2\mathrm{MnO}_{4}^{-} + \mathrm{Cl}_{2} + 2\mathrm{H_2O}\)

Step-by-step solution

Identify the components of each K expression

In each given expression, we can find different chemical species, either with partial pressure (P) or molar concentration ([]). Identify the species for each expression, and pay attention to the exponents as they will give us information about the stoichiometric coefficients in the equations. (a) \(K=\frac{(P_{\mathrm{CO}})^2(P_{\mathrm{H}_{2}})^5}{(P_{\mathrm{C}_{2}\mathrm{H}_{6}})(P_{\mathrm{H}_{2} \mathrm{O}})^2}\) Chemical species: CO, H2, C2H6, H2O (b) \(K=\frac{(P_{\mathrm{NH}})^4(P_{\mathrm{O}_{2}})^5}{(P_{\mathrm{NO}})^4(P_{\mathrm{H}_{2}\mathrm{O})^6}\) Chemical species: NH, O2, NO, H2O (c) \(K=\frac{([\mathrm{ClO}_{3}^{-}])^2([\mathrm{Mn}^{2+}])^2}{(P_{\mathrm{Cl}_{2}})([\mathrm{MnO}_{4}^{-})^2([\mathrm{H}^{+}])^4}\) Chemical species: ClO3-, Mn2+, Cl2, MnO4-, H+

Write the balanced chemical equations

Now, considering the stoichiometric coefficients to be the same as the exponents in the K expressions, we can write the balanced chemical equations for each equilibrium system. (a) \(2\mathrm{CO} + 5\mathrm{H_2} \rightleftharpoons \mathrm{C_2H_6} + 2\mathrm{H_2O}\) (b) \(4\mathrm{NH} + 5\mathrm{O_2} \rightleftharpoons 4\mathrm{NO} + 6\mathrm{H_2O}\) (c) \(2\mathrm{ClO}_{3}^{-} + 2\mathrm{Mn}^{2+} + 4\mathrm{H}^{+} \rightleftharpoons 2\mathrm{MnO}_{4}^{-} + \mathrm{Cl}_{2} + 2\mathrm{H_2O}\) (Here, we added the liquid water as a product as mentioned in the exercise)