Write the \(K_{e q}\) expressions for the following reactions: $$\begin{array}{c}{\mathbf{a} \cdot 4 \mathrm{H}_{3} \mathrm{O}^{+}(a q)+2 \mathrm{Cl}^{-}(a q)+\mathrm{MnO}_{2}(s) \rightleftarrows} \\\ {\mathrm{Mn}^{2+}(a q)+6 \mathrm{H}_{2} \mathrm{O}(l)+\mathrm{Cl}_{2}(g)} \\\ {\text { b. } \mathrm{As}_{4} \mathrm{O}_{6}(s)+6 \mathrm{H}_{2} \mathrm{O}(l) \rightleftarrows 4 \mathrm{H}_{3} \mathrm{AsO}_{3}(a q)}\end{array}$$

The equilibrium constant, represented as \(K_{eq}\), is a fundamental element in understanding chemical equilibrium. It is a dimensionless number that provides crucial insight into the relative proportions of reactants and products at equilibrium. The expression for \(K_{eq}\) is derived by taking the concentration of products, raised to the power of their stoichiometric coefficients, and dividing by the concentration of reactants, raised similarly.

For example, in reaction a, the \(K_{eq}\) is \[\frac{[Mn^{2+}][Cl_{2}]}{[H_{3}O^{+}]^{4}[Cl^{-}]^{2}}\]. Notably, the \(K_{eq}\) expression excludes pure liquids and solids; this is because their concentrations do not change during the reaction, and they have a constant activity of 1.

Understanding \(K_{eq}\) offers a predictor of the reaction's direction. If \(K_{eq}\) is much greater than 1, the reaction favors the formation of products. Conversely, if \(K_{eq}\) is much less than 1, the reactants are favored. \(K_{eq}\) remains constant at a given temperature, reflecting that equilibrium is a dynamic but stable state.

The reaction quotient \(Q\) is a similar concept to equilibrium constant but serves a different purpose. \(Q\) uses the same form as the \(K_{eq}\) expression, but it is evaluated with the initial concentrations or partial pressures of the reactants and products, not those at equilibrium.

For a general reaction, the expression is \[ Q = \frac{[\text{products}]}{[\text{reactants}]} \], with each concentration raised to their stoichiometric coefficients. Comparing \(Q\) to \(K_{eq}\) helps predict the shift of the reaction; if \(Q < K_{eq}\), the forward reaction is favored to reach equilibrium, and if \(Q > K_{eq}\), the reverse reaction is favored. \(Q\) allows one to assess if a system is at equilibrium and, if not, in which direction it will proceed to reach equilibrium.

An aqueous solution is a homogenous mixture where water acts as the solvent. In the context of chemical equilibrium, species in aqueous solutions are of interest because they are actively involved in the equilibrium process. Their concentrations change with the progress of the reaction and are therefore included in the equilibrium constant expression.

For instance, the ions \(H_{3}O^{+}(aq)\) and \(Cl^{-}(aq)\) in our first reaction will feature in the \(K_{eq}\) expression because their concentration varies during the reaction. They directly influence the position of equilibrium. In solutions, these aqueous ions participate in the dynamic equilibrium, consistently interchanging between reactants and products.

Solids and liquids play unique roles in chemical equilibrium. Since their concentrations are defined by their density and not affected by the amount present, they are not included in the equilibrium constant expression. We say that pure solids and liquids have an activity of 1, which has no effect on the \(K_{eq}\) value.

In the given reactions, for example, \(MnO_{2}(s)\) and \(H_{2}O(l)\) are omitted from the \(K_{eq}\) expression. This principle simplifies the calculation of \(K_{eq}\) and reflects the real behavior of these substances in a reaction mixture. Even though they are part of the reaction, their constant concentrations mean they do not drive the reaction towards or away from equilibrium.