Equal volumes of \(0.12 M \mathrm{AgNO}_{3}\) and \(0.14 \mathrm{M} \mathrm{ZnCl}_{2}\) solution are mixed. Calculate the equilibrium concentrations of \(\mathrm{Ag}^{+}, \mathrm{Cl}^{-}, \mathrm{Zn}^{2+},\) and \(\mathrm{NO}_{3}^{-}\)

To begin, one must determine initial concentrations of the ions when the solutions of \(\mathrm{AgNO}_{3}\) and \(\mathrm{ZnCl}_{2}\) are mixed. Since volumes are equal, the initial concentration of Ag\(^+\) is \(0.12M\) and that of Cl− is \(0.14M\). For both \(\mathrm{NO}_{3}^{-}\) and \(\mathrm{Zn}^{2+}\) the concentrations remain same as \(\mathrm{AgNO}_{3}\) and \(\mathrm{ZnCl}_{2}\) respectively throughout the process, because they do not participate in the precipitation reaction.

Next, one should consider the reaction between Ag+ and Cl− ions, which produces AgCl precipitate. This reaction proceeds until one of the reagents is exhausted. In this case, Ag+ is the limiting reactant as it is in less concentration. For each mole of Ag+ that reacts, one mole of Cl− also reacts, and one mole of AgCl is produced. So, the decrease in the concentration of Ag+ and Cl− will be equal to the concentration of Ag+ ions initially present, which is \(0.12M\). Thus, the concentration of Cl− after reaction would be \(0.14M - 0.12M = 0.02M\). Precipitation halts when Ag+ is exhausted, so its final concentration is 0.

Knowing that AgCl is a slightly soluble solid, a small amount of it will dissolve in water, producing a certain concentration of Ag+and Cl− ions in equilibrium with the precipitate. This concentration can be found using the solubility product constant for AgCl, which is written as \([Ag+][Cl-] = K_{sp}\). The equilibrium concentrations of \(Ag^{+}\) and \(Cl^{-}\) will therefore be equal and can be determined by finding the square root of the \(K_{sp}\). However, since the \(K_{sp}\) of AgCl, \(1.77 × 10^{−10}\), is very small, the concentration of Ag+ ions contributed by the dissolution of AgCl is extremely low compared to the Cl− concentration remaining after reaction, which is \(0.02M\). Hence, we can assume the equilibrium concentration of Ag+ equals \(\sqrt{K_{sp}}\) and that of Cl− remains \(0.02M\).

Finally, The equilibrium concentrations of \(\mathrm{NO}_{3}^{-}\) and \(\mathrm{Zn}^{2+}\) remain equal to their initial concentrations as defined in step 1.