On the basis of Le Châtelier's principle, explain how the addition of solid \(\mathrm{NH}_{4} \mathrm{Cl}\) to a beaker containing solid \(\mathrm{Mg}(\mathrm{OH})_{2}\) in contact with water is able to cause the \(\mathrm{Mg}(\mathrm{OH})_{2}\) to dissolve. Write equations for all of the chemical equilibria that exist in the solution after the addition of the \(\mathrm{NH}_{4} \mathrm{Cl}\).

Chemical equilibrium is a fundamental concept in chemistry where the rate of forward reaction equals the rate of the reverse reaction, and the concentrations of reactants and products remain constant over time. It's akin to a see-saw perfectly balanced with equal weights on both sides.

In a dynamic equilibrium, reactions still occur, but there's no net change in the composition of the system. This is because the quantities of reactants converted into products and products reconverted into reactants are the same. An important aspect to grasp is that equilibria can be disturbed by modifying conditions such as concentration, pressure, and temperature. Le Châtelier's Principle explains how the system responds to these changes: it adjusts to minimize the disturbance and re-establish equilibrium.

In our textbook exercise, when solid \(\mathrm{NH}_{4} \mathrm{Cl}\) is added to \(\mathrm{Mg}(\mathrm{OH})_{2}\) in water, it introduces new ions into the system, thus disturbing the equilibrium. The reaction would respond to this disturbance by trying to consume these added ions, shifting the equilibrium to counteract the change.

Solubility equilibrium is a specific type of chemical equilibrium that occurs when a solute dissolves in a solvent. This results in a saturated solution if the maximum amount of solute has been dissolved. At this point, any additional solute will not dissolve and remains in its solid form. The dissolution and precipitation processes are in balance, no net change occurs, and the solute concentration remains stable.

The equilibrium for \(\mathrm{Mg}(\mathrm{OH})_{2}\) in contact with water is represented by the following equilibrium equation: \[\mathrm{Mg}(\mathrm{OH})_{2} (s) \rightleftharpoons \mathrm{Mg}^{2+} (aq) + 2\mathrm{OH}^- (aq)\]. When the equilibrium is shifted due to changes, such as the introduction of a common ion, the solubility of the solute will be affected as the system tries to reestablish equilibrium.

The common ion effect is an application of Le Châtelier’s Principle in solutions. It refers to the decrease in solubility of an ionic compound when a salt with a common ion is added to the solution. For instance, in the exercise context, the addition of \(\mathrm{NH}_{4} \mathrm{Cl}\) to the solution with \(\mathrm{Mg}(\mathrm{OH})_{2}\) introduces chloride ions, which don't react with anything present, and ammonium ions, which react with hydroxide ions, leading to a decrease in hydroxide ion concentration.

This reaction is \[\mathrm{NH}_{4}^+ (aq) + \mathrm{OH}^- (aq) \rightleftharpoons \mathrm{NH}_3 (aq) + \mathrm{H}_2\mathrm{O} (l)\]. As the ammonium ions react with hydroxide ions, removing the hydroxide from the solution, the solubility equilibrium of \(\mathrm{Mg}(\mathrm{OH})_{2}\) is affected. There are now fewer hydroxide ions available to remain in solution, so more \(\mathrm{Mg}(\mathrm{OH})_{2}\) must dissolve to satisfy the equilibrium condition, resulting in more magnesium hydroxide dissolving than it would have without the addition of \(\mathrm{NH}_{4} \mathrm{Cl}\). This effectively increases the solubility of magnesium hydroxide in the presence of a common ion—a beautifully orchestrated dance of ions conforming to the rhythm of Le Châtelier's Principle.