Calculate the molar solubility of \(\mathrm{Mg}(\mathrm{OH})_{2}\) \(\left(K_{\mathrm{sp}}=1.8 \times 10^{-11}\right)\) in (a) pure water; (b) \(0.0862 \mathrm{M}\) \(\mathrm{MgCl}_{2} ;\) (c) \(0.0355 \mathrm{M} \mathrm{KOH}(\mathrm{aq})\).

The solubility product constant, often represented as K_sp, is a numerical expression that correlates with a compound's solubility in water. It is specific to sparingly soluble salts and is used in predicting the extent to which a salt can dissolve. For instance, in the exercise with magnesium hydroxide, Mg(OH)₂, the dissolution process is considered at equilibrium when the solid phase is in dynamic balance with its ions in solution.

This balance can be expressed as a chemical equation like this: Mg(OH)₂ (s) ↔ Mg²⁺(aq) + 2 OH⁻(aq). The K_sp is calculated by multiplying the molar concentrations of the resulting ions, each raised to the power of their coefficients in the balanced equation. It's important to note that solids are not included in the K_sp expression as their concentration is constant under given conditions. Understanding the solubility product constant is crucial for predicting solubility in different scenarios and is pivotal in the practice of chemistry.

Chemical equilibrium occurs when the rates of the forward and reverse reactions are equal, leading to no net change in the concentrations of the reactants and products over time. It is a dynamic state because reactions continue to occur, but these processes are balanced out. In the context of solubility, equilibrium describes the point at which the rate of dissolution of a solid equals the rate at which it precipitates.

In the exercise provided, magnesium hydroxide reaches equilibrium in water when the rate at which it dissolves to form Mg²⁺ and OH⁻ ions equals the rate at which these ions come together to form the solid compound. The concept of equilibrium is vital in understanding how disturbances, such as adding more solute or changing the temperature, can shift the position of the equilibrium, a foundation of Le Chatelier’s principle.

Electrolytes are substances that dissociate into ions when dissolved in water, and their dissociation is a key factor affecting solubility. The process is influenced by the nature of the electrolyte, whether it is a strong or weak one. Strong electrolytes, such as MgCl₂ and KOH as shown in the exercise, completely dissociate into their ions in aqueous solution. This means that each molecule of MgCl₂ yields one Mg²⁺ ion and two Cl⁻ ions.

Understanding the dissociation of electrolytes is crucial when calculating the solubility of a salt in a solution that already contains common ions. For example, if Mg²⁺ is already present from the dissociation of MgCl₂, the solubility of Mg(OH)₂ will be affected, a phenomenon known as the common ion effect.

Strong electrolytes are substances that completely dissociate into ions when dissolved in water, resulting in a solution that conducts electricity well. This contrasts with weak electrolytes, which only partially dissociate. In the context of the exercise, both MgCl₂ and KOH are examples of strong electrolytes. Their complete dissociation influences the molar solubility of Mg(OH)₂ in their respective solutions, due to an increase in the ion concentration which shifts the equilibrium position.

The concept of strong electrolytes in solution helps to explain a variety of phenomena, including conductivity, the behavior of acids and bases, and physiological processes. It also elucidates the principles of solubility and is essential in predicting the outcome when mixing different ionic compounds in solution.