Problem 82 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

For a sparingly soluble salt \(\mathrm{A}_{\mathrm{p}} \mathrm{B}_{\mathrm{q}}\), the relationship of its solubility product \(\left(L_{\mathrm{s}}\right)\) with its solubility \((S)\) is (a) \(L_{\mathrm{s}}=S^{p+q} \cdot p^{p} \cdot q^{q}\) (b) \(L_{\mathrm{s}}=S^{p+q} \cdot p^{q} \cdot q^{p}\) (c) \(L_{\mathrm{S}}=S^{p q} \cdot p^{p} \cdot q^{q}\) (d) \(L_{\mathrm{s}}=S^{p q} \cdot(p q)^{p+q}\)

Step-by-Step Solution

Answer

The correct expression for the solubility product in terms of solubility is option (a) \(L_{\mathrm{s}}=S^{p+q} \cdot p^{p} \cdot q^{q}\).

Step 1: Understand the Concept of Solubility Product

The solubility product, denoted as \(L_s\), is an equilibrium constant and thus relates to the extent a salt dissolves in water. For a general salt \(\mathrm{A}_{\mathrm{p}} \mathrm{B}_{\mathrm{q}}\), which dissociates into \(p\) cations of \(A\) and \(q\) anions of \(B\), the solubility product is the product of the concentrations of the ions, each raised to the power of their stoichiometric coefficient in the balanced dissolution equation.

Step 2: Write the Dissolution Equation

The salt \(\mathrm{A}_{\mathrm{p}}\mathrm{B}_{\mathrm{q}}\) dissociates into \(p\) ions of \(A\) and \(q\) ions of \(B\). The chemical equation is \(\mathrm{A}_{\mathrm{p}}\mathrm{B}_{\mathrm{q}} \rightleftharpoons p\mathrm{A}^{+} + q\mathrm{B}^{-}\).

Step 3: Express Solubility Product in Terms of Solubility

If the solubility of the salt is \(S\) moles per liter, then the concentration of \(A^{+}\) ions is \(pS\) and that of \(B^{-}\) ions is \(qS\). The solubility product \(L_s\) is then given by \(\left( pS \right)^p \left( qS \right)^q = S^{p+q} \cdot p^p \cdot q^q\).

Step 4: Match the Correct Option

Comparing the expression from Step 3 with the given options, it is clear that the correct relationship between solubility product and solubility is given by option (a) \(L_{\mathrm{s}}=S^{p+q} \cdot p^{p} \cdot q^{q}\).

Key Concepts

Equilibrium Constant

The concept of the equilibrium constant is central to understanding chemical reactions that can occur in both directions, reaching a state where the rates of the forward and reverse reactions are equal. This constant, symbolized as K for a generic chemical reaction, provides valuable information about the proportion of reactants and products at equilibrium.

For solubility equilibrium, in particular, this constant is referred to as the solubility product, denoted as L_s. It signifies the level at which a compound can dissolve in a solution before it stops dissolving and maintains a constant concentration. Determining L_s involves the use of stoichiometry and the concentrations of the dissolved ions to understand the dissolution process of a sparingly soluble salt.

Dissolution Equation

The dissolution equation represents the process by which a solid dissolves in a solvent to form a solution. For ionic compounds, this involves breaking down into individual ions. The equation provides a reaction formula that helps visualize the stoichiometry of the dissolution process.

An example of this can be seen with a salt A_pB_q, which upon dissolution in water generates p cations of A⁺ and q anions of B^-. Each ion's coefficient in the balanced equation illustrates the amount of ions in the solution. This information is critical when calculating ionic concentrations and the solubility product of the substance.

Ionic Concentration

Ionic concentration refers to the amount of ions of a particular charge present in a solution. When a salt dissolves, it dissociates into its constituent ions, and the concentration of these ions is directly proportional to the salt's solubility, denoted by S.

For instance, if a salt A_pB_q has a solubility of S, the concentration of A⁺ in the solution will be pS and that of B⁻ will be qS. It’s essential to accurately determine these concentrations because they're used to calculate the solubility product, a critical parameter in predicting the extent of a salt's solubility in water.

Stoichiometry

Stoichiometry is the quantitative relationship between reactants and products in a chemical reaction. It's a fundamental principle of chemistry that allows chemists to predict the amounts of substances consumed and produced in a reaction.

In the context of solubility, stoichiometry is used to relate the solubility of a compound (in moles per liter) to the amount of ions it produces in solution. For a compound A_pB_q, stoichiometry helps determine that for every mole of compound that dissolves, p moles of A⁺ and q moles of B⁻ ions are released. This relationship is crucial when we calculate the solubility product, using not just the concentrations of these ions, but also their stoichiometric coefficients as exponents in the formula.