Question

Mercury(I) chloride, \(\mathrm{Hg}_{2} \mathrm{Cl}_{2}\), was formerly administered orally as a purgative. Although we usually think of mercury compounds as highly toxic, the \(K_{\mathrm{sp}}\) of mercury(I) chloride is small enough \((1.3 \times\) \(10^{-18}\) ) that the amount of mercury that dissolves and enters the bloodstream is tiny. Calculate the concentration of mercury(I) ion present in a saturated solution of \(\mathrm{Hg}_{2} \mathrm{Cl}_{2}\).

Short answer

The concentration of mercury(I) ion present in a saturated solution of \(\mathrm{Hg}_{2} \mathrm{Cl}_{2}\) is approximately \(0.00091 \ \mathrm{M}\).

Step-by-step solution

Write the balanced equation for the dissolution of mercury(I) chloride.

The dissolution of mercury(I) chloride can be represented as: \[ \mathrm{Hg}_{2} \mathrm{Cl}_{2} \longleftrightarrow 2 \mathrm{Hg}^+ + 2 \mathrm{Cl}^- \]

Step 2:Write the expression for the solubility product constant (\(K_{sp}\)).

The \(K_{sp}\) expression for the dissolution of mercury(I) chloride is given by: \[ K_{sp} = [\mathrm{Hg}^+]^2 [\mathrm{Cl}^-]^2 \]

Relate the concentrations of \(\mathrm{Hg}^+\) and \(\mathrm{Cl}^-\) ions.

In the balanced equation, the stoichiometric coefficients of \(\mathrm{Hg}^+\) and \(\mathrm{Cl}^-\) ions are both 2. Therefore, the concentration of \(\mathrm{Cl}^-\) ions is equal to that of \(\mathrm{Hg}^+\) ions. We can represent this relationship as: \[ [\mathrm{Cl}^-] = [\mathrm{Hg}^+] \]

Substitute the relationship from Step 3 into the \(K_{sp}\) expression and solve for the concentration of \(\mathrm{Hg}^+\) ion.

Substitute the relationship between \(\mathrm{Hg}^+\) and \(\mathrm{Cl}^-\) concentrations into the \(K_{sp}\) expression: \[ K_{sp} = ([\mathrm{Hg}^+])^2 ([\mathrm{Hg}^+])^2 \] With \(K_{sp}=1.3 \times 10^{-18}\), we have: \[ 1.3 \times 10^{-18} = ([\mathrm{Hg}^+])^4 \] To find \([\mathrm{Hg}^+]\), we take the fourth root of both sides of the equation: \[ [\mathrm{Hg}^+] = (1.3 \times 10^{-18})^{1/4} \]

Calculate the concentration of Mercury(I) ion.

Compute the fourth root of the solubility product constant: \[ [\mathrm{Hg}^+] \approx 0.00091 \ \mathrm{M} \] Therefore, the concentration of mercury(I) ion present in a saturated solution of \(\mathrm{Hg}_{2} \mathrm{Cl}_{2}\) is approximately \(0.00091 \ \mathrm{M}\).

Key concepts

Chemical Equilibrium

Understanding chemical equilibrium is critical when dealing with reactions that can occur in both forward and reverse directions, which in essence is the nature of solubility. In such a dynamic state, the rate of the forward reaction, where a compound dissolves, equals the rate of the reverse reaction, where the dissolved species combine to form the undissolved compound. For the mercury(I) chloride, denoted as Hg₂Cl₂, this equilibrium process can be visualized.

When Hg₂Cl₂ dissolves, it dissociates into mercury(I) ions (Hg⁺) and chloride ions (Cl^-). At equilibrium, the rate at which Hg₂Cl₂ forms again from Hg⁺ and Cl^- is equal to the rate at which it dissolves back into these ions. Because of this delicate balance, the concentration of mercury(I) ion in a saturated solution is limited and can be predicted by the solubility product constant (K_sp).

Solubility Calculations

Calculating the solubility of compounds involves using the solubility product constant, K_sp, which is specific to each compound at a given temperature. Solubility calculations often follow a systematic approach.

Initially, one writes the balanced dissolution equation, then formulates the K_sp expression based on the stoichiometry of the dissolution process. For mercury(I) chloride, this involves squaring the concentrations of both mercury(I) and chloride ions, as both ions are produced in equal amounts. By understanding these stoichiometric relationships and applying the K_sp value, we can solve for the unknown concentration of ions in a saturated solution, providing valuable insight into the solubility of the compound.

Mercury(I) Chloride Solubility

Mercury(I) chloride's solubility is exceptionally low, which is indicated by its very small K_sp value of 1.3 x 10^-18. Because of this low solubility, mercury(I) chloride was historically used as a purgative, under the assumption that its limited dissolution in the body's aqueous environment would result in minimal absorption into the bloodstream.

From the exercise, we have learned that the solubility of mercury(I) chloride can be calculated by deriving the concentrations of its ions in a saturated solution. By manipulating the K_sp expression and considering the stoichiometric 1:1 ratio of mercury(I) to chloride ions, one can determine the solubility of Hg⁺ in molar concentration. The low molar solubility, calculated at approximately 0.00091 M, underscores the compound's limited ability to dissolve, thereby reflecting its low intrinsic toxicity when used in the past as a medication.