Question

Consider the complex ion \(\mathrm{Hg}(\mathrm{Cl})_{4}^{2-}\). Its \(K_{\mathrm{f}}\) value is \(1.2 \times 10^{15} .\) At what concentration of \(\mathrm{Cl}^{-}\) is \(58 \%\) of the \(\mathrm{Hg}^{2+}\) converted to \(\mathrm{Hg}(\mathrm{Cl})_{4}^{2-}\) ?

Short answer

Answer: The concentration of Cl⁻ ions after the conversion cannot be determined without more information about the initial concentrations of Hg²⁺ and/or the relationship between the initial concentrations of the two ions. However, the expression: y = 4(0.58x) + (√[⁴]{1/(0.42x(1.2x10¹⁵/0.58))}) represents how the required concentration depends on the initial concentration of the ions in the solution.

Step-by-step solution

Write the reaction equation and determine the initial concentrations

The reaction for the formation of the complex ion can be described by the following equation: $$\mathrm{Hg}^{2+} + 4\mathrm{Cl}^{-} \rightarrow \mathrm{Hg}(\mathrm{Cl})_{4}^{2-}$$ Given that \(58\%\) of \(\mathrm{Hg}^{2+}\) is converted, we can say that the initial concentration of \(\mathrm{Hg}^{2+}\) is \(100\%\), and we are left with \(42\%\) after the reaction. Let's represent the initial concentrations of \(\mathrm{Hg}^{2+}\) and \(\mathrm{Cl}^-\) as \(x\) and \(y\), respectively. The initial concentration of \(\mathrm{Hg}(\mathrm{Cl})_{4}^{2-}\) is 0. After the complex ion is formed, the final concentrations can be represented as follows: $$[\mathrm{Hg}^{2+}] = 0.42x$$ $$[\mathrm{Cl}^{-}] = y - 4(0.58x)$$ $$[\mathrm{Hg}(\mathrm{Cl})_{4}^{2-}] = 0.58x$$

Calculate the concentration of Cl− by applying the Kf value

We are given that \(K_{\mathrm{f}} = 1.2 \times 10^{15}\). As per the reaction equation, we can define \(K_{\mathrm{f}}\) as follows: $$K_{\mathrm{f}}=\frac{[\mathrm{Hg}(\mathrm{Cl})_{4}^{2-}]}{[\mathrm{Hg}^{2+}][\mathrm{Cl}^-]^4}$$ Now we can substitute the final concentrations from Step 1: $$1.2 \times 10^{15} =\frac{0.58x}{(0.42x)((y - 4(0.58x))^4)}$$ We need to solve for \(y\) in terms of \(x\). We can simplify the equation by dividing both sides by \(0.58x\): $$\frac{1.2 \times 10^{15}}{0.58} = \frac{1}{(0.42)((y - 4(0.58x))^4)}$$ Notice that only \(y\) is unknown in this equation. We can solve for it: $$y = 4(0.58x) + \sqrt[4]{\frac{1}{0.42 \times \frac{1.2 \times 10^{15}}{0.58}}}$$ However, we cannot obtain a numerical value for \(y\) – we need more information about the initial concentration of \(\mathrm{Hg}^{2+}\) and/or the relationship between the initial concentrations of the two ions. In conclusion, without more information, we cannot determine the concentration of \(\mathrm{Cl}^-\) when \(58\%\) of \(\mathrm{Hg}^{2+}\) is converted to \(\mathrm{Hg}(\mathrm{Cl})_{4}^{2-}\). The expression above, nevertheless, represents how the required concentration depends on the initial concentration of the ions in the solution.

Key concepts

Complex Ion Formation

Complex ions are species formed from a metal ion bonded to one or more ligands, which can be anions or neutral molecules, through coordinate covalent bonds. These complex ions are ubiquitous in nature and technology, including in our bodies, where they play a role in oxygen transportation (e.g., hemoglobin).

The formation of a complex ion involves a reversible chemical reaction where a central metal ion gets surrounded by a certain number of ligands. For example, the mercury complex ion, \(\mathrm{Hg}(\mathrm{Cl})_{4}^{2-}\), forms when mercury ions (\(\mathrm{Hg}^{2+}\)) react with chloride ions (\(\mathrm{Cl}^{-}\)) in solution:
\[\mathrm{Hg}^{2+} + 4\mathrm{Cl}^{-} \rightarrow \mathrm{Hg}(\mathrm{Cl})_{4}^{2-}\]
This reaction is dynamic and reversible, meaning it can proceed in both forward (formation of the complex ion) and reverse (dissociation) directions. In a saturated solution, these two processes occur at the same rate, leading to a state of equilibrium.

Equilibrium Constant (Kf)

The equilibrium constant (\(K_{f}\)) for the formation of a complex ion is a quantitative measure of its stability in a solution. It is defined for the forward reaction, where the complex ion is formed from its constituent ions.

Mathematically, it's expressed as the ratio of the product of the concentrations of the products to the concentrations of the reactants, each raised to the power of its stoichiometric coefficient. For the reaction forming \(\mathrm{Hg}(\mathrm{Cl})_{4}^{2-}\):
\[K_{f} = \frac{[\mathrm{Hg}(\mathrm{Cl})_{4}^{2-}]}{[\mathrm{Hg}^{2+}][\mathrm{Cl}^-]^4}\]
The equilibrium constant provides insight into how much complex ion is formed under specific conditions. A large value of \(K_{f}\) indicates that the formation of the complex ion is highly favorable, and at equilibrium, most of the metal ions will exist as part of the complex ion rather than as free ions. Conversely, a smaller \(K_{f}\) would suggest that the complex ion is less stable, and fewer metal ions would be tied up in the complex at equilibrium.

Chemical Reaction Stoichiometry

Chemical reaction stoichiometry deals with the quantitative relationships between reactants and products in a chemical reaction. It tells us how much of each reactant is needed to form a certain amount of product, assuming the reaction goes to completion.

In the context of complex ion equilibria, stoichiometry plays a critical role in determining the concentrations of all species at equilibrium. For the mercury complex formation, the stoichiometry is 1 mole of \(\mathrm{Hg}^{2+}\) reacting with 4 moles of \(\mathrm{Cl}^{-}\) to form 1 mole of \(\mathrm{Hg}(\mathrm{Cl})_{4}^{2-}\).

Understanding stoichiometry allows us to predict how the concentration of each reactant affects the concentration of the complex ion. When calculating the equilibrium concentration of \(\mathrm{Cl}^{-}\) in our exercise, we need to remember that for each mole of \(\mathrm{Hg}^{2+}\) turned into the complex ion, four moles of \(\mathrm{Cl}^{-}\) are used. This is why the stoichiometry informs our equilibrium expressions and why the exponent in the equilibrium constant expression is tied to the number of chloride ions involved in the reaction. This quantitative aspect is key to solving problems related to the formation and stability of complex ions.