Question

The overall formation constant for \(\mathrm{HgI}_{4}{ }^{2-}\) is \(1.0 \times 10^{30} .\) That is, $$ 1.0 \times 10^{30}=\frac{\left[\mathrm{HgI}_{4}{ }^{2-}\right]}{\left[\mathrm{Hg}^{2+}\right]\left[\mathrm{I}^{-}\right]^{4}} $$ What is the concentration of \(\mathrm{Hg}^{2+}\) in \(500.0 \mathrm{~mL}\) of a solution that was originally \(0.010 \mathrm{M} \mathrm{Hg}^{2+}\) and \(0.78 \mathrm{M} \mathrm{I}^{-}\) ? The reaction is $$ \mathrm{Hg}^{2+}(a q)+4 \mathrm{I}^{-}(a q) \rightleftharpoons \mathrm{HgI}_{4}^{2-}(a q) $$

Short answer

The final concentration of Hg^{2+} in the solution can be approximated as: \[\left[\mathrm{Hg}^{2+}\right] \approx 0.010 - 4.136 \times 10^{-10} M\]

Step-by-step solution

1. Formation constant expression

The overall formation constant for this reaction is given by the following expression: \[K_f=\frac{\left[\mathrm{HgI}_{4}{ }^{2-}\right]}{\left[\mathrm{Hg}^{2+}\right]\left[\mathrm{I}^{-}\right]^{4}}\]

2. Set up an equation using initial concentrations

We are given the initial concentrations: Initial Hg^{2+}: 0.010 M Initial I^{-}: 0.78 M Final volume: 500.0 mL Since we don't have the concentration of the complex ion HgI_4^{2-} initially, we make the following assumptions and changes in concentrations: Change in HgI_4^{2-}: +x Change in Hg^{2+}: -x Change in I^{-}: -4x Now the equation will look as follows: \[1.0 \times 10^{30} = \frac{x}{(0.010-x)(0.78-4x)^4}\]

3. Solve the equation for the final concentration of Hg^{2+}

Solving this equation may not be straightforward, but since K_f is very large, we can assume that the numerator x will be much smaller compared to 0.010 and 0.78 in the denominator. So, we can make an approximation: \[1.0 \times 10^{30} = \frac{x}{(0.010)(0.78)^4}\] Now, we can solve for x: \[x = (1.0 \times 10^{30})(0.010)(0.78)^4\] By calculating the value of x, we get: \[x = 4.136 \times 10^{-10} M\] Therefore, the final concentration of Hg^{2+} in the solution can be approximated as: \[\left[\mathrm{Hg}^{2+}\right] = 0.010 - x \approx 0.010 - 4.136 \times 10^{-10} M\]

Key concepts

Chemical Equilibrium

The concept of chemical equilibrium is fundamental in understanding how reactions proceed and reach a state of balance. At the microscopic level, chemical equilibrium occurs when the rate of the forward reaction equals the rate of the reverse reaction, resulting in no net change in the concentrations of reactants and products over time. This does not mean the reactions have stopped; instead, they continue to occur with reactants converting to products and vice versa at equal rates.

When a complex reaction involving the formation of a complex ion reaches equilibrium, as seen in the provided exercise, the equilibrium expression is derived from the rate constants of the forward and reverse reactions. In the context of complex ion formation, such as \(\mathrm{HgI}_{4}^{2-}\), the equilibrium constant is known as the formation constant, or \(K_f\). The magnitude of \(K_f\) gives insight into the stability of the complex ion where a larger \(K_f\) value implies a higher concentration of the complex ion at equilibrium, favoring the forward reaction.

Complex Ion Concentration

The concentration of complex ions in a solution is critical when determining chemical behavior and reaction extent. As observed with \(\mathrm{HgI}_{4}^{2-}\), the compound is a complex ion formed from the constituent ions \(\mathrm{Hg}^{2+}\) and \(\mathrm{I}^{-}\). The formation constant \(K_f\) helps calculate the equilibrium concentration of the complex ion in relation to the concentrations of these individual ions.

In the exercise, we used \(K_f\) and initial ionic concentrations to find out how much complex ion was formed and thus determine the remaining concentration of \(\mathrm{Hg}^{2+}\). This calculation is central to various applications, including biochemistry, medical diagnostics, and industrial processes where metal ion availability or sequestration is a factor. The concept of complex ion concentration also underscores the nature of ionic interactions and how they can be manipulated under varying conditions.

Chemical Reaction

Chemical reactions involve the transformation of substances through the breaking and forming of chemical bonds. The reaction between \(\mathrm{Hg}^{2+}\) and \(\mathrm{I}^{-}\) to form \(\mathrm{HgI}_{4}^{2-}\) illustrates a complexation reaction where a central metal ion (\(\mathrm{Hg}^{2+}\)) binds with several anions (\(\mathrm{I}^{-}\)).

In this case, the reaction is reversible, indicating that the products can revert to reactants. Such reversibility is common in many chemical systems and is described by an equilibrium constant that quantifies the extent of the reaction at equilibrium. Understanding the nuances of chemical reactions helps in predicting the outcome of reactions and in the synthesis of desired products, whether in industrial processes, pharmaceutical development, or environmental systems.