Problem 81 - Chemistry: An Atoms-Focused Approach

Question

The complexation of mercury(II) ion with methionine $$\mathrm{Hg}^{2+}+\text { methionine } \rightleftharpoons \mathrm{Hg}(\text { methionine })^{2+}$$ has a formation constant of \(\log K_{\mathrm{f}}=14.2,\) whereas the formation constant for the \(\mathrm{Hg}^{2+}\) complex with penicillamine $$\mathrm{Hg}^{2+}+\text { penicillamine } \rightleftharpoons \mathrm{Hg} \text { (penicillamine) }^{2+}$$ is \(\log K_{\mathrm{f}}=16.3 .\) Calculate the equilibrium constant for the reaction Hg(methionine) \(^{2+}+\) penicillamine \(\rightleftharpoons\) Hg(penicillamine) \(^{2+}+\) methionine

Step-by-Step Solution

Answer

Answer: The equilibrium constant for the reaction is K = 100.

Step 1: Identify the reactants and products

In the given reaction, Hg(methionine)\(^{2+}\) and penicillamine are the reactants, and Hg(penicillamine)\(^{2+}\) and methionine are the products.

Step 2: Write the equilibrium expression using the formation constants

Using the general rule, the equilibrium constant (\(K\)) can be expressed as the ratio of the formation constants (\(K_f\)) of the products to the reactants. In this case, we have: $$K = \frac{K_f(\mathrm{Hg(penicillamine)^{2+}})}{K_f(\mathrm{Hg(methionine)^{2+}})}$$

Step 3: Substitute the given formation constants into the expression for \(K\)

We are given \(\log K_f(\mathrm{Hg(methionine)^{2+}}) = 14.2\) and \(\log K_f(\mathrm{Hg(penicillamine)^{2+}}) = 16.3\). To find the \(K_f\) values from the given \(\log K_f\) values, we need to use the equation: $$K_f = 10^{\log K_f}$$ Now, we can substitute these \(K_f\) values into the equation for \(K\): $$K = \frac{10^{16.3}}{10^{14.2}}$$

Step 4: Calculate the equilibrium constant \(K\)

By dividing the numbers, we get: $$K = 10^{(16.3 - 14.2)}$$ which simplifies to: $$K = 10^2$$ Thus, the equilibrium constant for the given reaction is \(K = 100\).

Key Concepts

Formation Constant

The formation constant, often denoted as \( K_f \), is vital for understanding the stability of complex ions in chemical reactions. It represents the equilibrium constant for the formation of a complex from its constituent ions. When a metal ion like \( \text{Hg}^{2+} \) bonds with a ligand such as methionine or penicillamine, they form a complex ion.Key points to remember about formation constants:

• Formation constants are usually expressed in their logarithmic form, \( \log K_f \), due to the wide range of values they can take.
• A higher \( K_f \) value indicates a more stable and tightly bound complex.
• The formation constant is specific to a given metal-ligand complex under certain conditions (e.g., temperature, pH).

This concept helps chemists predict the behavior of metal ions in solutions, particularly in biological or environmental systems where complexation plays a significant role.

Complexation

Complexation refers to the process by which a central atom or ion, usually a metal, forms a complex compound with molecules or ions, known as ligands. Understanding complexation is crucial for grasping how species like \( \text{Hg}^{2+} \) interact with organic molecules such as methionine and penicillamine.During complexation:

• The central metal ion forms coordinate covalent bonds with ligands, which donate electron pairs.
• This results in a new entity, the complex ion, which has different properties than its individual components.

Complexation plays a significant role in different chemical processes, such as:
- The transport and bioavailability of metal ions in biological systems.- The detoxification processes for metals in organisms.- The development of new materials, where metal-ligand binding properties are exploited.

Equilibrium Constant Calculation

Calculating the equilibrium constant, \( K \), for a chemical reaction involves understanding the concentrations of reactants and products at equilibrium. For reactions involving complex ions, such as the given equations with mercury and methionine or penicillamine, the equilibrium constant is linked to the formation constants of the complexes.Here's a simplified method to calculate \( K \):
The equilibrium constant is determined by the ratio of the formation constants of the products to the reactants, based on the principle:\[K = \frac{K_f(\text{Product})}{K_f(\text{Reactant})}\]In the given exercise, we're dealing with formation constants expressed in logarithmic form:
- \( \log K_f(\text{Hg(methionine)}^{2+}) = 14.2 \)- \( \log K_f(\text{Hg(penicillamine)}^{2+}) = 16.3 \)To find \( K \), convert \( \log K_f \) values to \( K_f \) using:\[K_f = 10^{\log K_f}\]Substituting back, we get:\[K = \frac{10^{16.3}}{10^{14.2}}\]Simplify to find:
\(K = 10^{2}\)
Thus, the equilibrium constant of the reaction is 100, indicating the relative stability of the product complex compared to the reactant complex.