Silver ion reacts with excess CN- to form a colorless complex ion, [Ag(CN)2]-, which has a formation constant Kf = 3.0 x 10^20. Calculate the concentration of Ag+ in a solution prepared by mixing equal volumes of 2.0 x 10^-3 M AgNO3 and 0.20 M NaCN.

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Step 1: Write the balanced chemical equation for the formation of the complex ion: \[ \text{Ag}^+ + 2\text{CN}^- \rightleftharpoons [\text{Ag(CN)}_2]^- \]

Step 2: Use the initial concentrations of AgNO3 and NaCN to determine the initial concentrations of \( \text{Ag}^+ \) and \( \text{CN}^- \) in the mixed solution. Since equal volumes are mixed, the concentrations are halved: \( [\text{Ag}^+]_0 = 1.0 \times 10^{-3} \text{ M} \) and \( [\text{CN}^-]_0 = 0.10 \text{ M} \).

Step 3: Set up the expression for the formation constant \( K_f \) for the complex ion: \[ K_f = \frac{[\text{Ag(CN)}_2]^-}{[\text{Ag}^+][\text{CN}^-]^2} \]

Step 4: Assume that \( x \) is the change in concentration of \( \text{Ag}^+ \) that forms the complex ion. Then, express the equilibrium concentrations in terms of \( x \): \( [\text{Ag}^+] = 1.0 \times 10^{-3} - x \), \( [\text{CN}^-] = 0.10 - 2x \), and \( [\text{Ag(CN)}_2]^- = x \).

Step 5: Substitute the equilibrium concentrations into the \( K_f \) expression and solve for \( x \), which represents the concentration of \( [\text{Ag(CN)}_2]^- \). Use this to find the concentration of \( \text{Ag}^+ \) at equilibrium.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Complex Ion Formation

Complex ion formation occurs when a metal ion, such as Ag+, binds with ligands, like CN-, to create a more stable species. In this case, the silver ion forms the complex ion [Ag(CN)2]-, which is colorless. The stability of this complex is quantified by the formation constant (Kf), indicating the extent to which the complex forms in solution.

Equilibrium and Kf

The formation constant (Kf) is a measure of the equilibrium position for the formation of a complex ion from its constituent ions. A high Kf value, such as 3.0 x 10^20 for [Ag(CN)2]-, suggests that the complex is favored in solution, meaning that the concentration of free Ag+ ions will be very low when the complex is formed. Understanding Kf is crucial for calculating the concentration of uncomplexed ions.

Dilution and Concentration Calculations

When mixing solutions, the concentration of each component changes due to dilution. In this problem, equal volumes of 2.0 x 10^-3 M AgNO3 and 0.20 M NaCN are mixed, which requires calculating the new concentrations after dilution. This involves using the dilution formula (C1V1 = C2V2) to find the initial concentrations before considering the formation of the complex.

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