Answered

Which represents the correct equilibrium constant expression for the reaction below?



$Cu ( s )+2 Ag ^{+}( aq ) \longleftrightarrow Cu ^{2+}( aq )+2 Ag ( s )$


A$K _{e q}=\frac{\left[ Ag ^{+}\right]^2}{\left[ Cu ^{2+}\right]}$
B$K _{\text {eq }}=\frac{\left[ Cu ^{2+}\right]}{\left[ Ag ^{+}\right]^2}$
C$K _{e q}=\frac{\left[ Cu ^{2+}\right][ Ag ]^2}{[ Cu ]\left[ Ag ^{+}\right]^2}$
D$K _{e q}=\frac{[ Cu ]\left[ Ag ^{+}\right]^2}{\left[ Cu ^{2+}\right]\left[ Ag ^2\right]^2}$

Verified Answer :

  • Identify the reactants and products in the reaction: \(Cu(s)\) and \(Ag^+(aq)\) are reactants, \(Cu^{2+}(aq)\) and \(Ag(s)\) are products.
  • Note that pure solids like \(Cu(s)\) and \(Ag(s)\) are excluded from the equilibrium constant expression because their activity is 1.
  • Apply the law of mass action to the aqueous species: the product concentration \([Cu^{2+}]\) goes in the numerator, and the reactant concentration \([Ag^+]^2\) goes in the denominator.
  • Conclude that the correct expression is \(\boxed{b. K_{eq} = \frac{[Cu^{2+}]}{[Ag^{+}]^2}}\).

Explanation

  1. Analyzing the Problem and Rules of Equilibrium Expressions

The problem asks for the correct equilibrium constant expression for the reaction:
\[Cu(s) + 2Ag^{+}(aq) \longleftrightarrow Cu^{2+}(aq) + 2Ag(s)\]
The equilibrium constant \(K_{eq}\) for a reversible reaction is defined as the ratio of the products of the concentrations of the products to the reactants, where each concentration is raised to the power of its stoichiometric coefficient in the balanced equation. However, a crucial rule in thermodynamics is that the activities of pure solids and pure liquids are considered to be 1. Therefore, their concentrations do not appear in the equilibrium expression.

  1. Constructing the Equilibrium Expression

For the given reaction:

  • The products are \(Cu^{2+}(aq)\) and \(Ag(s)\).
  • The reactants are \(Cu(s)\) and \(Ag^{+}(aq)\).
  • The stoichiometric coefficients are 1 for \(Cu(s)\), 2 for \(Ag^{+}(aq)\), 1 for \(Cu^{2+}(aq)\), and 2 for \(Ag(s)\).

The standard form of the equilibrium expression, without yet accounting for the states, is:
\[K_{eq} = \frac{[Cu^{2+}]^{1} \cdot [Ag]^{2}}{[Cu]^{1} \cdot [Ag^{+}]^{2}}\]
Applying the rule that pure solids have an activity of 1, we replace \([Cu(s)]\) and \([Ag(s)]\) with 1:
\[K_{eq} = \frac{[Cu^{2+}] \cdot 1^2}{1 \cdot [Ag^{+}]^2}\]
Simplifying this expression, we get:
\[K_{eq} = \frac{[Cu^{2+}]}{[Ag^{+}]^2}\]

  1. Selecting the Correct Option

Comparing our derived expression \(K_{eq} = \frac{[Cu^{2+}]}{[Ag^{+}]^2}\) with the given options:

A\(K_{eq} = \frac{[Ag^{+}]^2}{[Cu^{2+}]}\)
B\(K_{eq} = \frac{[Cu^{2+}]}{[Ag^{+}]^2}\)
C\(K_{eq} = \frac{[Cu^{2+}][Ag]^2}{[Cu][Ag^{+}]^2}\)
D\(K_{eq} = \frac{[Cu][Ag^{+}]^2}{[Cu^{2+}][Ag]^2}\)

Option b matches our derived expression exactly. Thus, option b is the correct expression.

Examples

In industrial chemistry, understanding the equilibrium constant is vital for maximizing yield. For example, in the extraction of silver from copper-containing solutions, engineers use this specific reaction. By calculating the equilibrium constant \(K_{eq}\), they can predict whether the reaction will proceed to a sufficient degree to recover the maximum amount of silver effectively. If \(K_{eq}\) is large, the reaction favors the products, ensuring an efficient recovery process.

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