Correct Equilibrium Constant Expression For Cu(s) + 2Ag+(aq) ⇌ Cu2+(aq) + 2Ag(s)

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

A. $K_{eq} = \frac{[Cu][Ag^+]^2}{[Cu^{2+}][Ag]^2}$ B. $K_{eq} = \frac{[Cu^{2+}][Ag]^2}{[Cu][Ag^+]^2}$ C. $K_{eq} = \frac{[Cu^{2+}]}{[Ag^+]^2}$ D. $K_{eq} = \frac{[Cu^{2+}][Ag]^2}{[Ag^+]^2}$ E. $K_{eq} = \frac{[Cu][Ag^+]^2}{[Cu^{2+}]}$

Introduction to Equilibrium Constant Expressions

In the realm of chemical kinetics, the equilibrium constant ($K_{eq}$) stands as a cornerstone for understanding the extent to which a reversible reaction proceeds. This constant provides a quantitative measure of the ratio between products and reactants at equilibrium, offering insights into the favored direction of the reaction. Accurately determining the equilibrium constant expression is crucial for predicting reaction outcomes and manipulating reaction conditions to achieve desired results. In this article, we will delve into the concept of equilibrium constant expressions, focusing on the specific reaction between copper and silver ions, and identify the correct representation of $K_{eq}$.

Defining the Equilibrium Constant ($K_{eq}$)

The equilibrium constant, denoted as $K_{eq}$, is a numerical value that describes the ratio of products to reactants at equilibrium for a reversible reaction. Equilibrium is the state where the rates of the forward and reverse reactions are equal, and the net change in concentrations of reactants and products is zero. The magnitude of $K_{eq}$ indicates the extent to which a reaction will proceed to completion. A large $K_{eq}$ suggests that the reaction favors product formation, while a small $K_{eq}$ indicates that the reaction favors reactant formation.

Constructing the Equilibrium Constant Expression

To construct the equilibrium constant expression, we follow a simple yet crucial rule: the concentrations of products are placed in the numerator, and the concentrations of reactants are placed in the denominator. Each concentration is raised to the power of its stoichiometric coefficient in the balanced chemical equation. For a general reversible reaction:

$aA + bB \rightleftharpoons cC + dD$

The equilibrium constant expression is given by:

$K_{eq} = \frac{[C]^c[D]^d}{[A]^a[B]^b}$

Where:

• [A], [B], [C], and [D] represent the equilibrium concentrations of reactants A and B, and products C and D, respectively.
• a, b, c, and d are the stoichiometric coefficients from the balanced chemical equation.

It is essential to remember that the concentrations of pure solids and pure liquids are not included in the equilibrium constant expression because their activities are considered to be unity. This simplification significantly impacts the $K_{eq}$ expression for heterogeneous reactions involving solids or liquids.

Analyzing the Given Reaction: Copper and Silver Ions

Let's consider the specific reaction provided:

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

This is a redox reaction where solid copper ($Cu$) reacts with silver ions ($Ag^+$) in aqueous solution to form copper(II) ions ($Cu^{2+}$) and solid silver ($Ag$). Understanding the phases of the reactants and products is crucial for constructing the correct equilibrium constant expression. Copper and silver exist in the solid phase, while silver ions and copper(II) ions are in the aqueous phase.

Applying the Rules for $K_{eq}$ Expression

Following the rules for constructing the $K_{eq}$ expression, we place the concentrations of the products in the numerator and the concentrations of the reactants in the denominator. However, since copper and silver are solids, their concentrations are not included in the expression. Therefore, the equilibrium constant expression should only include the concentrations of the aqueous species, $Ag^+$ and $Cu^{2+}$.

Evaluating the Answer Choices

Now, let's evaluate the given answer choices based on the correct methodology for constructing the equilibrium constant expression.

A. $K_{eq} = \frac{[Cu][Ag^+]^2}{[Cu^{2+}][Ag]^2}$

This option incorrectly includes the concentrations of solid copper and solid silver in the expression. It does not account for the fact that the activities of pure solids are considered unity and should not appear in the $K_{eq}$ expression. Thus, this option is incorrect.

B. $K_{eq} = \frac{[Cu^{2+}][Ag]^2}{[Cu][Ag^+]^2}$

Similar to option A, this choice also includes the concentrations of solid copper and solid silver, making it an incorrect representation of the equilibrium constant expression. The presence of $[Cu]$ and $[Ag]$ in the expression indicates a misunderstanding of how to treat solids in equilibrium expressions.

C. $K_{eq} = \frac{[Cu^{2+}]}{[Ag^+]^2}$

This option correctly excludes the concentrations of the solid species, copper and silver. It places the concentration of the product, $Cu^{2+}$, in the numerator and the concentration of the reactant, $Ag^+$, in the denominator. The concentration of $Ag^+$ is correctly raised to the power of its stoichiometric coefficient, which is 2. This option aligns with the rules for constructing the equilibrium constant expression and is therefore the correct choice.

D. $K_{eq} = \frac{[Cu^{2+}][Ag]^2}{[Ag^+]^2}$

This option incorrectly includes the concentration of solid silver in the numerator. While it correctly excludes solid copper, the presence of $[Ag]^2$ makes this option incorrect. The concentration of solid silver should not be part of the equilibrium constant expression.

E. $K_{eq} = \frac{[Cu][Ag^+]^2}{[Cu^{2+}]}$

This option includes the concentration of solid copper in the numerator, which is a fundamental error in constructing the equilibrium constant expression. It also omits the solid silver from the denominator, but the presence of solid copper renders this option incorrect.

Correct Answer and Explanation

The correct answer is:

C. $K_{eq} = \frac{[Cu^{2+}]}{[Ag^+]^2}$

Detailed Explanation

The equilibrium constant expression accurately reflects the ratio of products to reactants at equilibrium, considering only the aqueous species. The expression is derived from the balanced chemical equation:

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

The correct equilibrium constant expression excludes the solid species ($Cu$ and $Ag$) and includes only the aqueous species ($Ag^+$ and $Cu^{2+}$). The concentration of $Cu^{2+}$ is placed in the numerator, and the concentration of $Ag^+$ is placed in the denominator, raised to the power of its stoichiometric coefficient (2). This results in the expression:

$K_{eq} = \frac{[Cu^{2+}]}{[Ag^+]^2}$

This expression correctly represents the equilibrium constant for the given redox reaction, providing a quantitative measure of the extent to which the reaction proceeds towards product formation at equilibrium.

Significance of the Equilibrium Constant

The equilibrium constant is a powerful tool in chemistry, offering insights into the behavior of reversible reactions. A large $K_{eq}$ value indicates that the reaction favors the formation of products, meaning that at equilibrium, the concentration of products will be significantly higher than the concentration of reactants. Conversely, a small $K_{eq}$ value suggests that the reaction favors the reactants, and at equilibrium, the concentration of reactants will be higher than that of products.

Applications of $K_{eq}$

Understanding the equilibrium constant is crucial for various applications in chemistry, including:

1. Predicting Reaction Direction: By comparing the reaction quotient ($Q$) with $K_{eq}$, we can predict the direction in which a reaction will shift to reach equilibrium. If $Q < K_{eq}$, the reaction will proceed forward to form more products. If $Q > K_{eq}$, the reaction will proceed in reverse to form more reactants. If $Q = K_{eq}$, the system is already at equilibrium.
2. Calculating Equilibrium Concentrations: Knowing the value of $K_{eq}$ and the initial concentrations of reactants, we can calculate the equilibrium concentrations of all species involved in the reaction. This is particularly useful in industrial processes where optimizing product yield is essential.
3. Manipulating Reaction Conditions: The equilibrium constant is temperature-dependent, and understanding this relationship allows us to manipulate reaction conditions (such as temperature and pressure) to favor the formation of desired products. This is guided by Le Chatelier's principle, which describes how a system at equilibrium responds to changes in conditions.
4. Assessing Reaction Feasibility: The magnitude of $K_{eq}$ can provide insights into the feasibility of a reaction. A very large $K_{eq}$ suggests that the reaction is highly favorable and will proceed to near completion, while a very small $K_{eq}$ indicates that the reaction is unlikely to proceed to a significant extent.

Factors Affecting Equilibrium

Several factors can influence the equilibrium position of a reaction, and understanding these factors is crucial for manipulating reactions to achieve desired outcomes. Key factors include:

1. Temperature: The equilibrium constant is temperature-dependent. For endothermic reactions (those that absorb heat), increasing the temperature favors product formation, while for exothermic reactions (those that release heat), increasing the temperature favors reactant formation. This relationship is quantified by the van't Hoff equation.
2. Pressure: For reactions involving gases, changes in pressure can affect the equilibrium position. Increasing the pressure favors the side of the reaction with fewer moles of gas, while decreasing the pressure favors the side with more moles of gas.
3. Concentration: Changing the concentration of reactants or products will shift the equilibrium position to counteract the change. Adding reactants will shift the equilibrium towards product formation, while adding products will shift the equilibrium towards reactant formation.
4. Catalysts: Catalysts increase the rate of both the forward and reverse reactions equally, thereby speeding up the attainment of equilibrium but not affecting the equilibrium position itself. Catalysts lower the activation energy required for the reaction to occur, allowing the reaction to reach equilibrium more quickly.

Conclusion: Mastering Equilibrium Constant Expressions

In conclusion, understanding and correctly constructing equilibrium constant expressions is fundamental to grasping chemical kinetics and equilibrium. For the reaction $Cu(s) + 2Ag^+(aq) \rightleftharpoons Cu^{2+}(aq) + 2Ag(s)$, the correct equilibrium constant expression is $K_{eq} = \frac{[Cu^{2+}]}{[Ag^+]^2}$. This expression accurately represents the ratio of products to reactants at equilibrium, considering the phases of the species involved and excluding the concentrations of pure solids. Mastering the principles of equilibrium constants allows chemists and scientists to predict reaction outcomes, manipulate reaction conditions, and optimize chemical processes across a wide range of applications.

By adhering to the rules of $K_{eq}$ construction and understanding the factors that influence equilibrium, we can effectively harness the power of chemical reactions to achieve desired results in both laboratory and industrial settings. The equilibrium constant remains a vital concept in chemistry, bridging the gap between theoretical principles and practical applications.