Problem 20 - Chemistry and Chemical Reactivity

Question

Which of the following correctly relates the equilibrium constants for the two reactions shown? \(A+B \rightleftarrows 2 C \quad K_{1}\) \(\mathrm{C} \rightleftarrows 1 / 2 \mathrm{A}+1 / 2 \mathrm{B} \quad K_{2}\) (a) \(K_{2}=1 /\left(K_{1}\right)^{1 / 2}\) (c) \(K_{2}=K_{1}^{2}\) (b) \(K_{2}=1 / K_{1}\) (d) \(K_{2}=-K_{1}^{1 / 2}\)

Step-by-Step Solution

Answer

The correct relation is option (a): \(K_{2} = 1 / \left(K_{1}\right)^{1/2}\).

Step 1: Identify the Relationship Between the Reactions

The first reaction is given as \(A+B \rightleftharpoons 2C\) with an equilibrium constant \(K_1\). The second reaction is \(C \rightleftharpoons \frac{1}{2}A + \frac{1}{2}B\) with an equilibrium constant \(K_2\). Notice that the second reaction is the reverse of the first reaction, with stoichiometric coefficients halved.

Step 2: Relate the Equilibrium Constants by Reaction Modification

For reversible reactions, if reaction number 1 is reversed, the equilibrium constant of the new reaction will be the reciprocal of \(K_1\). Additionally, if the equation's stoichiometry is changed by a factor, the equilibrium constant is raised to the power of that factor.

Step 3: Calculate the New Equilibrium Constant

Since the first reaction \(A + B \rightleftharpoons 2C\) is halved when it is reversed to obtain \(C \rightleftharpoons \frac{1}{2}A + \frac{1}{2}B\), we calculate the equilibrium constant \(K_2\) as:\[ K_2 = \left(\frac{1}{K_1}\right)^{1/2} = \frac{1}{\sqrt{K_1}} \]

Step 4: Select the Correct Option

From the above expressions, the correct relation between \(K_1\) and \(K_2\) is option (a): \(K_2 = \frac{1}{\left(K_1\right)^{1/2}}\).

Key Concepts

Reversible Reactions

In chemistry, many reactions can proceed in both forward and reverse directions. These are known as reversible reactions. This means that the products can react to form the reactants again. The idea behind reversible reactions is important because it helps explain why reactions can reach a state of balance, known as equilibrium. When writing reversible reactions, we use a double-headed arrow \( \rightleftharpoons \) to indicate that both forward and reverse reactions occur.

Stoichiometry Adjustment

Stoichiometry refers to the quantitative relationships between reactants and products in a chemical reaction. Sometimes, the stoichiometry of a reaction may need adjusting. For instance, if all coefficients in a balanced equation are multiplied by a common factor, the equilibrium constant changes as well. If you multiply or divide the coefficients by a factor, say \( n \) , the equilibrium constant itself must be adjusted by raising it to the power of \( n \). For example, if the reaction is halved, the equilibrium constant becomes the square root of the original. This adjustment ensures that the stoichiometry is correctly represented for the reaction at equilibrium.

Reciprocal Relationship

The equilibrium constants of reversible reactions are interconnected. If you reverse a reaction, the equilibrium constant of this reversed reaction becomes the reciprocal of the original equilibrium constant. This means \( K_{reverse} = \frac{1}{K_{forward}} \). The concept of reciprocity helps understand how changes in the direction of a reaction affect its equilibrium constant. Knowing these relationships allows chemists to predict how the concentrations of reactants and products can influence the reaction under different conditions.

Chemical Equilibrium

Chemical equilibrium is a state where the concentrations of reactants and products remain constant over time, even though the reaction continues. At equilibrium, the rates of the forward and reverse reactions are equal. When a reversible reaction reaches equilibrium, it doesn't mean the reactants and products are equal, but that they've settled into a steady state. The equilibrium is described by the equilibrium constant \( K \), which provides valuable information about the ratio of product concentrations to reactant concentrations at equilibrium. This value is crucial, as it indicates whether products or reactants are favored in the equilibrium mix.