Problem 70 - Chemistry The Central Science

Question

If \(K_{c}=1\) for the equilibrium \(2 \mathrm{~A}(g) \rightleftharpoons \mathrm{B}(g)\), what is the relationship between \([\mathrm{A}]\) and \([\mathrm{B}]\) at equilibrium?

Step-by-Step Solution

Answer

The relationship between the concentrations of A and B at equilibrium is \[[A]^2 = [B]\], given that the equilibrium constant \(K_c = 1\) for the reaction \(2A(g) \rightleftharpoons B(g)\).

Step 1: Write the equilibrium expression.

The equilibrium expression for the reaction 2A(g) ⇌ B(g) is given by: \[K_c = \frac{[B]}{[A]^2}\] This expression gives us the relation between the concentrations of A, B, and the equilibrium constant Kc at equilibrium.

Step 2: Substitute the given values.

We are given that the equilibrium constant Kc is equal to 1. Now, we can substitute this value into the equilibrium expression along with the unknown equilibrium concentrations of A (represented by [A]) and B (represented by [B]): \[1 = \frac{[B]}{[A]^2}\]

Step 3: Solve the equation for the relationship between the concentrations of A and B at equilibrium.

The goal of this step is to find the relationship between [A] and [B] at equilibrium. To do this, we can rearrange the equation from step 2: \[1 = \frac{[B]}{[A]^2}\] Multiplying both sides by \([A]^2\): \[[A]^2 = [B]\] This is the relationship between the concentrations of A and B at equilibrium.

Key Concepts

Equilibrium Constant (Kc)

In chemistry, the equilibrium constant, denoted as \( K_c \), is a crucial concept when analyzing reactions at equilibrium. It expresses the ratio of the concentrations of the products to the reactants, each raised to the power of their stoichiometric coefficients. In the context of a chemical reaction such as \( 2A(g) \rightleftharpoons B(g) \), the equilibrium expression can be established as:

• \[ K_c = \frac{[B]}{[A]^2} \]

This formula indicates that at equilibrium, the concentration of the product \( [B] \) divided by the square of the reactant \( [A] \) equals the equilibrium constant value.
Knowing the \( K_c \) is significant as it reflects the extent to which a reaction proceeds at equilibrium: if \( K_c \) is high, the products predominate, while a low \( K_c \) favors the reactants. Here, \( K_c = 1 \) suggests an equal balance between the reactants and products, meaning neither the reactants nor the products are favored in this reaction at equilibrium.

Concentration Relationship

Exploring the relationship between concentration values at equilibrium involves examining how reactant and product concentrations relate under a given \( K_c \). In our example of \( 2A(g) \rightleftharpoons B(g) \) with \( K_c = 1 \), we rearranged:

• \[ 1 = \frac{[B]}{[A]^2} \]

Multiplying both sides by \([A]^2\) yields \([A]^2 = [B]\).
This formulation provides a clear picture of the relationship between \([A]\) and \([B]\) at equilibrium: the concentration of \( B \) is equal to the square of the concentration of \( A \).
Thus, when you walk through solving equilibration problems, understanding these direct relationships helps decipher complex reaction behaviors.

Le Chatelier's Principle

Le Chatelier's Principle is a powerful guideline for predicting how a change in conditions can affect a reaction at equilibrium. It states that if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium shifts to counteract the change.
In terms of our reaction \( 2A(g) \rightleftharpoons B(g) \), adjustments in the system such as concentration shifts, pressure modifications, or temperature changes could result in different equilibrium concentrations. For instance, increasing \([A]\) would potentially favor the formation of more \( B \) to restore balance.

• If concentration of \( A \) is increased: more \( B \) will form.
• If concentration of \( B \) is increased: the system will shift to produce more \( A \).

This principle is not only vital for understanding how and why equilibria change under various circumstances but also for practical applications in industrial processes, ensuring the reaction conditions yield maximum efficiency.