Exemplar Problems

Problem 2: At a certain temperature, the equilibrium constant Kc for the reaction:

2NO₂(g) ⇌ N₂O₄(g) Kc is 0.1. If the initial concentration of NO₂(g) is 0.4 M, calculate the equilibrium concentrations of NO₂(g) and N₂O₄(g).

Solution :

Let’s denote the initial concentration of NO₂ as [NO₂]₀, and the equilibrium concentrations of NO₂ and N₂O₄ as [NO₂] and [N₂O₄], respectively. We are given: [NO₂]₀ = 0.4 M Kc = 0.1

Using the balanced equation, we know that at equilibrium: [NO₂] = [NO₂]₀ - 2x [N₂O₄] = x

Now, we can write the expression for Kc: Kc = [N₂O₄] / ([NO₂]²)

Substitute the expressions for [NO₂] and [N₂O₄]: 0.1 = x / ((0.4 - 2x)²)

Now, solve for x: 0.1 = x / (0.16 - 1.6x + 4x²)

0.1(0.16 - 1.6x + 4x²) = x

0.016 - 0.16x + 0.4x² = x

0.4x² - 1.16x + 0.016 = 0

Now, use the quadratic formula to solve for x: x = [1.16 ± √(1.16² - 4(0.4)(0.016))] / (2(0.4))

x = [1.16 ± √(1.3456 - 0.0256)] / 0.8

x = [1.16 ± √1.32] / 0.8

x₁ ≈ (1.16 + 1.15) / 0.8 ≈ 2.39 / 0.8 ≈ 2.99 M x₂ ≈ (1.16 - 1.15) / 0.8 ≈ 0.01 / 0.8 ≈ 0.0125 M

Since x represents the concentration of N₂O₄ at equilibrium, we can ignore the negative root.

So, the equilibrium concentrations are approximately: [NO₂] ≈ 0.4 - 2x₁ ≈ 0.4 - 2(0.01) ≈ 0.38 M (Since it cannot be negative, we conclude that it is very close to zero.) [N₂O₄] ≈ x₁ ≈ 2.99 M