Manipulating Equilibrium Reactions

Core Concept

The equilibrium constant is tied to the way a reaction is written. If you reverse an equation, multiply its coefficients, or add equations together, the value of K changes in a predictable way. Mastering these three rules lets you calculate K for almost any reaction from a small set of tabulated constants — the same logic behind Hess's law for ΔH.

Key Ideas

• Reverse a reaction → take the reciprocal: K' = 1/K.
• Multiply all coefficients by n → raise K to the nth power: K' = Kⁿ.
• Add two reactions → multiply their K values: K' = K₁ × K₂.
• These follow directly from the algebraic form of the equilibrium expression.

Equations and Variables

For a reaction with equilibrium constant K:

• Reverse: K_rev = 1/K
• Multiply by n: K_new = Kⁿ
• Sum of reactions 1 and 2: K_net = K₁ · K₂

For the general expression K = [C]^c[D]^d / [A]^a[B]^b:

• Reversing swaps numerator and denominator.
• Multiplying coefficients by n raises every exponent by n.
• Adding reactions multiplies the corresponding product/reactant ratios (species appearing on both sides cancel).

How It Works

Each rule is just algebra on the equilibrium expression. Reversing N₂O₄(g) ⇌ 2 NO₂(g) (K = 4.6 × 10⁻³) gives 2 NO₂(g) ⇌ N₂O₄(g), whose expression is [N₂O₄]/[NO₂]² — the reciprocal, K = 1/(4.6 × 10⁻³) = 217 ≈ 216.

Doubling the coefficients of N₂O₄ ⇌ 2 NO₂ gives 2 N₂O₄ ⇌ 4 NO₂, with expression [NO₂]⁴/[N₂O₄]² = ([NO₂]²/[N₂O₄])², so K = (4.6 × 10⁻³)² = 2.1 × 10⁻⁵.

When reactions are added, their expressions multiply, and any species appearing on both sides (as a product of one step and reactant of the next) cancels out of the net expression — just as in Hess's law. This is why the equilibrium constant of an overall reaction equals the product of the K values of the steps that sum to it.

Worked Example

Given these two equilibria at 25 °C:

• (1) N₂(g) + O₂(g) ⇌ 2 NO(g), K₁ = 4.1 × 10⁻³¹
• (2) 2 NO(g) + O₂(g) ⇌ 2 NO₂(g), K₂ = 4.5 × 10¹⁵ (value inverted from the tabulated reverse)

Find K for the net reaction N₂(g) + 2 O₂(g) ⇌ 2 NO₂(g).

Step 1 — Add the equations. Reaction (1) + reaction (2) gives:

N₂ + O₂ + 2 NO + O₂ ⇌ 2 NO + 2 NO₂ → N₂(g) + 2 O₂(g) ⇌ 2 NO₂(g)

The 2 NO cancels (product of step 1, reactant of step 2).

Step 2 — Multiply the K values:

K_net = K₁ × K₂ = (4.1 × 10^-31)(4.5 × 10¹⁵) = 1.8 × 10^-15

Step 3 — Demonstrate the reverse rule. If instead the net reaction were written backward, 2 NO₂ ⇌ N₂ + 2 O₂, its K would be 1/(1.8 × 10⁻¹⁵) = 5.6 × 10¹⁴ — a dramatic illustration of how direction changes K.

How it works

Each rule is just algebra on the equilibrium expression. Reversing N₂O₄(g) ⇌ 2 NO₂(g) (K = 4.6 × 10⁻³) gives 2 NO₂(g) ⇌ N₂O₄(g), whose expression is [N₂O₄]/[NO₂]² — the reciprocal, K = 1/(4.6 × 10⁻³) = 217 ≈ 216.

Doubling the coefficients of N₂O₄ ⇌ 2 NO₂ gives 2 N₂O₄ ⇌ 4 NO₂, with expression [NO₂]⁴/[N₂O₄]² = ([NO₂]²/[N₂O₄])², so K = (4.6 × 10⁻³)² = 2.1 × 10⁻⁵.

When reactions are added, their expressions multiply, and any species appearing on both sides (as a product of one step and reactant of the next) cancels out of the net expression — just as in Hess's law. This is why the equilibrium constant of an overall reaction equals the product of the K values of the steps that sum to it.

Common confusions

• Forgetting to square K when doubling coefficients. K' = Kⁿ, not n·K.
• Adding K values when reactions are added. You multiply, not add.
• Not canceling common species when summing equations.
• Confusing this with Hess's law for ΔH. ΔH adds (and reverses sign); K multiplies (and inverts).

Quick review

1. What is K for the reverse of a reaction with constant K?
2. What is K when all coefficients are tripled?
3. How do you combine K values for summed reactions?
4. For 2 NO₂ ⇌ N₂O₄ (K = 216), what is K for N₂O₄ ⇌ 2 NO₂?
5. Why must intermediates cancel when equations are added?