Combining Equilibrium Constants

7.6 Properties of K

Combining Equilibrium Constants

Reverse or scale each source reaction, then stack them until the sum matches a target. K_overall builds step by step as K_1^a × K_2^b across 7 puzzles.

Reverse & Scale7 PuzzlesK_overallSpecies Cancel

Topic 7.6

Properties of the Equilibrium Constant

Represent a multistep process with an overall equilibrium expression, using the constituent K expressions for each individual reaction.

Three algebraic rules, plus one clarification:

Reverse a reaction → K is inverted: K′ = 1/K.
Multiply the coefficients by a factor c → K is raised to the power c: K′ = K^c.
Add reactions together → the overall K is the product of the individual K values.
Because K and Q have identical mathematical forms, all valid algebraic manipulations of K also apply to Q.

Compare to Hess's law (6.9) — the structure is the same but the operation is different:

Operation	ΔH	K
Reverse	change sign	take reciprocal
Multiply by c	multiply by c	raise to power c
Add reactions	add ΔH values	multiply K values

The pattern is that anything additive for ΔH is multiplicative for K. That is not a coincidence: ΔG° = −RT ln K, and a logarithm turns products into sums.

Where this shows up: the K_a of a polyprotic acid's overall ionization is the product of the stepwise K_a values, and K_a × K_b = K_w for a conjugate pair (8.3) is exactly this rule applied to two reactions that sum to the autoionization of water.

Key points

Reverse → 1/K. Scale by c → K^c. Add reactions → multiply K values.
Every rule for K applies identically to Q.
Additive for ΔH means multiplicative for K.
K_a × K_b = K_w is this rule in disguise.

Equations

$K_{reverse} = \frac{1}{K}$
not on the sheet
$K_{scaled by c} = K^{c}$
not on the sheet
$K_{overall} = K_{1} \times K_{2} \times \dots$
not on the sheet

Common mistakes

Adding K values for combined reactions. Multiply them.
Multiplying K by c instead of raising it to the power c.
Negating K when reversing. K is always positive; take the reciprocal.
Halving a reaction and halving K. Halving coefficients means taking the square root.

Worked example

Given (1) 2 NO(g) ⇌ N₂(g) + O₂(g) K₁ = 1.0 × 10³⁰ (2) 2 NO(g) + O₂(g) ⇌ 2 NO₂(g) K₂ = 6.4 × 10⁹ calculate K for N₂(g) + 2 O₂(g) ⇌ 2 NO₂(g).

Target: N₂ + 2 O₂ ⇌ 2 NO₂

Place N₂: it must be a reactant, but in (1) it is a product. Reverse (1):
N₂(g) + O₂(g) ⇌ 2 NO(g) K = 1/K₁ = 1.0 × 10⁻³⁰

Place NO₂: equation (2) already has it as a product with coefficient 2. Use as is:
2 NO(g) + O₂(g) ⇌ 2 NO₂(g) K = 6.4 × 10⁹

Add the two equations:
N₂ + O₂ + 2 NO + O₂ ⇌ 2 NO + 2 NO₂
NO cancels, and the two O₂ combine:
N₂ + 2 O₂ ⇌ 2 NO₂ ✓

Multiply the constants:
K = (1.0 × 10⁻³⁰)(6.4 × 10⁹) = 6.4 × 10⁻²¹

Extremely reactant-favored — which is why the nitrogen and oxygen in the atmosphere do not spontaneously form NO₂ at room temperature.

Full notes for topic 7.6 →