Combining Equilibrium Reactions

Introduction

Some desired equilibrium reactions are sums of simpler reactions whose constants are known. When equations are added, their thermodynamic constants multiply. Intermediates that cancel from the net equation also cancel algebraically from the product of the individual equilibrium expressions under matching conditions.

Core explanation

Suppose reaction 1 is A ⇌ B with K₁ = a B/a A, and reaction 2 is B ⇌ C with K₂ = a C/a B. Adding equations cancels B and gives A ⇌ C. Multiplying expressions gives K₁K₂ = (a B/a A)(a C/a B) = a C/a A, exactly the expression for the net reaction. Thus K net = K₁K₂ at the same temperature with consistent activity standards.

Before multiplying, match equations algebraically to the target reaction. A step may need reversing, which reciprocates its K, or coefficient scaling, which raises K to a power. Only after these transformations should constants be multiplied. The written reaction sum is the audit: every species and coefficient on the combined sides must reduce to the requested equation.

For acid-base and complexation chemistry, this rule is especially useful. An acid dissociation step combined with a base protonation step can yield a net proton-transfer reaction. Its K may be built from the component constants after canceling H⁺ or another intermediate. The method is thermodynamic and does not claim that the system literally proceeds by the two isolated steps in sequence.

K values must refer to the same temperature. Multiplying a constant at 25 °C by one measured at 60 °C does not yield a meaningful net constant for either temperature. Also keep standard-state conventions consistent; otherwise intermediate activity factors might not cancel in the assumed way.

The product rule is easier to remember through logarithms: standard reaction Gibbs energies add when reactions add, and ln K is proportional to negative standard reaction Gibbs energy. Addition of energies therefore becomes addition of ln K, which becomes multiplication of K. The simple algebraic cancellation is usually sufficient for introductory problems.

Step-by-step reasoning

1. Write each given equation and the desired net equation. 2. Reverse or scale given equations as needed, transforming their K values. 3. Add equations and cancel identical species on opposite sides. 4. Multiply transformed constants and verify the resulting K expression.

Visual explanation

Write A ⇌ B and B ⇌ C on two stacked lines. Cross out B on opposite sides after addition, then show a B canceling when K₁ and K₂ multiply.

Real-world analogy

Converting dollars to euros and euros to yen uses two conversion ratios. Multiplying them cancels euros and gives a direct dollar-to-yen ratio without an intermediate currency remaining.

Real-world example

Coupled aqueous equilibria may be supplied as separate acid dissociation and complex-formation steps. Their constants can be combined to evaluate a net reaction that is difficult to measure directly.

Why?

Why multiply rather than add constants? Each K is a ratio of activity products; multiplying those ratios cancels intermediate activities and reproduces the net expression.

Common misconception

“Because equations are added, their K values are added.” Reaction free energies add; K values multiply after the necessary reverse and scaling transformations.

Worked example

Given A ⇌ B with K₁ = 2.0 and B ⇌ C with K₂ = 5.0, the net A ⇌ C has K = 2.0 × 5.0 = 10. If the target instead is C ⇌ A, reverse the net equation and use K = 0.10. The intermediate B does not appear in the net equilibrium expression.

Quick check

1. If two equations add to a net reaction and have K values 3 and 4, what is the net K? Answer: 12, provided both equations are used exactly as written at the same temperature and convention.

Exam focus

Show equation cancellation, not just numerical multiplication. Apply reciprocal and exponent rules before the product rule when a step is reversed or scaled.

Advanced insight

Reaction networks have thermodynamic cycle constraints: multiplying K values around a closed sequence of reactions must give one for a consistent set of standard states, because the net reaction is zero.

Summary

Adding reactions multiplies their equilibrium constants after any needed reversal or coefficient scaling. Intermediate activity factors cancel, leaving the constant for the final net equation.

Practice questions

1. A ⇌ B has K = 2; B ⇌ C has K = 3. Find K for A ⇌ C. Answer: 2 × 3 = 6. 2. What if the first equation is used in reverse instead? Answer: Replace its constant by 1/2 before multiplying with the other constant. 3. Must the component constants be at the same temperature? Answer: Yes. Each K is temperature-dependent and must describe one consistent net condition.