Worked example 1
For 2 SO₂(g) + O₂(g) ⇌ 2 SO₃(g), K = 4.0 × 10²⁴. Find K for 2 SO₃(g) ⇌ 2 SO₂(g) + O₂(g).
Try it first: Identify which of the three operations has been applied.
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EquilibriumCourse level: AP Chemistry
Manipulating Equilibrium Constants: Reversing, Scaling and Adding
What you'll be able to do: Find K for a target reaction by reversing, scaling and adding reactions whose constants are known.
Introduction
Tables rarely list the exact reaction you need. Three simple rules let you build its K from reactions that are listed, in the same spirit as Hess's law.
These are recommended, not required. You can start this lesson at any time.
Learning objectives
- Invert K when a reaction is reversed
- Raise K to a power when coefficients are scaled
- Multiply constants when reactions are added
- Assemble K for a target reaction from tabulated reactions
Reversing a reaction
Swapping products and reactants swaps the numerator and denominator of the expression, so the new constant is the reciprocal of the old one. If K = 100 forward, the reverse reaction has K = 0.010.
Reversing A ⇌ B changes K = aB/aA into Kreverse = aA/aB = 1/K. A product-favored forward equation therefore becomes a reactant-favored reverse equation; the physical equilibrium mixture is the same. Never negate K: positive activities give a positive equilibrium constant. A negative ΔG° and a negative K are not analogous ideas.
K(reverse) = 1 / K(forward)
Scaling a reaction
Doubling every coefficient doubles every exponent, which squares the whole expression. Halving the coefficients takes the square root. In general, multiplying by n raises K to the n.
For 2 A ⇌ B, K = aB/aA². The halved equation A ⇌ 1/2 B has K′ = aB1/2/aA = √K. This square root applies to the whole constant, not to one concentration chosen from a later mixture. Combining reversal and scaling by m gives K′ = K−m. Fractional reaction coefficients are valid bookkeeping, just as in formation-enthalpy equations.
K(new) = K(old)ⁿ
Halving coefficients means n = 1/2, so take the square root, not half of K.
Adding reactions
When two reactions are added, the intermediate cancels and the two expressions multiply together, leaving the expression for the overall reaction.
For A ⇌ B with K₁ = aB/aA and B ⇌ C with K₂ = aC/aB, multiplying gives K₁K₂ = aC/aA because aB cancels. This is why constants multiply when equations add. With K₁ = 4 and K₂ = 9, A ⇌ C has K = 36; reversing and doubling gives 2 C ⇌ 2 A with K = 36⁻² = 1/1296. Confirm the target direction before entering a power on the calculator.
K(overall) = K₁ x K₂
Working like Hess''s law
The procedure is the same as for enthalpy: manipulate each given reaction until adding them reproduces the target, then combine the constants. The only difference is the arithmetic. Enthalpies are added and their signs flipped; constants are multiplied and inverted.
All supplied constants must refer to the same temperature and compatible standard-state conventions. You cannot multiply a 298 K constant by a 500 K constant to obtain a definite constant for one unspecified temperature. If a target cannot be formed by cancelling the supplied species, the data do not determine it by this method.
The connection to Hess's law is exact through ΔrG° = −RT ln K. Adding ΔrG° values adds their logarithms, which multiplies K values. Scaling ΔrG° scales ln K, which raises K to a power. This also explains why adding numerical K values would give the wrong result.
Add ΔH values. Multiply K values.
Order of operations
Do the reversal and the scaling on each individual reaction first, then multiply the adjusted constants together. Scaling a reaction that has already been reversed is fine as long as both operations are applied to the same constant.
Apply reversal and scaling in sequence
| Equation operation | Operation on the equilibrium constant |
|---|---|
| When a reaction is reversed, | replace K by 1/K. |
| When every coefficient is multiplied by a factor m, | raise K to the mth power. |
| When reactions are added, | multiply their constants. |
Scroll horizontally if needed to read all columns.
Apply these operations in the same order as the equation manipulations and verify the net equation before computing.
Reverse → reciprocal; double → square; add → multiply.
Key ideas
Rule
Reverse
K becomes 1/K.
Rule
Multiply coefficients by n
K becomes Kⁿ.
Rule
Add reactions
Constants multiply.
Key concept
Compare with Hess's law
Enthalpies add; equilibrium constants multiply.
Key concept
Sanity check
A reaction that is barely product-favored forward must be reactant-favored in reverse.
Equation
Reversed reaction
K' = 1/K
- K' = constant of the reversed reaction
Equation
Scaled reaction
K' = Kⁿ
- n = factor applied to every coefficient
Equation
Added reactions
K(total) = K₁ x K₂
- K₁, K₂ = constants of the reactions being added
Worked examples
Worked example 2
For N₂(g) + 3 H₂(g) ⇌ 2 NH₃(g), K = 6.0 × 10⁵. Find K for (1/2) N₂(g) + (3/2) H₂(g) ⇌ NH₃(g).
Try it first: Compare coefficients to find the scaling factor.
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Worked example 3
Given A ⇌ B with K₁ = 2.0 and B ⇌ C with K₂ = 30, find K for A ⇌ C.
Try it first: Check that adding the two reactions cancels B.
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Worked example 4
A ⇌ B has K₁=4 and B ⇌ C has K₂=9. Find K for 2 C ⇌ 2 A.
Try it first: Build A ⇌ C first, then reverse and double.
0 of 3 steps revealed.
Common mistakes
| Mistake | Why it's wrong | Check instead |
|---|---|---|
| Negating K when reversing a reaction. | Equilibrium constants are never negative; the reverse constant is the reciprocal. | Take 1/K. |
| Halving K when the coefficients are halved. | Coefficients act as exponents, so halving means a square root. | Raise K to the power n. |
| Adding constants when reactions are added. | Enthalpies add, but the expressions for K multiply. | Multiply the constants. |
| Scaling before reversing and then applying only one of the two changes. | Both operations must act on the same reaction's constant. | Adjust each reaction fully before combining. |
Scroll horizontally if needed to read all columns.
After the lesson · Question bank
Independent practice
Work through new problems on your own. The lesson's quick checks and worked examples stay above if you need to review a step.
34 topic questions across 3 difficulty levels.
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What you should now know
Reversing a reaction inverts its constant, multiplying a reaction by a factor n raises the constant to the power n, and adding two reactions multiplies their constants. These rules follow directly from the algebra of the equilibrium expression. They mirror Hess's law, but where enthalpies are added, equilibrium constants are multiplied, and where an enthalpy sign flips, a constant is inverted. Applying the rules in the right order lets you assemble K for almost any target reaction from tabulated data.
- Reversing a reaction inverts K
- Scaling coefficients by n raises K to the power n
- Adding reactions multiplies their constants
- Adjust each reaction first, then combine
- Same logic as Hess's law, but multiplicative
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