7.6 Properties of the Equilibrium Constant

When you change a reaction, the equilibrium constant $K$ changes in predictable ways. Reverse a reaction and $K$ becomes $\frac{1}{K}$, multiply the coefficients by a factor and $K$ is raised to that power, and add reactions together and you multiply their $K$ values. For AP Chemistry, identify the reaction manipulation before changing the equilibrium constant.

Why This Matters for the AP Chemistry Exam

This topic gives you the algebra you need to handle equilibrium constants that aren't handed to you directly. On the exam you may get K values for individual steps and have to combine them into an overall K, or you may need to find K for a reversed or rescaled version of a reaction. Being able to identify what manipulation was done and apply the matching rule to K is the core skill here. These same rules also connect to free energy later in the course through the relationship between ΔG° and K, so getting comfortable now pays off in Unit 9.

Key Takeaways

• Reversing a reaction inverts K: the new constant is 1/K.
• Multiplying all coefficients by a factor c raises K to the power c.
• Adding reactions together multiplies their individual K values to give the overall K.
• These rules mirror Hess's Law for enthalpy, so the same flip, scale, and add logic applies.
• Because K and Q have the same mathematical form, every valid manipulation of K works for Q too.
• Most multistep problems require combining two or three rules, so identify each change before calculating.

Connection to Hess's Law

In Unit 6 you used Hess's Law to find ΔH by flipping, multiplying, and adding reactions. The properties of K follow the same structure, with one important difference: where Hess's Law uses addition and sign changes, equilibrium constants use multiplication and exponents.

Quick reminder of the three Hess's Law moves:

1. Reversing a reaction flips the sign of ΔH.
2. Multiplying a reaction by n multiplies ΔH by n.
3. Adding reactions adds their ΔH values.

If you were comfortable with Hess's Law, these K rules will feel familiar.

Properties of K

Reversing a Reaction

Reversing a reaction means starting from the products and ending with the reactants. To see what happens to K, write out the expression for both directions.

If the original reaction is A ⇌ B, then K = [B]/[A]. For the reverse reaction B ⇌ A, the forward direction is now what used to be the reverse. Rewriting the expression gives:

K₂ = [A]/[B] = 1/K

So reversing a reaction gives a new constant of 1/K.

Example: Inverting K

Consider the reaction N₂ + 3H₂ ⇌ 2NH₃ with K = 0.118. Find the equilibrium constant for the reaction 2NH₃ ⇌ N₂ + 3H₂.

The new reaction is the reverse of the original, so:

K = 1/0.118 = 8.47

Think of this as Final K = 1/Initial K.

Multiplying Coefficients by a Factor

Multiplying a reaction by a factor means taking something like A ⇌ B and scaling it to nA ⇌ nB. The reaction stays balanced, but K changes.

For A ⇌ B, K = [B]/[A].

For 2A ⇌ 2B:

K = [B]² / [A]² = ([B]/[A])² = K²

Generalizing to any factor n for nA ⇌ nB:

K = [B]ⁿ / [A]ⁿ = ([B]/[A])ⁿ = Kⁿ

So when you multiply the coefficients by a factor, you raise K to that same power. Multiplying a reaction by 3 cubes K, and multiplying by 1/2 takes the square root.

Example: Raising K to a Power

Consider the same reaction N₂ + 3H₂ ⇌ 2NH₃ with K = 0.118. Find the equilibrium constant for (1/2)N₂ + (3/2)H₂ ⇌ NH₃.

Only the coefficients changed, and they were all multiplied by 1/2. So raise the original K to the 1/2 power:

K = (0.118)¹ᐟ² = 0.343

Adding Reactions Together

Here is what happens when two reactions are added. Using a general example:

A ⇌ B : K₁ = [B]/[A]

C ⇌ D : K₂ = [D]/[C]

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A + C ⇌ B + D : K = [B][D] / [A][C]

Notice that the overall expression is K₁ times K₂. Adding the two reactions multiplied their equilibrium constants. So when you add reactions, you multiply their K values to get the overall K.

Summary Table

Here are the three rules side by side with their Hess's Law counterparts.

One more point worth remembering: since Q has the same mathematical form as K, all of these same manipulations apply to Q as well.

How to Use This on the AP Chemistry Exam

Problem Solving

A typical multistep problem gives you K values for two or three reactions and asks for the K of a target reaction. Work through it in order:

1. Compare each given reaction to the target. Decide whether it needs to be reversed, scaled, or both.
2. Apply the matching K rule for each change (invert, raise to a power).
3. Once the reactions line up to add into the target, multiply the adjusted K values together.

Worked Practice Problem

Using the two reactions below, find the equilibrium constant for N₂ + 2O₂ ⇌ 2NO₂.

• Reaction 1: (1/2)N₂ + (1/2)O₂ ⇌ NO, K₁ = 6.55 × 10⁻¹³
• Reaction 2: 2NO + O₂ ⇌ 2NO₂, K₂ = 6.9 × 10⁵

Start by getting each reaction into a form that adds up to the target.

Multiply Reaction 1 by 2 to get N₂ + O₂ ⇌ 2NO. Raise K₁ to the power 2:

(6.55 × 10⁻¹³)² = 4.3 × 10⁻²⁵

Now add the scaled Reaction 1 to Reaction 2. The NO terms cancel, the N₂ stays, and the O₂ terms combine into 2O₂, giving the target N₂ + 2O₂ ⇌ 2NO₂. Multiply the adjusted K values:

K = (4.3 × 10⁻²⁵)(6.9 × 10⁵) = 3.0 × 10⁻¹⁹

Common Trap

There is no single fixed order for these problems. Look at the target reaction first, figure out what each step needs, then apply the rules. Practicing several problems builds the instinct for when to reverse, scale, or add.

Common Misconceptions

• Reversing a reaction does not change the sign of K the way it changes the sign of ΔH. K can never be negative. Reversing inverts it to 1/K.
• Multiplying coefficients does not multiply K. It raises K to a power. Doubling the coefficients squares K, it does not double it.
• Adding reactions does not add their K values. It multiplies them. Addition is for ΔH, multiplication is for K.
• A larger coefficient does not automatically mean a larger K. If K is less than 1, raising it to a higher power makes it smaller.
• Fractional coefficients are allowed when you scale a reaction. Multiplying by 1/2 means taking the square root of K, not dividing K by 2.

Related AP Chemistry Guides

• 7.1 Introduction to Equilibrium
• Unit 7 Overview: Equilibrium
• 7.2 Direction of Reversible Reactions
• 7.3 Reaction Quotient and Equilibrium Constant
• 7.4 Calculating the Equilibrium Constant
• 7.11 Introduction to Solubility Equilibria