# Properties of the Equilibrium Constant

> AP Chemistry · AP Chemistry 2024-2027 Curriculum
> Source: https://www.owlsprep.com/study/ap-chemistry-u7-properties-of-the-equilibrium-constant/

This module covers core rules for modifying equilibrium constant values when reactions are reversed, scaled, or summed, plus the mathematical relationship between Kc and Kp for gaseous systems.

**Prerequisites:** [Writing correct K expressions for balanced homogeneous and heterogeneous reactions](https://www.owlsprep.com/study/ap-chemistry-u7-writing-equilibrium-expressions/); [Basic stoichiometry of balanced chemical equations](https://www.owlsprep.com/study/ap-chemistry-u4-balancing-chemical-equations/)

## Learning objectives

- Predict how reversing a reaction affects the value of K
- Calculate the new K value when a reaction is scaled by a constant factor
- Derive the combined K value for multiple sequential reactions
- Relate Kp and Kc using the ideal gas law relationship

## Rule 1: Reversing a Chemical Reaction

When you reverse a balanced chemical reaction, the products of the original reaction become the reactants of the reversed reaction, and vice versa. This flips the numerator and denominator of the original K expression entirely.

$$K_{\text{reversed}} = \frac{1}{K_{\text{original}}} = K_{\text{original}}^{-1}$$

**Worked example:** The equilibrium constant for the reaction N₂(g) + 3 H₂(g) ⇌ 2 NH₃(g) is K = 4.2 × 10⁸ at 25°C. Calculate K for the reverse reaction: 2 NH₃(g) ⇌ N₂(g) + 3 H₂(g).

1. Identify that the target reaction is the exact reverse of the given reference reaction.
2. $$K_{\text{reverse}} = \frac{1}{K_{\text{forward}}} = \frac{1}{4.2 \times 10^8}$$
3. $$K_{\text{reverse}} = 2.4 \times 10^{-9}$$

> **tip**
>
> If the forward K is very large, the reverse K will be very small, which matches the expectation that a reaction that goes almost to completion barely proceeds in the backwards direction.

**Check your understanding**

Test your understanding of the reverse K rule

1. What is K for the reverse of a reaction with K=0.032?

   *Why:* You simply take 1 divided by the original K value to get the reversed K.

## Rule 2: Scaling a Reaction by a Constant Factor

If you multiply every stoichiometric coefficient in a balanced reaction by a constant factor n, each concentration term in the K expression is raised to the power of n. This means the new K value is the original K raised to the power of n.

$$K_{\text{scaled}} = (K_{\text{original}})^n$$

**Worked example:** Given the reference reaction H₂(g) + I₂(g) ⇌ 2 HI(g) has K = 54.3 at 400°C, calculate K for the reaction ½ H₂(g) + ½ I₂(g) ⇌ HI(g).

1. Identify the scaling factor n: every coefficient in the target reaction is ½ of the reference reaction, so n = 0.5.
2. $$K_{\text{new}} = (K_{\text{original}})^{0.5} = \sqrt{54.3}$$
3. $$K_{\text{new}} = 7.37$$

> **mnemonic**
>
> Coefficients become exponents on K: multiply reaction by n → raise K to n, divide reaction by n → take nth root of K.

> **Exam tip:** AP exam questions often ask for K for a reaction with fractional coefficients, do not forget to take the root instead of dividing the original K by n.

## Rule 3: Combining Multiple Sequential Reactions

When you add two or more individual chemical reactions to get an overall net reaction, you multiply their respective equilibrium constants together to get the K value for the net reaction.

$$K_{\text{net}} = K_1 \times K_2 \times K_3 \times ...$$

**Worked example:** Given Reaction 1: 2 C(s) + O₂(g) ⇌ 2 CO(g) K₁ = 1.2 × 10¹⁶; Reaction 2: 2 CO(g) + O₂(g) ⇌ 2 CO₂(g) K₂ = 2.3 × 10⁹⁰. Calculate K for the net reaction C(s) + O₂(g) ⇌ CO₂(g).

1. Add Reaction 1 and Reaction 2 first: 2 C(s) + 2 O₂(g) ⇌ 2 CO₂(g). The K for this sum is K₁ × K₂.
2. $$K_{\text{sum}} = (1.2 \times 10^{16}) \times (2.3 \times 10^{90}) = 2.76 \times 10^{106}$$
3. Divide all coefficients of the summed reaction by 2 to get the target net reaction, so raise K_sum to the power of ½.
4. $$K_{\text{net}} = \sqrt{2.76 \times 10^{106}} = 1.7 \times 10^{53}$$

## Relationship Between Kc and Kp

For reactions containing gaseous species, you can convert between Kc (concentration-based) and Kp (pressure-based) using the ideal gas law, which relates partial pressure to molar concentration.

$$K_p = K_c (RT)^{\Delta n}$$

**\Delta n** — Total moles of gaseous products minus total moles of gaseous reactants in the balanced reaction. Solids and liquids are not counted.

*Notation:* Δn

**Worked example:** For the reaction 2 SO₃(g) ⇌ 2 SO₂(g) + O₂(g), Kc = 4.2 × 10⁻³ at 700 K. Calculate Kp for this reaction.

1. Calculate Δn: total moles of gas products = 2 + 1 = 3; total moles of gas reactants = 2. Δn = 3 - 2 = 1.
2. Substitute values into the Kp equation: R = 0.0821 L·atm/(mol·K), T = 700 K, Δn = 1.
3. $$K_p = (4.2 \times 10^{-3}) \times (0.0821 \times 700)^1$$
4. $$K_p = 0.24$$

## Common pitfalls

- **Wrong:** Adding K values together when summing reactions
  - Why it fails: Students confuse K rules with enthalpy rules (which do add directly)
  - Correct: Multiply individual K values when combining reactions, only add ΔH values.
- **Wrong:** Dividing K by the scaling factor n instead of raising K to the power of n
  - Why it fails: Treating K as a linear value instead of an exponential ratio
  - Correct: If you multiply reaction coefficients by n, raise the original K to the nth power.
- **Wrong:** Counting moles of solid or liquid species when calculating Δn
  - Why it fails: Forgetting that pure solids and liquids do not appear in K expressions at all
  - Correct: Only count moles of gaseous reactants and products to calculate Δn for Kp/Kc conversion.
- **Wrong:** Using temperature in Celsius instead of Kelvin for the Kp/Kc equation
  - Why it fails: Missing that the ideal gas law requires absolute temperature units
  - Correct: Always convert Celsius temperature to Kelvin before substituting into the RT term.

## Cheatsheet

| Operation on Reaction | Corresponding Operation on K |
| --- | --- |
| Reverse full reaction | Take reciprocal: K_new = 1/K_old |
| Multiply all coefficients by n | Raise K to power n: K_new = K_old^n |
| Add two reactions together | Multiply K values: K_net = K1 * K2 |
| Convert Kc to Kp | Kp = Kc(RT)^Δn |

## What's next

Mastering these properties of K is a critical prerequisite for solving all equilibrium calculation problems on the AP Chemistry exam, including ICE table setups, reaction quotient comparisons, and Le Chatelier's principle predictions. You will use these manipulation rules repeatedly across Unit 7, especially when calculating equilibrium constants for reactions that are not given directly in exam prompts. These skills together make up ~10-15% of your total AP Chemistry exam score, so ensure you can apply all K manipulation rules without error before moving forward.

- [Reaction Quotient Q and Comparing Q vs K](https://www.owlsprep.com/study/ap-chemistry-u7-reaction-quotient-q/)

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