Partial pressure

The partial pressure of a gas is the pressure it would exert if it filled the container on its own. In a mixture, each gas contributes its share, and the shares add up to the total pressure. To find one, multiply the gas's mole fraction by the total pressure: p_A = x_A × p_total. At A-level, partial pressures are what go into the equilibrium constant K_p.

Dalton's law of partial pressures

OpenStax states Dalton's law like this: the total pressure of a mixture of ideal gases is equal to the sum of the partial pressures of the component gases. For a mixture of gases A, B and C:

It works because, in an ideal gas, the molecules don't attract or repel each other. Each gas hits the walls as if the others weren't there, so its pressure depends only on how many of its own molecules there are.

Partial pressure formula

The mole fraction of a gas is its share of the total number of moles:

Mole fractions have no units and always add up to 1, which is a handy check. Partial pressures have the same unit as the total pressure, usually kPa or Pa. AQA's A-level specification asks you to derive partial pressure from mole fraction and total pressure, and OCR A uses both terms in its equilibrium topic.

Example: a cylinder holds 0.20 mol of oxygen and 0.80 mol of nitrogen at a total pressure of 250 kPa. The mole fractions are 0.20 ÷ 1.00 = 0.20 and 0.80 ÷ 1.00 = 0.80, so p(O₂) = 0.20 × 250 = 50 kPa and p(N₂) = 0.80 × 250 = 200 kPa. Check: 50 + 200 = 250 kPa.

Kp: the equilibrium constant from partial pressures

For a reversible reaction between gases, the equilibrium constant can be written with partial pressures instead of concentrations. AQA defines K_p as the equilibrium constant calculated from partial pressures for a system at constant temperature. For a general reaction aA + bB ⇌ cC + dD, all gases:

Products go on top and reactants underneath. The number in front of each gas in the balanced equation becomes the power on its partial pressure. It is the same pattern as K_c in dynamic equilibrium, with partial pressures in place of square-bracket concentrations.

Writing Kp for the Haber process

For N₂(g) + 3H₂(g) ⇌ 2NH₃(g):

Units: pressure² on top and pressure⁴ underneath leaves pressure⁻², so with partial pressures in kPa the unit is kPa⁻². Work the units out for each reaction; when the powers top and bottom are equal, K_p has no units.

How to calculate Kp: worked example

An equilibrium mixture for N₂ + 3H₂ ⇌ 2NH₃ contains 0.60 mol of N₂, 1.80 mol of H₂ and 0.40 mol of NH₃ at a total pressure of 14.0 MPa. Calculate K_p.

1. Total moles = 0.60 + 1.80 + 0.40 = 2.80 mol.
2. Mole fractions: N₂ = 0.60 ÷ 2.80 = 0.214, H₂ = 1.80 ÷ 2.80 = 0.643, NH₃ = 0.40 ÷ 2.80 = 0.143. They add up to 1.
3. Partial pressures: N₂ = 0.214 × 14.0 = 3.00 MPa, H₂ = 0.643 × 14.0 = 9.00 MPa, NH₃ = 0.143 × 14.0 = 2.00 MPa.
4. K_p = 2.00² ÷ (3.00 × 9.00³) = 4.00 ÷ 2187 = 1.83 × 10⁻³ MPa⁻².

Using Kp to find a partial pressure

Rearrange the expression. At the same temperature, a different mixture has p(N₂) = 4.0 MPa and p(H₂) = 10.0 MPa at equilibrium. Then p(NH₃)² = K_p × p(N₂) × p(H₂)³ = 1.83 × 10⁻³ × 4.0 × 1000 = 7.32, so p(NH₃) = √7.32 = 2.7 MPa.

The Rates and Equilibrium Calculator handles K_c calculations, which follow the same pattern.

What changes Kp?

Only temperature changes the value of K_p. If the forward reaction is exothermic, raising the temperature lowers K_p; if it is endothermic, raising the temperature increases it. When the total pressure changes, the mole fractions shift until the partial pressures fit the same K_p again. AQA's specification says a catalyst affects how fast equilibrium is reached but not the value of the equilibrium constant.

Heterogeneous equilibria

If a reaction involves solids as well as gases, only the gases go into K_p. For CaCO₃(s) ⇌ CaO(s) + CO₂(g), K_p = p(CO₂). OCR A includes K_p for heterogeneous equilibria; AQA's K_p topic covers homogeneous systems.

Common mistakes

• Using moles instead of partial pressures. Convert moles to mole fractions, then to partial pressures, before they go into K_p.
• Forgetting the powers. 3H₂ means p(H₂)³ in the expression.
• Using square brackets. Square brackets mean concentration. K_p uses p for partial pressure.
• Including solids. Only gases appear in K_p.
• Wrong units. Work them out from the expression every time; they differ between reactions.
• Saying pressure changes Kp. Pressure can move the equilibrium position, but only temperature changes the constant.

Kp is A-level only in AQA's specification, so practise it on today's A-level Chemistry: Year 2 short timed paper. For how gas pressure relates to moles and temperature, see the ideal gas law guide.

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