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Partial pressures and Kp

Key idea: Find each gas's partial pressure from its mole fraction and total pressure. Kp uses these equilibrium partial pressures raised to the equation coefficients, including only reacting gases.

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  • GCE A-Level H2 Chemistry 2027
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For a reaction between gases, it is often easier to measure pressures than concentrations. The equilibrium constant Kₚ is written using the partial pressure of each gas. This lesson shows how to find partial pressures from the composition of a mixture, and how to write and calculate Kₚ.

Partial pressures

For an ideal gas mixture, each gas contributes part of the total pressure. The partial pressure of a gas is the pressure it would exert if it alone occupied the whole volume at the same temperature. The partial pressures of all the gases add up to the total pressure.

The partial pressure of a gas X is its mole fraction multiplied by the total pressure:

pX = xX × Pₜₒₜₐₗ, xX = nX/nₜₒₜₐₗ

nₜₒₜₐₗ includes every gas in the mixture, including any unreactive gas.

Worked example 1

A mixture with helium

Problem

A gas mixture contains 2.00 mol of A, 1.00 mol of B and 1.00 mol of helium at a total pressure of 100.0 kPa. Calculate the partial pressure of each gas.
Show full solution

  1. Total amount of gas

    Method

    Add the amounts of all three gases.

    Reason

    Helium does not react, but it still contributes to the total pressure.

    Working

    Answer

    nₜₒₜₐₗ = 2.00 + 1.00 + 1.00 = 4.00 mol

  2. Partial pressures

    Method

    Multiply each mole fraction by the total pressure.

    Reason

    pX = xX × Pₜₒₜₐₗ

    Working

    Answer

    pA = 2.00/4.00 × 100.0 = 50.0 kPa; pB = p_He = 1.00/4.00 × 100.0 = 25.0 kPa

  3. Check

    Working

    Answer

    50.0 + 25.0 + 25.0 = 100.0 kPa, the total pressure.

Check your understanding 1

A mixture contains 0.600 mol of A, 0.200 mol of B and 0.200 mol of helium at 150.0 kPa. Calculate the total amount, A’s mole fraction and A’s partial pressure.

Show the steps Optional working
Unit: mol
Unit: kPa
Show answer

Total amount = 0.600 + 0.200 + 0.200 = 1.000 mol. A’s mole fraction is 0.600 and its partial pressure is 0.600(150.0) = 90.0 kPa. B and helium each exert 30.0 kPa, giving 150.0 kPa in total.

The equilibrium constant Kp

For a reaction between gases,

aA(g) + bB(g) ⇌ cC(g) + dD(g)
Kₚ = (pC)^c(pD)^d/(pA)^a(pB)^b

Kₚ is written in the same way as Kc, but with the equilibrium partial pressure of each gas instead of its concentration:

  • only gases that take part in the reaction appear; solids, liquids and unreactive gases do not;
  • each partial pressure is raised to the power of its coefficient in the balanced equation;
  • every pressure must be in the same unit.

To find the units of Kₚ, replace each partial pressure by its unit and cancel, as for Kc. For example, for N₂(g) + 3H₂(g) ⇌ 2NH₃(g) with pressures in kPa:

(kPa)²/(kPa)(kPa)³ = kPa⁻²

Guided practice 2

Writing Kp

About 4 min

Problem

Write an expression for Kₚ for 2SO₂(g) + O₂(g) ⇌ 2SO₃(g), and give its units when the pressures are in kPa.

Try this before viewing the solution

Hints

Hint 1

top and bottom

Which species is the product?
Hint 2

powers

Use the coefficients 2, 1 and 2 as powers.
Show solution step by step

  1. Expression

    Method

    Put the product over the reactants, using the coefficients as powers.

    Reason

    Kₚ follows the balanced equation, like Kc.

    Working

    Answer

    Kₚ = (p_SO₃)²/((p_SO₂)² p_O₂)

  2. Units

    Working

    Answer

    (kPa)²/(kPa)²(kPa) = kPa⁻¹

Calculating Kp

To calculate Kₚ, find the partial pressure of each gas at equilibrium, then substitute into the expression.

Worked example 3

Kp from mole fractions

Problem

An equilibrium mixture of H₂, I₂ and HI has mole fractions 0.200, 0.200 and 0.600 respectively, at a total pressure of 100.0 kPa. Calculate Kₚ for H₂(g) + I₂(g) ⇌ 2HI(g).
Show full solution

  1. Partial pressures

    Method

    Multiply each mole fraction by the total pressure.

    Reason

    Kₚ needs the partial pressure of each gas.

    Working

    Answer

    p_H₂ = 20.0 kPa, p_I₂ = 20.0 kPa, p_HI = 60.0 kPa

  2. Substitute

    Method

    Use the expression with the coefficient powers.

    Reason

    The coefficient of HI is 2.

    Working

    Answer

    Kₚ = (p_HI)²/(p_H₂ p_I₂) = (60.0)²/(20.0 × 20.0)

  3. Result

    Working

    Answer

    Kₚ = 9.00. The units cancel, so Kₚ has no units.

Exam-style question 1

Kp for ammonia formation

3 marks

Problem

For N₂(g) + 3H₂(g) ⇌ 2NH₃(g), the equilibrium partial pressures are p_N₂ = 20.0 kPa, p_H₂ = 40.0 kPa and p_NH₃ = 10.0 kPa. Calculate Kₚ, giving its units. [3 marks]

Try this before viewing the solution

Show the steps Optional working
Unit: kPa²
Unit: kPa⁴
Units of Kp
Show solution step by step

  1. Expression

    1 mark

    Method

    Write Kₚ before substituting.

    Reason

    The coefficients give the powers.

    Working

    Answer

    Kₚ = (p_NH₃)²/p_N₂(p_H₂)³

  2. Substitute

    1 mark

    Reason

    All the pressures are in kPa.

    Working

    Answer

    Kₚ = (10.0)²/(20.0 × (40.0)³) = 100/(1.28 × 10⁶)

  3. Value and units

    1 mark

    Working

    Answer

    Kₚ = 7.81 × 10⁻⁵ kPa⁻²

Try it yourself 4

Dinitrogen tetroxide and nitrogen dioxide

Minimal support

Problem

An equilibrium mixture contains only N₂O₄ and NO₂, with mole fractions 0.750 and 0.250 respectively, at a total pressure of 200.0 kPa. Calculate Kₚ for N₂O₄(g) ⇌ 2NO₂(g), giving its units.

Try this before viewing the solution

Show the steps Optional working
Unit: kPa
Units of Kp

Hints

Hint 1

partial pressures

How do mole fraction and total pressure give each partial pressure?
Hint 2

the power

The partial pressure of NO₂ is squared.
Show solution step by step

  1. Partial pressures

    Method

    Use the mole fractions.

    Reason

    pX = xX × Pₜₒₜₐₗ

    Working

    Answer

    p_N₂O₄ = 0.750 × 200.0 = 150 kPa; p_NO₂ = 0.250 × 200.0 = 50.0 kPa

  2. Kp

    Method

    Substitute into the expression.

    Reason

    The coefficient of NO₂ is 2.

    Working

    Answer

    Kₚ = (p_NO₂)²/p_N₂O₄ = (50.0)²/150 = 16.7

  3. Units

    Working

    Answer

    (kPa)²/kPa = kPa, so Kₚ = 16.7 kPa.

Check your understanding 2

For the dinitrogen tetroxide mixture above, write your two partial pressures, your Kₚ expression and the cancellation that gives its units.

Show answer

Compare these three decisions: p_N₂O₄ = 0.750(200.0) = 150 kPa and p_NO₂ = 0.250(200.0) = 50.0 kPa; Kₚ = (p_NO₂)²/p_N₂O₄; units kPa²/kPa = kPa. Thus Kₚ = 16.7 kPa. Using mole fractions alone would miss the remaining factor of total pressure.

Check your understanding 3

A reacting gas mixture also contains helium. A student includes helium when working out the mole fractions, but leaves it out of the Kₚ expression. Is this correct?

Show hint
Which quantities depend on all the gases present, and which depend only on the reaction?
Show answer

Yes. Helium contributes to the total pressure, so it must be counted in nₜₒₜₐₗ when working out mole fractions. It does not take part in the reaction, so it has no term in Kₚ.

Common mistakes

  • Using the total pressure in place of each gas’s partial pressure.
  • Leaving an unreactive gas out of nₜₒₜₐₗ when working out mole fractions.
  • Including an unreactive gas, a solid or a liquid in the Kₚ expression.
  • Substituting mole fractions into Kₚ instead of partial pressures.
  • Leaving out the coefficient powers, or mixing pressure units.
  • Leaving out the units of Kₚ, or working them out from the wrong powers.

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