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Ideal Gas

Key idea: Learn the ideal gas equation of state pV = NkT and how to use it in calculations with moles, particles and SI units (A Level Physics).

  • GCE A-Level H2 Physics 2027
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Learning objectives

  • Use ideal-gas equations with particles, moles and SI units.

1. Definitions (Must Know)

A. Ideal gas

An ideal gas is a model of a gas that obeys the ideal gas equation of state: pV = NkT

B. Symbols and units (must know)

  • Pressure, p (Pa)
  • Volume, V (m³)
  • Thermodynamic temperature, T (K)
  • Number of particles, N (dimensionless)
  • Amount of substance, n (mol)
  • Avogadro constant, N_A = 6.02 × 10²³ mol⁻¹
  • Boltzmann constant, k (J K⁻¹)
  • Molar gas constant, R (J mol⁻¹ K⁻¹)

2. Key Ideas (What Earns Marks)

  • Use kelvin in every ideal gas calculation.
  • Use SI units by default: Pa, m³, K.
  • Two equivalent forms of the equation of state:
    • pV = NkT
    • pV = nRT
  • Connect particles to moles:
    • N = nN_A
    • Nk = nR
  • Ideal-gas behaviour is a good approximation at low pressure and high temperature (particles far apart; interactions less important).
Exam pitfall: Celsius used in gas equations

Use absolute temperature in kelvin for pV = nRT and pV/T ratios. Convert first, then substitute; never plug Celsius temperatures directly.

3. Detailed Explanations

A. What “equation of state” means

An equation of state links the macroscopic variables p, V and T for a fixed amount of gas.

For an ideal gas: pV = NkT

B. From particles to moles: pV = nRT

You will often be given the amount of gas in moles, n.

Using N = nN_A, substitute into pV = NkT: pV = (nN_A)kT

Since R = N_Ak: pV = nRT

C. When the ideal gas model is valid (exam language)

Real gases behave more ideally when:

  • pressure is low (particles are far apart),
  • temperature is high (particles have higher kinetic energy, so attractions matter less).

4. Common Mistakes

  • Using T in °C instead of K.
  • Mixing units (e.g. using V in litres without converting to m³).
  • Confusing N (number of particles) with n (number of moles).
  • Using R with N, or k with n (wrong constant with wrong “amount of gas”).

5. Exam Tips

  • Write the equation first, then substitute with units, then give the final answer with a unit.
  • Convert common volumes:
    • 1 L = 10⁻³ m³
    • 1 cm³ = 10⁻⁶ m³
  • Convert common pressures:
    • 1 atm ≈ 1.01 × 10⁵ Pa (use only if needed; SI Pa is preferred)

6. Worked Examples

Modelled example 1

Use pV = nRT to find number of moles

Core

Problem

A cylinder contains an ideal gas at p = 2.0 × 10⁵ Pa and V = 3.0 × 10⁻³ m³ at T = 300 K. Find n. Take R = 8.31 J mol⁻¹K⁻¹.
Study the worked solution

  1. Choose the mole form

    Method

    Use pV = nRT.

    Reason

    The amount is required in moles and the molar constant R is supplied.

    Working

    pV = nRT

  2. Rearrange and substitute

    Method

    Make n the subject and insert the SI state variables.

    Reason

    Pressure, volume and temperature are already in Pa, m³ and K.

    Working

    n = pV/RT = ((2.0 × 10⁵)(3.0 × 10⁻³))/(8.31)(300)

  3. Evaluate

    Method

    The cylinder contains approximately 0.24 mol.

    Reason

    The calculation gives 600/2490 ≈ 0.241.

    Working

    n ≈ 0.24 mol

Guided practice 2

Use N = nN_A to find number of particles

About 3 min

Problem

How many molecules are in 0.50 mol of an ideal gas? Take N_A = 6.02 × 10²³ mol⁻¹.

Try this before viewing the solution

Unit: molecules

Hints

Hint 1: link moles and particles
Use N = nN_A.
View solution step by step

  1. Select the conversion

    Method

    Multiply the amount in moles by Avogadro’s constant.

    Reason

    N_A is the number of particles per mole.

    Working

    N = nN_A

  2. Calculate

    Method

    The sample contains 3.01 × 10²³ molecules.

    Reason

    The mole unit cancels against mol⁻¹.

    Working

    N = (0.50)(6.02 × 10²³) = 3.01 × 10²³

Common misconception 3

Solve for temperature from pV = NkT

Find and correct the mistake

Learner claim

A gas has N = 1.2 × 10²³ molecules, V = 2.0 × 10⁻³ m³ and p = 1.0 × 10⁵ Pa. A learner plans to use pV = NRT. Diagnose the equation choice and find T using k = 1.38 × 10⁻²³ J K⁻¹.

Try this before viewing the solution

Unit: K

View solution step by step

  1. Correct the pairing

    Method

    Use pV = NkT, not pV = NRT.

    Reason

    N counts particles and therefore pairs with the per-particle constant k; R pairs with moles n.

    Working

    N ↔ k, n ↔ R

  2. Rearrange

    Method

    Make temperature the subject.

    Reason

    All other quantities in the particle equation are supplied.

    Working

    T = pV/Nk

  3. Calculate

    Method

    The temperature is approximately 1.2 × 10² K.

    Reason

    Substitution gives 200/1.656 ≈ 121.

    Working

    T = ((1.0 × 10⁵)(2.0 × 10⁻³))/((1.2 × 10²³)(1.38 × 10⁻²³)) ≈ 1.2 × 10² K

Examiner practice 4

Find pressure from pV = nRT

4 marks

Examination question

An ideal gas has n = 0.10 mol in volume V = 2.0 L at T = 350 K. Find the pressure. [4 marks]

Try this before viewing the solution

Unit: Pa

View solution step by step

  1. Convert volume

    1 mark

    Method

    2.0 L = 2.0 × 10⁻³ m³.

    Reason

    The SI gas constant requires volume in cubic metres when pressure is reported in pascals.

    Working

    1 L = 10⁻³ m³

  2. Choose and rearrange

    1 mark

    Method

    Use the mole form and make pressure the subject.

    Reason

    The amount is in moles and R is the matching constant.

    Working

    p = nRT/V

  3. Substitute

    1 mark

    Method

    Insert all quantities in SI units.

    Reason

    This keeps the result in pascals.

    Working

    p = (0.10)(8.31)(350)/(2.0 × 10⁻³)

  4. Report the result

    1 mark

    Method

    The pressure is approximately 1.45 × 10⁵ Pa.

    Reason

    The result has the expected atmospheric order of magnitude.

    Working

    p ≈ 1.45 × 10⁵ Pa

Challenge 5

Find volume from pV = NkT

Minimal support

Independent transfer

A gas has N = 3.0 × 10²² molecules at T = 400 K and p = 1.0 × 10⁵ Pa. Find the volume in both m³ and litres. Take k = 1.38 × 10⁻²³ J K⁻¹.

Try this before viewing the solution

Hints

Hint 1: choose by amount representation
The gas amount is a particle count, so start from pV = NkT.
View solution step by step

  1. Rearrange the particle form

    Method

    Make volume the subject of pV = NkT.

    Reason

    The prompt gives N rather than a mole amount.

    Working

    V = NkT/p

  2. Calculate in SI units

    Method

    The volume is 1.66 × 10⁻³ m³.

    Reason

    Using Pa, K and the SI value of k produces cubic metres.

    Working

    V = ((3.0 × 10²²)(1.38 × 10⁻²³)(400))/(1.0 × 10⁵) = 1.66 × 10⁻³ m³

  3. Change representation

    Method

    The same volume is 1.66 L.

    Reason

    Each cubic metre contains 10³ litres.

    Working

    1.66 × 10⁻³ m³ = 1.66 L

7. Mind Stretchers

Mind stretcher 1: Mixing gases, same containerExtension

A rigid container has volume V and temperature T. It contains n₁ moles of gas 1 and n₂ moles of gas 2 (both ideal). Write an expression for the total pressure p.

Show Answer

Total number of moles is n = n₁ + n₂.

For an ideal mixture, the total pressure is: pV = (n₁ + n₂)RT ⇒ p = ((n₁ + n₂)RT)/V

Mind stretcher 2: Pressure ratio with Celsius given (trap)Extension

A fixed mass of ideal gas is in a rigid container. Its pressure is p₁ at 27°C.

What is the new pressure p₂ if the temperature is increased to 127°C?

Show Answer

In a rigid container, V and n are constant so p ∝ T.

Convert to kelvin: T₁ = 27 + 273 = 300 K, T₂ = 127 + 273 = 400 K p₂/p₁ = T₂/T₁ = 400/300 = 4/3 So p₂ = (4/3)p₁.

Mind stretcher 3: Optional (Enrichment)Extension

A. Combined gas law (derived from pV = nRT)

If n is constant, then pV/T is constant: p₁V₁/T₁ = p₂V₂/T₂

This is often tested as “combined gas law” questions.

B. Internal energy of an ideal gas (beyond this lesson)

In Thermodynamic Systems, you may meet results like U ∝ T for an ideal gas. That relies on kinetic theory and how internal energy is defined, so keep it separate from the equation-of-state skills in this lesson.

8. Practice (Quiz)

Practice (Quiz)

Practice pV = nRT and pV = NkT questions:

A Level Temperature & Ideal Gases Quiz

Continue with the next resource in this course.

Course and syllabus information
Course
GCE A-Level H2 Physics
Edition
GCE A-Level H2 Physics 2027