Ideal-gas equations, particles and moles
Key idea: H2 Physics lessons on temperature, ideal gases, internal energy and thermodynamic systems.
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The core idea
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Big question: How do the macroscopic gas laws connect particles, moles and state variables?
An ideal gas obeys pV = nRT = NkT. Use n for amount in moles and N for number of molecules, linked by N = nNA. The model assumes negligible molecular volume and intermolecular forces except during brief elastic collisions.
Choose one particle-count language
The ideal-gas equation can be written pV = nRT for n moles or pV = NkT for N molecules. The bridges are N = nNₐ and R = Nₐk. Both forms describe the same state, so never use N with R or n with k.
Use absolute pressure, volume in m³ and temperature in K with SI constants. A change question is often clearest as p₁V₁/T₁ = p₂V₂/T₂ for a fixed amount of gas, but this comes from the equation rather than replacing it.
Check your understanding: A sample contains 0.25 mol. How many molecules does it contain?
N = nNₐ = 0.25(6.02 × 10²³) = 1.51 × 10²³ molecules.
Know what 'ideal' assumes
An ideal gas consists of many point particles in random motion. Their own volume is negligible compared with the container, intermolecular forces are negligible except during brief collisions, and collisions are perfectly elastic.
The model works best at low density and sufficiently high temperature. At high pressure, molecular volume matters; at low temperature, attractive forces matter. State the relevant departure instead of saying only that a gas is 'not ideal'.
Check your understanding: Why can a real gas deviate strongly at high pressure?
Particles are crowded, so their finite molecular volume is no longer negligible compared with the gas volume.
Key ideas to keep
- Pressure must be in pascals, volume in cubic metres and temperature in kelvin for SI constants.
- Do not mix the gas constant R with Boltzmann's constant k.
- A straight p–1/V graph at fixed T tests Boyle behaviour.
See the reasoning
Worked example
Move consistently between particle and mole forms
Question: A vessel contains 3.01 × 10²² molecules at 400 K and 1.00 × 10⁵ Pa. Find the volume using k = 1.38 × 10⁻²³ J K⁻¹, then verify the mole form.
Step 1: Choose the particle equation
Why: The question supplies a number of molecules and Boltzmann's constant.
Working: Use pV = NkT, so V = NkT/p.
Step 2: Substitute in SI units
Why: Pressure, temperature and k are already in compatible SI units.
Working: V = (3.01×10²²)(1.38×10⁻²³)(400)/(1.00×10⁵) = 1.66×10⁻³ m³.
Step 3: Verify with moles
Why: The two ideal-gas forms must describe the same state.
Working: n = N/Nₐ = 0.0500 mol; nRT/p gives the same volume because R = Nₐk.
Answer: V = NkT/p = 1.66 × 10⁻³ m³. N/Nₐ = 0.0500 mol, and nRT/p gives the same volume because R = Nₐk.
Check: The volume is about 1.7 L, a reasonable scale for 0.050 mol near atmospheric pressure.
Use a hint if needed
Practise with support
Try this
At constant volume, an ideal gas changes from 1.2 × 10⁵ Pa at 300 K to 450 K. Find its new pressure.
Hint: Start with pV = NkT and identify what remains fixed.
Check your answer
For fixed N and V, p/T is constant. p₂ = 1.2 × 10⁵(450/300) = 1.8 × 10⁵ Pa.
Now work without the hint
Practise independently
Your turn
A 2.00 mol ideal gas at 350 K occupies 0.0400 m³. Find pressure and show how N, n, k, R and Nₐ are related.
Check your answer
p = nRT/V = 1.45 × 10⁵ Pa. N = nNₐ and R = Nₐk, so Nk = nNₐk = nR.
Avoid these traps
Common mistakes
Common mistake
Celsius can be used directly in pV = NkT.
What is wrong with this reasoning?
Show better thinking
Ideal-gas equations require absolute thermodynamic temperature in kelvin.
Common mistake
The particle and mole forms mix N with R or n with k.
What is wrong with this reasoning?
Show better thinking
Use pV = NkT or pV = nRT, linked by N = nNₐ and R = Nₐk.
Write for the examiner
Exam guidance
List the state variables before and after, then choose one consistent form of the ideal-gas equation.
Exam-style practice [6 marks]
Use pV = NkT to find N for p = 2.0 × 10⁵ Pa, V = 3.0 × 10⁻³ m³ and T = 290 K; then find n.
Plan before you answer
- Select pV = NkT for particle count.
- Rearrange before substituting.
- Convert N to n using Nₐ.
Mark your answer and compare the model
Marking points
Tick each point only if your answer states it clearly.
Model answer
N = pV/(kT) = 1.50 × 10²³ molecules. With Nₐ = 6.02 × 10²³ mol⁻¹, n = 0.249 mol.
Come back in three days
Check what stayed with you
Recall question
A fixed amount of ideal gas doubles both its kelvin temperature and volume. State the pressure factor.
Check the answer
From pV = NkT, p ∝ T/V. Both numerator and denominator double, so pressure is unchanged.
Syllabus and review details
This lesson covers the listed H2 Physics 9478 outcomes. Temperature and ideal-gas ideas lead into the first law of thermodynamics. Use ΔU = Q + W, where W is work done on the system; for expansion against constant external pressure, work done by the gas is pΔV and work done on the gas is −pΔV.
- GCE A-Level H2 PhysicsTopic 12(c) / Topic 12(d) · 2027Checked against the syllabus · partial topic coverageOfficial 9478 syllabus
Course and syllabus information
- Course
- GCE A-Level H2 Physics
- Edition
- GCE A-Level H2 Physics 2027