Use the ideal-gas equation of state
Recognise the ideal-gas proportionality, use both equation-of-state forms and relate R to the Boltzmann constant.
Section 15 · Ideal gases
A Level; Cambridge International Physics 9702, 2025–2027 syllabus, Topic 15. These original explanations and calculations are study aids; full timed assessment and human subject-teacher review remain outstanding.
Before you begin
- Use pressure, volume, temperature in kelvins, energy and scientific notation.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
For a fixed amount of ideal gas, which temperature belongs in pV ∝ T?
What you will learn
- Recognise an ideal gas from pV ∝ T for fixed amount of gas
- Use pV = nRT and pV = NkT
- Use k = R/NA
An ideal gas has a simple equation of state
For a fixed amount of gas, an ideal gas obeys pV ∝ T, where T is thermodynamic temperature in kelvins. The proportionality fails if the amount changes, so keep n fixed when using this form as a relationship between states.
The equation of state is pV = nRT, where p is pressure in pascals, V is volume in cubic metres, n is amount in moles, R is the molar gas constant, and T is in kelvins.
Count particles or use moles consistently
The same equation can be written pV = NkT, with N the number of molecules and k the Boltzmann constant. Use either n with R or N with k; do not mix the two forms.
The constants are related by k = R/NA. With R = 8.31 J K⁻¹ mol⁻¹ and NA = 6.02 × 10²³ mol⁻¹, k is about 1.38 × 10⁻²³ J K⁻¹.
Pause and explain
Which equation matches particle count N?
Worked example
A gas at 300 K has pressure 1.00 × 10⁵ Pa and volume 0.0249 m³. Estimate the amount of gas.
Show the worked solution
- Use n = pV/(RT).
- Substitute the SI values and R = 8.31 J K⁻¹ mol⁻¹.
- Round to a sensible number of significant figures.
Answer n = (1.00 × 10⁵ × 0.0249)/(8.31 × 300) ≈ 1.00 mol.
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Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A gas has p = 1.00 × 10⁵ Pa, V = 8.31 × 10⁻³ m³ and T = 300 K. Find n using R = 8.31 J K⁻¹ mol⁻¹.
Give me a hint
Use n = pV/(RT) and retain SI units.
Compare my reasoning
- n = pV/(RT).
- pV = 831 J and RT = 2493 J mol⁻¹.
- Divide and state moles.
n ≈ 0.333 mol.
Look for these in your work
- Equation rearranged
- pV and RT evaluated
- Amount unit stated
Estimate k from R = 8.31 J K⁻¹ mol⁻¹ and NA = 6.02 × 10²³ mol⁻¹.
Give me a hint
Use k = R/NA; the mole units cancel.
Compare my reasoning
- k = 8.31/(6.02 × 10²³).
- Evaluate the power of ten.
- Give J K⁻¹.
k ≈ 1.38 × 10⁻²³ J K⁻¹.
Look for these in your work
- Constant relation used
- Mole units cancel
- Boltzmann unit stated
A fixed sample of ideal gas is heated from 300 K to 450 K at constant volume. Compare its final and initial pressures.
Give me a hint
At fixed volume and fixed amount, p is proportional to absolute temperature.
Compare my reasoning
- p₂/p₁ = T₂/T₁.
- p₂/p₁ = 450/300 = 1.5.
- The final pressure is 1.5 times the initial pressure.
The pressure increases by a factor of 1.5.
Look for these in your work
- Fixed state variables identified
- Kelvin ratio used
- Pressure ratio interpreted
Common mistakes
- Using Celsius temperature in the gas equation
- Changing the amount while applying pV ∝ T
- Combining N with R or n with k
What can you explain now?
For a fixed amount of ideal gas, which temperature belongs in pV ∝ T?
Compare with the explanation
Thermodynamic temperature in kelvins
The proportionality uses absolute temperature.
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Make it stick
When should you use pV = nRT, and how is it related to pV = NkT?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.