Determine a gas Mr from pV = nRT
Combine gas volume data with a measured sample mass to determine relative molecular mass.
Section 4 · States of matter
AS Level; Cambridge International Chemistry 9701, 2025–2027 syllabus. Keep pressure, volume, amount and absolute temperature consistent in gas calculations. Written practice is self-checked
Before you begin
- Rearrange a formula and convert between common metric volume units.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
After calculating gas amount n and sample mass m, how do you find molar mass?
What you will learn
- Find amount from pressure, volume and temperature
- Calculate Mr from sample mass and gas amount
- Show consistent unit conversion and significant figures
Find amount before relative molecular mass
Use pV = nRT to calculate the amount of gas in the measured volume. The pressure must be absolute, the temperature in kelvin and the volume compatible with R.
Then use n = m/M to find molar mass M = m/n. Relative molecular mass Mr is dimensionless and has the same numerical value as molar mass in g mol⁻¹ at the precision given.
Check the sample state and units
The gas sample's mass and p, V, T must refer to the same sample. Convert grams to the mass units consistent with the molar mass calculation, and round only the final result to a justified number of significant figures.
If a question supplies a different value or unit for R, use the matching pressure, volume and temperature units rather than silently substituting an SI convention.
Pause and explain
Does relative molecular mass Mr have a unit?
Worked example
A 2.90 g gas sample occupies 2.493 × 10⁻³ m³ at 100 000 Pa and 300 K. Find Mr. Use R = 8.31 J mol⁻¹ K⁻¹.
Show the worked solution
- n = pV/(RT) = 100 000 × 0.002493/(8.31 × 300) = 0.100 mol.
- M = m/n = 2.90/0.100 = 29.0 g mol⁻¹.
- Mr has the same numerical value without units.
Answer Mr = 29.0.
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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 1.60 g gas occupies 1.662 × 10⁻³ m³ at 100 000 Pa and 300 K. Calculate Mr using R = 8.31 J mol⁻¹ K⁻¹.
Give me a hint
Calculate n from the gas equation, then divide sample mass by n.
Compare my reasoning
- n = pV/(RT) = 100 000 × 0.001662/(8.31 × 300) ≈ 0.0667 mol.
- M = 1.60/0.0667 ≈ 24.0 g mol⁻¹.
- Report Mr without a unit.
Mr ≈ 24.0.
Look for these in your work
- Gas amount calculated
- Mass divided by amount
- Dimensionless Mr stated
A gas sample has n = 0.0500 mol and mass 2.20 g. Find its molar mass and Mr.
Give me a hint
Use M = m/n and distinguish the unit-bearing result from the ratio.
Compare my reasoning
- M = 2.20/0.0500.
- M = 44.0 g mol⁻¹.
- Mr has the same numerical value without the unit.
M = 44.0 g mol⁻¹ and Mr = 44.0.
Look for these in your work
- Molar mass correct
- Unit shown for M
- No unit on Mr
Explain why a valid Mr calculation needs the measured mass and p, V, T to describe the same gas sample.
Give me a hint
The equation first turns that sample's state into an amount.
Compare my reasoning
- p, V and T determine the amount of the gas in the measured volume.
- Molar mass is mass divided by amount.
- Using a different sample's mass or state would pair unrelated quantities.
The gas equation calculates the amount for that specific sample. Dividing a mass from another sample by it would not give a valid mass per mole.
Look for these in your work
- Gas amount linked to its state
- M = m/n relationship stated
- Same-sample condition explained
Common mistakes
- Reporting units after dimensionless Mr
- Using mass divided by pV without first finding amount
- Combining mass and gas data from different samples
- Rounding n too early
What can you explain now?
After calculating gas amount n and sample mass m, how do you find molar mass?
Compare with the explanation
M = m/n
Mass divided by amount gives mass per mole.
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Make it stick
How do p, V and T provide n, and how does n combine with mass to provide Mr?
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.