Use mean-square and root-mean-square speed
Interpret mean-square speed, calculate root-mean-square speed and use the kinetic-theory pressure relation.
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
How do you calculate root-mean-square speed from mean-square speed?
What you will learn
- Interpret mean-square molecular speed ⟨c²⟩
- Use cᵣₘₛ = √⟨c²⟩
- Relate rms speed and pressure through kinetic theory
Mean-square speed averages the squared molecular speeds
For a collection of molecules with speeds c₁, c₂, …, the mean-square speed ⟨c²⟩ is the average of c₁², c₂², and so on. It is not the square of the average speed in general.
Its unit is m² s⁻². In the pressure relation pV = ⅓Nm⟨c²⟩, this average captures the contribution of all molecules in the gas.
Root-mean-square speed is the square root of the mean square
The root-mean-square speed is cᵣₘₛ = √⟨c²⟩. It has units m s⁻¹ and is a useful single speed that represents the spread of molecular motion in kinetic-theory calculations.
To recover mean-square speed from cᵣₘₛ, square it: ⟨c²⟩ = cᵣₘₛ². Keep the square root and the square in the correct direction.
Pause and explain
If ⟨c²⟩ = 9.0 × 10⁴ m² s⁻², what is cᵣₘₛ?
Worked example
A gas has mean-square molecular speed 1.296 × 10⁵ m² s⁻². Calculate cᵣₘₛ.
Show the worked solution
- Use cᵣₘₛ = √⟨c²⟩.
- Take the square root of 1.296 × 10⁵.
- Give m s⁻¹.
Answer cᵣₘₛ = 360 m s⁻¹.
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Test this topic →From guided practice to a new situation
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.
Calculate cᵣₘₛ when ⟨c²⟩ = 2.5 × 10⁵ m² s⁻².
Give me a hint
Take the square root of both the coefficient and the power of ten.
Compare my reasoning
- cᵣₘₛ = √(2.5 × 10⁵).
- √(2.5 × 10⁵) ≈ 500.
- State m s⁻¹.
cᵣₘₛ = 500 m s⁻¹ (to 2 significant figures).
Look for these in your work
- Square root applied
- Magnitude sensible
- Speed unit stated
A gas has cᵣₘₛ = 420 m s⁻¹. Find its mean-square speed.
Give me a hint
Square the rms speed and keep the squared-speed unit.
Compare my reasoning
- ⟨c²⟩ = cᵣₘₛ².
- ⟨c²⟩ = 420².
- Give m² s⁻².
⟨c²⟩ = 1.764 × 10⁵ m² s⁻².
Look for these in your work
- Rms relation rearranged
- Speed squared
- Squared-speed unit stated
Two gases have the same molecule count, molecular mass and volume. Gas B has twice the rms speed of gas A. Compare their pressures.
Give me a hint
Pressure is proportional to mean-square speed, which is the square of rms speed.
Compare my reasoning
- ⟨c²⟩ = cᵣₘₛ².
- Doubling cᵣₘₛ makes ⟨c²⟩ four times as large.
- With N, m and V fixed, gas B has four times the pressure.
The pressure of gas B is four times that of gas A.
Look for these in your work
- Rms and mean square linked
- Square factor applied
- Pressure ratio derived
Common mistakes
- Reporting mean-square speed in m s⁻¹
- Squaring when the question asks for rms speed
- Treating rms speed as the arithmetic mean speed
What can you explain now?
How do you calculate root-mean-square speed from mean-square speed?
Compare with the explanation
Take the square root
cᵣₘₛ = √⟨c²⟩.
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Make it stick
What is the difference between mean-square speed and root-mean-square speed?
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