Kinetic model, pressure derivation and molecular energy
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 countless molecular collisions create a steady pressure?
Gas pressure arises from molecular momentum changes at the walls. Summing elastic collisions for random motion gives pV = (1/3)Nm⟨c²⟩. Combining this with pV = NkT shows mean translational kinetic energy is (3/2)kT, so absolute temperature measures mean molecular kinetic energy in the ideal model.
Follow one molecule to one wall
For a molecule of mass m with x-component cₓ in a cube of side L, an elastic collision reverses cₓ and changes its momentum by 2mcₓ. It returns to the same wall after time 2L/cₓ, so its mean force contribution is mcₓ²/L.
Summing over N molecules and dividing by wall area L² gives pV = Nm⟨cₓ²⟩. Squaring matters: molecules moving in opposite directions give the same positive pressure contribution.
Check your understanding: Why does the derivation contain speed squared?
Impulse is proportional to cₓ and collision frequency is also proportional to cₓ, so their product is proportional to cₓ².
Use random motion in three dimensions
For isotropic random motion, ⟨cₓ²⟩ = ⟨cᵧ²⟩ = ⟨c_z²⟩ and their sum is ⟨c²⟩. Each component therefore contributes one third, giving pV = (1/3)Nm⟨c²⟩.
Comparing with pV = NkT yields ½m⟨c²⟩ = 3kT/2. This is mean translational kinetic energy per molecule. If T quadruples, mean energy quadruples but root-mean-square speed only doubles.
Check your understanding: At one temperature, do all molecules have the same speed?
No. The equation concerns a mean squared speed over a distribution of molecular speeds.
Key ideas to keep
- Use mean square speed ⟨c²⟩, not the square of mean speed.
- Individual molecules have a range of speeds even at one temperature.
- The factor one third comes from three-dimensional random motion.
See the reasoning
Worked example
Derive pressure from collision mechanics
Question: Derive the pressure contribution of molecules moving between opposite walls of a cube, then obtain the three-dimensional result.
Step 1: Analyse one wall collision
Why: Pressure begins with momentum transferred to a wall.
Working: Elastic reversal changes x-momentum by 2mcₓ; successive hits on that wall are 2L/cₓ apart.
Step 2: Find mean force and pressure
Why: Mean force is impulse rate and pressure is force per area.
Working: F̅ = mcₓ²/L. Summing and dividing by L² gives pV = Nm⟨cₓ²⟩.
Step 3: Apply isotropy
Why: Random motion has equal mean-square components in three perpendicular directions.
Working: ⟨cₓ²⟩ = ⅓⟨c²⟩, hence pV = ⅓Nm⟨c²⟩.
Answer: For one molecule, a wall collision changes x-momentum by 2mcₓ and successive hits on that wall are 2L/cₓ apart, giving mean force mcₓ²/L. Summing and dividing by wall area L² gives pV = Nm⟨cₓ²⟩. Random isotropic motion gives equal component means and ⟨cₓ²⟩ = ⅓⟨c²⟩, hence pV = ⅓Nm⟨c²⟩.
Check: The squared component prevents opposite molecular velocities from cancelling their pressure contributions.
Use a hint if needed
Practise with support
Try this
The thermodynamic temperature of a monatomic ideal gas quadruples. State the factors for mean translational kinetic energy and rms speed.
Hint: Separate a squared speed from speed.
Check your answer
Mean translational kinetic energy quadruples because it is 3kT/2. Since ½m⟨c²⟩ ∝ T, rms speed √⟨c²⟩ doubles.
Now work without the hint
Practise independently
Your turn
Starting from a molecule's elastic wall collision, explain why gas pressure depends on mean squared speed rather than mean speed.
Check your answer
Each impulse is proportional to cₓ and collision frequency is also proportional to cₓ, so force is proportional to cₓ². Summing yields pV = Nm⟨cₓ²⟩ = ⅓Nm⟨c²⟩; opposite velocities do not cancel when squared.
Avoid these traps
Common mistakes
Common mistake
Random molecular velocities cancel, so gas pressure is zero.
What is wrong with this reasoning?
Show better thinking
Opposite velocities cancel in the mean velocity, but pressure depends on mean squared components, which are positive.
Common mistake
Mean molecular speed is directly proportional to T.
What is wrong with this reasoning?
Show better thinking
Mean translational kinetic energy and mean squared speed are proportional to T; rms speed is proportional to √T.
Write for the examiner
Exam guidance
In a derivation, state the assumptions and explain each averaging step rather than quoting only the final equation.
Exam-style practice [8 marks]
State the kinetic-theory assumptions and derive pV = ⅓Nm⟨c²⟩ far enough to identify the isotropy step.
Plan before you answer
- State the molecular assumptions.
- Follow one collision to a mean wall force.
- Identify the isotropy step explicitly.
Mark your answer and compare the model
Marking points
Tick each point only if your answer states it clearly.
Model answer
Brief elastic wall collisions give pV = Nm⟨cₓ²⟩. Random isotropic motion makes ⟨cₓ²⟩ = ⟨cᵧ²⟩ = ⟨c_z²⟩ and their sum ⟨c²⟩, so each is one third. The model assumes point particles, negligible forces between collisions and random Newtonian motion.
Come back in three days
Check what stayed with you
Recall question
At 300 K, find mean translational kinetic energy per molecule using k = 1.38 × 10⁻²³ J K⁻¹.
Check the answer
Mean energy = 3kT/2 = 1.5(1.38 × 10⁻²³)(300) = 6.21 × 10⁻²¹ J.
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(e) / Topic 12(f) / Topic 12(g) · 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