Kinetic Theory & Equipartition (IPhO Thermo/Stat)
IPhO thermo/stat lesson on kinetic-theory tools: pressure from momentum transfer, thermal speeds, degrees of freedom, and quick heat-capacity estimates.
On this page
Kinetic theory is the bridge between “microscopic particles” and macroscopic laws. In olympiad problems it is mainly a set of fast conversion formulas: pressure to mean-square speed, temperature to typical speed, degrees of freedom to heat capacity, and mean free path to transport scales.
- Number density: n = N/V.
- Pressure-speed link: P = (1/3)nm⟨v²⟩.
- Temperature link: (1/2)m⟨v²⟩ = (3/2)kT so vᵣₘₛ = square root of (⟨v²⟩) = square root of (3kT/m).
- Per mole: replace kN_A by R.
1. Definitions (Must Know)
- Ideal gas: dilute gas where interactions are negligible except during brief collisions; equation of state PV = nRT.
- Thermal speeds: mean speed ⟨v⟩, most probable speed vₘₚ, and rms speed vᵣₘₛ. In IPhO estimates, any of these is usually acceptable up to factors of order 1.
- Degrees of freedom (dof): independent quadratic energy terms in the Hamiltonian (translations, rotations, vibrations in the classical limit).
- Equipartition theorem (classical): each quadratic dof contributes average energy (1/2)kT per molecule.
- Mean free path λ: typical distance between collisions; for hard spheres, λ∼1/(square root of 2 nσ) with collision cross-section σ∼ π d².
2. Key Ideas (What Earns Marks)
- Convert macro to micro fast: from PV = nRT get n = N/V = P/(kT), then plug into P = (1/3)nm⟨v²⟩.
- Use equipartition to get U and heat capacities: count active dof, then U = (f/2)NkT and C_V = (f/2)Nk.
- Know when equipartition fails: at low temperatures, rotational and vibrational modes can freeze out. If the problem hints quantum effects, do not blindly count dof.
- Scaling beats exact integrals: many IPhO questions want proportionalities, not a full Maxwell distribution derivation.
3. Detailed Explanations
A. Pressure from momentum transfer (the one derivation worth owning)
Derivation sketch: P = (1/3) n m mean(v^2)
Consider molecules in a cube of side L so V = L³. For a molecule with x-velocity component vₓ, an elastic collision with the wall reverses vₓ, giving momentum change magnitude Δ pₓ = 2mvₓ.
The time between successive hits on the same wall is Δ t = 2L/|vₓ|, so the average force contribution from that molecule on the wall is Fₓ = (Δ pₓ)/(Δ t) = 2mvₓ/(2L/|vₓ|) = (m vₓ²)/L.
Sum over molecules and divide by wall area A = L²: P = F/A = 1/L³∑ᵢ m v_(x,i)² = n m⟨vₓ²⟩.
For an isotropic gas, ⟨vₓ²⟩ = ⟨v_y²⟩ = ⟨v_z²⟩ = 1/3⟨v²⟩, so P = (1/3)nm⟨v²⟩.
B. Equipartition and heat capacities (classic IPhO table)
If f quadratic degrees of freedom are active: U = (f/2)NkT, C_V = ((∂ U)/(∂ T))_V = (f/2)Nk, and per mole: U = (f/2)nRT, C_V = (f/2)nR, C_P = C_V + nR.
Typical classical counts:
- Monatomic: f = 3 (translations) so C_V = (3/2)R, γ = 5/3.
- Diatomic (room temperature, vibrations frozen): f = 5 (3 translational + 2 rotational) so C_V = (5/2)R, γ = 7/5.
C. Mean free path and collision time
Once you have a typical speed v: τ∼λ/v is a typical collision time, and ν∼ 1/τ is a collision frequency. These show up in viscosity, diffusion, and thermal conduction scaling.
4. Common Mistakes
- Mixing k and R (per molecule vs per mole). Decide early: molecule language uses k, mole language uses R.
- Using vᵣₘₛ but calling it ⟨v⟩ (usually not fatal, but do not mix them inside the same calculation).
- Counting vibrational modes for diatomic molecules at ordinary temperatures without justification.
- Treating mean free path as a geometric size; it depends strongly on density (pressure).
- Forgetting that P = (1/3)nm⟨v²⟩ uses ⟨v²⟩, not ⟨v⟩².
5. Exam Tips
- If you only need a scaling, write v∼ square root of (kT/m) and move on.
- For mixtures at the same temperature, ⟨(1/2)mv²⟩ is the same for all species (translation), so heavier molecules have smaller typical speeds.
- When asked for γ, count dof and use γ = C_P/C_V = 1 + 2/f (for an ideal gas in the classical limit).
- If a question looks like transport, expect λ and a “typical speed” to appear.
6. Worked Examples
Example 1: Typical molecular speed from temperature
Estimate the rms speed of helium atoms at temperature T = 300 K. Use m = 4u and u ≈ 1.66 × 10⁻²⁷ kg.
Worked solution (v_rms = sqrt(3kT/m))
Helium atom mass: m ≈ 4(1.66 × 10⁻²⁷) kg = 6.64 × 10⁻²⁷ kg.
Use vᵣₘₛ = square root of (3kT/m) .
With k ≈ 1.38 × 10⁻²³ J/K and T = 300 K: 3kT ≈ 3(1.38 × 10⁻²³)(300) ≈ 1.24 × 10⁻²⁰ J.
Therefore
Example 2: Speed ratio in a mixture at one temperature
A mixture contains hydrogen molecules (H₂) and oxygen molecules (O₂) at the same temperature. What is the ratio of their rms speeds?
Worked solution (same T, so v_rms scales as 1/sqrt(m))
Since vᵣₘₛ = square root of (3kT/m) at fixed T, vᵣₘₛ(H₂)/vᵣₘₛ(O₂) = square root of (m_O₂/m_H₂) .
Using molecular masses m_H₂ ∝ 2 and m_O₂ ∝ 32, vᵣₘₛ(H₂)/vᵣₘₛ(O₂) = square root of (32/2) = square root of 16 = 4.
Example 3: Heat capacity and gamma from degrees of freedom
Assume a diatomic ideal gas at moderate temperature has f = 5 active degrees of freedom. Find C_V, C_P, and γ per mole.
Worked solution (equipartition)
Per mole: C_V = (f/2)R = (5/2)R, C_P = C_V + R = (7/2)R.
Therefore γ = C_P/C_V = (7/2)/(5/2) = 7/5.
7. Mind Stretchers
- Equipartition predicts that vibrational modes add kT per mode (kinetic plus potential). Why does this fail at low temperature, and what would you expect C_V(T) to do as temperature rises?
- Use kinetic theory to argue why gas pressure depends on ⟨v²⟩, not on ⟨v⟩.
- In a gravitational field, gases have density gradients. Why does thermal equilibrium still have a single temperature (not a temperature gradient)?
Mind-stretcher (solution idea): why heavy species move slower at the same T
Temperature in kinetic theory is tied to average translational kinetic energy per molecule: ⟨(1/2)mv²⟩ = (3/2)kT.
At the same T, the left-hand side must match for all species. Therefore ⟨v²⟩ ∝ 1/m, meaning typical speeds scale as v∼ 1/square root of m.
This is why, in mixtures, lighter molecules have higher thermal speeds and tend to diffuse faster.
8. Practice
- Drill the conversions: PV = nRT, P = (1/3)nm⟨v²⟩, and vᵣₘₛ = square root of (3kT/m).
- For any gas in a problem, decide early: monatomic or diatomic (with rotations), then write f, C_V, and γ on the page.
Syllabus and review details
No official syllabus alignment is listed for this lesson.