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Kinetic Theory

Gas laws, molecular speeds and degrees of freedom. Lighter than before; the degrees-of-freedom table answers a large part of it.

Questions in the bank
110
q/paper in 2025–26
0.68
Numeric answer
10%
Notes pages
5

Tier: Long tail

When you’ll see it

The molecules of a gas: pressure or kinetic energy from a temperature, an rms or average speed, a mean free path, degrees of freedom, or γ of a mixture.

How this chapter is tested

Kinetic Theory sits in the long tail and has shrunk on the recent papers. Most questions are short: change every temperature to kelvin, then apply one proportion, such as pressure with temperature, kinetic energy with temperature, or the rms speed with √(T/M).

The second half counts degrees of freedom. Each one holds ½kT, so f fixes Cv = fR/2, Cp = Cv + R and γ = 1 + 2/f, and a vibrational mode adds two, not one. The longer questions track moles through joined vessels, or replace a mixture by one equivalent gas whose f or Cv is a mole-weighted average, with γ found last.

The chapter feeds Thermodynamics directly: the same Cv gives ΔU there, and the same γ sets the adiabat.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • Ideal gas laws

    PV = nRT = NkT with T in kelvin; P₁V₁/T₁ = P₂V₂/T₂ for a fixed amount; when vessels are joined, add the moles, not the pressures.

  • Pressure and kinetic energy

    P = ⅓ρv²rms, so PV is two-thirds of the translational energy; the mean energy per molecule, (3/2)kT, depends on T alone.

  • Molecular speeds and mean free path

    Speeds go as √(T/M): rms √(3RT/M), average √(8RT/πM), most probable √(2RT/M); the mean free path is 1/(√2 π d² n).

  • Degrees of freedom

    Count translations, rotations by the molecule's shape, and two for each vibrational mode; then Cv = fR/2, Cp = Cv + R and γ = 1 + 2/f.

  • Internal energy and mixtures

    U = n(f/2)RT; a mixture's f and Cv are mole-weighted averages, and gases mixed at different temperatures share their total energy, weighted by nf.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • Celsius in a ratio

    Going from 27 °C to 54 °C is 300 K to 327 K, not a doubling. Convert to kelvin before every ratio.

  • A vibrational mode counted once

    A vibration stores kinetic and potential energy, so one mode adds 2 to f. Counting it as 1 gives the wrong Cv and γ.

  • Averaging γ directly

    For a mixture, average f or Cv by moles first, then find γ. The mean of the two γ values is close to the right answer but not equal to it.

  • Rms speed for average speed

    The average speed is √(8RT/πM) and the rms speed √(3RT/M). They are close, and both appear among the options.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each page. Read the notes once, then drill page by page below.

Kinetic Theory notes

Drill every Kinetic Theory question

110 questions from the bank, across 5 subtopics.

Drill the 110 questions

Drill one subtopic at a time

The 5 subtopics, in teaching order.

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