3.6.2.2ThermalStretch

Ideal Gases and Kinetic Theory

Watch pollen grains under a microscope and you’ll see them jitter about randomly, for no obvious reason — direct, visible proof that the water around them is made of invisibly small, constantly-moving molecules. From this single idea, a full mathematical model of gas behaviour follows: one that connects the everyday gas laws to the microscopic motion of individual molecules, and reveals that temperature itself is just a measure of how fast those molecules are moving.

45 min Video by Science Shorts 3.6.2 Thermal physics

Kinetic Theory of Gases — A-level PhysicsWatch the full walkthrough before the notes below.

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What you'll be able to do

Use Boyle’s law and the ideal gas equation $p V = n R T$
State the basic assumptions of the kinetic theory model of an ideal gas
Use the kinetic theory equation $p V = \frac{1}{3} N m ⟨ c^{2} ⟩$
Relate absolute temperature to the average kinetic energy of a gas molecule
Explain Brownian motion as evidence for the kinetic theory model

The ideal gas equation

For a fixed mass of gas at constant temperature, $Boyle’s law$ states that pressure and volume are inversely proportional ( $p V = constant$ ). Combining this with how pressure and volume separately depend on temperature gives the full $ideal gas equation$ , valid for a fixed amount of gas behaving ideally.

p V = n R T

n = number of moles, R = molar gas constant (8.31 J mol⁻¹ K⁻¹), T in kelvin.

1Rearrange

p V = n R T

V = \frac{n R T}{p}

V = \frac{2.0 \times 8.31 \times 300}{1.0 \times 1 0 ^{5}}

V = \frac{4986}{1.0 \times 1 0 ^{5}}

Answer

V \approx 0.050

m³ (50 litres)

Tip — Temperature MUST be in kelvin in every gas-law equation — always convert from Celsius first (T = θ + 273) before substituting.

The kinetic theory model

Kinetic theory explains gas pressure and the gas laws entirely from the mechanics of individual molecules colliding with the walls of their container, based on a set of simplifying assumptions: a large number of identical molecules in random, rapid motion; the volume of the molecules themselves is negligible compared with the volume of the container; collisions between molecules (and with the walls) are perfectly elastic (no kinetic energy is lost overall); the time of a collision is negligible compared with the time between collisions; and there are no intermolecular forces except during collisions.

From these assumptions, the pressure a gas exerts can be derived directly from the momentum change of molecules bouncing off the container walls, giving the kinetic theory equation — connecting the macroscopic, measurable pressure and volume directly to the number, mass and mean square speed of the underlying molecules.

p V = \frac{1}{3} N m ⟨ c^{2} ⟩

N = number of molecules, m = mass of one molecule, ⟨c²⟩ = mean square speed.

1Rearrange

p V = \frac{1}{3} N m ⟨ c^{2} ⟩

p = \frac{N m ⟨ c ^{2} ⟩}{3 V}

p = \frac{3.0 \times 1 0 ^{23} \times 5.0 \times 1 0 ^{- 26} \times 2.5 \times 1 0 ^{5}}{3 \times 0.020}

3Numerator:

3.0 \times 1 0^{23} \times 5.0 \times 1 0^{- 26} = 1.5 \times 1 0^{- 2}

; then

\times 2.5 \times 1 0^{5} = 3750

p = \frac{3750}{0.060} = 62500

Pa.

Answer

p \approx 6.25 \times 1 0^{4}

Temperature and molecular kinetic energy

Combining the kinetic theory equation with the ideal gas equation reveals something remarkable: the average kinetic energy of a single gas molecule depends $only$ on the absolute temperature, not on the type of gas, its pressure, or its volume. Absolute temperature is, quite literally, a direct measure of the average random kinetic energy of the particles in a substance.

\frac{1}{2} m ⟨ c^{2} ⟩ = \frac{3}{2} k T

k = Boltzmann constant (1.38 × 10⁻²³ J K⁻¹).

1Use

E_{k} = \frac{3}{2} k T

E_{k} = 1.5 \times 1.38 \times 1 0^{- 23} \times 300

Answer

E_{k} \approx 6.21 \times 1 0^{- 21}

Tip — This is exactly why absolute zero (0 K) is a genuine physical limit, not just an arbitrary reference point: it is the temperature at which the average kinetic energy of every particle would fall to zero.

Brownian motion

$Brownian motion$ is the continuous, random, jittery movement of small but visible particles (like smoke particles in air, viewed through a microscope, or pollen grains in water) caused by many, many random collisions with the much smaller, much faster-moving molecules of the surrounding fluid. Because these collisions arrive from random directions with random impact forces at any instant, the visible particle receives an unbalanced net force at random moments, producing the erratic zig-zag path observed.

Brownian motion is important historically and conceptually because it was the first direct, visible evidence that gases (and liquids) really are made of vast numbers of individual, constantly-moving molecules — a claim kinetic theory otherwise relies on purely as an assumption.

Tip — Don’t confuse Brownian motion (the visible jittering of a LARGE particle, caused by many unequal molecular impacts) with the invisible molecular motion itself — Brownian motion is the indirect evidence for that underlying molecular motion, not the same thing as it.

Equation recap

p V = n R T

Ideal gas equation.

p V = \frac{1}{3} N m ⟨ c^{2} ⟩

Kinetic theory equation.

\frac{1}{2} m ⟨ c^{2} ⟩ = \frac{3}{2} k T

Average molecular kinetic energy and absolute temperature.

Common mistakes to avoid

Using temperature in Celsius in any ideal gas law calculation.

Always convert to kelvin first (T = θ + 273) — using Celsius directly gives completely wrong answers, especially near 0°C.

Treating Brownian motion as the direct observation of molecules moving.

Brownian motion is the visible jittering of a much LARGER particle, caused by many random molecular collisions — it is indirect evidence for molecular motion, not molecules seen directly.

Assuming average kinetic energy of gas molecules depends on which gas it is.

At a given absolute temperature, average molecular kinetic energy (½m⟨c²⟩ = 3/2 kT) is the same for any ideal gas — only average SPEED differs, because lighter molecules need to move faster for the same kinetic energy.

Key takeaways

Ideal gas equation: pV = nRT (temperature always in kelvin).
Kinetic theory model assumptions: many molecules, negligible molecular volume, elastic collisions, negligible collision time, no intermolecular forces except during collisions.
Kinetic theory equation: pV = ⅓Nm⟨c²⟩, linking pressure to molecular mass and mean square speed.
Average molecular kinetic energy = 3/2 kT — absolute temperature is a direct measure of average molecular kinetic energy, independent of the type of gas.
Brownian motion (jittering of visible particles from random molecular collisions) is evidence for the constant random motion of molecules.

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Ideal Gases and Kinetic Theory

45 min Video by Science Shorts 3.6.2 Thermal physics