5.1ThermalStretch

Ideal Gases and Kinetic Theory

Watch smoke particles under a microscope and they jitter about randomly for no visible reason — direct evidence that the air around them is made of countless invisible, constantly-moving molecules. That single observation, Brownian motion, underpins an entire mathematical model connecting gas pressure to the speed of individual molecules.

45 min Video by ETphysics Module 5.1 Thermal physics

Kinetic Theory and Brownian Motion — A2 Ideal GasWatch the full walkthrough before the notes below.

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What you'll be able to do

Use the ideal gas equation $p V = n R T$
State the assumptions of the kinetic theory model
Use the kinetic theory equation $p V = \frac{1}{3} N m ⟨ c^{2} ⟩$
Relate absolute temperature to average molecular kinetic energy
Explain Brownian motion as evidence for the kinetic theory model

The ideal gas equation

For a fixed mass of gas, pressure, volume and temperature are linked by the ideal gas equation.

p V = n R T

n = number of moles, R = 8.31 J mol⁻¹ K⁻¹, T in kelvin.

1Rearrange:

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

V = \frac{1.5 \times 8.31 \times 280}{1.0 \times 1 0 ^{5}}

Answer

V \approx 0.035

m³

Tip — Temperature MUST be in kelvin — always convert from Celsius first.

The kinetic theory model

Kinetic theory derives gas pressure from molecular collisions, based on assumptions: a large number of molecules in rapid random motion; negligible molecular volume compared to the container; perfectly elastic collisions; negligible collision time; no intermolecular forces except during collisions.

p V = \frac{1}{3} N m ⟨ c^{2} ⟩

Kinetic theory equation.

1Rearrange:

p = \frac{N m ⟨ c ^{2} ⟩}{3 V}

2Numerator:

2.0 \times 1 0^{23} \times 4.8 \times 1 0^{- 26} \times 2.2 \times 1 0^{5} = 2.11 \times 1 0^{3}

p = \frac{2.11 \times 1 0 ^{3}}{3 \times 0.015}

Answer

p \approx 4.69 \times 1 0^{4}

Temperature and molecular kinetic energy

Combining the kinetic theory and ideal gas equations shows average molecular kinetic energy depends only on absolute temperature — the same for any ideal gas.

\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 350

Answer

E_{k} \approx 7.25 \times 1 0^{- 21}

Brownian motion

$Brownian motion$ is the random, erratic movement of small visible particles caused by countless unbalanced collisions with much smaller, faster-moving molecules of the surrounding fluid — the historical evidence that gases and liquids are made of constantly-moving molecules.

Tip — Brownian motion is the visible jittering of a LARGE particle from many small unequal impacts — not the same as the invisible molecular motion itself, which it provides indirect evidence for.

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

Molecular kinetic energy and temperature.

Common mistakes to avoid

Using Celsius temperature in the ideal gas equation.

Always convert to kelvin first — using Celsius directly gives badly wrong answers.

Treating Brownian motion as directly observing molecules.

Brownian motion is the jittering of a much LARGER particle from many random molecular collisions — indirect, not direct, evidence.

Assuming average kinetic energy depends on which gas it is.

At a given temperature, average molecular kinetic energy is the same for any ideal gas — only average speed differs, since lighter molecules move faster for the same energy.

Key takeaways

Ideal gas equation: pV = nRT (temperature always in kelvin).
Kinetic theory assumptions: many molecules, negligible volume, elastic collisions, negligible collision time, no intermolecular forces except during collisions.
Kinetic theory equation: pV = ⅓Nm⟨c²⟩.
Average molecular kinetic energy = 3/2 kT, the same for any ideal gas at a given temperature.
Brownian motion (jittering of visible particles) is evidence for constant random molecular motion.

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Ideal Gases and Kinetic Theory

45 min Video by ETphysics Module 5.1 Thermal physics