B.3 Kinetic theory of gases

IB Physics Higher Level· Theme B: Particulate nature of matter, Unit 2, Topic B.3· 15 min read

1. 1. Core Assumptions of the Kinetic Model★☆☆☆☆⏱ 3 min

The kinetic theory models an ideal gas as a collection of tiny particles moving according to Newtonian mechanics. The model relies on five key simplifying assumptions that hold well for real gases at low pressure and high temperature.

📘 Definition

Ideal Gas (Kinetic Model)

A gas that obeys all core assumptions of the kinetic model, and follows the ideal gas law $p V = n R T$ under all conditions.

Example:

Helium at room temperature and atmospheric pressure approximates closely to an ideal gas.

The number of molecules is large, so statistical averaging is valid.
The volume of individual molecules is negligible compared to the total volume of the gas.
Molecules move randomly at constant speeds between collisions.
All collisions (between molecules, and with container walls) are perfectly elastic.
There are no intermolecular forces between molecules except during collisions.

Exam tip:

Deviations from ideal gas behavior in real gases are always linked to failures of these assumptions: high pressure means molecular volume is not negligible, low temperature means intermolecular forces are significant.

2. 2. Derivation of the Kinetic Theory Equation★★★☆☆HL only⏱ 6 min

We can derive the equation relating pressure, volume and molecular speed by considering molecules bouncing between the walls of a cubical container. This full derivation is a common 5-6 mark exam question, so you need to memorize every step.

🔬 Derivation

Goal:

Derive the kinetic theory equation of state for an ideal gas

Starting from:

Newton's laws of motion for a colliding molecule

1
Consider one molecule of mass $m_{0}$ moving along the x-axis with speed $v_{x}$ towards a container wall of side length $L$ . The change in momentum for the molecule after an elastic collision with the wall is:
2
$\Delta p_{\text{molecule}} = -m_0 v_x - m_0 v_x = -2m_0 v_x$
3
By Newton's third law, the momentum transferred to the wall is $+ 2 m_{0} v_{x}$ . The time between successive collisions with the same wall is:
4
$\Delta t = \frac{2L}{v_x}$
5
Force is the rate of change of momentum, so force from the single molecule on the wall is:
6
$F_1 = \frac{\Delta p}{\Delta t} = \frac{2m_0 v_x}{2L / v_x} = \frac{m_0 v_x^2}{L}$
7
Sum over all $N$ molecules, and use symmetry: $⟨ v_{x}^{2} ⟩ = ⟨ v_{y}^{2} ⟩ = ⟨ v_{z}^{2} ⟩ = \frac{1}{3} ⟨ v^{2} ⟩$ .
8
Pressure $p = F / A$ , where $A = L^{2}$ and total container volume $V = L^{3}$ . Substitute to get:

Result:

pV = \frac{1}{3} N m_0 \langle v^2 \rangle

3. 3. Temperature and Average Molecular Kinetic Energy★★☆☆☆⏱ 4 min

We can connect the kinetic theory equation to the ideal gas law to find a fundamental relation between temperature and the kinetic energy of gas molecules.

📐 Worked Example

Show that the average kinetic energy of an ideal gas molecule is $\frac{3}{2} k_{B} T$ .

1
Start by writing both the kinetic theory equation and the ideal gas law in terms of Boltzmann constant $k_{B}$ (where $R = N_{A} k_{B}$ and $N = n N_{A}$ ):
2
$pV = \frac{1}{3} N m_0 \langle v^2 \rangle \quad \text{and} \quad pV = N k_B T$
3
Equate the two expressions for $p V$ and cancel the common term $N$ :
4
$k_B T = \frac{1}{3} m_0 \langle v^2 \rangle$
5
Rearrange to get average kinetic energy $⟨ E_{k} ⟩ = \frac{1}{2} m_{0} ⟨ v^{2} ⟩$ :
6
$\langle E_k \rangle = \frac{1}{2} m_0 \langle v^2 \rangle = \frac{3}{2} k_B T$

This is one of the most important results of kinetic theory: it proves that absolute temperature is a direct measure of the average random kinetic energy of the molecules of an ideal gas.

✓ Quick check

Test your understanding:

Two different ideal gases are at the same temperature. What must be true?
- Molecules of both gases have the same average speed
- Molecules of both gases have the same average kinetic energy
- Molecules of both gases have the same rms speed
- Molecules of both gases have the same mass
Reveal answer
1 —
Correct! Equal absolute temperature means equal average kinetic energy per molecule, regardless of gas type. Heavier molecules have lower average speed at the same temperature.

4. 4. Calculating Root-Mean-Square Speed★★☆☆☆⏱ 4 min

✓ Calculator OK

📘 Definition

Root-mean-square (rms) speed

v_{\text{rms}} = \sqrt{\langle v^2 \rangle}

A convenient statistical measure of the average speed of molecules in a gas, calculated from the mean of the squares of individual molecular speeds.

📐 Worked Example

Calculate the rms speed of nitrogen molecules at 20°C, given molar mass $M = 0.028 kg mol^{- 1}$ and $R = 8.31 J mol^{- 1} K^{- 1}$ .

1
First convert temperature to Kelvin:
2
$T = 20 + 273 = 293 \text{ K}$
3
Rearrange the kinetic theory equation to get $v_{rms}$ in terms of molar mass $M = N_{A} m_{0}$ :
4
$v_{\text{rms}} = \sqrt{\frac{3RT}{M}}$
5
Substitute values and calculate:
6
$v_{\text{rms}} = \sqrt{\frac{3 \times 8.31 \times 293}{0.028}} \approx 510 \text{ m s}^{-1}$

rms speed is proportional to the square root of temperature, and inversely proportional to the square root of molar mass. So lighter gases have much higher average molecular speeds at the same temperature.

5. Common Pitfalls

Wrong move:

Claiming all molecules in a gas have the same speed at a given temperature

Why:

Kinetic theory only fixes the average kinetic energy, individual molecules have a wide range of speeds

Correct move:

State that average kinetic energy (and thus average rms speed for a given gas) is fixed by temperature

Wrong move:

Using Celsius temperature instead of Kelvin in calculations

Why:

All kinetic theory equations rely on absolute (Kelvin) temperature; Celsius gives a drastically wrong result

Correct move:

Always add 273 to Celsius temperatures before substituting into any kinetic theory formula

Wrong move:

Using molar mass in $g mol^{- 1}$ instead of $kg mol^{- 1}$

Why:

SI units for energy and mass require kilograms, so this gives a speed ~30 times smaller than the correct value

Correct move:

Convert molar mass to $kg mol^{- 1}$ by dividing by 1000 before calculation

Wrong move:

Claiming temperature is proportional to total kinetic energy of a gas

Why:

Temperature is proportional to average kinetic energy per molecule, not total. A larger volume of gas has more total energy but the same temperature

Correct move:

Remember temperature is proportional to average kinetic energy per molecule, not total kinetic energy of the whole sample

6. Quick Reference Cheatsheet

Quantity	Formula	Key Notes
Kinetic theory equation	$p V = \frac{1}{3} N m_{0} ⟨ v^{2} ⟩$	N = number of molecules
Average KE per molecule	$⟨ E_{k} ⟩ = \frac{3}{2} k_{B} T$	Same for all gases at same T
rms speed	$v_{rms} = \frac{3 R T}{M}$	M = molar mass in kg mol⁻¹
Core assumptions	5 key assumptions	Common 1-2 mark exam question

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2025 · 1
Identify assumption of kinetic model
2024 · 2
Derive kinetic theory equation
2023 · 1
Calculate rms speed of molecules

Going deeper

SyllabusOfficial IB Physics 2025 Topic B.3Full syllabus requirements

What's Next

Kinetic theory of gases forms the microscopic foundation for all thermal physics topics in IB HL. It connects the random motion of individual particles to the macroscopic properties of pressure and temperature that you measure experimentally. The concepts you learned here will be extended when you study the Maxwell-Boltzmann molecular speed distribution, and are foundational for understanding thermodynamics, entropy, and phase changes later in the course. Mastery of the derivation of the kinetic theory equation and rms speed calculations is frequently tested in long answer questions on Paper 2, so it is important to practice these skills.