Kinetic Theory and Ideal Gases

Overview

This page develops the microscopic model of matter and applies it to gases. It explains how particle motion gives rise to temperature, pressure, internal energy and the gas laws.

This is one of the most important conceptual sections of Thermal Physics B.

Main skills:

• explain thermal behaviour using particle ideas
• apply gas laws
• use the ideal gas equation
• relate temperature to molecular kinetic energy
• calculate rms speed
• distinguish real gases from ideal gases

Related hub:

Thermal Physics B

Definition

Kinetic theory models matter in terms of particles in constant motion and uses that motion to explain measurable quantities such as pressure, temperature, and internal energy.

Why It Matters

This topic links the microscopic and macroscopic views of thermal physics. If the particle model is weak, gas laws and thermodynamic explanations often become memorised instead of understood.

Key Representations

1. Kinetic Theory of Matter

Core Idea

All matter is made of atoms or molecules in constant motion.

The observable properties of matter arise from:

• motion of particles
• collisions between particles
• intermolecular attractive forces
• particle spacing

Particle Behaviour in States of Matter

Solid

• particles closely packed
• vibrate about fixed positions
• strong intermolecular forces
• fixed shape and volume

Liquid

• particles close together
• move past one another
• weaker effective bonding than solids
• fixed volume, no fixed shape

Gas

• particles far apart
• rapid random motion
• negligible intermolecular forces (approximately)
• no fixed shape or volume

2. Internal Energy

Definition

Internal energy (U) is the total microscopic energy stored in a system.

It is the sum of:

• random kinetic energy of particles
• intermolecular potential energy

$$U=K{E}_{\text{random,total}}+P{E}_{\text{total}}$$

Important Clarification

Internal energy does not include:

• kinetic energy of the whole object moving through space
• gravitational potential energy of the whole object
• macroscopic elastic potential energy unless stated

State Property

Internal energy depends only on the state of the system:

• temperature
• pressure
• volume
• amount of substance

3. Random Kinetic Energy and Temperature

Temperature is related to the average random kinetic energy of particles.

Higher temperature means:

• particles move faster on average
• more energetic collisions
• larger average kinetic energy

For gases:

$$\text{average KE}\propto T$$

where (T) is in Kelvin.

4. Intermolecular Potential Energy

Potential energy depends on particle separation.

When Particles Move Further Apart

• attractive forces are overcome
• potential energy increases

When Particles Move Closer

• attraction stronger
• potential energy decreases

This explains why phase changes often involve changes in potential energy.

5. Particle View of Changes of State

Melting

During melting:

• particles gain energy
• vibrations become large enough to break fixed lattice arrangement
• particles can move past each other

Temperature remains constant during melting of a pure substance.

Energy supplied increases potential energy.

Boiling

During boiling:

• particles throughout liquid gain enough energy to separate widely
• bubbles form inside liquid

Temperature remains constant during boiling.

Why (L_v > L_f)

Specific latent heat of vaporisation is usually larger than fusion because:

• particles separate much further
• more intermolecular attraction must be overcome
• gas expansion may do work on surroundings

6. Cooling by Evaporation

Evaporation occurs from the liquid surface.

Fastest surface molecules escape first.

Therefore remaining liquid has lower average kinetic energy.

Hence temperature falls.

Everyday Examples

• sweating cools body
• perfume evaporates quickly
• alcohol wipes feel cold

Exam Tip

Evaporation can occur below boiling point.

Boiling occurs throughout liquid at fixed boiling temperature.

7. Ideal Gas Assumptions

An ideal gas is a simplified model gas.

Assumptions:

1. molecules are point particles with negligible volume
2. no intermolecular forces except during collision
3. collisions are perfectly elastic
4. molecules move randomly in straight lines between collisions
5. collision time is negligible

When Real Gases Behave Ideally

Most closely at:

• high temperature
• low pressure

8. Gas Pressure from Molecular Collisions

Gas molecules collide with container walls.

Each collision changes momentum.

Force is rate of change of momentum:

$$F=\frac{\Delta p}{\Delta t}$$

Pressure is force per unit area:

$$P=\frac{F}{A}$$

Therefore Pressure Increases If:

• molecules move faster (higher temperature)
• more molecules are present
• volume is smaller (more frequent collisions)

9. Boyle’s Law

For fixed mass of gas at constant temperature:

$$P\propto \frac{1}{V}$$ $$P_1{V}_1=P_2{V}_2$$

Microscopic View

Reducing volume causes:

• more frequent wall collisions
• pressure increases

10. Charles’ Law

For fixed mass of gas at constant pressure:

$$V\propto T$$ $$\frac{V_1}{T_1}=\frac{V_2}{T_2}$$

Microscopic View

Higher temperature gives faster molecules.

To maintain same pressure, volume increases.

11. Pressure Law

For fixed mass of gas at constant volume:

$$P\propto T$$ $$\frac{P_1}{T_1}=\frac{P_2}{T_2}$$

Microscopic View

Faster molecules hit walls harder and more often.

So pressure rises.

12. Ideal Gas Equation

Combining all gas laws:

$$PV=nRT$$

Where:

• (P): pressure
• (V): volume
• (n): number of moles
• (R): molar gas constant
• (T): Kelvin temperature

Alternative form:

$$PV=NkT$$

Where:

• (N): number of molecules
• (k): Boltzmann constant

13. Mole Concept

Avogadro Constant

$$N_A=6.02\times 10^{23}{\text{mol}}^{-1}$$

One mole contains (N_A) particles.

Conversion

$$N=n{N}_A$$

Constant Relation

Comparing the two gas equations:

$$PV=nRT$$ $$PV=NkT$$

gives:

$$R=N_{A}k$$

14. Internal Energy of an Ideal Gas

For an ideal gas:

• intermolecular potential energy is negligible

Hence internal energy is kinetic only.

For monatomic ideal gas:

$$U=\frac{3}{2}NkT=\frac{3}{2}nRT=\frac{3}{2}PV$$

Key Result

For fixed amount of ideal gas:

$$U\propto T$$

So if temperature is unchanged:

$$\Delta U=0$$

15. RMS Speed

Definition

Root mean square speed:

$$v_{\text{rms}}=\sqrt{\frac{v_1^2+v_2^2+\cdots +v_N^2}{N}}$$

Formula

$$\frac{1}{2}m{v}_{\text{rms}}^2=\frac{3}{2}kT$$

Hence:

$$v_{\text{rms}}=\sqrt{\frac{3kT}{m}}$$

or

$$v_{\text{rms}}=\sqrt{\frac{3RT}{M}}$$

Where:

• (m): mass of one molecule
• (M): molar mass

Trends

• higher temperature → larger rms speed
• lighter molecules → larger rms speed

16. Worked Examples

Example 1: Ideal Gas Equation

A gas occupies (2.5\times10^{-3},\text{m}^3) at pressure (1.5\times10^5,\text{Pa}) and (300,\text{K}).

Find number of moles.

$$PV=nRT$$ $$n=\frac{PV}{RT}=\frac{(1.5\times 10^5)(2.5\times 10^{-3})}{8.31(300)}=0.150\text{ mol}$$

Example 2: Boyle’s Law

Gas volume changes from (4.0\times10^{-3},\text{m}^3) to (2.0\times10^{-3},\text{m}^3) at constant temperature.

Initial pressure (1.0\times10^5,\text{Pa}).

$$P_1{V}_1=P_2{V}_2$$ $$P_2=\frac{P_1{V}_1}{V_2}=\frac{(1.0\times 10^5)(4.0)}{2.0}=2.0\times 10^5\text{ Pa}$$

Example 3: RMS Speed Ratio

If temperature rises from (300,K) to (1200,K),

$$v_{\text{rms}}\propto \sqrt{T}$$ $$\frac{v_2}{v_1}=\sqrt{\frac{1200}{300}}=2$$

RMS speed doubles.

Links

• Thermal Physics B
• First Law and Thermodynamic Processes
• p-V Diagrams and Cycles
• Thermal Physics B Common Exam Traps

Summary

Kinetic theory and the ideal-gas model explain why gases obey simple laws. The main discipline is to connect every macroscopic result back to particle motion, collisions, and temperature.