Kinetic theory: five rules for a perfect gas

The problem

A room full of air holds an unimaginable number of molecules, all moving, all colliding. Tracking even one of them is hopeless. So how can physics say anything exact about a gas at all?

The trick is to agree on a simple model first: five rules about what the molecules are like. A gas that follows them exactly is an ideal gas. From these rules alone, pressure and temperature can be worked out.

The kinetic theory of gases describes a gas as molecules in motion. Gas pressure arises from particles colliding with each other and with the walls of the container. The theory applies to ideal gases.

Key terms

Tiny and far apart

A gas consists of a very large number of tiny spherical particles, far apart compared with their size. The volume of the particles themselves is negligible compared with the volume of the gas.

Random, elastic motion

The particles are in constant, rapid motion in random directions. Their collisions with the container walls are perfectly elastic, so no kinetic energy is lost.

No forces between

There are no forces of attraction or repulsion between gas particles, except when they collide. Between collisions each particle moves in a straight line.

Energy and temperature

The average kinetic energy of the gas particles depends on the temperature of the gas. Heat the gas and the particles, on average, move faster.

Short questions with model answers

1. Q1. Which condition makes most gases behave nearly ideally: low or high temperature, low or high pressure? Explain using the assumptions.

   • Low pressure: particles far apart, their volume negligible.
   • High temperature: particles fast, attractions unimportant.

   High temperature and low pressure (MCQ 13.1, option b).

2. Q2. On reducing the volume of a gas at constant temperature, its pressure increases. Explain on the basis of kinetic theory.

   • Constant temperature → same average speed.
   • Smaller box → shorter trips → more wall collisions per second.

   More frequent collisions with the walls mean a larger force per unit area: higher pressure.

3. Q3. A gas enclosed in a container is heated. What is the effect on the gas molecules and its pressure?

   • Temperature rises → average K.E. rises.
   • Faster molecules hit the walls harder and more often.

   The molecules speed up, so the pressure increases.

Common mistakes

• ✗ Choosing “molecules are tiny hard spheres undergoing inelastic collisions” as a kinetic theory assumption.

  ✓ The collisions are perfectly elastic. If they were inelastic, the gas would lose kinetic energy with every hit and cool down by itself.

• ✗ Saying the volume of a gas is the total volume of its molecules.

  ✓ The gas volume is the space it fills, the container. The molecules' own volume is negligible compared with it.

• ✗ Answering “low temperature and low pressure” for ideal behaviour.

  ✓ At low temperature the slow molecules feel their attractions and the gas can even liquefy. Ideal behaviour needs high temperature and low pressure.

MCQs

1. 1. Which condition is necessary for most gases to behave nearly ideally?

   1. (a) low temperature and low pressure
   2. (b) high temperature and low pressure
   3. (c) constant temperature and low pressure
   4. (d) high temperature and constant pressure
   Show answer

   (b) High temperature makes attractions unimportant; low pressure makes the particles' volume negligible.

2. 2. Which statement is NOT a statement of the kinetic theory of gases?

   1. (a) molecules interact during collisions
   2. (b) molecules are in continuous random motion
   3. (c) collisions are of short duration
   4. (d) molecules are tiny hard spheres undergoing inelastic collisions
   Show answer

   (d) Inelastic collisions break the assumption of perfectly elastic collisions (MCQ 13.6).

3. 3. According to the kinetic theory, the average kinetic energy of gas particles depends on the gas's:

   1. (a) volume
   2. (b) temperature
   3. (c) colour
   4. (d) container shape
   Show answer

   (b) Temperature: this is assumption 5, and 13.4 turns it into ⟨K.E.⟩ = (3/2)k_B T.

4. 4. The mean free path is:

   1. (a) the size of a molecule
   2. (b) the average distance a particle covers before a collision that changes its direction or energy
   3. (c) the length of the container
   4. (d) the path of the fastest molecule
   Show answer

   (b) It is the average free run between collisions that really change the particle's motion.

5. 5. Between collisions, an ideal-gas particle moves in a straight line because:

   1. (a) the container is a cube
   2. (b) there are no forces between particles except in collisions
   3. (c) gravity is very strong
   4. (d) the particles are charged
   Show answer

   (b) With no attraction or repulsion between collisions, nothing changes the particle's velocity.

Quick revision

• Kinetic theory explains gas pressure as collisions of moving particles, and it applies to ideal gases.
• Five assumptions: tiny far-apart particles, random rapid motion, perfectly elastic collisions, no forces except in collisions, average K.E. set by temperature.
• Real gases behave most ideally at high temperature and low pressure.