Kinetic theory of matter

Kinetic theory explains macroscopic quantities (pressure, temperature, internal energy) through the motion and interaction of invisible particles. Why does a gas press on the walls of its container, why does its pressure rise when heated, and what does temperature really measure? Kinetic theory answers these questions precisely. Gas laws, thermodynamics, states of matter and heat transfer are all built on this foundation.

§1. The three main ideas of kinetic theory

1. All substances are made of particles (molecules, atoms and ions), with gaps between them.
2. The particles are in random, continuous motion, called thermal motion.
3. The particles interact: at very short distances they repel, at slightly larger distances they attract.

Experimental evidence:

• Brownian motion: the endless random jiggling of tiny particles in a liquid (pollen grains, for example). Brown observed it in 1827; it is caused by unbalanced impacts of molecules from different sides. Einstein’s theory and Perrin’s experiments (early twentieth century) used it to prove firmly that molecules exist.
• Diffusion: the spontaneous mixing of substances (a smell spreading, sugar dissolving in tea). Diffusion speeds up when the temperature rises, so temperature is linked to how fast the particles move.
• Gases are easily compressed and the volume of mixed liquids can shrink, showing there are gaps between particles; solids resist stretching, showing attractive forces.

§2. Mass and size of molecules

Molecules are tiny and extremely numerous, so they are counted in moles:

• M is the molar mass (kg/mol): M = M_r·10⁻³. For example, M(H₂O) = 18·10⁻³ kg/mol, M(O₂) = 32·10⁻³ kg/mol.
• Mass of one molecule: m₀ = M/N_A.
• Number density, the number of molecules per m³: n = N/V (m⁻³).

Estimating molecular size: in a liquid the molecules are closely packed, so each occupies a volume V₀ = M/(ρN_A) and has a linear size d ≈ ∛V₀. Atoms and simple molecules are about 10⁻¹⁰ m across.

Worked example

How many molecules are there in 36 g of water, and how big is one water molecule?

1. ν = m/M = 36/18 = 2 mol, N = νN_A = 2·6.02·10²³ ≈ 1.2·10²⁴.
2. Volume per molecule: V₀ = M/(ρN_A) = 0.018/(1000·6.02·10²³) ≈ 3·10⁻²⁹ m³.
3. d ≈ ∛(3·10⁻²⁹) ≈ 3·10⁻¹⁰ m.

§3. The ideal gas and the kinetic theory equation

In the ideal gas model the molecules are point particles that interact only in (elastic) collisions. Gas pressure results from a huge number of molecular impacts on the wall. Each impact passes momentum to the wall; the number of impacts depends on the number density, the strength of each on the speed. In room air, for example, about 10²³ molecules hit each square centimetre of wall every second, so individual impacts are not noticed and the pressure seems steady. The calculation gives

This is the basic equation of kinetic theory: the macroscopic pressure is expressed through microscopic quantities (m₀, n, v̄²). The factor 1/3 arises because molecules are equally likely to move along any of the three directions of space.
Since ρ = m₀n, also p = ρv̄²/3.

Worked example

A gas has pressure 2.5·10⁵ Pa and density 12 kg/m³. Find the root-mean-square speed of its molecules.

1. p = ρv²/3 → v = √(3p/ρ) = √(3·2.5·10⁵/12).
2. v = √62 500.
3. v = 250 m/s.

§4. Temperature as a measure of molecular energy

The quantity that is equal for bodies in thermal equilibrium is temperature. Kinetic theory links it to the average kinetic energy of the molecules:

• At the same temperature the average kinetic energy of molecules of any gas is the same, whether they are light or heavy.
• The absolute (Kelvin) scale: T = t + 273. Absolute zero (0 K ≈ −273 °C) is the unreachable limit at which the energy of thermal motion is lowest.
• A temperature difference is the same in kelvins and degrees: ΔT = Δt.

From p = nkT: at equal pressure and temperature all gases have the same number density (Avogadro’s law). At standard conditions n ≈ 2.7·10²⁵ m⁻³.

Worked example

Find the average kinetic energy of a gas molecule at 27 °C and the number density at 10⁵ Pa.

1. T = 27 + 273 = 300 K.
2. Ē_k = 1.5·1.38·10⁻²³·300 ≈ 6.2·10⁻²¹ J.
3. n = p/(kT) = 10⁵/(1.38·10⁻²³·300) ≈ 2.4·10²⁵ m⁻³.

§5. Molecular speeds

From Ē_k = m₀v̄²/2 = 3kT/2 we get the root-mean-square speed:

• Speed is proportional to √T: if the temperature rises 4 times, the speed doubles.
• At the same temperature lighter molecules are faster: a hydrogen molecule is 4 times faster than an oxygen molecule (√(32/2) = 4).

Molecules have a range of speeds, some slow, some fast; Maxwell found the speed distribution theoretically. Stern (1920) measured molecular speeds directly with rotating cylinders. Although molecules are fast (hundreds of m/s), a smell spreads slowly across a room, because each molecule collides very often and follows a zigzag path.

Worked example

Find the rms speeds of hydrogen (M = 0.002 kg/mol) and oxygen (M = 0.032 kg/mol) molecules at 300 K.

1. Hydrogen: v = √(3·8.31·300/0.002) ≈ 1930 m/s.
2. Oxygen: v = √(3·8.31·300/0.032) ≈ 483 m/s.
3. The ratio ≈ 4 equals the square root of the ratio of molar masses.

§6. Internal energy of an ideal gas

The internal energy of a body is the total kinetic energy of the random motion of all its particles plus the potential energy of their interaction. In an ideal gas the molecules do not interact, so the internal energy is purely kinetic. For a monatomic gas (helium, neon, argon):

So the internal energy of an ideal gas depends only on temperature (and the amount of gas); if the volume changes at constant temperature, the internal energy stays the same. Polyatomic molecules also rotate, so their coefficient is larger (5/2 for a diatomic gas). This formula is the main tool when applying the first law of thermodynamics.

Worked example

What is the internal energy of 2 mol of helium at 300 K? By how much does it grow when heated by 100 K?

1. U = (3/2)·νRT = 1.5·2·8.31·300 ≈ 7480 J.
2. ΔU = (3/2)·νRΔT = 1.5·2·8.31·100 ≈ 2490 J.

Check yourself

1. What is 27 °C in kelvins?

2. How many atoms are in 4 g of helium (M = 4 g/mol)?

3. The absolute temperature rises 4 times. By what factor does v_rms change?

4. Mass of one oxygen molecule?

5. Ratio of the average kinetic energies of hydrogen and oxygen molecules at the same temperature?

6. The number density doubles at constant temperature. By what factor does the pressure change?

7. Number density of a gas at 0 °C and 10⁵ Pa?

8. What is the random motion of tiny particles in a liquid called?

Common mistakes

• Leaving the molar mass in g/mol: forgetting the factor 10⁻³.
• Putting the temperature in °C into the formulas.
• Thinking that at the same temperature lighter molecules have more kinetic energy: they are faster, but the energy is the same.
• Treating v_rms as proportional to temperature: it is proportional to √T.
• Thinking Brownian motion is the motion of molecules themselves: what we see is a large particle being hit by molecules.