Kinetic Theory and Equipartition

Blow up a balloon and it pushes back against your fingers with a steady pressure. Nothing inside is alive; there is no little engine. So what pushes? The answer is one of the great unifying pictures in physics: a gas is a swarm of tiny molecules in relentless, random motion, hammering the walls billions of times a second. Each hit is a feather-light tap, but the sheer number of taps averages into a smooth, constant force — the pressure. And the hotter the gas, the faster the molecules fly. Kinetic theory takes this cartoon seriously and does the arithmetic, and out drops something astonishing: temperature is nothing but the average kinetic energy of the molecules.

This is the microscopic meaning of the temperature we defined with thermometers, and of the gas pressure you met at school. By the end of the page you'll be able to connect a thermometer reading to how fast an air molecule is actually moving past your nose right now — roughly the speed of a rifle bullet, as it happens.

Pressure from a hail of collisions

Follow a single molecule of mass m as it flies at the right-hand wall of a box with a speed component v_x perpendicular to that wall. It hits the wall and bounces straight back (an elastic collision), so its x-velocity flips from +v_x to -v_x. Its change in momentum is

\Delta p = m v_x - (-m v_x) = 2 m v_x.

By Newton's third law the wall feels an equal and opposite kick. Now multiply up: a molecule in a box of side L returns to smack this wall every 2L/v_x seconds, so it delivers kicks at a steady rate; sum over the countless molecules, divide the total force by the wall's area, and the taps blur into a constant pressure. Reveal the figure to watch a single bounce.

Doing that sum carefully (averaging over all directions, so that the mean-square speed splits equally into x, y, z parts) gives the central result of kinetic theory:

pV = \tfrac{1}{3} N m \langle v^2 \rangle,

where N is the number of molecules and \langle v^2 \rangle is the average of the squared speed. Pressure times volume is set entirely by how many molecules there are and how fast they move.

Temperature is average kinetic energy

Compare the kinetic-theory result with the experimental ideal-gas law pV = N k_B T, where k_B = 1.38\times10^{-23}\ \text{J/K} is Boltzmann's constant — the tiny conversion factor between a degree of temperature and an amount of energy. Setting the two expressions for pV equal:

\tfrac{1}{3} N m \langle v^2 \rangle = N k_B T \quad\Longrightarrow\quad \tfrac{1}{2} m \langle v^2 \rangle = \tfrac{3}{2} k_B T.

The average translational kinetic energy of a gas molecule is \tfrac{1}{2} m \langle v^2 \rangle = \tfrac{3}{2} k_B T.
It depends only on the temperature, not on the molecule's mass. At the same temperature, heavy molecules simply move more slowly so that their average energy matches the light ones.
The root-mean-square speed follows: v_{\text{rms}} = \sqrt{\langle v^2 \rangle} = \sqrt{\dfrac{3 k_B T}{m}}.

This is the punchline. The abstract "temperature" a thermometer reads is, at the molecular level, literally a measure of the average jiggling energy of the particles. Absolute zero is where that jiggling (classically) stops. Warm the gas and you are, quite precisely, speeding up its molecules.

Equipartition: energy shared out fairly

Why the \tfrac{3}{2}? Because a molecule flying through space can move in three independent directions — x, y and z — and the energy splits evenly among them, \tfrac{1}{2}k_B T for each. Each independent way a molecule can store energy as a squared term (a quadratic degree of freedom) collects the same fair share. This is the equipartition theorem.

In thermal equilibrium, every quadratic degree of freedom — each independent term in the energy that goes as a square, like \tfrac{1}{2}mv_x^2 or a spring's \tfrac{1}{2}k x^2 — holds an average energy of \tfrac{1}{2} k_B T.
A single molecule with f active quadratic degrees of freedom therefore has average energy \tfrac{f}{2} k_B T.

Count the degrees of freedom and you predict the energy — and hence the heat capacity — of a gas:

A monatomic gas (helium, argon): just 3 translational directions, so f = 3 and each molecule averages \tfrac{3}{2} k_B T.
A diatomic gas (nitrogen, oxygen) at room temperature: 3 translational + 2 rotational (spinning about the two axes across the dumbbell), so f = 5 and each averages \tfrac{5}{2} k_B T.
At high temperatures a diatomic molecule also vibrates, adding 2 more (kinetic + potential) for f = 7 — but those modes "freeze out" when it's cold, a hint that nature is secretly quantum.

Worked examples

Example 1 — average energy of an air molecule. At room temperature T = 300\ \text{K}, the average translational kinetic energy of any gas molecule is

\tfrac{3}{2} k_B T = \tfrac{3}{2}(1.38\times10^{-23})(300) \approx 6.2\times10^{-21}\ \text{J}.

Tiny per molecule — but there are about 10^{25} of them in a breath.

Example 2 — how fast is a nitrogen molecule? A nitrogen molecule has mass m \approx 4.7\times10^{-26}\ \text{kg}. Its rms speed at 300\ \text{K} is

v_{\text{rms}} = \sqrt{\frac{3 k_B T}{m}} = \sqrt{\frac{3(1.38\times10^{-23})(300)}{4.7\times10^{-26}}} \approx 515\ \text{m/s}.

Over 1800\ \text{km/h} — the molecules of air around you are moving faster than a passenger jet.

Example 3 — internal energy of a monatomic gas. Two moles of argon (N = 2 N_A atoms) at 300\ \text{K}. With f = 3, the total internal energy is

U = N \cdot \tfrac{3}{2} k_B T = \tfrac{3}{2} n R T = \tfrac{3}{2}(2)(8.31)(300) \approx 7480\ \text{J},

using N_A k_B = R = 8.31\ \text{J/(mol·K)}, the gas constant.

Equipartition says every molecule in the air — light or heavy — carries the same average kinetic energy \tfrac{3}{2}k_B T. But equal energy does not mean equal speed. Since \tfrac{1}{2}m\langle v^2\rangle is fixed, a lighter molecule must move faster: v_{\text{rms}} \propto 1/\sqrt{m}. A hydrogen molecule (mass 2) rockets along about \sqrt{28/2}\approx3.7 times faster than a nitrogen molecule (mass 28). Way out at the top of the atmosphere, the fleetest hydrogen molecules routinely exceed Earth's escape velocity and trickle away into space, while the sluggish nitrogen stays gravitationally bound. Over geological time Earth has lost most of its free hydrogen for exactly this reason — and the Moon, with feeble gravity, couldn't hold on to any gas. It is the same reason helium party balloons go flat and the same reason gaseous diffusion (lighter isotopes move faster) was once used to enrich uranium.

Two traps. First: v_{\text{rms}} = \sqrt{\langle v^2\rangle} is the root-mean-square speed, and it is not the same as the plain average speed — because squaring weights the fast molecules more heavily, v_{\text{rms}} comes out a little larger than the mean speed. Don't use one where a problem asks for the other. Second, and more common: temperature in \tfrac{3}{2}k_B T is always in kelvin. Plugging in Celsius gives nonsense — and near 0\,^\circ\text{C} it would even suggest negative or zero molecular energy. Finally, remember \tfrac{3}{2}k_B T is only the translational energy; a diatomic molecule stores more in rotation, so don't use it for the total energy of anything but a monatomic gas.