Foundation lesson

Temperature and thermal energy

In equilibrium at a given temperature, every kind of particle, molecule or grain, has the same average energy of motion along each direction it can move: half of Boltzmann's constant times the temperature. The temperature fixes that average over many particles. It is not the energy of any one of them.

Written for this edition, not translated from Einstein. Editorial review pending.

What does a temperature say about the jostling of molecules and grains?

The air in a room at 20 °C is a crowd of molecules moving in every direction and colliding billions of times a second. Warm the room and, on average, they move faster. The kinetic theory of heat reads temperature this way: through the average energy of motion of the particles, taken over very many of them.

For any particle in equilibrium with its surroundings, the average energy of motion along each direction is the same:

\left\langle \frac{1}{2}\,m\,v_x^{2}\right\rangle = \frac{1}{2}\,k_B\,T

The average of one half m v x squared equals one half k B T.

Here m is the particle's mass, $v_x$ its speed along x, T the absolute temperature, and $k_B$ = 1.38 × 10⁻²³ joules per kelvin is Boltzmann's constant. It equals R/N, the gas constant shared out per molecule. The Brownian paper writes RT/N where a modern text writes $k_BT$ ; the paper's own letter k means the viscosity, not this constant.

The average is the point. One molecule's energy of motion changes at every collision, from almost nothing to several times the average. The temperature fixes how energy is shared out over many particles. It is not the energy of any one of them.

Nothing in the rule mentions size. A grain ten billion times heavier than a molecule has the same average energy of motion, so it moves more slowly, by the square root of the mass ratio. That is why a suspended grain belongs to the same thermal story as a dissolved molecule, as §1 of the paper insists.

Worked example: A nitrogen molecule and a half-micrometre grain at 20 °C

At T = 293 K, $k_BT$ = 1.381 × 10⁻²³ × 293 = 4.05 × 10⁻²¹ J, so the average energy of motion along one direction is half of that, 2.02 × 10⁻²¹ J.
A nitrogen molecule has a mass of 4.65 × 10⁻²⁶ kg. Setting $\frac{1}{2} m v_x^2$ equal to $\frac{1}{2} k_BT$ gives a typical speed along one direction of √(4.05 × 10⁻²¹ ÷ 4.65 × 10⁻²⁶), about 295 metres a second.
A grain of radius 0.5 μm and density 1200 kg/m³ has a mass of 6.3 × 10⁻¹⁶ kg, about 1.4 × 10¹⁰ times the molecule's. The same average energy gives it about 2.5 millimetres a second.
In water that motion is turned about within some 70 nanoseconds: the grain's mass divided by its drag coefficient 6πηa. That is far too brief to follow, so what a microscope sees is the net result of the zigzag.

This is why the Brownian paper asks for a displacement over a time rather than a speed: the thermal speed is real, but it turns about far faster than any observer can watch.

JavaScript is off, so the controls below cannot change anything. The construction shows its first setting, and its closing paragraph, “What it shows, in words”, describes the rest.

Try it: one spread of speeds, two scales

At any moment, the speeds of particles along one axis are spread in a bell curve. The temperature sets its width, and the width depends on the particle's mass. Label the same curve for a molecule or for a grain.

Label the axis for

At 293 K, a nitrogen molecule moves along one axis with a spread of about 295 m/s. The curve does not change; only the numbers on its axis do, by a factor of about 120 000.

Shaded in the middle, under half a width either way: the particles that have, at that moment, less than a quarter of the average energy of motion along the axis, about 38 per cent of them. Outlined at the two ends, beyond two widths: those with more than four times the average, about 5 per cent. Collisions move each particle about the whole curve.

What it shows, in words

At 293 kelvin a nitrogen molecule's speed along one axis is spread with a width of about 295 metres a second, and a half-micrometre grain's with a width of about 2.5 millimetres a second. Measured in its own width the spread is the same bell curve for both, with the same average energy of motion along the axis, 2.02 × 10⁻²¹ joules. At any moment about 38 per cent of the particles have less than a quarter of that average and about 5 per cent more than four times it. The temperature fixes the spread, not the energy of any one particle.

Where this lesson stops

This lesson stops at the average energy of motion. How the jostling of many molecules makes a grain wander, and how far, is §4 of the Brownian paper.

If you came here from a passage, Back returns you to the exact place you left.

All foundation lessons · Read as Markdown