Temperature and kinetic theory

Hold a mug of coffee and it warms your hands. Leave it on the desk and an hour later it is the same cool temperature as the room. Something moved from the coffee into the air, and it always moves that one direction. To see what that something is, we have to zoom in past what any thermometer shows, down to the molecules, and ask what is actually different about a hot thing and a cold thing.

Play: a box of nothing but bouncing dots

Here is a gas with the story stripped down to its bones: a box of point particles that do nothing but fly in straight lines and bounce off the walls. There is no "pressure" variable hidden in the code. Drag temperature up and the dots speed up; slide the amber wall inward to shrink the box. Watch the pressure readout, which is nothing more than the wall impacts counted per second.

Two things should jump out. Heating the gas raises the pressure even though you never touched the walls. And shrinking the box raises the pressure too, while the product of pressure and volume stays about level whenever you leave temperature alone. Both are consequences of one picture: pressure is just molecules drumming on the walls, and anything that makes them hit harder or more often pushes the number up.

Temperature is average jiggle energy

Back in energy and conservation a moving mass carried kinetic energy $\frac{1}{2}m{v}^2$. A gas is a swarm of tiny masses, each with its own. They are not all moving at the same speed - some crawl, some race - so the honest single number describing the swarm is the average kinetic energy per molecule. Temperature is exactly that average, on a rescaled dial:

This is stated, not derived here - pinning down the exact factor of $\frac{3}{2}$ takes a careful count over all directions of motion. But the shape is the whole idea: temperature $T$ is proportional to the average of $\frac{1}{2}m{v}^2$. Double the absolute temperature and you double the average kinetic energy of the jiggle. The constant $k\approx 1.38\times 10^{-23}$ joules per kelvin is only a unit converter between the temperature humans invented and the energy nature actually cares about.^[1]

Where pressure comes from

You felt this in the widget already. Each time a molecule bounces off a wall, it reverses direction, and reversing a molecule's motion takes a shove from the wall - which, by the interaction-pair rule, means the molecule shoves the wall back just as hard. Pressure is the sum of those countless tiny shoves, spread over the wall's area.

Count what raises it. More molecules means more hits. A smaller box means each molecule reaches a wall sooner, so it hits more often. And higher temperature means faster molecules, which hit both more often and harder. Turn that counting into algebra and, for an ideal gas, it collapses to one line:

The left side is what you measured in the widget: pressure times volume. The right side says it is set entirely by how many molecules $N$ you have and how hot they are $T$. Hold temperature fixed and $PV$ is constant, so squeezing the box (less $V$) forces the pressure up - exactly the trade the amber wall showed. Heat a sealed rigid container and $V$ and $N$ cannot change, so the pressure alone must rise. That is why a car's tire pressure reads higher after a long fast drive: the road and flexing rubber heat the trapped air, and with nowhere to expand, its pressure climbs.^[2]

The railing and the bench, resolved

Now the predict probe pays off. The metal railing and the wooden bench sat out together all night, so they reached the same temperature - the same average molecular jiggle. What differs is how fast each one pulls heat out of your warm hand. Metal shuttles energy away quickly, so it drains heat from your skin fast and feels cold; wood is a sluggish conductor, so your hand barely notices. "Feels colder" is a statement about rate of heat flow, not about temperature. Your skin never measures temperature - it measures how fast it is losing heat.

Because temperature is defined as an average, we can compute a gas's typical molecular speed straight from it. Rearranging $\frac{1}{2}m⟨v^2⟩=\frac{3}{2}kT$ gives the root-mean-square speed:

import math

k = 1.380649e-23        # Boltzmann constant, J/K
T = 293.0               # room temperature, kelvin (about 20 C)

def v_rms(mass_kg):
    # From (1/2) m <v^2> = (3/2) k T  ->  v_rms = sqrt(3 k T / m)
    return math.sqrt(3 * k * T / mass_kg)

m_n2 = 4.65e-26         # one nitrogen molecule, kg
m_o2 = 5.31e-26         # one oxygen molecule, kg

print(round(v_rms(m_n2)))   # ~511  m/s  (nitrogen)
print(round(v_rms(m_o2)))   # ~478  m/s  (oxygen)

The air in the room is not still. Its molecules are tearing around at roughly the speed of sound, colliding billions of times a second. "Room temperature" is a calm-sounding average laid over microscopic chaos.

Check yourself