Module 4 · an excursion into thermal physics
Boyle, Charles and the rest measured how gases behave and wrote down rules. A century later Maxwell and Boltzmann explained the rules by imagining the gas as a swarm of tiny balls bouncing around — and every rule fell out of Newton's laws applied to the balls. This module builds the swarm first, and lets the laws fall out in front of you.
Here is a gas: a few hundred molecules in a box, bouncing off the walls, ignoring each other. The right-hand wall is a piston you can push in. Pressure is measured exactly the way the real thing is — by adding up how hard the molecules hammer the walls.
Keep the temperature fixed and push the piston in until the volume is half what it was. The pressure:
Three things to notice, in order.
Now do what the simulation did, with algebra. One molecule of mass m moving with velocity component vx toward a wall. It bounces elastically: vx becomes −vx, so the molecule's momentum changes from +mvx to −mvx — a change of 2mvx. Momentum is conserved (Module 2), so the wall picked up that 2mvx. Force is momentum per second (Module 3), so the question is: how many such bounces per second, per square metre of wall?
Multiply both sides by the volume. Since nV is just the total number of molecules — call it N:
Boyle's law is now one line away — provided we grant that fixed temperature means fixed ⟨½mv²⟩, which Bench 3 will earn properly. Take that on trust and the right-hand side of PV = ⅔N⟨½mv²⟩ is a fixed number, so PV is constant. Squeeze the volume to half and the pressure must double — not because of any law about gases, but because there are twice as many molecules in the way of each square centimetre of wall.
Breathing is Boyle's law. Your diaphragm drops, your chest volume rises by half a litre, the pressure inside falls a fraction below atmospheric, and air is pushed in from outside — you do not suck air in; the atmosphere pushes it. A scuba diver at 30 m is under four atmospheres; a lungful of air taken there and held on the way up expands fourfold, which is why the first rule of diving is "never hold your breath ascending." A syringe, with your finger sealing the nozzle, is Bench 2 in its purest form — pushing the plunger walks you along one isotherm.
Hold the pressure fixed instead — a gas under a freely sliding piston with a weight on it — and heat it. The volume grows. Charles found in 1787 that the growth is a straight line: heat the gas by one degree Celsius and it expands by about 1/273 of its 0 °C volume; heat it by ten degrees and it expands by ten of those steps. Strangest of all, the same fraction — 1/273 — comes out for every gas you try.
Draw volume against Celsius temperature for any gas and extend the straight line backward. It hits zero volume at about −273 °C. What does that mean?
Now look at the kinetic-theory equation again with fresh eyes. PV = ⅔N⟨½mv²⟩. Charles says V ∝ (T°C + 273) at fixed P. Combining, the average kinetic energy per molecule is proportional to (T°C + 273). Define a new temperature T measured from −273 °C — this is the kelvin, T(K) = T(°C) + 273 — and let the proportionality constant be ³⁄₂k, with the ³⁄₂ chosen so that it cancels the ⅔ in PV = ⅔N⟨½mv²⟩ and the gas law comes out clean:
This is what temperature is: the average kinetic energy of the molecules. Not a fluid, not a substance, and not the same thing as heat — heat is a separate accounting Module 8 will take up. The freezing and boiling points of water come out 100 apart because the kelvin was picked to make them so; the size of one degree is the only arbitrary part. The Celsius scale is a historical accident; the kelvin scale is the one the molecules actually use. And absolute zero is not "very cold" — it is the reading at which the molecules have no kinetic energy to lose. There is no colder, in the same way that there is no emptier than empty.
A real gas does not reach −273 °C with any volume at all; long before that it condenses to a liquid, and the whole model — non-interacting balls — has stopped applying. The line's intercept is not a prediction about what the gas does there; it is a property of the law, revealed by extrapolation. Physicists trust such intercepts when every substance points at the same one. That coincidence — helium, nitrogen, argon, all agreeing to the fraction of a degree — is what convinced people that −273.15 °C was a fact about the universe and not about any particular gas. Quantum mechanics later confirmed it, and also revealed that even at absolute zero the molecules are not quite still. That is Module 22, where it is called the zero-point energy.
If ⟨½mv²⟩ = ³⁄₂kT, you can compute how fast air molecules are moving right now. The bench below compares that speed with two others: the speed of sound, and Earth's escape velocity — the upward speed something needs at the ground to leave Earth for good, about 11 km/s. Rearranged for the typical speed:
The gate below leans on "the tail" of the speed distribution without saying what shape it has. Here is the shape. Bench 1's box is two-dimensional, so each molecule's kinetic energy — call it E — is ½mv², and it can only be shared between two directions of motion instead of three. Counting the ways that energy can be spread among the molecules — the same kind of argument Module 8 makes with entropy — shows that the chance of a molecule carrying speed at least u times the typical (rms) speed falls off as a clean exponential:
Earth's atmosphere has almost no free hydrogen or helium, though both are constantly produced (helium from radioactive decay in rocks). Jupiter's atmosphere is mostly hydrogen. Why?
Back to Bench 1 with the thermostat off. A wall moving inward at speed u meets a molecule coming at it at v. In the wall's own frame the collision is elastic — the molecule closes at v + u and rebounds at v + u; back in the box's frame that means it now moves outward at v + 2u. Every bounce off the advancing piston adds speed. That is the mechanism by which work done on a gas becomes heat in the gas — no mystery, and no need for the old idea of caloric, an invisible weightless liquid once thought to flow from hot things to cold ones; just Module 2's relative-velocity argument applied to a moving wall. Pull the piston out and the reverse happens: molecules come off slower, the gas cools, and this is how a refrigerator works.
A diesel engine has no spark plug. It compresses air by a factor of about 18, quickly enough that no heat escapes, and the temperature climbs past 500 °C — hot enough that fuel sprayed in ignites on its own. A pressure cooker is Gay-Lussac's law used to push water's boiling point above 100 °C so that dal cooks in a third of the time. A hot-air balloon is Charles's law: heat the air, it expands, the same mass now occupies more volume and is less dense than the cold air around it. Tyre pressure warnings on a hot afternoon in Chennai are the same law read the other way.
1. Two boxes at the same temperature, same volume, same pressure: one holds hydrogen, the other oxygen (16 times heavier). Which box holds more molecules?
2. A sealed rigid steel tank of air is heated from 27 °C to 327 °C. The pressure:
3. In the Bench 1 simulation, the molecules never collide with each other, only with the walls — yet the gas laws come out right. Why is that a legitimate simplification?