Kinetic Theory: Why Gases Push - Physics for Pilot Assessments

This lesson is the mechanism behind everything else in the module. By the end of it you should be able to describe what the particles of a gas are actually doing, explain gas pressure as the accumulated effect of those particles striking the walls rather than as the weight of the gas, say what temperature means at the particle level, and predict which way the pressure moves when you change the volume, the temperature or the number of particles.

Everything is made of particles, and they never stop moving

Matter is made of particles — single atoms, or atoms bound together into molecules — and in a gas those particles are far apart compared with their own size, so the container is mostly empty space. They are also moving, constantly and fast, in every direction at once, colliding with each other and with the walls and rebounding. Nothing keeps them going: there is no friction to speak of between them, so a particle that bounces off a wall carries on until it meets something else. At ordinary room temperature the average speed of a nitrogen molecule in the air around you is roughly 500 metres per second, which is faster than a rifle bullet, and the reason a smell takes seconds rather than microseconds to cross a room is that each molecule is knocked off course billions of times per second and makes very slow progress in any one direction.

You can see the consequences without any equipment. A gas released into a room spreads through all of it and never collects in a corner, which is what you expect from particles moving in random directions and nothing you would expect from particles sitting still. Look at smoke in a beam of light under a microscope and the visible specks jitter about erratically; they are being knocked by air molecules too small to see, arriving unevenly from all sides. That jittering is called Brownian motion, and it is the most direct evidence there is that the particle picture is not a metaphor.

Where gas pressure comes from

Now follow one particle to the wall. It arrives, it is stopped, and it is sent back the other way — and reversing a moving object's direction takes a force, so the wall must have pushed on it. From the mechanics module, forces come in pairs acting in opposite directions on the two objects involved, so the particle pushed on the wall exactly as hard as the wall pushed on the particle. One collision therefore delivers one tiny outward push to the wall, over an extremely short time. On its own it is far too small to detect. But a cubic centimetre of air at ordinary pressure contains something like twenty billion billion molecules, and the wall is struck an unimaginable number of times every second, so the individual pinpricks blur into a steady, perfectly smooth outward force. Divide that force by the area of wall it is spread over and you have the pressure, exactly the same definition of pressure used for solids and liquids earlier in the course.

Pressure

p = \frac{F}{A}

$p$: pressure, in pascals (Pa) — one pascal is one newton per square metre
$F$: the average force the particle impacts deliver perpendicular to the wall, in newtons (N)
$A$: the area of wall the force is spread over, in square metres (m²)

Read that as a bookkeeping statement about collisions and two predictions follow immediately: anything that increases the number of impacts per second increases the pressure, and so does anything that makes each impact harder.

Every particle that strikes a wall gives it a tiny push, and no single impact matters. There are so many, arriving so often, that the wall feels a smooth steady push — and that push divided by the wall's area is what a pressure gauge reads. More impacts, or harder impacts, means more pressure.

It is not the weight of the gas

The single most persistent wrong idea about gas pressure is that it is the weight of the gas pressing down, and it is worth demolishing carefully because it feels so reasonable. Three observations kill it. First, a sealed cylinder of gas presses just as hard outwards on its top face as on its bottom face, and weight only ever acts downwards. Second, turn the cylinder over, or lay it on its side, and the gauge does not move — whereas anything caused by weight would change as soon as you changed which face was underneath. Third, the numbers are nowhere close: the air in an ordinary room weighs something like six hundred newtons in total, while atmospheric pressure acting on the floor of that room amounts to millions of newtons. Gas pressure comes from motion, not from weight, and it acts equally in every direction because the particles are travelling in every direction.

There is one honest qualification. A gas does have weight, and in a tall enough column of it the lower layers are squeezed by the weight of everything above, so the pressure at the bottom of the atmosphere really is greater than the pressure high up. That effect needs kilometres of gas to be noticeable and is dealt with properly later in the module; inside a cylinder, a syringe or a room it is far too small to matter.

Three levers on the pressure

Given that pressure is impacts per second multiplied by the force per impact and divided by the wall area, there are exactly three things you can change. Push more particles into the same space and there are more of them available to hit the walls, so the impact rate rises in proportion: double the number of particles in a container and you double the pressure, which is what a bicycle pump does to a tyre. Shrink the space with the same particles inside and each one has a shorter distance to travel between one wall and the next, so it comes back more often; halve the volume and each particle hits twice as often, doubling the pressure. Raise the temperature and the particles move faster, which does two things at once — they cross the container more often, and each collision reverses a faster particle and so delivers a bigger push.

Temperature is average particle kinetic energy

That last lever needs the link between temperature and particle motion stated properly. From the energy module, an object of mass m moving at speed v carries kinetic energy equal to half of m times v squared. In a gas the particles are not all travelling at the same speed — some are much faster than average, some much slower, and any individual particle's speed changes at every collision — so what matters is the average kinetic energy across all of them. That average is what temperature measures, and the relationship is a direct proportionality to the absolute temperature in kelvin.

Temperature and particle energy

E_\text{k(average)} \propto T

$E_\text{k(average)}$: the average kinetic energy of one particle, in joules (J)
$\propto$: "is proportional to" — double one and the other doubles
$T$: the absolute temperature, in kelvin (K) — never in degrees Celsius

The insistence on kelvin is not a convention, it is the content of the statement. A proportionality means that when the temperature reaches zero the average kinetic energy reaches zero too, and the temperature at which particle energy runs out is absolute zero — which is 0 K, not 0 °C. Water freezing has nothing to do with particles stopping. Put 0 °C into that proportionality and you claim that ice-cold air has motionless molecules, which is plainly false; put 0 K in and you have the definition of absolute zero from the previous lesson, arrived at from a completely different direction.

One more consequence, because it explains a common surprise. Since it is the energy that is proportional to temperature, and the energy depends on the square of the speed, doubling the absolute temperature does not double the particle speed. Doubling the energy multiplies the square of the speed by two, so the speed itself is multiplied by the square root of two — taking a square root asks what number multiplied by itself gives you this one, and the square root of two is about 1.41. So a gas at 600 K has particles moving about 1.4 times as fast as the same gas at 300 K, not twice as fast, even though each particle carries twice the energy.

Example 1 — halve the volume, keeping the temperature fixed

A sealed cylinder holds gas at 100 kilopascals. The piston is pushed in until the gas occupies half its original volume, slowly enough that the temperature ends up back where it started. What happens to the pressure, and why?

Nothing has escaped, so the same number of particles is present. The temperature is unchanged, so their average speed is unchanged too, which means each individual impact is exactly as hard as before.
What has changed is the distance a particle travels between hitting one wall and hitting the next: it has halved. Travelling at the same speed over half the distance, each particle arrives at a wall twice as often.
Twice as many impacts per second, each of the same strength, spread over the same wall area, is twice the pressure.
So the pressure rises from 100 kilopascals to 200 kilopascals.
Notice that the reasoning used only the particle picture. The formal law behind it comes later in the module, and it will say exactly this.

Example 2 — double the temperature, keeping the volume fixed

The same cylinder is clamped so its volume cannot change, and the gas is heated from 300 K to 600 K. What happens to the pressure?

Doubling the absolute temperature doubles the average kinetic energy of each particle, because the two are proportional.
Since kinetic energy depends on the square of the speed, doubling the energy multiplies the speed by the square root of two, about 1.41.
That raises the pressure twice over. Particles moving 1.41 times as fast reach the walls 1.41 times as often, and each collision reverses a 1.41-times-faster particle, so each impact delivers 1.41 times the push.
Multiply the two effects: $1.41 \times 1.41 = 2$ So the pressure doubles, from 100 kilopascals to 200 kilopascals.
The pressure is proportional to the absolute temperature, then — and the working shows why it is the kelvin figure and not the Celsius one that matters, because it was the doubling of energy that drove everything.

Example 3 — pump in more particles

The cylinder is restored to its original volume, again holding gas at 100 kilopascals. A pump now forces in extra air until there are twice as many particles inside, and the gas is left until its temperature has settled back to where it started. What is the pressure now?

The volume is unchanged and the temperature has returned to its starting value, so the average particle speed is what it always was: each impact on a wall is exactly as hard as before, and each individual particle strikes the walls just as often as before.
What has changed is how many particles are doing the striking. Twice as many particles, each hitting just as often, means every wall takes twice as many impacts per second.
Twice the impacts, each of the same strength, spread over the same wall area, is twice the pressure: it rises from 100 kilopascals to 200 kilopascals.
This is the lever a bicycle pump pulls. Pumping a tyre firms it up not by squeezing the air already inside but by adding more particles to share the striking.
Notice that the three levers are independent. This example changed only the number of particles, the first changed only the volume, and the second only the temperature — and each one alone was enough to move the pressure.

Why it matters in aviation

Reading pressure as a count of molecular impacts is what explains the loss of engine output with height. Climb, and there are fewer molecules in every cubic metre of air. An engine can only burn fuel with the oxygen molecules that actually reach it, and can only push against the molecules arriving at its intake and its blades, so the thrust or power available falls away steadily with altitude although nothing whatever has changed about the engine itself.

The same count explains why the air at cruising level cannot sustain a person. The proportions of the mixture are much as they are at the surface, so that air is not short of oxygen as a percentage; what has fallen is the number of oxygen molecules arriving at the lungs with each breath, and it is that number the body responds to. A pressurised cabin restores it by holding more molecules in the same space than the outside air contains, and supplemental oxygen restores it by raising the proportion instead. Either way the quantity being fixed is molecules per breath — something the particle picture makes visible and a bare pressure figure does not.

What the test asks

The reliable distractor is the weight one: an option explaining gas pressure as the gas pressing down under its own weight, sometimes with the plausible-sounding addition that a heavier gas therefore produces more pressure. Anything that makes gas pressure directional is wrong, because the particles are not directional. Expect also an item that asks what happens to the particles when a gas is heated and offers "they expand" or "they get bigger" as an option: the particles themselves are unchanged, and it is their speed and their spacing that alter — the same trap the states-of-matter material sets. And watch for the speed-and-energy step in Example 2: an item that doubles the absolute temperature and offers doubled particle speed is exploiting the square, since the speed rises only by a factor of about 1.4.