You already met Boyle's Law: squeeze a trapped gas into a smaller space, at the same temperature, and its pressure climbs in exact opposite step, so pV stays fixed. But Boyle's Law froze one thing deliberately still — the temperature — precisely so it could ignore it. Let it move, and a whole new set of behaviours wakes up.

Heat a sealed tin and it can burst. Warm the air in a balloon and it swells. Leave a car in the summer sun and its tyres read a higher pressure by evening. A gas juggles three quantities at once — pressure p, volume V and temperature T — and the gas laws are the three simple rules that tie them together. Hold one of the three fixed, and the other two dance in a fixed pattern:

• Fix temperature → pV=\text{constant} (Boyle's Law — the one you know).
• Fix pressure → \dfrac{V}{T}=\text{constant} (Charles's Law).
• Fix volume → \dfrac{p}{T}=\text{constant} (the Pressure Law).

Every one of them is the same particle picture from before — tiny particles drumming on the walls — seen from a different angle. But first there is a trap to disarm, because the T in those two new laws is not the temperature your thermometer reads in degrees Celsius.

First: temperature has to start at true zero

Charles's Law says volume is proportional to temperature. Proportional means: double the temperature, double the volume — and halve the temperature, halve the volume. That only makes sense if the temperature scale has a genuine zero, a point where there is nothing left. The Celsius scale fails this test badly: 0\,^\circ\text{C} is just the freezing point of water, an ordinary chilly day, nowhere near "no heat at all." A gas at 0\,^\circ\text{C} is still buzzing with motion. So you cannot say a gas at 20\,^\circ\text{C} has "twice" the temperature of one at 10\,^\circ\text{C} — the numbers lie about the physics.

The fix is the kelvin scale (symbol K). It counts from the coldest anything can ever be: absolute zero, the temperature at which particle motion drops to its absolute minimum and a gas would exert no pressure at all. There is nothing colder — you cannot take energy out of something that has none left. Absolute zero sits at -273\,^\circ\text{C}, and a kelvin is the same size of step as a Celsius degree, so the two scales are simply shifted by 273:

So 0\,^\circ\text{C}=273\,\text{K}, 27\,^\circ\text{C}=300\,\text{K}, and a comfy room at 20\,^\circ\text{C} is 293\,\text{K}. Now the scale has a real zero, doubling the kelvin temperature really does double the average energy of motion, and "proportional" finally means what it says. Every gas-law calculation on this page begins by turning °C into K.

The Kelvin (absolute) temperature scale

• Absolute zero — the lowest possible temperature, where particle motion is a minimum — is 0\,\text{K} = -273\,^\circ\text{C}.
• A kelvin is the same size as a Celsius degree, so convert with T(\text{K}) = \theta(^\circ\text{C}) + 273 (and back with \theta = T - 273).
• The gas laws \tfrac{V}{T} and \tfrac{p}{T} hold only when T is in kelvin.

Charles's Law: heat a free gas and it grows

Trap a gas but let it push a wall freely — a piston that slides, or the stretchy skin of a balloon — so the pressure inside always just matches the air outside. This is constant pressure. Now warm the gas. Its particles speed up, batter the moving wall harder and more often, and shove it outward; the gas expands until the collisions, spread over the now-larger wall, settle back to the same steady push as before. Warmer gas, at the same pressure, takes up more room.

Measure carefully and the volume turns out to be directly proportional to the absolute temperature: double the kelvin temperature and the volume doubles. That is Charles's Law:

A graph of volume against temperature in kelvin is therefore a straight line heading through the origin — and if you extend that line backwards, every gas would reach zero volume at 0\,\text{K}. (Real gases turn to liquid long before then, but the line pointing at absolute zero is one of the clues that told scientists absolute zero was real.)

That second line is the working tool. A hot-air balloon flies on exactly this: burn the flame, heat the trapped air, and by Charles's Law it expands — the same mass of air now filling more space, so each cubic metre is lighter than the cool air outside, and up the balloon floats.

The Pressure Law: heat a sealed gas and it pushes harder

Now do the opposite — clamp the gas in a rigid, sealed box so its volume cannot change. This is constant volume. Heat it, and the particles again speed up and slam the walls harder and more often — but this time the walls can't move to relieve them. So all that extra collision goes straight into pressure. Warmer gas, sealed in a fixed space, pushes harder on every wall.

Again the rise is proportional to the absolute temperature: double the kelvin temperature and the pressure doubles. This is the Pressure Law (sometimes called Gay-Lussac's law):

This is why a car tyre reads a higher pressure after a long motorway run than it did cold in the morning — the friction-warmed air, sealed at (almost) fixed volume, pushes harder. And it is why a sealed tin thrown on a fire can explode: heat drives its inside pressure up and up until the walls give way. Same three quantities, same particle picture — just with the volume pinned instead of the pressure.

See both laws in one box

Here is the same gas twice over. Drag the temperature slider (in kelvin) and watch. Choose constant pressure and the right wall is a free piston: heating pushes it outward, so the volume grows while the pressure gauge holds steady — that is Charles's Law. Choose constant volume and the box is sealed rigid: now heating leaves the size alone but drives the pressure gauge up — that is the Pressure Law. Cool it back down and everything reverses.

Notice that both laws are the same speeding-up of particles. What changes is only which wall is allowed to give: a free wall lets the gas trade the extra push for extra room (volume up), while a fixed wall forces the extra push to show up as pressure.

Worked examples — always convert to kelvin first

Example 1 — a balloon warms up (Charles's Law). A balloon holds 2.0\ \text{m}^3 of air at 27\,^\circ\text{C}. It is warmed to 327\,^\circ\text{C} at the same (atmospheric) pressure. What is its new volume?

Step 1 — convert both temperatures to kelvin. T_1 = 27 + 273 = 300\ \text{K} and T_2 = 327 + 273 = 600\ \text{K}. Step 2 — write Charles's Law and rearrange. \dfrac{V_1}{T_1} = \dfrac{V_2}{T_2}, so V_2 = V_1\,\dfrac{T_2}{T_1}. Step 3 — put the numbers in.

Doubling the kelvin temperature doubled the volume. Notice the trap this dodges: the Celsius numbers went from 27 to 327 — twelve times bigger — which would have given a wildly wrong answer. In kelvin the true ratio is a clean 2.

Example 2 — a tyre heats on the motorway (Pressure Law). A sealed tyre holds air at 200\ \text{kPa} when cold at 17\,^\circ\text{C}. After a hard drive the air reaches 77\,^\circ\text{C}, and the tyre barely changes size (constant volume). What is the new pressure?

Step 1 — kelvin. T_1 = 17 + 273 = 290\ \text{K}, T_2 = 77 + 273 = 350\ \text{K}. Step 2 — Pressure Law. \dfrac{p_1}{T_1} = \dfrac{p_2}{T_2}, so p_2 = p_1\,\dfrac{T_2}{T_1}. Step 3 — numbers.

A modest-sounding 60-degree warming raises the pressure by about a fifth — enough to matter, and exactly why tyre pressures are meant to be checked cold.

Example 3 — cooling a gas (Charles's Law backwards). A gas occupies 500\ \text{cm}^3 at 127\,^\circ\text{C} at constant pressure. It is cooled to 27\,^\circ\text{C}. Find its volume.

In kelvin, T_1 = 400\ \text{K} and T_2 = 300\ \text{K}, so

Cooling shrank the gas — as it must, since less energetic particles need less room to keep the same pressure.