Charles's Law

Hold the pressure on a gas constant and its volume follows its absolute temperature in strict proportion: warm it by 10% and it swells by 10%. The reason is worth having rather than memorising. Temperature is a measure of how fast the molecules are moving; pressure is how hard their impacts push on each square metre of wall. If the gas is to keep pushing with the same force while its molecules move faster, the only thing it can do is spread out, so that each patch of wall is struck less often. Volume rises exactly as fast as temperature to keep that balance, and $V_1/T_1 = V_2/T_2$ is that sentence in symbols.

A 2.5 L balloon leaves a 22 °C room and goes into a −18 °C freezer. In kelvin those are 295.15 and 255.15, so $V_2 = 2.5 \times 255.15/295.15 = 2.16\ \text{L}$ — it loses about 14% of its volume and visibly puckers. Bring it back out and it recovers. Nothing left the balloon; the same molecules simply stopped needing as much room.

Jacques Charles found this with hydrogen balloons around 1787 and never published it. Joseph Louis Gay-Lussac did the careful work and published in 1802, and generously named the result after Charles. The interesting part is what came out of extending the straight line. Plot volume against Celsius temperature and you get a line that, extrapolated backwards, hits zero volume at about −273 °C. No one in 1802 could reach anywhere near that temperature, yet the graph pointed straight at it. That extrapolation is how absolute zero was first located, and it is why William Thomson could propose an absolute scale in 1848 by simply moving the origin to where the gases were pointing. The modern value, −273.15 °C, is the zero of the kelvin scale.

Which is exactly why a ratio in Celsius is meaningless here. Going from 20 °C to 40 °C does not double the volume; in kelvin that is 293.15 to 313.15, a rise of under 7%. Worse, a Celsius ratio breaks outright when the temperature crosses zero — 0 °C in the denominator gives infinity, and a negative Celsius temperature gives a negative volume. This page converts your °C or °F entries to kelvin before it does anything, but the trap is worth recognising when you meet the equation off-screen.

Two smaller conditions do real work. The pressure must actually be constant: a gas sealed in a rigid tank obeys Gay-Lussac's law instead, where pressure rises and volume does not move at all. And the gas must stay a gas. Cool steam through 100 °C and the relation does not merely become inaccurate, it stops applying — the vapour condenses and the volume collapses by a factor of about 1600. A balloon is also only approximately constant-pressure, since the stretched rubber adds a little tension of its own, which is why the real shrinkage runs slightly under what the arithmetic predicts.