Why a Balloon Shrinks in the Cold and Swells in the Sun

A football pumped up at noon goes soft on a cold morning, a sealed bottle bulges in a hot car, and a pressure cooker whistles as steam builds up. Gases respond to changes in pressure, volume and temperature in simple, predictable ways that a few laws — and a picture of fast-moving molecules — explain.

This lesson covers the gas laws and the ideal gas equation, the kinetic molecular theory, and the speeds of gas molecules.

**Boyle's law says pressure and volume are inversely proportional at constant temperature, Charles's law says volume is proportional to absolute temperature at constant pressure, Avogadro's law says volume is proportional to the number of moles, and together they give the ideal gas equation $PV=nRT$.

The laws:

- Boyle's law** — $P_1{V}_1=P_2{V}_2$ at constant T and n
- Charles's law — $\frac{V_1}{T_1}=\frac{V_2}{T_2}$ at constant P and n, with T in kelvin
- Avogadro's law — $V\propto n$ at constant T and P; one mole of an ideal gas occupies 22.7 L at 273.15 K and 1 bar
- Ideal gas equation — $PV=nRT$, with $R=8.314$ J mol${}^{-1}$ K${}^{-1}$ or $0.0821$ L atm mol${}^{-1}$ K${}^{-1}$

Worked example 1 — Boyle's law. A gas at 1.0 bar occupies 2.5 L. Compressed at constant temperature to 0.50 L:

$$P_2=\frac{1.0\times 2.5}{0.50}=5.0\text{ bar}$$

Worked example 2 — Charles's law. 2.0 L of gas at 300 K is heated to 450 K at constant pressure:

$$V_2=\frac{2.0\times 450}{300}=3.0\text{ L}$$

Worked example 3 — ideal gas equation. 2.0 mol of gas in a 10 L container at 300 K exerts

$$P=\frac{nRT}{V}=\frac{2.0\times 0.0821\times 300}{10}=4.93\text{ atm}$$

Density and molar mass. Rearranging gives $d=\frac{PM}{RT}$, so heavier gases are denser at the same temperature and pressure.

An everyday example. Tyre pressure checked at a petrol pump after a long drive reads higher, because the air inside has warmed while its volume stayed almost the same.

The substance. Temperature must always be in kelvin — using degrees Celsius in Charles's law would absurdly predict zero volume at 0 °C.

The kinetic molecular theory explains gas behaviour by assuming that a gas is made of tiny, widely spaced particles in constant random motion, colliding elastically, with no forces between them and an average kinetic energy proportional to absolute temperature.

Postulates:

- A gas consists of a large number of identical, very small particles whose own volume is negligible compared with the container
- There are no attractive or repulsive forces between the particles
- Particles move constantly and randomly in straight lines, colliding with each other and with the walls
- Collisions are perfectly elastic, so total kinetic energy is conserved
- At any moment, particles have different speeds and so different kinetic energies
- The average kinetic energy of the particles is directly proportional to absolute temperature

How it explains the laws:

- Pressure — caused by particles striking the walls of the container
- Boyle's law — halving the volume doubles how often particles strike the walls, doubling the pressure
- Charles's law — hotter particles move faster and push the walls out to a larger volume at constant pressure

Worked example. The average kinetic energy of one molecule is $\frac{3}{2}kT$. At 300 K, with $k=1.38\times 10^{-23}$ J K${}^{-1}$:

$$\overline{KE}=\frac{3}{2}\times 1.38\times 10^{-23}\times 300=6.21\times 10^{-21}\text{ J}$$

An everyday example. The smell of hot jalebis spreading through a sweet shop shows gas molecules moving randomly and mixing with the air.

The substance. The postulates describe an ideal gas — real molecules do attract each other and take up space, which is why real gases deviate at high pressure and low temperature.