Volume and the rate of collisions

1. Consider a fixed mass of gas held at constant temperature, so the number of particles and their average speed both stay the same.
2. Reducing the volume brings the walls closer together, so each particle travels a shorter distance between one wall collision and the next.
3. The particles therefore reach the walls more often, so the number of collisions each second on every square metre of wall increases.
4. The force of each individual collision is unchanged, because the temperature and therefore the average speed of the particles has not changed.
5. More collisions of the same strength on each square metre give a greater total force on each square metre, so the pressure increases.
6. Increasing the volume reverses every step, so the particles hit the walls less often and the pressure decreases.
7. Squeezing a sealed balloon becomes harder as you go, because the pressure inside climbs as the volume of the trapped gas falls.
8. An air bubble released at the bottom of a deep tank grows as it rises, because the pressure of the water around it falls and the bubble expands in response.

The pressure and volume equation

1. For a fixed mass of gas at constant temperature the relationship is $p_1{V}_1=p_2{V}_2$.
2. In this equation $p_1$ and $V_1$ are the pressure and volume before the change, and $p_2$ and $V_2$ are the pressure and volume after it.
3. The units only need to be consistent, so both pressures must be in the same unit and both volumes must be in the same unit.
4. The equation says that the product of pressure and volume stays constant, so pressure and volume are inversely proportional.
5. Halving the volume therefore doubles the pressure, and doubling the volume halves it.
6. Rearranging for the new pressure gives $p_2=\frac{p_1{V}_1}{V_2}$.
7. Rearranging for the new volume gives $V_2=\frac{p_1{V}_1}{p_2}$.
8. The equation fails if the temperature changes or if gas leaks in or out, so check both conditions before using it.

Graphs of pressure and volume

1. A graph of pressure against volume is a curve that falls steeply at small volumes and flattens at large ones, never touching either axis.
2. That curved shape is the signature of an inverse relationship rather than a proportional one.
3. A graph of pressure against $\frac{1}{V}$ is a straight line through the origin, which is the standard way of showing that the relationship really is inverse.
4. The gradient of that straight line equals the constant product $pV$ for the sample of gas being tested.
5. Multiplying each pressure reading by its volume reading is a quick numerical check, because every pair should give the same product within experimental uncertainty.