Heating a gas at constant volume raises its pressure

1. Start with a fixed mass of gas sealed in a rigid container, such as a steel can, so no particles get in or out and the volume cannot change.
2. Heating the gas raises the average kinetic energy of its particles, so on average they move faster.
3. Faster particles cross the container in less time, so they strike the walls more often.
4. Each collision is also harder, because a faster particle rebounds with a bigger change of velocity and pushes the wall more strongly.
5. More frequent and harder collisions mean a greater total force on the same area of wall, so the pressure rises.
6. Cooling the same sealed can reverses every step, so the pressure falls, which is why an aerosol left in a cold shed sprays weakly.

Squeezing a gas at constant temperature also raises its pressure

1. Now take the same fixed mass of gas in a syringe with the nozzle blocked, and push the plunger in slowly so the temperature stays constant.
2. Because the temperature has not changed, the particles are moving at the same average speed as before, so each individual collision is just as hard as it was.
3. The smaller volume gives each particle a shorter distance to travel before it meets a wall again.
4. The same number of particles therefore makes more collisions per second on each square metre of wall, and the pressure rises.
5. Pulling the plunger back out increases the volume, spreads the collisions thinner, and the pressure falls again.
6. Notice the difference between the two cases: heating changes both how often and how hard, while squeezing changes only how often.

At constant temperature, pressure times volume stays the same

1. Halving the volume doubles the pressure, and trebling the volume cuts the pressure to a third, so pressure and volume are inversely proportional.
2. Two conditions must hold for the equation to work: the mass of gas is fixed and the temperature is constant.
3. Any consistent units work, because the same units appear on both sides, so pressure in $kPa$ with volume in ${cm}^3$ is fine as long as you do not switch part-way through.
4. A graph of $p$ against $V$ is a smooth curve falling steeply at first and then flattening, never touching either axis.
5. A graph of $p$ against $1/V$ is a straight line through the origin, and that straight line is the usual way of showing that the relationship really is inverse proportion.