Pressure.

Step 1 of 7

Hits on the walls

The air around you presses on every square centimetre of your skin with the weight of about a kilogram, and you don’t feel it, because it presses equally from every direction, and from inside you too. You feel it only when it is unbalanced: ears popping in a lift, or a can crushed when the air is pumped out of it.

Pressure is force spread over an area:

$p = \frac{F}{A}$⁠

It is measured in pascals, one newton on every square metre. The air’s pressure at sea level is about 101 kPa, one atmosphere. The Scene has gas in a cylinder under a piston, and Show the wall hits is On: each flash is a particle striking the wall. The gauge beside the cylinder shows the pressure.

Step 2 of 7

Squeeze it

Push the piston down and the same particles have less room: they cross the cylinder faster, reach the walls more often, and there is less wall to share the hits over. Robert Boyle found in the 1660s that at a fixed temperature the pressure times the volume stays the same:

$p_1V_1 = p_2V_2$⁠

The Temperature is fixed at 300 K now, and the piston is at the top: Volume is 100%.

Step 3 of 7

Heat it

Heat the gas and the particles move faster. Not all at one speed: some dawdle and some race. But the temperature in kelvin grows in step with their average kinetic energy, $\tfrac{1}{2}mv^2$⁠. The gas here has been heated to 600 K, twice 300 K, and the Temperature is fixed.

Step 4 of 7

The gas law

Heated at a fixed volume, the particles hit the walls both more often and harder, and the two together double the pressure when the temperature doubles. Put both laws together and you have the gas law:

$\frac{pV}{T} = \text{const}$$\frac{pV}{T} =$$\text{const}$⁠

with the temperature in kelvin. That is why a tyre’s pressure is higher after a long drive, and why a sealed can heated in a fire can explode.

Step 5 of 7

Two atmospheres

Now it is up to you: both the Volume and the Temperature are free.

Step 6 of 7

Pressure with depth

Liquids press too, and their pressure grows with depth, because each layer carries the weight of everything above it:

$p = p_0 + \rho g h$⁠

$p_0$⁠ is the air’s pressure on the surface, $\rho$⁠ the liquid’s density and $h$⁠ the depth. Ten metres of water add almost exactly one more atmosphere. That is why dams are thickest at the base, and why divers rise slowly.

Step 7 of 7

Hits in a hot gas

Back to the gas in the cylinder. Show the wall hits marks every hit of a particle on the wall with a flash. The gas has been heated to 600 K, and the volume has not changed.