Q.What is the expression between pressure and kinetic energy of a gas molecule ?

Kinetic Theory Explanation

Imagine you're sitting in a quiet room. The air around you feels still, but it isn't. Every second, billions of tiny particles — molecules of nitrogen and oxygen — are zipping past you at hundreds of metres per second. You don't feel them because they're too small, and they're moving in every direction at once. But if you put your hand near a hot stove, you suddenly feel heat. Why?

The answer is the kinetic theory of matter. It's a way of explaining what we observe at the human scale (temperature, pressure, heat) by thinking about what's happening at the molecular scale.

The core idea

The kinetic theory says three simple things:

1. All matter is made of tiny particles (atoms or molecules) that are in constant, random motion.
2. The particles collide with each other and with the walls of their container — these collisions are perfectly elastic (no energy is lost).
3. The average kinetic energy of these particles is directly proportional to the temperature of the substance.

That's it. Everything else — pressure, diffusion, the way a gas expands when heated — follows from these three statements.

Building intuition

Think of a single gas molecule bouncing around inside a box. It hits a wall, bounces off, and keeps going. Each time it hits the wall, it exerts a tiny force. Now multiply that by billions of molecules hitting every square centimetre of wall every second. That constant, collective force is what we measure as pressure.

Now heat the box. The molecules move faster — their average kinetic energy increases. They hit the walls harder and more often. Pressure goes up. If the walls can move (like a piston), the gas expands until the pressure inside equals the pressure outside.

This is why a bicycle tyre feels hot after you pump it: you're doing work on the air, compressing it, which increases the average kinetic energy of the molecules — and that's exactly what temperature is.

The precise statement

For an ideal gas (a gas where intermolecular forces are negligible and collisions are perfectly elastic), the kinetic theory gives us a direct mathematical link:

where $k_B$ is Boltzmann's constant ($1.38 \times 10^{-23} \, \text{J/K}$) and $T$ is the absolute temperature in Kelvin.

This means that at the same temperature, all gas molecules — regardless of their mass — have the same average kinetic energy. A light hydrogen molecule moves faster than a heavy oxygen molecule at the same temperature, but their average kinetic energies are equal.

From this, we can derive the ideal gas law:

where $P$ is pressure, $V$ is volume, $N$ is the number of molecules, $m$ is the mass of one molecule, and $\overline{v^2}$ is the mean square speed.

What the theory explains

The kinetic theory isn't just abstract. It explains everyday phenomena:

• Evaporation cools you: The fastest molecules escape from a liquid surface, leaving behind slower ones. The average kinetic energy drops — so the temperature drops.
• Diffusion: Molecules spread out because they're constantly moving and colliding, gradually mixing with neighbouring molecules.
• Brownian motion: Pollen grains jitter under a microscope because they're being bombarded unevenly by invisible water molecules. …