Kinetic Theory of Gases and Radiation

Have you ever wondered what is actually happening inside a gas? When you pump air into a bicycle tyre, the pressure increases. When you heat a sealed container, the pressure also increases. But what is pressure, really, at the microscopic level? The kinetic theory of gases answers this beautifully: it connects the macroscopic world of pressure, temperature, and volume that we can measure with instruments to the microscopic world of countless molecules zipping around in constant, random motion. It is one of the greatest intellectual achievements of physics — a bridge between the visible and the invisible. The fundamental picture is this: a gas consists of an enormous number of molecules (about 2.7 × 10²⁵ molecules per cubic metre at room temperature and pressure) moving in all directions with a wide range of speeds. These molecules collide with each other and with the walls of the container. Each collision with a wall exerts a tiny force. Add up the forces from billions upon billions of collisions every second, and you get the macroscopic pressure we measure. The kinetic theory makes several simplifying assumptions: gas molecules are treated as point particles (their size is negligible compared to the distances between them); there are no intermolecular forces except during collisions; collisions are perfectly elastic (no kinetic energy is lost on average); and the motion is completely random. From these assumptions, we can derive the ideal gas equation PV = nRT, which was originally discovered experimentally. The derivation is elegant and illuminating. Consider a cubical container of side length L containing N molecules of a gas, each of mass m. A molecule moving with velocity v_x towards one wall collides, reverses its x-component of velocity, and undergoes a change in momentum of 2mv_x. The time between successive collisions with the same wall is 2L/v_x, so the force exerted by this molecule on the wall is Δp/Δt = (2mv_x)/(2L/v_x) = mv_x²/L. Summing over all N molecules and dividing by the area of the wall (L²), we get pressure P = (Nmv²_rms)/(3V). Since the average kinetic energy per molecule is (½)mv²_rms = (3/2)kT, we arrive at PV = NkT = nRT. The temperature T is nothing but a measure of the average kinetic energy of the molecules — higher temperature means faster-moving molecules. This microscopic picture explains so much. It tells us why the pressure of a gas increases when you compress it (molecules hit the walls more frequently because they travel shorter distances between walls). It explains why heating a gas increases pressure (molecules move faster, hitting walls harder and more often). It predicts Graham's law of effusion (lighter molecules escape faster through a tiny hole). It gives us the Maxwell-Boltzmann distribution of molecular speeds, showing that not all molecules move at the same speed — some are slow, some are fast, and the distribution depends on temperature. It even explains transport phenomena like viscosity, thermal conductivity, and diffusion in gases. For real gases at high pressures or low temperatures, the assumptions break down, and we need corrections — leading to the van der Waals equation. But as a first approximation, the kinetic theory is remarkably powerful and forms the foundation of our understanding of gases.