Kinetic Theory

The previous chapter treated a gas as a black box obeying laws about pressure, volume and temperature. This chapter opens the box. Starting from a single idea — that a gas is an enormous number of tiny molecules in ceaseless random motion — it derives the gas laws instead of assuming them, and along the way explains what temperature actually is. It begins with the evidence that molecules exist at all, from Dalton's laws of proportion and Avogadro's hypothesis to Brownian motion seen under a microscope, and with the scale of things: molecules an angstrom across, spaced tens of angstroms apart in a gas, travelling thousands of angstroms between collisions. The ideal gas equation is then set out in all four of its forms — $PV = \mu RT$, $PV = Nk_BT$, $P = nk_BT$ and $P = \frac{\rho RT}{M_0}$ — with Boyle's law, Charles' law, the pressure law and Dalton's law of partial pressures falling out as special cases, and with an honest account of where real gases depart from the model and why. The heart of the chapter is the derivation of $$P = \frac{1}{3}nm\overline{v^2} = \frac{1}{3}\rho\,\overline{v^2}$$ from nothing but elastic collisions and Newton's laws, and its comparison with the gas equation, which yields the result the whole subject rests on: the average translational kinetic energy of a molecule is $\frac{3}{2}k_BT$, independent of pressure, volume and the nature of the gas. Temperature, it turns out, is molecular kinetic energy. From there come the root mean square speed $v_{rms} = \sqrt{\frac{3RT}{M_0}}$ and the full Maxwell distribution of molecular speeds, with the most probable and average speeds and the fixed ratio between all three. Counting degrees of freedom and applying the law of equipartition of energy — $\frac{1}{2}k_BT$ for every quadratic term, so a vibrational mode counts twice — then predicts the molar specific heats of gases from molecular structure alone, giving $C_v = \frac{f}{2}R$, $C_p - C_v = R$ and $\gamma = 1 + \frac{2}{f}$, and the Dulong-Petit value $3R$ for solids. The chapter closes with the mean free path $l = \frac{1}{\sqrt{2}n\pi d^2}$, which explains why a gas whose molecules move faster than sound still takes minutes to diffuse across a room, and with Graham's law for the rates at which different gases do it. Topics the rationalised syllabus trimmed — the Maxwell speed distribution, the formulas for the most probable and average speeds, Brownian motion, the pressure law and Graham's law — are restored in full, because Boards, JEE Main and NEET ask about them every year. Dedicated JEE and NEET Corners follow, each with a full-length exam-pattern practice drill.