Kinetic Theory

1. What this chapter covers

Thermodynamics, the previous chapter, never once asked what a gas is made of — it worked entirely with macroscopic quantities like pressure and temperature. This chapter opens the box. It rebuilds pressure and temperature from a molecular picture, and in doing so explains why the macroscopic laws from Chapter 10 and Chapter 11 look exactly the way they do.

2. The molecular picture, and why gases are the easy case

Matter consists of molecules in constant, random motion — the Atomic Hypothesis, which Richard Feynman considered the single most information-dense sentence in physics. Solids and liquids have molecules packed only a couple of angstroms apart, close enough that intermolecular forces dominate their behaviour.

Gases are different. At ordinary pressure and temperature, gas molecules sit tens of angstroms apart — far enough that the forces between them are negligible except during the brief moment of a collision. This single fact is why kinetic theory can treat a gas as a swarm of essentially free, independently moving particles, and why gases are mathematically the simplest state of matter to model even though solids look more orderly.

Avogadro's law states that equal volumes of all gases at equal temperature and pressure contain the same number of molecules — independent of what the gas actually is. Combined with Dalton's atomic theory, this explains Gay-Lussac's law of combining volumes. The number of molecules in one mole (22.4 litres of any gas at STP) is the Avogadro number, $N_A=6.02\times 10^{23}$.

The atomic hypothesis is often credited to John Dalton, but it was proposed independently, centuries earlier, in more than one place. The Vaisheshika school in India, founded by Kanada around the sixth century BCE, described atoms (paramanu) as eternal and indivisible, and even estimated atomic size by conjecture to a value close to the modern $10^{-10}$ m.

In Greece, Democritus argued a few centuries later that atoms differ from each other in shape and size, and that this alone explains why different substances behave differently. Neither tradition had the quantitative experiments to test these ideas — that had to wait for Dalton's laws of definite and multiple proportions roughly two thousand years later.

A worked estimate, in the style of the chapter's own examples. Water's density is $1000{\text{kg/m}}^3$ as a liquid, where molecules sit essentially touching, but only $0.6{\text{kg/m}}^3$ as vapour at $100^{\circ }\text{C}$ and 1 atm. Since a fixed mass occupies a volume inversely proportional to its density, the vapour occupies $1000/0.6\approx 1667$ times more volume than the same mass of liquid water.

If the molecules themselves fill essentially all of the liquid's volume, they fill only $1/1667\approx 6\times 10^{-4}$, or about 0.06%, of the vapour's volume — the rest is empty space between molecules. This exact style of reasoning, run in reverse, is what Exercise 12.1 asks for.

3. The ideal gas equation, and where it comes from

For a gas at low pressure and high temperature — well above the point where it would liquefy — pressure, volume, and temperature satisfy

$PV=\mu RT$

where $\mu$ is the number of moles and $R=8.314{\text{J mol}}^{-1}{\text{K}}^{-1}$ is the universal gas constant. An equivalent molecular form replaces moles with molecule count: $PV=k_{B}NT$, where $k_B=R/N_A=1.38\times 10^{-23}{\text{J K}}^{-1}$ is the Boltzmann constant — the same constant that converts between a macroscopic, per-mole description and a microscopic, per-molecule one throughout this chapter.

A gas that obeys $PV=\mu RT$ exactly at every pressure and temperature is called an ideal gas — a theoretical idealisation that no real gas fully reaches, though real gases approach it closely at low pressure and high temperature, where molecules are far enough apart that their mutual interactions barely matter.

Three familiar gas laws all fall out of this one equation as special cases:

Dalton's law deserves a closer look: for a mixture of non-interacting ideal gases sharing a container, $P_1={\mu }_{1}RT/V$ is exactly the pressure gas 1 would exert if it occupied the container alone. The total pressure of the mixture is simply the sum of these individual partial pressures — a direct consequence of each gas's molecules not caring that other molecules are present at all.

4. Deriving pressure from molecules hitting a wall

This is the derivation the whole chapter is building toward: a formula for pressure that never mentions "gas" as anything except a swarm of particles.

Picture a gas enclosed in a cube of side $l$, with a molecule of velocity $(v_x,v_y,v_z)$ striking a wall perpendicular to the x-axis. The collision is elastic, so the molecule rebounds with $v_y$ and $v_z$ unchanged but $v_x$ reversed — a momentum change of $2m{v}_x$ transferred to the wall.

In a short time $\Delta t$, only molecules within a distance $v_x\Delta t$ of the wall can reach it, and on average half of those are moving toward the wall. So the number of molecules with velocity component $v_x$ striking the wall in time $\Delta t$ is $\frac{1}{2}nA{v}_x\Delta t$, where $n$ is the number density and $A$ is the wall's area. Multiplying by the momentum each transfers, then dividing by $A\Delta t$ to get pressure:

$P=nm{v}_x^2$

Real gases have a whole distribution of speeds, not one shared $v_x$, so this becomes an average: $P=nm\overline{v_x^2}$. Since the gas is isotropic — no direction is preferred — $\overline{v_x^2}=\overline{v_y^2}=\overline{v_z^2}=\frac{1}{3}\overline{v^2}$, giving the chapter's central result:

$P=\frac{1}{3}nm\overline{v^2}$

Neither the container's shape nor $A$ and $\Delta t$ survive into this final formula — a hint that the result is genuinely general, not an artefact of choosing a cube. The derivation also ignores collisions between molecules entirely; that turns out not to matter, because in a steady state, any molecule knocked out of a given velocity is statistically replaced by another molecule knocked into it, leaving the average $\overline{v_x^2}$ unaffected.

5. The kinetic meaning of temperature

Combining $P=\frac{1}{3}nm\overline{v^2}$ with the ideal gas equation gives the chapter's real payoff. Multiplying through by $V$ and comparing with $PV=\mu RT=k_{B}NT$:

$\frac{1}{2}m\overline{v^2}=\frac{3}{2}{k}_{B}T$

The average translational kinetic energy of a single molecule is proportional to absolute temperature alone — not to pressure, volume, or which gas it is. A helium molecule and a uranium hexafluoride molecule at the same temperature carry exactly the same average kinetic energy; the heavier molecule simply moves slower to compensate. This is the kinetic interpretation of temperature that gives the chapter its name.

The square root of $\overline{v^2}$ is the root mean square speed, $v_{rms}=\sqrt{3k_{B}T/m}$. For nitrogen at 300 K this works out to about 516 m/s — comparable to the speed of sound in air, which is not a coincidence, since sound itself propagates through exactly these molecular collisions.

One consequence worth stating explicitly: since $\frac{1}{2}m\overline{v^2}=\frac{3}{2}{k}_{B}T$ depends only on $T$, at a fixed temperature a lighter molecule must have a larger $v_{rms}$ to carry the same average kinetic energy as a heavier one. This single fact drives isotope-separation techniques and explains why light gases like hydrogen and helium escape a planet's atmosphere far more easily than heavier ones.

6. The law of equipartition of energy

A molecule's kinetic energy is a sum of squared terms — $\frac{1}{2}m{v}_x^2$, $\frac{1}{2}m{v}_y^2$, $\frac{1}{2}m{v}_z^2$ for translation, and similar squared terms for rotation and vibration if the molecule has those motions available. Each such independent squared term is called a degree of freedom.

Law of equipartition of energy: In thermal equilibrium at temperature $T$, the total energy is shared equally among all available degrees of freedom, with each contributing an average energy of $\frac{1}{2}{k}_{B}T$.

The textbook explicitly states this without proof — "the proof of the law of equipartition of energy is beyond the scope of this book" — so treat it as a fact to apply, not derive.

Counting degrees of freedom by molecule type:

• Monatomic (e.g. argon): 3 translational only. Total: 3.
• Diatomic, rigid (e.g. ${\text{O}}_2$, ${\text{N}}_2$ at moderate temperature): 3 translational + 2 rotational (rotation about the bond axis itself has negligible moment of inertia and does not contribute). Total: 5.
• Diatomic with vibration (e.g. CO at higher temperature): the 5 above, plus one vibrational mode. A vibrational mode counts as two degrees of freedom, not one — it carries both kinetic and potential energy terms. Total: 7.
• Polyatomic: 3 translational + 3 rotational + $f$ vibrational modes (each worth 2). Total: $6+2f$.

7. Specific heat capacities, predicted from degrees of freedom alone

Each degree of freedom contributes $\frac{1}{2}{k}_{B}T$ of energy per molecule, so a mole of gas with $N$ degrees of freedom has internal energy $U=\frac{N}{2}RT$, giving $C_v=\frac{N}{2}R$ directly — no separate experiment needed once the degree-of-freedom count is known.

$C_p-C_v=R$ holds across every row — it is a property of being an ideal gas, not of any particular degree-of-freedom count. These predictions match measured specific heats well for most gases at ordinary temperature; gases like ${\text{Cl}}_2$ that measure noticeably higher than predicted are exactly the ones where a vibrational mode has become active and was left out of the simple count.

The same reasoning, applied to a solid where each of $N$ atoms vibrates in three dimensions (2 degrees of freedom per dimension, since vibration counts double), predicts $C=3R$ — the Dulong-Petit law already met in Chapter 11, now derived rather than merely quoted.

8. Mean free path: why a "fast" gas diffuses so slowly

Gas molecules travel at hundreds of metres per second, yet a smell takes minutes to cross a room. The reason is collisions: a molecule cannot travel far in a straight line before colliding with another and being deflected onto a new, random path.

Model molecules as spheres of diameter $d$. A molecule with average speed $⟨v⟩$ sweeps out a cylindrical volume $\pi d^2⟨v⟩\Delta t$ in time $\Delta t$; any molecule whose centre lies in that volume causes a collision. With $n$ molecules per unit volume, the collision rate is $n\pi d^2⟨v⟩$, so the average time between collisions is $\tau =1/(n\pi ⟨v⟩d^2)$.

The average distance covered between two successive collisions is the mean free path:

$l=⟨v⟩\tau =\frac{1}{\sqrt{2}n\pi d^2}$

The $\sqrt{2}$ enters once the derivation properly accounts for every other molecule also moving, not sitting still.

For air at STP, working out $n$ and using $d\approx 2\times 10^{-10}$ m gives $l\approx 2.9\times 10^{-7}$ m — about 1500 times the molecular diameter, and roughly 100 times the average interatomic spacing. That gap between "how far apart molecules typically are" and "how far a molecule actually travels before colliding" is the real reason diffusion is so slow despite molecular speeds being so high.

9. Where "work done in compressing a gas" actually belongs

CBSE's own Unit IX syllabus line for this chapter reads: "Equation of state of a perfect gas, work done in compressing a gas." The equation of state is covered above in Section 3 — but this chapter never derives a work-done formula. The quantitative work formulas for compressing or expanding a gas, $W=\mu RT\ln (V_2/V_1)$ for isothermal and $W=\mu R(T_1-T_2)/(\gamma -1)$ for adiabatic processes, belong to Chapter 11, Thermodynamics, and were derived there.

What this chapter does supply is the missing piece for using those formulas correctly on a real gas: the value of $\gamma$ to plug in, which depends on the gas's degrees of freedom (Section 7). A question that asks for the work done compressing a diatomic gas adiabatically is really asking you to combine Chapter 11's formula with this chapter's $\gamma =7/5$.

Summary

• Gases are the simplest state of matter to model because their molecules sit far enough apart, tens of angstroms, that intermolecular forces are negligible except during a collision.
• The ideal gas equation $PV=\mu RT=k_{B}NT$ reduces to Boyle's law, Charles' law, and Dalton's law of partial pressures as special cases.
• Kinetic theory derives $P=\frac{1}{3}nm\overline{v^2}$ from molecules making elastic collisions with a wall, without assuming anything about pressure in advance.
• Combining that with the ideal gas equation gives the kinetic meaning of temperature: $\frac{1}{2}m\overline{v^2}=\frac{3}{2}{k}_{B}T$ — average molecular kinetic energy depends on temperature alone, the same for every gas at a given $T$.
• The law of equipartition of energy (stated, not proved, in this book) assigns $\frac{1}{2}{k}_{B}T$ to each degree of freedom — 3 for a monatomic gas, 5 for a rigid diatomic, 7 once a vibrational mode activates, each vibrational mode counting as two.
• Specific heats follow directly from the degree-of-freedom count: $C_v=\frac{N}{2}R$, and $C_p-C_v=R$ always, for any ideal gas.
• The mean free path — roughly 1500 molecular diameters in air at STP — is why gas diffusion is slow despite molecular speeds being comparable to the speed of sound.
• CBSE's "work done in compressing a gas" line under this chapter's syllabus heading is answered using Chapter 11's work formulas together with this chapter's $\gamma$ values, not by anything derived here.