Kinetic Theory

Everything in this chapter rests on one working picture of a gas: a huge number of molecules, each a tiny point of mass, flying about in straight lines at random, colliding elastically with each other and with the walls of their container, and otherwise exerting no force on one another at all. This is the ideal gas model. No real gas obeys it exactly, but dilute gases well above their liquefaction temperature (air in this room, for instance) come astonishingly close, which is exactly why it is worth building a whole theory around.

Experimentally, long before anyone could watch a single molecule, this picture was already compressed into one empirical law relating a gas’s pressure $P$, volume $V$, and absolute temperature $T$:

Here $\mu$ is the number of moles, $N$ the total number of molecules, $R$ the universal gas constant (8.314 J/mol·K), and $k_B$ the Boltzmann constant, simply the gas constant rewritten per molecule instead of per mole:

The rest of this chapter’s job is to show that this equation, along with everything it implies about temperature and heat, is not a separate law of nature at all. It falls straight out of Newton’s laws applied to a swarm of colliding molecules, once you stop tracking individual molecules and start tracking their statistical averages.

Here is the central derivation of the chapter: where gas pressure actually comes from. Picture a cubical box of side $L$ containing $N$ identical molecules, each of mass $m$, moving randomly and never interacting with each other, only bouncing elastically off the walls.

Take one molecule with velocity component $v_x$ perpendicular to a particular wall. An elastic collision with a rigid wall simply reverses that component, $v_x \to -v_x$, leaving its speed unchanged, so the molecule’s momentum change is $-2mv_x$, and by Newton’s third law the wall receives momentum $+2mv_x$ on every such hit.

After bouncing off this wall, the molecule must cross the box, hit the opposite wall, and return, a round trip of distance $2L$, before it can strike this wall again. So it delivers a momentum kick of $2mv_x$ once every $2L/v_x$ seconds, which is exactly a steady average force on the wall of:

Summing this over all $N$ molecules gives the total force on that wall, $F = \dfrac{m}{L}\sum v_x^2 = \dfrac{Nm\langle v_x^2\rangle}{L}$, where $\langle v_x^2\rangle$ is the average of $v_x^2$ over all the molecules. Dividing by the wall’s area $L^2$ gives the pressure:

Nothing in the setup singled out the $x$-direction, so by symmetry $\langle v_x^2\rangle = \langle v_y^2\rangle = \langle v_z^2\rangle$, and since $v^2 = v_x^2+v_y^2+v_z^2$ for every molecule, each of these three equal pieces must be exactly one third of the total: $\langle v_x^2\rangle = \tfrac{1}{3}\langle v^2\rangle$. Substituting this in gives the chapter’s key result:

where $\rho = Nm/V$ is the gas’s mass density and $\langle v^2\rangle$ is the mean square speed, averaged over every molecule in the box. Notice what this derivation never assumed: it never needed molecules to push against each other, only against the walls. A gas of molecules that never collided with one another at all would still, by this argument, exert a perfectly ordinary pressure.

The derivation above is mechanics alone, it never mentioned temperature. The link appears the moment you compare it with the §12.1 ideal gas equation. Both expressions give $PV$, so they must be equal:

The $N$ cancels, and rearranging gives one of the most important results in all of thermal physics:

The left side is the average translational kinetic energy per molecule. The right side is a multiple of absolute temperature alone. In other words, temperature is not some separate quantity that happens to be correlated with molecular motion, it is a direct, literal measure of the average kinetic energy of molecular motion, and nothing else about the gas (its pressure, its volume, which gas it even is) enters this relation at all. Taking the square root of $\langle v^2\rangle$ gives the rms speed:

with $M = N_Am$ the molar mass. This is exactly the quantity the molecules-in-a-box simulation on the concept page tracks, and it is worth being precise about what it depends on and what it does not. $v_{rms}$ depends only on $T$ (for a fixed gas). It does not appear anywhere in terms of $P$ or $V$ separately, even though $P$ and $V$ were exactly what the §12.2 derivation was built from.

This is why squeezing the same gas into a smaller fixed box, at the same temperature, does not speed its molecules up at all. The pressure rises, but purely because the molecules now hit the (closer) walls more often, not because any molecule is moving any faster between hits. In the simulation, dragging the Volume slider visibly shrinks the box and visibly changes the pressure readout, while every single dot keeps exactly the speed it had before. Only dragging Temperature changes how fast the dots move. That separation, speed is set by $T$ alone, pressure is set by $T$ and the collision rate that $V$ controls, is the single most important idea in this section.

Section 12.3 derived $\tfrac{1}{2}m\langle v^2\rangle = \tfrac{3}{2}k_BT$ by counting three independent translational directions, $x$, $y$, and $z$, each contributing an equal share, $\tfrac{1}{2}k_BT$, to the total. The law of equipartition of energy generalises this counting argument far beyond translation:

In thermal equilibrium, the total energy of a system is shared equally among every independent (quadratic) degree of freedom available to it, each contributing an average of $\tfrac{1}{2}k_BT$, regardless of the nature of that degree of freedom or the temperature.

A “degree of freedom” here just means an independent way a molecule can store energy that enters its energy expression as a square, a velocity component, an angular velocity component, a spring-like displacement. Translation along $x$, $y$, and $z$ gives three such terms for any molecule; what changes between gases is what else is available besides translation.

Equipartition hands over the internal energy $U(T)$ of an ideal gas directly, and specific heats are just derivatives of that. The molar specific heat at constant volume is defined as the heat needed per mole per degree, with volume held fixed, so no work is done and all the heat goes into raising $U$:

The step $C_p = C_v + R$ used above, Mayer’s relation, is not specific to monatomic or diatomic gases, it holds for any ideal gas, and is worth deriving once in general. For $\mu=1$ mole, $PV=RT$, so at constant pressure, $P\,dV = R\,dT$. The first law gives $dQ = dU + P\,dV = C_v\,dT + R\,dT$, and since this same $dQ$ at constant pressure is by definition $C_p\,dT$, matching coefficients gives:

The extra $R$ is simply the work a gas does pushing back its surroundings as it expands at constant pressure, work a gas held at constant volume never has to do, so it always takes more heat to raise its temperature by one degree at constant pressure than at constant volume.

Section 12.3’s rms speeds come out to hundreds of metres per second, faster than sound. Yet open a bottle of perfume across a still room and the smell visibly takes minutes to arrive, not milliseconds. The resolution is that a molecule almost never travels in a straight line for long: real molecules have a finite size, and they constantly collide with each other, each collision flinging them off in a new, effectively random direction.

The average distance a molecule covers between one collision and the next is its mean free path, $\lambda$. For a gas of molecules of effective diameter $d$ and number density $n = N/V$, a reasonably careful count of how many other molecules lie within a molecule’s collision “tube” as it travels gives:

Using $n = P/(k_BT)$ from the ideal gas equation, this can equally be written in terms of pressure and temperature directly, which is often more convenient:

Two features are worth remembering. First, $\lambda$ shrinks as pressure rises (more molecules packed into the same space means more frequent collisions) and grows with temperature at fixed pressure. Second, $\lambda$ is typically hundreds of molecular diameters even at atmospheric pressure, a molecule travels a long way, relatively speaking, before each collision, but it does so in a random zig-zag, not a straight line, which is exactly why diffusion is so much slower than the rms speed alone would suggest. The worked exercise below puts real numbers to both $\lambda$ and the rate of collisions this implies.

Two of the chapter’s own classic exercise questions, with original worked solutions.