Kinetic Theory of Gases
Kinetic theory is the bridge from mechanics to heat. Its hypothesis is that a gas is an enormous number of molecules in rapid, disordered motion, and its programme is to compute the state variables of Classical Thermodynamics from the mechanics of Newtonian Dynamics: pressure from momentum transfer at the walls, temperature from the mean kinetic energy, and the transport coefficients from the mean free path between collisions. The programme succeeded so completely that it settled a question older than physics—whether matter is molecular—and put a number on it, Avogadro's constant.
This chapter carries the hypothesis from Bernoulli's first kinetic model to the Boltzmann equation and its H-theorem, together with the Maxwell–Boltzmann distribution and the elementary mean-free-path theory of viscosity, heat conduction and diffusion. The systematic solution theory of the Boltzmann equation—Chapman–Enskog and its successors—belongs to Nonequilibrium Thermodynamics and Transport, and the ensemble foundation that frees the results from molecular models to Statistical Mechanics.
Kinetic Theory of Gases: all derivations of this chapter are pending.
The kinetic hypothesis
From Bernoulli to Clausius
[Reserved: Bernoulli's derivation of Boyle's law from particle impacts in section ten of the Hydrodynamica [Bernoulli:1738]; the long eclipse of the idea under the caloric theory; Waterston's complete kinetic memoir, refused in 1845 and printed only in 1892 with Rayleigh's introduction [Waterston:1892]; Krönig's short revival [Kronig:1856]; Clausius's “kind of motion we call heat”, with its translational, rotational and vibrational shares [Clausius:1857].]
Pressure from molecular impacts
[Reserved: the elementary pressure formula \(P=\tfrac{1}{3}nm\langle v^{2}\rangle\) of [Bernoulli:1738] [Clausius:1857]; temperature as mean translational kinetic energy, and the kinetic reading of the ideal-gas law of Classical Thermodynamics; the virial theorem of Clausius [Clausius:1870] as the systematic version, and its later use for real gases.]
The mean free path
[Reserved: Buys Ballot's objection that gases interdiffuse far too slowly for molecules moving at hundreds of metres per second, and Clausius's answer—the mean free path, \(\ell\sim1/(n\sigma)\)—in the 1858 paper [Clausius:1858]; the concept is customarily dated to the 1857 paper [Clausius:1857], which does not contain it; collision frequency and effective cross-section; orders of magnitude for air at standard conditions, with \(\ell\) of order \(10^{-7}\,\mathrm{m}\).]
The Maxwell velocity distribution
Maxwell's derivation
[Reserved: the 1860 derivation of the equilibrium velocity distribution from isotropy and the independence of the components [Maxwell:1860]; mean, most probable and root-mean-square speeds; the stronger 1867 rederivation from detailed balance of collisions [Maxwell:1867]; the distribution as the stationary point of the collision term, anticipating Section 55.4.2.]
Molecular-beam verification
[Reserved: Stern's direct measurement of thermal molecular speeds in a silver beam [Stern:1920]; the velocity-selector measurements of Miller and Kusch on potassium and thallium beams, confirming the Maxwell distribution at the per-cent level [Miller:1955]; why a beam samples the flux-weighted distribution, and the correction that implies.]
Molecules leaving a small aperture in an oven held at temperature \(T\) arrive at a distant detector spread over a range of speeds, and the spread is the one Maxwell computed and no other. Stern measured it directly in a beam of silver atoms, by the deflection of the beam on a rotating drum [Stern:1920]; Miller and Kusch, passing beams of potassium and thallium through a slotted rotating cylinder that transmits one speed at a time, confirmed the predicted shape to the per-cent level [Miller:1955]. Inside the oven the number of molecules with speeds between \(v\) and \(v+\dd v\) is proportional to
while the escaping beam carries one further power of \(v\), because fast molecules reach the aperture more often than slow ones.
Derivation. Let \(f(\vect{v})\,\dd^{3}v\) be the number density of molecules with velocity within \(\dd^{3}v\) of \(\vect{v}\). Two properties of a gas at rest fix its form. Isotropy: no direction is distinguished, so \(f\) depends on \(\vect{v}\) only through \(v^{2}\). Independence: in a dilute gas the Cartesian components are uncorrelated, so \(f=\varphi(v_{x})\varphi(v_{y})\varphi(v_{z})\) with the same \(\varphi\) for each. Together,
Take logarithms in Equation (55.2) and differentiate with respect to \(v_{x}\) at fixed \(v_{y},v_{z}\): \(\varphi'(v_{x})/\varphi(v_{x})=2v_{x}F'/F\). The right-hand side is symmetric under permuting the three components, so \(\varphi'(v_{x})/\left(v_{x}\varphi(v_{x})\right)\) is one and the same constant \(-2\lambda\) for each of them, whence \(\varphi(v_{x})=A\exp(-\lambda v_{x}^{2})\) and \(f\propto\exp(-\lambda v^{2})\). Normalisability requires \(\lambda>0\).
The pressure fixes \(\lambda\). A wall normal to \(\hat{x}\) is reached in unit time, per unit area, by those molecules within a distance \(v_{x}\) of it that are moving towards it; each rebounds elastically with its normal momentum reversed, delivering \(2mv_{x}\), so
the last step by isotropy. Comparison with the observed equation of state \(P=nk_{\mathrm{B}}T\) of Classical Thermodynamics gives \(\avg{v^{2}}=3k_{\mathrm{B}}T/m\), while the Gaussian above has \(\avg{v^{2}}=3/(2\lambda)\); hence \(\lambda=m/(2k_{\mathrm{B}}T)\). Multiplying \(f\) by the area \(4\pi v^{2}\) of the shell of constant speed converts it into the distribution of speeds, which is Equation (55.1). Finally, the molecules crossing the aperture in unit time are those within a distance \(v_{z}\) of it, so their number carries an extra factor \(v_{z}\); averaging \(v_{z}\) over the forward hemisphere at fixed speed gives \(v/2\), and the beam distribution is proportional to \(v^{3}\exp(-mv^{2}/2k_{\mathrm{B}}T)\), as observed.
∎Equipartition and specific heats
The equipartition theorem
[Reserved: the equality of mean translational energies between components and between species in Maxwell's 1860 paper [Maxwell:1860]; Boltzmann's general equipartition theorem, with mean energy \(\tfrac{1}{2}k_{\mathrm{B}}T\) per quadratic degree of freedom [Boltzmann:1868]; what counts as a degree of freedom for monatomic and diatomic gases.]
Specific heats and the classical failure
[Reserved: predicted heat-capacity ratios against measurement; the Dulong–Petit law of solid specific heats [Petit:1819] as an equipartition statement; the failure for diatomic gases at low temperature and for solids—degrees of freedom frozen out, which classical mechanics cannot freeze—flagged by Maxwell himself [Maxwell:1860]; the resolution is quantum and is derived in Black-Body Radiation and Planck's Hypothesis and Phonons and Lattice Dynamics.]
At ordinary temperatures the molar heat capacity of a simple solid is close to \(3R\) and very nearly the same for every element—the law of Dulong and Petit [Petit:1819]—while the ratio \(\gamma=C_{P}/C_{V}\) that Laplace inferred for air from the speed of sound is close to \(7/5\) [Laplace:1816]. These are exactly the values equipartition predicts, and their agreement was the first quantitative success of the molecular hypothesis. Both fail on cooling: the heat capacities of solids and the rotational and vibrational shares of molecular gases fall away as the temperature drops, and no mechanical system with a fixed number of quadratic degrees of freedom can behave so. Maxwell named the discrepancy the outstanding difficulty of the theory [Maxwell:1860]; the low-temperature measurements are collected in Phonons and Lattice Dynamics and the resolution is quantum (Black-Body Radiation and Planck's Hypothesis, Quantum Statistics).
Derivation. By the equipartition theorem [Boltzmann:1868], every coordinate or momentum entering the energy quadratically carries a mean energy \(\tfrac{1}{2}k_{\mathrm{B}}T\) in equilibrium. A monatomic gas has three such terms per molecule, the translational momenta, so \(U=\tfrac{3}{2}nRT\) and \(C_{V}=\tfrac{3}{2}nR\). For any gas in the dilute limit of Classical Thermodynamics the two heat capacities differ by \(C_{P}-C_{V}=nR\) (Experiment: The Mechanical Equivalent of Heat), so
with \(f\) the number of quadratic terms: \(\gamma=5/3\) for \(f=3\). A diatomic molecule at ordinary temperature turns as well as translates, adding two rotational terms about the two axes perpendicular to the bond—the third has negligible moment of inertia—so \(f=5\) and \(\gamma=7/5\), the measured value for air. In a solid each atom oscillates about a fixed site in three directions, contributing three kinetic and three potential quadratic terms, so \(f=6\), \(U=3nRT\) and the molar heat capacity is \(3R\): the Dulong–Petit law.
The failure has no derivation here, and that is the point. \(f\) in Equation (55.4) counts degrees of freedom that exist or do not; classical mechanics offers no mechanism by which a rotation or a vibration present at one temperature can be absent at a lower one, whatever the interaction assumed. The observation is therefore not a defect of the estimate but a refutation of the mechanics used to make it.
∎The Boltzmann equation
The transport equation
[Reserved: the one-particle distribution function on phase space; the streaming term and the binary-collision integral of Boltzmann's 1872 memoir [Boltzmann:1872]; the molecular-chaos assumption (Stosszahlansatz), stated explicitly as the point where probability enters the mechanics; the collision invariants and the conservation laws they imply.]
The H-theorem
[Reserved: the functional \(H=\int f\ln f\,\dd^{3}v\) and Boltzmann's theorem that collisions can only decrease it [Boltzmann:1872]; stationarity exactly on the Maxwell distribution; \(-H\) as a kinetic entropy, and the second law of Classical Thermodynamics recovered for dilute gases.]
An isolated dilute gas prepared with any velocity distribution whatever relaxes, within a few mean times between collisions, to the Maxwell distribution Equation (55.1) at the temperature its energy fixes, and remains there: the equilibrium form is what every measurement on an undisturbed gas finds [Miller:1955]. The reverse has never been seen. No gas in equilibrium has ever been observed to sort itself spontaneously into a non-Maxwellian state, or to unmix after mixing—and this although the underlying molecular mechanics is exactly invariant under reversal of every velocity, so that the reverse of any observed history is itself a mechanically possible history. Boltzmann's transport equation reproduces the observed one-sidedness through the H-theorem [Boltzmann:1872].
The H-theorem: that the binary-collision integral of the Boltzmann transport equation makes the functional \(H=\int f\ln f\,\dd^{3}v\) non-increasing in time, with stationarity exactly on the Maxwell distribution; together with an explicit statement of the molecular-chaos assumption on which the monotonicity rests, and of what the reversibility and recurrence objections do and do not refute.
Reversibility and recurrence
[Reserved: Loschmidt's reversibility objection [Loschmidt:1876]; Poincaré's recurrence theorem [Poincare:1890] and Zermelo's recurrence objection [Zermelo:1896]; Boltzmann's probabilistic reply—the H-theorem holds for the overwhelming majority of microstates, not for all [Boltzmann:1877]; how the arrow of time enters through the Stosszahlansatz, with the full discussion deferred to Statistical Mechanics.]
The Maxwell–Boltzmann distribution
[Reserved: Boltzmann's 1877 combinatorial definition of the probability of a macrostate and the distribution over energies it yields [Boltzmann:1877]; the Boltzmann factor and the barometric formula; the 1877 counting as the origin of the statistical entropy developed in Statistical Mechanics.]
Transport phenomena
Viscosity
[Reserved: the mean-free-path estimate of the shear viscosity of a gas, and Maxwell's startling prediction that it is independent of density [Maxwell:1860]; his own Bakerian-lecture measurements confirming it [Maxwell:1866]; the refined transport theory of [Maxwell:1867]; hand-off of the continuum description to Fluid Dynamics.]
Rarefy a gas and its viscosity does not fall. Over a wide range of pressures the shear viscosity of a gas is independent of its density, and it grows with temperature—the opposite of a liquid, which thins on heating. Maxwell obtained the prediction from the mean free path before anyone had looked, called it the most startling consequence of the theory, and then measured it himself in oscillating-disc experiments [Maxwell:1860] [Maxwell:1866]. The independence holds only while the mean free path stays small against the apparatus; when the gas is rarefied past that point the viscosity does begin to fall with density, which is the free-molecular regime of [Knudsen:1909] and Section 55.6.2.
Derivation. A molecule of collision cross-section \(\sigma\) sweeps, in time \(t\), the volume \(\sigma\avg{v_{\mathrm{rel}}}t\), and meets in it every molecule whose centre lies inside; with the Maxwell distribution Equation (55.1) the mean relative speed of two molecules is \(\sqrt{2}\avg{v}\), so the number of collisions is \(\sqrt{2}n\sigma\avg{v}t\) and the mean free path is the distance travelled per collision,
Now let the gas be sheared, with a flow velocity \(u(z)\) along \(x\). Molecules crossing the plane \(z\) carry with them the mean \(x\) momentum of their last collision, one mean free path away, so the momentum transported per unit area and time is of order \(\tfrac{1}{3}n\avg{v}\cdot m\left[u(z-\ell)-u(z+\ell)\right]\), which identifies the viscosity as
The number density has cancelled between the density of carriers and the distance Equation (55.5) each one carries its momentum: halve the density and each molecule travels twice as far before delivering its load. Since \(\avg{v}=\sqrt{8k_{\mathrm{B}}T/\pi m}\) from Equation (55.1), the estimate also gives \(\eta\propto\sqrt{mT}/\sigma\), increasing with temperature. Only the numerical factor in Equation (55.6) is unreliable—it depends on how crudely the transport step is modelled—and fixing it is what the systematic expansion of Section 55.5.4 does. The cancellation itself does not depend on that factor, and it fails only when \(\ell\) ceases to be set by Equation (55.5), that is when it grows to the size of the vessel.
∎Heat conduction
[Reserved: Fourier's law of conduction from his analytical theory of heat [Fourier:1822]; the mean-free-path estimate of the thermal conductivity of a gas and its relation to the viscosity and the specific heat; density independence and the experimental range over which it holds.]
Diffusion
[Reserved: Graham's law of gaseous diffusion [Graham:1833]; Fick's phenomenological law [Fick:1855]; the mean-free-path estimate of the diffusion coefficient; Einstein's fluctuation relation connecting diffusion to molecular agitation [Einstein:1905d], with Brown's original observations [Brown:1828] and the decisive experiments of Experiment: Brownian Motion and Avogadro's Number.]
Chapman–Enskog and the road to hydrodynamics
[Reserved: the systematic expansion of the Boltzmann equation about local equilibrium by Chapman [Chapman:1916] and Enskog [Enskog:1917], which turns the estimates of this section into controlled transport coefficients and derives the Navier–Stokes equations of Fluid Dynamics; developed in Nonequilibrium Thermodynamics and Transport and only named here.]
Effusion and rarefied gases
Graham's law of effusion
[Reserved: effusion through an orifice small against the mean free path; the rate proportional to the inverse square root of the molar mass, established by Graham for diffusion [Graham:1833] and for effusion and transpiration [Graham:1846]; the kinetic effusion flux \(\tfrac{1}{4}n\langle v\rangle\); isotope separation named as the application.]
A gas escaping through an aperture small compared with the mean free path does so at a rate inversely proportional to the square root of its molar mass, the pressure and temperature being fixed; a mixture driven through such an aperture is thereby partially separated, the lighter component passing preferentially. Graham established the square-root law first for the passage of gases through a porous plug [Graham:1833] and then for effusion and for capillary transpiration [Graham:1846]. That the law involves the mass and not the chemical character of the molecule is the point: it is a statement about mechanics, not chemistry.
Derivation. The aperture being small against \(\ell\), the escaping molecules are simply those that would have struck that patch of wall, and their distribution inside is undisturbed. The number crossing unit area of the plane in unit time is
where \(F\) is the speed distribution normalised to \(\int F\,\dd v=n\), and the angular factor is the fraction \(\tfrac{1}{2}\sin\vartheta\,\dd\vartheta\) of solid angle at polar angle \(\vartheta\) times the obliquity \(\cos\vartheta\). Elementary Gaussian integrals over Equation (55.1) give \(\avg{v}=\sqrt{8k_{\mathrm{B}}T/\pi m}\), so
using \(P=nk_{\mathrm{B}}T\). At the same pressure and temperature the number density is the same for both gases, so the ratio of the two effusion rates is \(\sqrt{m_{2}/m_{1}}=\sqrt{M_{2}/M_{1}}\), which is Graham's law. The same factor \(\tfrac{1}{4}n\avg{v}\) governs the rate at which a surface is struck, and so the growth of an adsorbed layer in a vacuum system.
∎The Knudsen regime
[Reserved: the Knudsen number; Knudsen's laws of molecular flow through tubes at low pressure [Knudsen:1909]; thermal transpiration; where the continuum description of Fluid Dynamics fails and free-molecular flow replaces it, with vacuum technology as the working application.]
Molecular reality and Avogadro's constant
Loschmidt's count
[Reserved: Avogadro's hypothesis [Avogadro:1811] made quantitative: Loschmidt's 1865 estimate of the molecular diameter and the number density of air, from the mean free path and the density of the liquefied gas [Loschmidt:1865]; the modern Loschmidt constant, about \(2.687\times 10^{25}\,/\mathrm{m}^{3}\) at standard conditions [Mohr:2025].]
The number of molecules in a mole of substance can be inferred by methods sharing no apparatus and no theory beyond the molecular hypothesis itself—from the mean free path together with the density of the liquefied gas [Loschmidt:1865], from the Brownian displacements of suspended particles [Einstein:1905d] [Perrin:1909], from the charge carried in electrolysis, and from the counting of particles emitted by a radioactive source—and the answers agree. That convergence, and not any single determination, is what established that matter is molecular, since no one route is more than an estimate. The constant is now fixed by definition, \(N_{\mathrm{A}}=6.02214076\times 10^{23}\,/\mathrm{mol}\) [BIPM:2019] [Mohr:2025], and it is the measurements that are graded against it rather than the other way about.
Derivation. Loschmidt's route needs two measured numbers and one geometrical assumption. The first number is the mean free path \(\ell\). Read Equation (55.6) in the form \(\eta\approx\tfrac{1}{3}\rho\avg{v}\ell\), with \(\rho=nm\) the mass density of the gas, which is weighed directly, and \(\avg{v}=\sqrt{8RT/\pi M}\) fixed by the molar mass alone; then \(\ell=3\eta/(\rho\avg{v})\) follows from measurements that never mention the number of molecules. The second number is the condensation ratio \(\varepsilon\), the volume of liquid obtained from unit volume of gas. Take the molecules to be spheres of diameter \(d\), packed in the liquid so as to fill it; then per unit volume of gas
while Equation (55.5) with \(\sigma=\pi d^{2}\) gives \(\ell=1/(\sqrt{2}\pi d^{2}n)\). These are two equations in the two unknowns \(n\) and \(d\); substituting the second into Equation (55.9) leaves
and then \(n=1/(\sqrt{2}\pi d^{2}\ell)\) and \(N_{\mathrm{A}}=nV_{\mathrm{m}}\) with \(V_{\mathrm{m}}\) the molar volume. The packing assumption is crude, so the result is an order of magnitude and not a measurement; what makes it evidence is that the wholly unrelated Brownian and electrolytic routes land on the same order of magnitude.
∎From estimates to an exact constant
[Reserved: the convergence of independent determinations of Avogadro's constant—kinetic, Brownian (Perrin's programme [Perrin:1909], with the experiments of Experiment: Brownian Motion and Avogadro's Number), electrolytic and radiometric—as the historical proof of molecular reality; the 2019 redefinition of the SI, which fixes the constant exactly at \(6.02214076\times 10^{23}\,/\mathrm{mol}\) [BIPM:2019] [Mohr:2025].]