Experiment: The Kinetic Theory Tested

Contents

Stern: the direct measurement of molecular speeds (1920)
Miller and Kusch: the shape of the distribution (1955)
Maxwell: the viscosity of air and its density (1866)
Graham: diffusion, effusion and the transport coefficients (1833–1846)
Loschmidt: the size of an air molecule (1865)
Joule and Thomson: the porous plug and the forces between molecules (1852)
Summary of the evidence
Giauque: adiabatic demagnetization and the third law (1933)

Kinetic Theory of Gases makes a gas out of molecules and then makes predictions with them: a definite distribution of molecular speeds at a given temperature, a viscosity that does not fall when the gas is rarefied, three transport coefficients built from one and the same mean free path, a count of molecules reachable from macroscopic measurements alone, and—by what it omits—an equation of state that can be exact only if the molecules exert no forces on one another at a distance. None of that is self-evident. One item, the density independence of the viscosity, Maxwell himself called the most startling consequence of the theory, and he did not believe it until he had measured it. This chapter reports the measurements.

The six experiments below are ordered by what they test rather than by date. Two are molecular-beam experiments that record the distribution of speeds itself, which is the central assertion of the theory and the one that a purely thermodynamic argument can never reach. One is Maxwell's own measurement of the prediction he found hardest to believe. One collects the transport coefficients and checks the relations kinetic theory forces among them. One turns those same measurements into a count of molecules, which is how the molecular hypothesis first acquired a number. The last, the porous plug of Joule and Thomson, measures what the ideal gas leaves out. The Brownian evidence for the same hypothesis—the most direct of all, since it watches the fluctuations rather than the averages—is treated separately in Experiment: Brownian Motion and Avogadro's Number, and the thermodynamic reading of the porous-plug experiment in Experiment: The Mechanical Equivalent of Heat.

Stern: the direct measurement of molecular speeds (1920)

Kinetic theory asserts that the molecules of a gas at $300\,\mathrm{K}$ are moving at some hundreds of metres per second. Before 1920 that number had never been measured; it had only been inferred, from the pressure a gas exerts and from the speed of sound, by arguments that assume the molecular picture they are meant to support. Stern's molecular-beam apparatus measured a molecular speed directly, by timing a flight over a known distance [Stern:1920].

Apparatus

A platinum wire, coated with silver and heated electrically in high vacuum, evaporates silver atoms. The wire lies on the axis of two coaxial cylinders fixed to it: an inner cylinder of radius $r_{1}$ carrying a narrow slit parallel to the axis, and an outer cylinder of radius $r_{2}$ whose inner surface collects the atoms that pass the slit. The whole assembly—wire, slit and collector—can be spun about the wire axis at an angular speed $\omega$ of a few hundred radians per second. The vacuum must be good enough that the flight from slit to collector is collision-free, that is, the mean free path of Phenomenon 57.7 must exceed $r_{2}-r_{1}$, and the deposit is built up over a long run and then developed, as in the molecular-beam experiment of Experiment: Stern–Gerlach.

Procedure

With the assembly at rest the atoms travel radially and the deposit forms on the collector directly opposite the slit; that position is the zero of the measurement. With the assembly spinning, the collector moves while the atom is in flight, so the deposit is displaced along the circumference by an amount that depends on the time of flight and hence on the speed. Reversing the sense of rotation displaces it the other way; taking half the separation between the two deposits removes any error in locating the zero, which is the only quantity in the measurement that cannot be read off a scale. The oven temperature, the two radii and the angular speed complete the data.

Observations and data

[Reserved: the primary tables of [Stern:1920], to be reproduced here in SI—the two cylinder radii in metres, the angular speed in radians per second, the wire temperature in kelvin, the measured displacement of the deposit maximum in metres, and the mean molecular speed inferred from them with its uncertainty, presented per [JCGM:2008]. For orientation, silver at about $1300\,\mathrm{K}$ has $\avg{v}=\sqrt{8RT/\pi M}\approx5\times 10^{2}\,\mathrm{m}/\mathrm{s}$, so with centimetre-scale radii and $\omega$ of a few hundred radians per second the displacement is of order a millimetre—small, but far larger than the width of the undisplaced trace.]

Interpretation

The deposit moves, in the direction and by roughly the amount that a flight time $\left(r_{2}-r_{1}\right)/v$ requires with $v$ of the predicted size. It is also smeared, and the smear is not a defect of the apparatus: since the displacement varies as $1/v$, the deposit is a map of the distribution of speeds, stretched towards the zero for the fast molecules and away from it for the slow ones. This first version measured essentially one moment of that distribution; recording its shape is the business of Section 57.2.

One caution belongs with any beam measurement and is developed in Equation (57.3): what leaves an aperture is not a sample of the gas inside. Fast molecules reach the aperture more often than slow ones, so the beam is weighted by an extra power of the speed, and its mean speed exceeds the mean speed in the oven by the factor $3\pi/8$. Comparing a beam average against the in-oven $\avg{v}$ therefore misses by $18\,\mathrm{\%}$ before any experimental error is committed.

Primary references

[Stern:1920]. The method is the one Stern had developed from 1919 onward and applied two years later in Experiment: Stern–Gerlach; the diffraction of the same beams, which measures a wavelength rather than a speed, belongs to Matter Waves [Estermann:1930].

Phenomenon 57.1 (Molecules move at the speeds kinetic theory assigns them).

The atoms of a vapour in equilibrium at temperature $T$ cross a known distance in a measurable time, and the speed so obtained is of the order $\sqrt{k_{\mathrm{B}}T/m}$ that kinetic theory requires—some hundreds of metres per second for a heavy metal vapour at a thousand kelvin, and larger for lighter molecules in the same ratio $M^{-1/2}$. Stern measured it by the displacement of a silver deposit on a rotating collector [Stern:1920]. The measurement is direct in the sense that matters here: it uses a length and a time, and no property of a gas other than the fact that its molecules fly freely between collisions.

Derivation. Let the slit sit at radius $r_{1}$ and the collector at radius $r_{2}$ on the same rigid body, turning at angular speed $\omega$. An atom that passes the slit travels radially outward in a straight line—this is the assumption the vacuum has to earn—and reaches the collector after a time $t=\left(r_{2}-r_{1}\right)/v$. During that time the collecting surface has advanced along its own circumference by $\omega r_{2}t$, so the deposit lands a distance

\begin{equation}\tag{57.1} s=\frac{\omega r_{2}\left(r_{2}-r_{1}\right)}{v} \end{equation}

from the point opposite the slit. Everything in Equation (57.1) but $v$ is a laboratory constant, so the equation is a measurement of $v$; and since $s\propto1/v$, a spread of speeds becomes a spread of deposit positions, with the slow molecules thrown furthest. Reversing $\omega$ reverses the sign of $s$, so half the separation of the two deposits gives $\abs{s}$ without knowing where the zero lies.

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Miller and Kusch: the shape of the distribution (1955)

Between Stern's measurement and this one stand the rotating-drum velocity analysers of the early 1930s—Zartman's and Ko's—which replaced the single displaced deposit by a deposit profile and so recorded the shape of the distribution rather than one moment of it [Zartman:1931] [Ko:1934]. Miller and Kusch's slotted-rotor selector is the precision version of the same idea, and it is the experiment that turned the Maxwell distribution from a plausible functional form into a measured curve [Miller:1955].

Apparatus

An oven holding potassium or thallium at a controlled and measured temperature, effusing through a slit into high vacuum. Between the oven and the detector sits a velocity selector: a cylinder carrying many slots cut helically through its rim, spun about its axis, so that an atom passes only if its time of flight along the cylinder matches the time the entrance slot takes to rotate into the position of the exit slot. The transmitted band of speeds is therefore proportional to the rotation rate, and its width is set by the machining. Detection is by surface ionisation on a hot wire followed by ion collection, which counts atoms as a current rather than accumulating a deposit, and so turns the measurement from a photographic one into an electrical one.

Procedure

Hold the oven at a measured temperature, scan the rotor speed, and record the detector current at each setting. The result is the transmitted intensity as a function of the selected speed. The raw curve is not yet the distribution: it must be corrected for the finite width of the transmission window, for the higher-order pass bands at submultiples of the rotation rate, and for the response of the ioniser. The corrected curve is then compared with the flux-weighted Maxwell form Equation (57.2), with the oven temperature taken from the thermocouple and not adjusted to improve the fit—which is what makes the comparison a test rather than a fit.

Observations and data

[Reserved: the primary curves of [Miller:1955], to be tabulated here as transmitted intensity against selected speed in $\mathrm{m}/\mathrm{s}$ for potassium and for thallium at their stated oven temperatures in kelvin, together with the residuals against Equation (57.2) and their uncertainties per [JCGM:2008]. The headline result quoted in Kinetic Theory of Gases is agreement with the predicted shape at the per-cent level; the individual residuals are what show whether the disagreement is random or structured, and they are not reproduced here because the primary tables were not consulted.]

Interpretation

Two things are being tested at once, and they are worth separating. The first is the shape: an exponential in $v^{2}$ with a polynomial prefactor, over a range of speeds wide enough that the prefactor and the exponential are separately visible. The second is the parameter: the temperature that best describes the curve must be the temperature the thermocouple reads, since the distribution has no other adjustable constant. A theory that got the shape right with a free temperature would be much weaker evidence.

The composition of the vapour enters the interpretation and is not a detail: a dimer of twice the molecular mass has the same distribution with $v$ rescaled by $1/\sqrt{2}$, so an unrecognised dimer fraction imitates a distorted distribution of the monomer. Any beam experiment of this kind therefore owes an account of what is actually in the beam, which is one reason for measuring two species of very different mass.

Primary references

[Zartman:1931] [Ko:1934] [Miller:1955]. The rotating-drum analysers come first and the velocity selector of Miller and Kusch is the precision version of the same idea.

Phenomenon 57.2 (The distribution of speeds in a molecular beam).

The number of molecules per unit time leaving an oven at temperature $T$ with speeds between $v$ and $v+\dd v$ is proportional to $v^{3}\exp\left(-mv^{2}/2k_{\mathrm{B}}T\right)\dd v$, and the temperature that describes the measured curve is the temperature of the oven. Miller and Kusch established this for potassium and thallium beams, passing them through a slotted rotating selector, and found the predicted shape confirmed at the per-cent level [Miller:1955]. The cube of the speed, rather than the square that describes the gas inside the oven, is not an adjustable feature: it follows from the aperture being small against the mean free path.

Derivation. Inside the oven the number density of molecules with speeds in $\dd v$ about $v$ is $F(v)\,\dd v\propto v^{2}\exp\left(-mv^{2}/2 k_{\mathrm{B}}T\right)\dd v$, the factor $v^{2}$ being the area of the shell of constant speed in velocity space. A molecule crosses the aperture in unit time only if it lies within a distance $v_{z}$ of it and is moving towards it, so the flux carries one power of the speed more than the density does:

\begin{equation}\tag{57.2} \Phi(v)\,\dd v\propto v^{3}\exp\left(-\frac{mv^{2}}{2k_{\mathrm{B}}T}\right)\dd v\ep \end{equation}

The extra power is large. Write $a=m/2k_{\mathrm{B}}T$ and use $\int_{0}^{\infty}v^{4}\exp\left(-av^{2}\right)\dd v =\tfrac{3}{8}\sqrt{\pi}\,a^{-5/2}$ together with $\int_{0}^{\infty}v^{3}\exp\left(-av^{2}\right)\dd v =\tfrac{1}{2}a^{-2}$; then the mean speed of the molecules in the beam is

\begin{equation}\tag{57.3} \avg{v}_{\text{beam}}=\frac{3}{4}\sqrt{\frac{\pi}{a}} =\frac{3\pi}{8}\avg{v}\approx1.18\avg{v}\ec \end{equation}

since $\avg{v}=\sqrt{8k_{\mathrm{B}}T/\pi m}=2/\sqrt{\pi a}$ is the mean speed in the oven. The two differ by $18\,\mathrm{\%}$. The same argument shows why the aperture must be small against the mean free path: if it were not, the escaping molecules would collide on the way out, the gas would flow hydrodynamically through the hole, and the beam would carry a shifted rather than a weighted distribution.

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Maxwell: the viscosity of air and its density (1866)

In 1860 Maxwell derived from the mean free path a conclusion he plainly disliked: the shear viscosity of a gas should not depend on its density [Maxwell:1860]. Halve the number of carriers and each survivor carries its momentum twice as far, and the two effects cancel exactly. No experiment at the time contradicted it, because no experiment had looked. Maxwell looked, in the Bakerian Lecture of 1866 [Maxwell:1866].

Apparatus

Parallel glass discs mounted on a common vertical axis and hung from a torsion fibre, so that the assembly oscillates in its own plane between fixed discs a small distance away. The whole is enclosed in a vessel that can be evacuated to a controlled pressure and heated, the gas temperature being read by a thermometer inside and the pressure by a manometer. The angular amplitude is read optically, by a mirror on the suspension and a distant scale, so that nothing touches the moving system.

Procedure

Set the discs swinging and record the period and the decay of the amplitude. The gas in the narrow gaps resists the motion, and the logarithmic decrement of the oscillation—the constant ratio between successive amplitudes—measures that resistance; with the geometry of the gaps known, it yields the shear viscosity. Repeat at a series of pressures at fixed temperature, which is the density test, and at a series of temperatures at fixed pressure, which is the temperature test. The residual decrement of the suspension itself is measured with the vessel evacuated and subtracted.

Observations and data

[Reserved: the primary tables of [Maxwell:1866], to be reproduced here in SI—gas pressure in $\mathrm{Pa}$, temperature in $\mathrm{K}$, period in $\mathrm{s}$, logarithmic decrement, and the viscosity in $\mathrm{Pa}\,\mathrm{s}$ with its uncertainty, converted from Maxwell's units on port per the unit axiom of Measurement, SI Units, and the Theory of Errors. Two derived statements are what the table has to support: that the viscosity is constant along the pressure series, and that along the temperature series it rises as $T$ rather than as $\sqrt{T}$. For orientation, the viscosity of air near room temperature is about $18\,\mu\mathrm{Pa}\,\mathrm{s}$.]

Interpretation

The pressure series is the striking one. A liquid's viscosity is a property of how crowded it is; a gas's is not, and over the range of pressures the apparatus could span the measured viscosity did not change. That is the mean-free-path cancellation, observed. It is not unlimited: when the gas is rarefied until the mean free path is comparable with the width of the gaps, the molecules cross from one surface to the other without colliding, the internal friction ceases to be a property of the gas alone, and the viscosity does begin to fall with the density—the free-molecular regime of [Knudsen:1909]. The failure is as informative as the success, because it locates the length that the elementary theory says is responsible.

The temperature series is the more consequential one, and it went against the theory as Maxwell had first written it. Hard spheres give $\eta\propto\sqrt{T}$; the measurements gave something much closer to $\eta\propto T$. Maxwell drew the correct conclusion—that molecules are not hard spheres but centres of a repulsive force—and rebuilt the transport theory in 1867 around the one force law, the inverse fifth power, that makes the collision integrals tractable [Maxwell:1867]. This is the first quantitative information about an intermolecular force obtained from a bulk measurement, and it is the same programme that the porous plug of Section 57.6 continues at long range.

Primary references

[Maxwell:1866]; the prediction under test is from [Maxwell:1860] and the theory rebuilt on the result is [Maxwell:1867]. The low-pressure breakdown is [Knudsen:1909].

Phenomenon 57.3 (Rarefying a gas does not thin it).

Over a wide range of densities the shear viscosity of a gas is independent of its density, so that pumping gas out of a vessel does not reduce the drag on a body moving inside it. Maxwell predicted this from the mean free path before any measurement existed [Maxwell:1860] and then measured it himself with oscillating discs [Maxwell:1866]. The independence holds only while the mean free path remains small against the apparatus; below that the viscosity does fall with density [Knudsen:1909].

Derivation. Molecules crossing a plane in a sheared gas carry the mean flow momentum they acquired at their last collision, one mean free path $\ell$ away, and they cross at a rate of order $\tfrac{1}{3}n\avg{v}$ per unit area. The transported momentum per unit area and time is therefore of order $\tfrac{1}{3}nm\avg{v}\ell$ times the velocity gradient, so

\begin{equation}\tag{57.4} \eta\approx\tfrac{1}{3}\rho\avg{v}\ell =\frac{m\avg{v}}{3\sqrt{2}\,\sigma}\ec \end{equation}

using $\rho=nm$ and $\ell=1/\left(\sqrt{2}n\sigma\right)$ with $\sigma$ the collision cross-section. The number density has cancelled: it counts the carriers, and it also fixes how far each carrier goes. Nothing in Equation (57.4) depends on how crudely the transport step was modelled except the numerical factor, so the cancellation is robust where the estimate of the coefficient is not. It fails exactly when $\ell$ stops being given by the collisions—that is, when the vessel, not the gas, sets the distance a molecule travels between momentum exchanges.

∎

Phenomenon 57.4 (The viscosity of a gas rises with temperature, and too fast for hard spheres).

Heating a gas makes it more viscous—the opposite of a liquid—and for air the rise is close to proportional to the absolute temperature. Maxwell's oscillating-disc measurements gave a temperature dependence much nearer $\eta\propto T$ than the $\eta\propto\sqrt{T}$ that follows from treating molecules as hard spheres [Maxwell:1866], and it was this discrepancy, not any theoretical preference, that led him to replace the spheres by point centres repelling as the inverse fifth power of the distance [Maxwell:1867].

Derivation. Let the molecules repel along the line of centres with a force $K/r^{\nu}$. Two molecules approaching with relative kinetic energy of order $k_{\mathrm{B}}T$ are deflected through an angle of order unity when the potential energy at closest approach is comparable with that energy, $K/b^{\nu-1}\sim k_{\mathrm{B}}T$, where $b$ is the impact parameter. The effective collision radius therefore shrinks as the gas is heated,

\begin{equation}\tag{57.5} b\sim\left(\frac{K}{k_{\mathrm{B}}T}\right)^{1/(\nu-1)}\ec\qquad \sigma\sim\pi b^{2}\propto T^{-2/(\nu-1)}\ec \end{equation}

and substituting into Equation (57.4) with $\avg{v}\propto\sqrt{T}$ gives a pure power law,

\begin{equation}\tag{57.6} \eta\propto T^{s}\ec\qquad s=\frac{1}{2}+\frac{2}{\nu-1}\ep \end{equation}

Hard spheres are the limit $\nu\to\infty$, for which $s=\tfrac{1}{2}$: the cross-section is a constant of the molecule and the whole temperature dependence comes from $\avg{v}$. The observed $s\approx1$ requires $2/(\nu-1)=\tfrac{1}{2}$, that is $\nu=5$. So the measurement does not merely disagree with hard spheres; it returns a number, the exponent of the force law, and it is a bulk measurement of a molecular interaction.

∎

Graham: diffusion, effusion and the transport coefficients (1833–1846)

The kinetic hypothesis nearly foundered on an objection of Buys Ballot: if the molecules of the air move at hundreds of metres per second, a smell released at one end of a room should reach the other end at once, and it does not. Clausius's answer was the mean free path [Clausius:1858]. The quantitative form of the objection and of its answer is the diffusion coefficient, and Graham had measured the relevant rates before either was written down [Graham:1833] [Graham:1846].

Apparatus

For diffusion: a graduated glass tube closed at one end by a plug of plaster of Paris—a septum whose pores are small but very numerous—and standing with its open end under water or mercury, so that the change of volume as the enclosed gas exchanges with the outside air is read directly on the graduations. For transpiration: long capillary tubes of known bore through which the gas is driven by a measured pressure difference. For effusion: a fine aperture in a thin plate, small compared with the mean free path, which is the case the kinetic theory treats exactly.

Procedure

Fill the tube with the test gas, note the initial volume, and let the gas interchange with the atmosphere through the septum until the level has moved by a measured amount; time the process. Repeat with a series of gases under identical conditions of temperature, pressure and apparatus, so that everything except the molecular mass is held fixed and the ratios of the rates carry the whole result. The same discipline governs the capillary and aperture versions.

Observations and data

[Reserved: Graham's rate tables from [Graham:1833] and [Graham:1846], to be reproduced here as diffusion rates relative to air for hydrogen, oxygen, nitrogen, carbon dioxide and the other gases he used, set against $M^{-1/2}$, with the residuals and their uncertainties. To be tabulated alongside them, for the comparison of Equation (57.9), are the modern transport coefficients of air near $300\,\mathrm{K}$ and $100\,\mathrm{kPa}$: a viscosity of about $18\,\mu\mathrm{Pa}\,\mathrm{s}$, a thermal conductivity of about $26\,\mathrm{mW}/\mathrm{m}/\mathrm{K}$, and a self-diffusion coefficient of order $2\times 10^{-5}\,\mathrm{m}^{2}/\mathrm{s}$.]

Interpretation

Graham's law—rate proportional to $M^{-1/2}$ at fixed temperature and pressure—is a statement about mass and not about chemistry, and that is its whole force: nothing in a chemical account of gases would predict that carbon dioxide and nitrous oxide, which behave quite differently in every reaction, should diffuse at the same rate because they happen to have the same molecular mass.

The coefficients are the sharper test. Kinetic theory does not merely predict that gases diffuse; it predicts a diffusion coefficient built from the same $\avg{v}$ and the same $\ell$ that appear in the viscosity, and therefore predicts relations among the three transport coefficients with no free parameter at all. Those relations hold in order of magnitude, which is the content of Phenomenon 57.6, and fail in their numerical coefficients—the elementary argument gives a heat-conduction to viscosity ratio of $1$ where the monatomic gases give about $\tfrac{5}{2}$. The discrepancy is not a failure of the molecular picture but of the averaging: fast molecules carry more energy than slow ones but the same momentum, so the three transports do not sample the same molecules and cannot share one mean free path. Repairing that is the systematic expansion of Chapman and Enskog [Chapman:1916] [Enskog:1917] [Chapman:1970], developed in Nonequilibrium Thermodynamics and Transport, whose continuum limit is the hydrodynamics of Fluid Dynamics.

Primary references

[Graham:1833] [Graham:1846]; the phenomenological law of diffusion is [Fick:1855] and that of heat conduction [Fourier:1822]. The mean free path that reconciles the rates with the speeds is [Clausius:1858].

Phenomenon 57.5 (Gases interdiffuse far more slowly than they move).

The molecules of air travel at some hundreds of metres per second, yet a gas released at one point of a still room takes hours to spread across it by diffusion alone. The measured interdiffusion coefficients of common gases at standard conditions are of order $2\times 10^{-5}\,\mathrm{m}^{2}/\mathrm{s}$ [Graham:1833] [Fick:1855], some ten orders of magnitude below the product of a molecular speed and a room. The discrepancy is not evidence against the kinetic hypothesis but a measurement of the mean free path [Clausius:1858].

Derivation. A molecule does not travel in a straight line but executes a random walk of step $\ell$ at speed $\avg{v}$, so the transport is diffusive with

\begin{equation}\tag{57.7} D\approx\tfrac{1}{3}\avg{v}\ell\ec \end{equation}

by the same carrier argument that gave Equation (57.4), with the number density of a labelled species in place of the momentum density. With $\avg{v}\approx470\,\mathrm{m}/\mathrm{s}$ and $\ell\approx10^{-7}\,\mathrm{m}$ for air near $300\,\mathrm{K}$ and $100\,\mathrm{kPa}$—the values established in Phenomenon 57.7—this gives $D\approx1.6\times 10^{-5}\,\mathrm{m}^{2}/\mathrm{s}$, which is the observed order. The time to cross a distance $L$ follows from $L^{2}\approx2Dt$: about a hundredth of a second for a millimetre, but some hours for a metre, since the time grows as the square of the distance. The suppression relative to free flight is the ratio $\ell/L$, which is about $10^{-7}$ for a metre—and that, and not any slowness of the molecules, is the answer to Buys Ballot.

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Phenomenon 57.6 (The three transport coefficients of a gas are not independent).

Viscosity, thermal conductivity and diffusion in a dilute gas are carried by the same molecules over the same mean free path, so kinetic theory predicts that the dimensionless combinations $\rho D/\eta$ and $\kappa m/\left(\eta c_{V}^{(1)}\right)$ are pure numbers of order unity, independent of the gas, of its density and of its temperature. Measurement confirms that they are numbers of order unity, and shows that the elementary estimate of those numbers is wrong: for the monatomic gases the second ratio is close to $\tfrac{5}{2}$ rather than to $1$ [Maxwell:1866] [Eucken:1913] [Chapman:1916] [Enskog:1917].

Derivation. Each coefficient is the same transport step applied to a different carried quantity. A molecule crossing a plane carries the value its last collision gave it, one mean free path away, so for any transported density $\chi$ the flux is of order $-\tfrac{1}{3}\avg{v}\ell\,\dd\chi/\dd z$. Taking $\chi$ to be the momentum density, the energy density and the number density of a labelled species in turn,

\begin{equation}\tag{57.8} \eta\approx\tfrac{1}{3}\rho\avg{v}\ell\ec\qquad \kappa\approx\tfrac{1}{3}nc_{V}^{(1)}\avg{v}\ell\ec\qquad D\approx\tfrac{1}{3}\avg{v}\ell\ec \end{equation}

with $c_{V}^{(1)}$ the heat capacity of a single molecule. The mean free path and the mean speed—the two quantities that are hard to measure and easy to get wrong—cancel between them, leaving

\begin{equation}\tag{57.9} \frac{\rho D}{\eta}\approx1\ec\qquad f:=\frac{\kappa m}{\eta c_{V}^{(1)}}\approx1\ep \end{equation}

That these are pure numbers is a prediction with no adjustable content whatever, and it is confirmed. That they are not equal to unity is equally informative. The elementary argument gave one $\ell$ to all three transports, but the molecules that dominate the energy flux are faster than those that dominate the momentum flux, and a faster molecule has both a longer free path and a larger weight; the correct averages therefore differ, and by a factor of order unity that only a solution of the transport equation can supply.

∎

Derivation pending.

The Chapman–Enskog transport coefficients: that the first-order solution of the Boltzmann transport equation about local equilibrium fixes the numerical coefficients left undetermined by the mean-free-path argument, and in particular gives the ratio of thermal conductivity to viscosity times specific heat as 5/2 for a monatomic gas, against the value 1 of the elementary estimate.

Loschmidt: the size of an air molecule (1865)

Every measurement so far in this chapter has been of a bulk property. Loschmidt's determination is the step that converts two of them into a molecular length and a molecular count, and it is the first time anyone put a number on the size of a molecule [Loschmidt:1865]. It is included among the experiments, although it built no apparatus of its own, because its inputs are measurements and its output is a number about the world; the honest description of it is a determination, not a theory.

Apparatus

None of its own. The determination consumes two independent laboratory results: the viscosity of the gas, from an experiment of the kind described in Section 57.3, and the density of the same substance in its condensed state, from weighing a known volume of the liquid. Both were available in 1865, and neither had been made with molecules in mind.

Procedure

From the measured viscosity and mass density obtain the mean free path, $\ell\approx3\eta/\left(\rho\avg{v}\right)$, with $\avg{v}$ fixed by the molar mass and the temperature alone. From the volume of liquid obtained from unit volume of gas obtain the condensation ratio $\varepsilon$. Assume the molecules are spheres that fill the liquid when packed. Then $\ell$ and $\varepsilon$ are two equations in the two unknowns—the molecular diameter $d$ and the number density $n$—and both follow.

Observations and data

[Reserved: Loschmidt's own inputs and outputs from [Loschmidt:1865], converted to SI—the viscosity and condensation ratio he used, the diameter he obtained (about $1\,\mathrm{nm}$), and the number density that follows. The comparison to record beside them: on modern inputs the same equations give a collision diameter near $3\times 10^{-10}\,\mathrm{m}$, and the Loschmidt constant, the number density of an ideal gas at $273.15\,\mathrm{K}$ and $101325\,\mathrm{Pa}$, is $2.6867\times 10^{25}\,/\mathrm{m}^{3}$ [Mohr:2025], while the Avogadro constant has been exact since 2019 at $6.02214076\times 10^{23}\,/\mathrm{mol}$ [BIPM:2019].]

Interpretation

The packing assumption is crude and the answer is correspondingly crude—Loschmidt's diameter is several times the modern collision diameter, and since $n$ varies as $d^{-2}$ his number density is low by about an order of magnitude. That is not the point, and treating it as a failed precision measurement misreads what happened. Before 1865 the molecular hypothesis had no scale at all: molecules could as well have been $10^{10}$ times smaller. Afterwards it had one, obtained from two measurements that mention no molecules, and the scale was small enough to explain why molecules had never been seen and large enough to be consistent with the mean free path.

What made the number believable was not its accuracy but its corroboration by methods sharing neither apparatus nor theory: the Brownian determinations of Experiment: Brownian Motion and Avogadro's Number [Perrin:1909] [Perrin:1913], the electrolytic route through the elementary charge [Millikan:1913], and the counting of particles from a radioactive source. Agreement among routes with nothing in common but the hypothesis is what an inference of this kind can be worth, and it is why the molecular constitution of matter stopped being an opinion around 1910.

Primary references

[Loschmidt:1865]; the viscosity input is of the kind measured in [Maxwell:1866] and the mean free path is [Clausius:1858]. The modern values are [Mohr:2025] and [BIPM:2019].

Phenomenon 57.7 (The mean free path is a measurable length).

The average distance a molecule travels between collisions is not a theoretical fiction: it is fixed by measurements that never mention molecules, since inverting the viscosity relation expresses it through the measured viscosity, the weighed mass density and the molar mass alone [Clausius:1858] [Maxwell:1866]. For air near $300\,\mathrm{K}$ and $100\,\mathrm{kPa}$ it comes out of order $10^{-7}\,\mathrm{m}$: about three hundred times the molecular diameter, and seven orders of magnitude below the size of any vessel—which is why the gas behaves as a continuum in Fluid Dynamics and ceases to when it does not [Knudsen:1909].

Derivation. Solve Equation (57.4) for the mean free path,

\begin{equation}\tag{57.10} \ell\approx\frac{3\eta}{\rho\avg{v}}\ec\qquad \avg{v}=\sqrt{\frac{8RT}{\pi M}}\ec \end{equation}

and note what stands on the right: $\eta$ from a viscometer, $\rho$ from a balance, and $\avg{v}$ from the molar mass and the thermometer. With $\eta\approx18\,\mu\mathrm{Pa}\,\mathrm{s}$, $\rho\approx1.2\,\mathrm{kg}/\mathrm{m}^{3}$ and $M\approx0.029\,\mathrm{kg}/\mathrm{mol}$ at $300\,\mathrm{K}$, so that $\avg{v}\approx470\,\mathrm{m}/\mathrm{s}$, this gives $\ell\approx10^{-7}\,\mathrm{m}$. The cross-section follows from $\ell=1/\left(\sqrt{2}n\sigma\right)$ with $n=P/k_{\mathrm{B}}T\approx 2.4\times 10^{25}\,/\mathrm{m}^{3}$, giving $\sigma\approx3\times 10^{-19}\,\mathrm{m}^{2}$ and a collision diameter $d=\sqrt{\sigma/\pi}\approx3\times 10^{-10}\,\mathrm{m}$. Loschmidt's second equation then closes the system: taking the molecules to be spheres of diameter $d$ packed so as to fill the liquid, the condensation ratio is $\varepsilon=n\pi d^{3}/6$, and eliminating $n$ between it and $\ell=1/\left(\sqrt{2}\pi d^{2}n\right)$ leaves $d=6\sqrt{2}\,\varepsilon\ell$—a molecular diameter from a condensation ratio and a viscosity. The Avogadro constant is then $N_{\mathrm{A}}=nV_{\mathrm{m}}$ with $V_{\mathrm{m}}$ the molar volume. Every step but the packing assumption is exact; that one step is what makes the result an order of magnitude rather than a measurement.

∎

Joule and Thomson: the porous plug and the forces between molecules (1852)

The ideal gas of Classical Thermodynamics is the gas whose molecules do not interact except in the instant of collision. Kinetic theory needs to know whether that is true, and if not, by how much and with what sign. The porous-plug experiment answers all three questions with one thermometer, because the effect it measures vanishes identically for a gas of non-interacting molecules [Joule:1852].

Apparatus

A long tube, thermally lagged and of low conductivity, with a plug of cotton wool or unspun silk packed tightly into the bore so as to present a large flow resistance without any moving part. Compressed gas is supplied steadily from a pump through a coil immersed in a bath of known temperature, so that the gas arrives at the plug at a controlled temperature; thermometers sit in the stream immediately upstream and downstream of the plug, and pressure gauges read the pressure on each side.

Procedure

Establish a steady flow and hold it until the temperatures cease to drift, which is what makes the analysis a steady-flow one and lets the enthalpy balance of Experiment: The Mechanical Equivalent of Heat apply. Read the upstream and downstream temperatures and pressures. The plug is essential and is not merely a convenient resistance: it destroys the kinetic energy of the stream, so that no part of the temperature change can be attributed to the gas being set in bulk motion. Repeat for different gases, different pressure drops and different bath temperatures, the last being what maps the change of sign.

Observations and data

[Reserved: the primary series of [Joule:1852] and its sequels, to be reproduced here in SI—upstream and downstream pressures in $\mathrm{Pa}$, bath temperature in $\mathrm{K}$, and the measured temperature drop in $\mathrm{K}$ with its uncertainty, for each gas and each pressure drop, together with the coefficient $\mu=\left(\pp T/\pp P\right)_{H}$ in $\mathrm{K}/\mathrm{Pa}$ derived from them. The qualitative pattern the table must show: air at room temperature cools by a fraction of a kelvin per atmosphere of pressure drop, carbon dioxide cools by several times more, and hydrogen warms instead of cooling.]

Interpretation

The kinetic reading of the result is the one that concerns this chapter. A gas of point molecules that never interact has an internal energy depending on temperature alone and an equation of state $PV=nRT$, and for such a gas the throttling coefficient is identically zero—not small, but exactly zero, as the derivation below shows. Every kelvin measured here is therefore a direct measurement of the interactions the ideal gas omits. Better, the sign separates them: cooling means that work has been done against attraction as the molecules were pulled apart, warming that the molecules were already so crowded that their own volume dominates. A gas that does both, at different temperatures, is telling the experimenter that both a long-range attraction and a short-range repulsion exist, and where they balance.

This complements Section 57.3 exactly. The viscosity measurement probes the repulsive core, because it is collisions that transport momentum, and returned an exponent for the repulsive force. The porous plug probes the attractive tail, because it is the work done against attraction on expansion that cools the gas. Between them they outline the intermolecular potential from a thermometer and a torsion fibre—a fact worth pausing over, since neither apparatus can resolve anything smaller than a millimetre.

Primary references

[Joule:1852]. The cruder free expansion that failed to see the effect is [Joule:1845a]; the thermodynamic treatment of both, with the enthalpy argument and the inversion curve, is in Experiment: The Mechanical Equivalent of Heat and Classical Thermodynamics.

Phenomenon 57.8 (Molecules exert forces on one another at a distance).

Forcing a real gas steadily through a porous plug from a higher to a lower pressure, along a thermally insulated tube, changes its temperature by a small but definite amount, and the sign of the change depends on the gas and on its temperature: most gases at room temperature emerge cooler, hydrogen and helium emerge warmer, and each gas has an inversion temperature separating the two behaviours. Joule and Thomson established the effect in the series beginning in 1852 [Joule:1852]. For a gas of molecules exerting no forces at a distance the effect would be exactly zero at every temperature and every pressure, so its magnitude and sign measure the intermolecular potential directly.

Derivation. The measured quantity is $\mu=\left(\pp T/\pp P\right)_{H}$, since throttling conserves enthalpy—the steady-flow first law of Experiment: The Mechanical Equivalent of Heat—and by the chain rule together with a Maxwell relation,

\begin{equation}\tag{57.11} \mu=\frac{1}{C_{P}} \left[T\left(\pdv{V}{T}\right)_{P}-V\right]\ep \end{equation}

For the ideal gas $V=nRT/P$ gives $T\left(\pp V/\pp T\right)_{P}=V$ identically, so $\mu\equiv0$: nothing whatever is measured, at any pressure or temperature. This is what makes the experiment a clean test. Now let the molecules attract weakly at long range and exclude a volume $b$ per mole at short range, so that to first order in the departures from ideality $V_{\mathrm{m}}\approx RT/P+b-a/RT$. Then $T\left(\pp V_{\mathrm{m}}/\pp T\right)_{P} =RT/P+a/RT$, and

\begin{equation}\tag{57.12} \mu\approx\frac{1}{C_{P,\mathrm{m}}} \left(\frac{2a}{RT}-b\right)\ep \end{equation}

The two molecular effects enter with opposite signs, exactly as the observation requires: the attraction $a$ cools, the excluded volume $b$ warms, and the attraction loses its advantage as the gas is heated because the molecules pass too quickly to be held. The coefficient vanishes at the inversion temperature

\begin{equation}\tag{57.13} T_{\mathrm{inv}}=\frac{2a}{Rb}\ec \end{equation}

above which throttling warms the gas. Hydrogen and helium have small $a$ and therefore inversion temperatures far below room temperature, which is why they warm on throttling and must be precooled before a Joule–Thomson stage will liquefy them—an inference from Equation (57.13) that the liquefaction industry has tested more thoroughly than any laboratory. Finally, $b$ is the same excluded volume that appears as the collision cross-section in Phenomenon 57.7: two independent measurements of one molecular size, which is the consistency check that makes the whole scheme more than a fit.

∎

Summary of the evidence

Experiment	Prediction tested	Result
Stern 1920	thermal speeds	flight time over a known distance gives speeds of the predicted order, scaling as $\sqrt{T/M}$
Miller–Kusch 1955	shape of the distribution	beam intensity follows $v^{3}\exp(-mv^{2}/2k_{\mathrm{B}}T)$ at the per-cent level, at the oven temperature
Maxwell 1866	viscosity independent of density	confirmed over the accessible pressure range; and $\eta\propto T$, not $\sqrt{T}$, giving the force exponent $\nu=5$
Graham 1833–1846	transport from one mean free path	rates scale as $M^{-1/2}$; the coefficients obey the predicted relations in order of magnitude, with the numbers fixed only by Chapman–Enskog
Loschmidt 1865	a molecular scale exists	a diameter and a number density from viscosity plus a condensation ratio, correct to an order of magnitude
Joule–Thomson 1852	the ideal gas omits forces	a nonzero throttling coefficient of both signs, where non-interacting molecules give exactly zero

Tests of the kinetic theory of gases reported in this chapter. The middle column names the prediction of Kinetic Theory of Gases that the experiment puts at risk; in every case a purely thermodynamic account of gases predicts nothing at all for the quantity measured, which is what makes these experiments evidence for molecules rather than merely for heat.

No entry in Table 57.1 can be obtained from thermodynamics, and none of them contains an adjustable parameter beyond the one molecular size and one force law that the whole set shares. Taken together with the Brownian measurements of Experiment: Brownian Motion and Avogadro's Number, which watch the fluctuations that these experiments only average over, they are the experimental case for the hypothesis of Kinetic Theory of Gases: that a gas is a very large number of small bodies in disordered motion, obeying the mechanics of Newtonian Dynamics. What they do not settle is the statistical foundation—why the equilibrium distribution is the one observed, independently of any model of the molecules—which is the work of Statistical Mechanics, nor the failure of equipartition at low temperature, which no classical mechanics can repair and which Phonons and Lattice Dynamics and Black-Body Radiation and Planck's Hypothesis resolve.

Giauque: adiabatic demagnetization and the third law (1933)

This last section is an addendum to the six tests summarised in Table 57.1 and not a seventh of them: it puts no prediction of Kinetic Theory of Gases at risk. It is placed here because the paragraph above names the boundary at which the classical account of matter fails, and because crossing that boundary was itself an experiment. The third law is stated in Classical Thermodynamics in the form Nernst gave it [Nernst:1906] [Nernst:1911], and the unattainability of absolute zero is deduced there as a theorem of thermodynamics. Both halves are also measurements, and the measurements came out of one programme: the cooling of paramagnetic salts by adiabatic demagnetization, proposed independently by Debye in 1926 and by Giauque in 1927, achieved by Giauque and MacDougall in 1933, and recognised by the Nobel Prize in Chemistry for 1949. What the technique opened was not merely a lower temperature but a region in which the entropy of matter can be watched approaching its limit.

Apparatus

A paramagnetic salt in which the magnetic ions are held far apart by water of crystallisation: Giauque and MacDougall used hydrated gadolinium sulfate, $\mathrm{Gd}_{2}(\mathrm{SO}_{4})_{3}\cdot8\mathrm{H}_{2}\mathrm{O}$, whose $\mathrm{Gd}^{3+}$ ion has a half-filled shell, and therefore no orbital moment to be quenched, and a ground multiplet of eight equally spaced states. The salt hangs in a vacuum space inside a bath of liquid helium—available at all only since [Onnes:1908]—pumped below its normal boiling point to reach somewhat under $1\,\mathrm{K}$. An electromagnet supplies a flux density of order a tesla over the sample. The thermal switch between sample and bath is helium exchange gas, admitted to couple them and pumped away to isolate the sample; there is no mechanical link, because at these temperatures any solid support conducts far too well. Thermometry is the sample itself: two coaxial coils around the vacuum space read its magnetic susceptibility by mutual inductance, and the Curie law of Magnetism in Matter converts a susceptibility into a temperature.

That last step is the subtle one, and the apparatus has to answer for it. The Curie law is the high-temperature limit of the paramagnetic response [Curie:1895] [Langevin:1905] [Blundell:2001], and it fails exactly where the experiment is going; what the bridge reads is therefore a magnetic temperature $T^{*}$ defined by $\chi=C/T^{*}$, which coincides with the thermodynamic temperature only while the ions remain independent. Converting one into the other needs a separate calorimetric measurement: feed the isolated salt a measured quantity of energy, follow the resulting change in $T^{*}$, and recover the thermodynamic temperature from $T^{-1}=\left(\pp S/\pp U\right)$. A magnetic temperature quoted as a thermodynamic one is the characteristic error of this field, and it is an error in the direction of claiming too much.

Procedure

Four steps, run as a cycle. First, with exchange gas in place and the salt anchored to the bath at $T_{\mathrm{i}}$, raise the field to $B_{\mathrm{i}}$: the moments align, the magnetic entropy falls, and the heat of magnetization passes into the helium, which is what makes this step isothermal. Second, pump the exchange gas away, isolating the salt. Third, lower the field to zero, as slowly as the eddy-current heating of the metal parts requires, and no more slowly than the heat leak allows; the salt cools. Fourth, read the susceptibility bridge repeatedly as the salt warms back up, which yields the final temperature, the heat capacity, and—from the slope of the warm-up—the size of the parasitic heat input that has to be subtracted from it. The whole is repeated over a series of initial fields and initial bath temperatures, because the prediction under test is a relation between a ratio of temperatures and a ratio of fields, not a single number.

Observations and data

[Reserved: the primary series of Giauque and MacDougall (1933) and of de Haas, Wiersma and Kramers in the same year, to be reproduced here in SI—initial bath temperature in $\mathrm{K}$, initial flux density in $\mathrm{T}$, the susceptibility before and after, the magnetic temperature $T^{*}$ derived from it, and the thermodynamic temperature with its uncertainty stated per [JCGM:2008]; alongside them the calibration run, as energy input in $\mathrm{J}$ against entropy in $\mathrm{J}/\mathrm{mol}/\mathrm{K}$, from which the zero-field curve $S(T)$ is built. Three orientation figures, computed here from stated inputs rather than taken from the papers: the ground multiplet of $\mathrm{Gd}^{3+}$ carries a molar entropy $R\ln8=17.3\,\mathrm{J}/\mathrm{mol}/\mathrm{K}$ per mole of ions; the Bohr magneton divided by the Boltzmann constant is $0.67\,\mathrm{K}/\mathrm{T}$ [Mohr:2025] [BIPM:2019], so at $1\,\mathrm{T}$ that multiplet is spread over about $9\,\mathrm{K}$ and a bath near $1\,\mathrm{K}$ is cold enough to order it; and the lattice entropy competing with it is smaller by orders of magnitude, by the Debye law used in Equation (57.15). The first demagnetizations reached a few tenths of a kelvin, and the more dilute salts used later reach a few millikelvin, which is the regime Experiment: Parity Violation works in.]

Interpretation

Two facts make the method work, and neither is incidental. The first is that at $1\,\mathrm{K}$ a solid has almost no entropy left in its lattice: by the Debye law [Debye:1912a] the vibrational entropy falls as $T^{3}$, so for a salt of Debye temperature of order $10^{2}\,\mathrm{K}$ it stands four or more orders of magnitude below the magnetic entropy. Essentially all the disorder that remains is the orientation of the magnetic ions, and that is precisely the disorder a magnetic field can address. The second is that the ions are dilute: the water of crystallisation holds them far enough apart that they behave independently well below the temperature of the bath, which is what makes the magnetic entropy a function of $B/T$ alone and gives the cooling the simple law of Equation (57.16).

The same dilution sets the limit, and it is there that the experiment stops being a refrigeration technique and becomes a statement about the third law. The ions are not perfectly independent: each sits in the dipolar field of its neighbours and in the electric field of the crystal, and the residual internal field that results cannot be switched off [Kramers:1930] [Kramers:1934]. Demagnetizing to zero applied field therefore leaves the moments in that residual field rather than in nothing, and the temperature falls only in the corresponding ratio. Stage a second cycle below the first and the same factor applies again, so the temperature after several stages is a geometric sequence with a positive ratio: it has zero as a limit and never as a value. That is the unattainability statement, and here it is not a postulate but the operating characteristic of a machine, measurable stage by stage.

It is worth pausing on the fact that the two halves of the third law are one fact seen from two sides. Were the ions exactly independent, the residual field would vanish, a single stage would reach absolute zero, and the entropy at zero temperature would be $R\ln\left(2J+1\right)$ per mole in zero field and zero in any field however small—so Nernst's theorem, which requires the limiting entropy to be independent of the other variables, would fail in the same breath in which unattainability failed. The interactions that put a floor under the refrigerator are the interactions that give the salt a non-degenerate ground state. A thermodynamic law that reads like a prohibition is, microscopically, the statement that matter has a ground state.

The approach of the entropy to a constant was established by a second route in the same programme, and the route matters because at the bottom it uses no thermodynamics at all. The entropy of a substance can be obtained calorimetrically, by integrating $C_{P}/T$ upward from the lowest attainable temperature and closing the gap below it with the Debye law; and it can be computed, for a gas, from spectroscopic constants alone by way of the partition function of Statistical Mechanics. These are independent determinations of the same quantity, and the calorimetric one is complete only if the entropy left below the integration limit is zero. For most substances the two agree. Where they do not, the discrepancy is not scatter but a number of a particular kind—close to $R\ln w$ for a small $w$, with $w$ counting the arrangements a molecule can be frozen into—and that is a far stronger confirmation of the third law than agreement alone would be, because it says what the limiting entropy is and not merely that there is one.

Finally, the technique outlived the question it was built for. Cooling a paramagnetic salt to a few millikelvin puts the hyperfine field of a nucleus embedded in it to work: once the nuclear Zeeman splitting is comparable with $k_{\mathrm{B}}T$, the nuclei align, and an ensemble of oriented nuclei is a laboratory sample with a definite direction in it. That is exactly how the cobalt-60 of Experiment: Parity Violation was polarized—the active layer grown on a cerium magnesium nitrate crystal, demagnetized, and left to warm through the useful window [Wu:1957] [Ambler:1957]. A method devised to test a law of thermodynamics is what made a symmetry of the weak interaction measurable, which is the kind of debt between fields that a treatise organised by subject tends to hide.

Primary references

The primary sources of this section are named here in full because they are not yet in the bibliography of this treatise; the works cited by key below are the supporting ones. The proposal is P. Debye, “Einige Bemerkungen zur Magnetisierung bei tiefer Temperatur”, Annalen der Physik 386 (1926) 1154–1160, and, independently, W. F. Giauque, “A Thermodynamic Treatment of Certain Magnetic Effects. A Proposed Method of Producing Temperatures Considerably Below $1^{\circ}$ Absolute”, Journal of the American Chemical Society 49 (1927) 1864–1870. The achievement is W. F. Giauque and D. P. MacDougall, “Attainment of Temperatures Below $1^{\circ}$ Absolute by Demagnetization of $\mathrm{Gd}_{2}(\mathrm{SO}_{4})_{3}\cdot8\mathrm{H}_{2}\mathrm{O}$”, Physical Review 43 (1933) 768, confirmed within months by W. J. de Haas, E. C. Wiersma and H. A. Kramers in Physica 1 (1933). The residual entropy of ice is measured in W. F. Giauque and J. W. Stout, Journal of the American Chemical Society 58 (1936), and counted in L. Pauling, “The Structure and Entropy of Ice and of Other Crystals with Some Randomness of Atomic Arrangement”, Journal of the American Chemical Society 57 (1935) 2680–2684, whose argument is reproduced in [Pauling:1960]. The nuclear extension is N. Kurti, F. N. H. Robinson, F. Simon and D. A. Spohr, “Nuclear Cooling”, Nature 178 (1956) 450–453.

[Reserved: the eight references named in full above are to be entered in the bibliography and the prose replaced by citations. Until then the supporting keys are [Nernst:1906] [Nernst:1911] for the heat theorem, [Debye:1912a] for the lattice entropy, [Curie:1895] [Langevin:1905] [Blundell:2001] for the paramagnetism, [Kramers:1930] [Kramers:1934] for the interactions that set the floor, [Onnes:1908] for the helium that preceded the technique, and [Wu:1957] [Ambler:1957] for the use later made of it.]

Phenomenon 57.9 (Demagnetizing an isolated paramagnetic salt cools it).

A paramagnetic salt magnetized isothermally in contact with a bath near $1\,\mathrm{K}$, then thermally isolated and demagnetized, ends colder than the bath, and the drop is fixed by the ratio of the fields rather than by any property of the apparatus: the final temperature is the initial one multiplied by the ratio of the residual field to the applied field. Giauque and MacDougall reached a few tenths of a kelvin from hydrated gadolinium sulfate in 1933, and de Haas, Wiersma and Kramers did the same within months. With more dilute salts the method reaches a few millikelvin [Wu:1957] and, applied to nuclear rather than electronic moments, the microkelvin range. Before it, the lowest temperatures in existence were those of pumped liquid helium [Onnes:1908].

Derivation. Take $N$ non-interacting ions of angular momentum $J$ and Landé factor $g$ in a flux density $B$. Each level of the multiplet has energy $E_{m}=g\mu_{\mathrm{B}}Bm$ with $m=-J,\dots,J$, so the single-ion partition function

\begin{equation}\tag{57.14} Z=\sum_{m=-J}^{J}\exp\left(-\frac{g\mu_{\mathrm{B}}Bm} {k_{\mathrm{B}}T}\right) \end{equation}

depends on $B$ and $T$ only through their ratio: the field scales every energy in the same proportion, and the temperature is measured against energy. The magnetic entropy inherits the property, since with $x=g\mu_{\mathrm{B}}B/k_{\mathrm{B}}T$ the free energy is $-Nk_{\mathrm{B}}T\ln Z(x)$ and

\begin{equation}\tag{57.15} S_{\mathrm{m}}=Nk_{\mathrm{B}}\left(\ln Z-x\,\dv{\ln Z}{x}\right) =S_{\mathrm{m}}\!\left(B/T\right)\ep \end{equation}

A reversible demagnetization of a thermally isolated sample holds the total entropy fixed. The lattice carries almost none of it—by the Debye law the vibrational entropy per mole of atoms is $\left(4\pi^{4}/5\right)R\left(T/\Theta_{\mathrm{D}}\right)^{3}$ [Debye:1912a], which near $1\,\mathrm{K}$ is smaller than $R\ln\left(2J+1\right)$ by orders of magnitude—so holding the total entropy fixed holds $S_{\mathrm{m}}$ fixed, and by Equation (57.15) it holds $B/T$ fixed:

\begin{equation}\tag{57.16} \frac{T_{\mathrm{f}}}{T_{\mathrm{i}}} =\frac{B_{\mathrm{f}}}{B_{\mathrm{i}}}\ep \end{equation}

Nothing in Equation (57.16) refers to the particular salt, which is what makes it a prediction rather than a fit; the salt enters only through how small $B_{\mathrm{f}}$ can be made and how much entropy $R\ln\left(2J+1\right)$ there was to remove. The two assumptions are equally visible: the ions must be independent, which fails once $k_{\mathrm{B}}T$ falls to their interaction energy, and the process must be slow enough to be reversible, which eddy currents in any metal near the sample threaten.

∎

Phenomenon 57.10 (The entropy of a substance tends to a constant, and the constant counts ground states).

The entropy of a substance measured calorimetrically—by integrating $C_{P}/T$ upward from the lowest attainable temperature—agrees, for most substances, with the entropy computed from spectroscopic data through the partition function; and the agreement requires that the entropy remaining below the integration limit be zero. Where the two determinations disagree, the discrepancy is not scattered but close to $R\ln w$ with $w$ a small number of arrangements per molecule frozen in as the substance is cooled. For ice it is near $R\ln\left(3/2\right)=3.37\,\mathrm{J}/\mathrm{mol}/\mathrm{K}$, which is the count of proton placements consistent with two protons per oxygen [Pauling:1960]; for carbon monoxide, whose two ends are nearly indistinguishable, it is of the order of $R\ln2=5.76\,\mathrm{J}/\mathrm{mol}/\mathrm{K}$. The entropy therefore tends as $T\to0$ to a constant fixed by the degeneracy of the ground state and not by the substance's chemistry, which is Equation (54.13) in its sharp form.

Derivation. The calorimetric statement is Equation (54.14): the integral $\int_{0}^{T}C_{x}\,\dd T'/T'$ converges, so it defines $S(T)-S(0)$ and the calorimetric route measures entropy differences only. The absolute value it is compared against comes from Boltzmann's counting [Boltzmann:1877], $S=k_{\mathrm{B}}\ln W$, which at zero temperature counts ground states. If each molecule can be frozen into any of $w$ arrangements independently of its neighbours then $W=w^{N}$ and

\begin{equation}\tag{57.17} S(0)=Nk_{\mathrm{B}}\ln w=R\ln w \end{equation}

per mole: a discrete set of admissible values, which is why a discrepancy between the two routes is diagnostic rather than merely disappointing.

Pauling's count for ice is the worked case, and it needs only the two rules the hydrogen bonds impose. Each oxygen has four tetrahedral neighbours and keeps two protons close to it, which it may place on any two of the four bonds: $\binom{4}{2}=6$ arrangements per molecule, hence $6^{N}$ if the molecules were free of one another. Each of the $2N$ bonds must carry exactly one proton; of the four ways the two molecules sharing a bond may independently place a proton on it or not, two satisfy that rule and two do not, so each bond costs a factor $\tfrac{1}{2}$. Treating the bonds as independent,

\begin{equation}\tag{57.18} W\approx6^{N}\left(\tfrac{1}{2}\right)^{2N} =\left(\tfrac{3}{2}\right)^{N}\ec\qquad S(0)=R\ln\tfrac{3}{2}\ep \end{equation}

The independence of the bonds is an approximation, and the exact counting on the ice lattice exceeds Equation (57.18) by well under a per cent—far less than the calorimetric uncertainty. The comparison therefore tests the physical claim, that the protons are disordered, rather than the combinatorics.

∎

Phenomenon 57.11 (Absolute zero is unattainable, and the unattainability is an operating characteristic).

Every stage of a demagnetization cycle multiplies the temperature by a ratio smaller than one but bounded away from zero, because the applied field cannot be reduced below the field the magnetic ions exert on one another. A finite number of stages therefore leaves a finite temperature, and the successive stages of every cooling technique are observed to yield diminishing returns, exactly as Phenomenon 54.7 requires. The statement is not a postulate of thermodynamics tested at one remove: the bounding ratio is a measurable property of the salt [Kramers:1930] [Kramers:1934], and the machine displays the prohibition in its own performance.

Derivation. Write the field at an ion as the applied flux density combined with an internal field of magnitude $b$, arising from the dipolar field of the neighbours and from the crystal-field splitting of the multiplet; $b$ is a property of the lattice and does not respond to the magnet. The smallest field a demagnetization can leave is therefore $b$, and Equation (57.16) gives

\begin{equation}\tag{57.19} T_{\mathrm{f}}\geq\frac{b}{B_{\mathrm{i}}}\,T_{\mathrm{i}}\ec \end{equation}

so that $n$ identical stages reach $T_{n}\geq\left(b/B_{\mathrm{i}}\right)^{n}T_{\mathrm{i}}>0$ for every finite $n$. A geometric sequence of positive ratio has zero as a limit and never as a value, which is unattainability in the form the apparatus states it.

The two halves of the third law meet here. Suppose $b$ were zero, so that the bound vanished and one stage sufficed. The ions would then be exactly independent at every temperature, the zero-temperature entropy would be $R\ln\left(2J+1\right)$ per mole in zero field and zero in any field however small, and Equation (54.13)—the requirement that the limiting entropy not depend on the other variables—would fail in the same breath. Conversely, an interaction strong enough to order the moments discharges $R\ln\left(2J+1\right)$ before absolute zero is reached and makes the limiting entropy independent of the field, at the price of a floor under the cooling. Unattainability and Nernst's theorem are one statement about the ground state, read once as a prohibition and once as a limit.

∎

Derivation pending.

The residual internal field of a paramagnetic salt: that the dipolar sum over the lattice of magnetic ions, together with the crystal-field splitting of the ground multiplet, fixes the ordering temperature and hence the floor under a single demagnetization stage, so that the cooling ratio is a computable property of the salt rather than a fitted parameter.

Derivation pending.

The spectroscopic entropy of a gas: that the translational, rotational, vibrational and electronic partition functions, evaluated from band-spectrum constants and molecular masses, give the absolute molar entropy against which the calorimetric integration is compared, together with the symmetry number and the nuclear-spin terms that cancel between the two routes.