New Foundations Laid by Maxwell and Boltzmann
(1860 - 1872)

The kinetic theory of Clausius was quickly taken up and developed into a powerful mathematical research instrument by the Scottish physicist James Clerk Maxwell (1831-1879).

Clausius had used probability concepts in his derivation of the mean-free-path formula, but it was Maxwell who converted the kinetic theory of gases into a fully statistical doctrine. Clausius and earlier kinetic theorists had assumed that all molecules in a a homogeneous gas at a given temperature have the same speed, but Maxwell asserted that the collisions among molecules will instead produce a statistical distribution of speeds. To describe this distribution he borrowed a mathematical formula from the social sciences. The crucial step was his translation of the "normal distribution law" or "law of errors," discovered by Adrain and Gauss and extensively applied by Quetelet, into a distribution law for molecular velocities. Maxwell's velocity distribution law, and its extension by Boltzmann to include the effects of forces, will be discussed in Chapter 15.2. Here we need note only that according to Maxwell's law, the average speed is proportional to T as in earlier kinetic theories, but a gas at a high temperature contains molecules moving at all speeds, including very low speeds, and a gas at a low temperature also contains (in different proportions) molecules moving at very high speeds.

Maxwell greatly extended the scope of the kinetic theory by showing how it could be used to calculate not only the thermal and mechanical properties of gases in equilibrium, but also their "transport properties": diffusion, viscosity, and heat conduction.

In his first paper, published in 1860, he used the Clausius mean-free-path idea to obtain unexpected results for the viscosity of a gas; and he analyzed the collisions of systems of spherical or nonspherical bodies, attempting to find a theoretical model that could account for the observed ratios of specific heats of gases.

Maxwell calculated the viscosity of a gas by estimating on the mutual friction of neighbouring layers of gas moving at different speeds. One might expect, on the basis of experience with liquids, that a fluid will have higher viscosity (will flow less freely) at lower temperatures, and that a denser fluid will be more viscous than a rarer fluid, since in both cases the motion will be more strongly obstructed by intermolecular forces. Maxwell showed that if the kinetic theory of gases is correct, both expectations will be wrong, because the mechanism that produces viscosity is different. In a gas, viscous force originates not in the forces between neighboring molecules but in the transfer of momentum that occurs when a molecule from a faster-moving stream wanders over to a slower-moving stream and collides with a molecule there. The rate of momentum transfer increases with the average molecular speed, so (1) the viscosity increases with temperature.

The momentum transferred by each collision is also proportional to the difference in each speeds of the two layers, and if we assume that this difference is simply proportional to the distance between the layers (linear velocity gradient) then it is proportional to the mean free path between collisions. The rate of collisions is proportional to the density, but the mean free path is inversely proportional to the density according to Clausius' formula (see above), hence the density term cancels out and (2) the viscosity is independent of density. It is, like the mean free path, inversely proportional to the molecular diameter.

At that time the only experimental data on gas viscosity appeared to indicate that it increases with density. Maxwell therefore expected, when he published his theoretical calculation in 1859, that it would lead to a refutation of the kinetic theory.

Maxwell also obtained disappointing results from his analysis of the distribution of energy in systems of colliding nonspherical particles. In such systems he found that a generalized equipartition theorem should apply: the average kinetic energy of translational motion of the particles should be equal to the average kinetic energy of rotation around each of the three principal axes of the particle. This led to a ratio of specific heats equal to 4/3, clearly different from the observed value of about 1.4 for common gases. Hence, he concluded, "a system of such particles could not possibly satisfy the known relation between the two specific heats of all gases" (Maxwell 1860, p. 318).

According to the kinetic theory, assuming equipartition of energy, the ratio of specific heats should be c_p/c_v = (2+n)/n, where n is the number of mechanical "degrees of freedom." Thus a point-mass has 3 degrees of freedom because it can move in any of the 3 spatial dimensions, so its specific heat ratio should be 5/3; the ratios for monatomic gases such as mercury and argon actually have this value. A nonspherical body, or a diatomic molecule composed of two point- masses bound together by conservative forces, has an additional 3 degrees of freedom, so its ratio should be 4/3, whereas diatomic molecules such as hydrogen and oxygen have the ratio 1.4 or 1.41.

Boltzmann later suggested that a diatomic molecule may really have only 5 effective degrees of freedom because rotation around the axis of symmetry (the line joining the two atoms) is not changed by collisions between molecules and therefore does not contribute to the specific heat. This would make the ratio 1.4. Maxwell did not accept this hypothesis and continued to regard the problem as unsolved; fortunately the refutation of his initial predictions about specific heats and viscosity did not discourage him from pursuing the kinetic theory. (The specific heat problem was not satisfactorily resolved until the 20th century, when it was found to be a quantum effect.)

Maxwell himself initiated the experimental test of his predictions for the effect of temperature and density on viscosity. He found that the viscosity coefficient of air is indeed constant over a wide range of densities, contrary to the alleged experimental facts mentioned above. Later it was pointed out by 0. E. Meyer and others that the analysis of the pendulum data had been based on the assumption -- natural enough before Maxwell's theory was known -- that the viscosity coefficient does go smoothly to zero as the density goes to zero. While it is true that its value must be zero at zero density (if there is no gas left, it can't exert any viscous resistance to the swing of the pendulum), the drop occurs quite suddenly at a density lower than that reached in most experiments before 1850.

Crudely speaking, the theory refuted the experiment in this case. One might argue that it was not the experiment itself but only its interpretation that was refuted. But in ordinary scientific discourse the term "experimental fact" is commonly applied to data inferred from experiments with the help of some kind of interpretation or theoretical assumption. As Einstein observed (quoted by Heisenberg 1971, p. 63), our theories determine what we observe; hence experiments can never furnish a completely impartial test of a theory.

The investigation of the temperature dependence of viscosity did not yield such a clear-cut result. According to the original "billiard ball" model (elastic spheres with no forces except at contact) the viscosity coefficientB5 ought to be proportional to the square root of the absolute temperature. But Maxwell and others found a stronger temperature variation, T^z, where z ranges from about .75 to 1.O.

At about the same time (early 1860s) Maxwell developed a much better formulation of transport theory, which avoided the mean-free-path approximation. All the results of the new theory depended on the velocity distribution function for a gas not in thermal equilibrium -- a function that Maxwell was unable to determine -- except in the special case of repulsive forces inversely proportional to the 5th power of the distance between two particles. For this case the velocity-distribution function did not have to be known, and Maxwell found that the viscosity coefficient is directly proportional to the absolute temperature, T. Maxwell's own experimental results agreed with this, and so he concluded by 1866 that the kinetic theory gave completely accurate predictions for gas viscosity. The viscosity coefficient is still independent of density for the inverse 5th-power force law, as long as the density is not so high or so low that the gas properties do not depend mainly on binary collisions of particles; at very low densities interactions with the surface of the container dominate the flow behavior, whereas at very high densities simultaneous interactions of three or more particles must be taken into account.

A few years later, the Austrian physicist Ludwig Boltzmann (1844-1906) developed a different version of transport theory, equivalent to Maxwell's theory insofar as it leads to the same formulas for the coefficients of diffusion, viscosity and heat conduction, but more convenient for some other applications. Rather than trying to eliminate the non-equilibrium velocity distribution function by choosing a special form of the force law, Boltzmann used that function as the primary object of study. We will write it f(v,x,t) to indicate that it depends not only on the molecular velocity (v) but also on spatial position (x) and time (t). Boltzmann computed the change in f(v,x,t) resulting from all relevant physical parameters, including especially the collisions that changed the numbers of particles having specified velocities. The result, published in 1872, was an integrodifferential equation for f(v,x,t), now called "Boltzmann's transport equation" or simply "the Boltzmann equation." It plays a major role in 20th century kinetic theory, including theories of ionized gases (plasma physics) and in calculations of neutron flow in nuclear reactors.

In the 1910s, the Swedish physicist David Enskog (1884-1947) developed a general solution of Boltzmann's transport equation, while the British geophysicist Sydney Chapman (1880-1970) worked out an equivalent general solution for Maxwell's transport equations. Enskog and Chapman could then derive formulae for the transport coefficients for a wide variety of force laws; they also uncovered a new transport process, "thermal diffusion," predicted by kinetic theory and later used as one of the processes for separating isotopes in the development of the atomic bomb. The experimentally-determined temperature dependence of transport coefficients can now be used to draw conclusions about which law most nearly represents the actual force between atoms.

But in the 1860s, when the mere existence of an atomic structure of matter was no more than a plausible hypothesis, Maxwell's theory was used to accomplish a major advance: the first reliable estimate of the size of an atom. For this purpose the earlier result relating M to the particle diameter d was most useful:

M ~ L ~ V/Nd^2

(L = Clausius' mean free path, N = number of molecules in volume V.) Josef Loschmidt (1821-1895), an Austrian physicist and chemist, pointed out in 1865 that this relation could be used to determine d if one other equation for N and d were known. In particular, he suggested that the volume occupied by the gas molecules themselves, if they were closely packed, should be approximately the volume of the substance condensed to the liquid state,

V_{liq} = Nd^3.

If the density of a substance is known in both the liquid and gaseous states, the ratio or "condensation coefficient" V/V_{liq}=V/Nd^3 could be combined with the mean free path (L ~ V/Nd^2) to obtain a value for d.

In this way Loschmidt concluded that the diameter of an "air molecule" is about d ~ 10^{-7} cm. This value is about four times too large according to modern data, but considerably better than any other well-founded estimate available at the time.

The corresponding value of the number of molecules in a cubic centimeter of an ideal gas at standard conditions (O C, 1 atm pressure) would be

N_L ~ 2 x 10^{18}. Although Loschmidt himself did not give this result explicitly in his 1865 paper, it can easily be deduced from his formula, and so this number is now sometimes called "Loschmidt's number." Its modern value is 2.687 x 10^{19}. It should not be confused with the related constant, "Avogadro's number," defined as the number of molecules per gram-mole, equal to

N_A = N_L/V_0 = 6.02 x 10^{23},

where V_0 = 22420.7 cm^3 atm mole^{-1}. Avogadro himself did not give any estimate of this number, but only postulated that it should have the same value for all gases.

During the next few years, other scientists (the most influential being William Thomson, Lord Kelvin) made similar estimates of atomic sizes and other parameters with the help of the kinetic theory of gases. As a result, the atom came to be regarded as no longer a merely hypothetical concept but a real physical entity, subject to quantitative measurement, even though it could not be "seen." This was one of the most important contributions of the kinetic theory to 19th-century science; yet it was carelessly brushed aside by skeptics like Ernst Mach and Wilhelm Ostwald, who argued at the end of the century that we still have no convincing evidence for the existence of the atom and should therefore banish it from the elite company of established physical theories.

Updated 24 August 2001