In the paper “On the Dynamical Evidence of the Molecular Constitution of Bodies,” published in the Journal of the Chemical Society, June 1875, Maxwell introduces Clausius’s conception of the “virial” of a system, a quantity of which great use has been made in the discussion of molecular theories since a name has been given to it by Clausius. This quantity is the sum of the products of the attractions between each pair of molecules into the distance between them. Maxwell commences by explaining Clausius’s equation, which states that the energy of a quantity of gas is the sum of two quantities, one of which is the virial, and the other is proportional to the product of the pressure and the volume. Now Joule showed that when a gas expands into a vacuum its temperature remains sensibly unaffected, though no energy is communicated to or taken from it, and since the gas obeys Boyle’s law the product of its pressure and volume is constant. Hence the energy being unchanged, and the product of pressure and volume remaining the same, it follows that the virial must he unaffected by expansion. But the virial must depend on the density if there he any sensible forces between the molecules when at sensible distances, unless the force between two molecules varies inversely as the distance between them, a law which Newton showed to be inadmissible in the case of molecular forces. We must therefore conclude that the force between two molecules is sensibly zero, unless the distance between them be very small compared with the average distance between molecules, and that the virial is therefore sensibly zero in a gas, so that the energy and pressure of a gas depend on the motion of the molecules and not on the forces between them.

The experiments of Regnault indicated that in most gases as the density increases the pressure falls below that indicated by Boyle’s law, which shows that when the particles approach very near together the virial is positive, or there is attraction between the molecules. When the pressure is made still greater the gas reaches a condition in which a small increase of density is accompanied by an enormous increase of pressure, so that the virial is negative, and the forces between the particles repulsive. This is the case in the liquid state. Maxwell infers that as the particles of a gas approach each other the forces between them become attractive, attain a maximum, then diminish, and when they are within a certain distance become repulsive, the repulsion increasing so rapidly “that no attainable force can reduce the distance of the particles to zero.” The same conclusion as far as the attraction between the particles at small distances is concerned, was indicated by the experiments which Joule conducted, in company with Sir William Thomson, on the expansion of air into a vacuum, when very careful measurements indicated that a slight cooling effect took place.

The greatest difficulty which the kinetic theory of gases has to face is the observed relation between the specific heats of gases at constant pressure and at constant volume. Boltzmann showed that if each molecule possess n “degrees of freedom” the ratio of the kinetic energy of translation to the whole kinetic energy of the system must be equal to the ratio of 3 to n. Now in a rigid body capable of rotating in any manner n is equal to 6, and this makes the total energy equal to twice the energy of translation. This requires that the ratio of the two specific heats should be 1·33 instead of 1·408, and the observed ratio therefore disproves the hypothesis of hard bodies. If we suppose the molecules to be material points, incapable of rotation or vibration n is equal to 3, and the energy of translation is the whole of the kinetic energy possessed by the molecules. This would make the ratio of the specific heats to be 1·66, which is too great for any real gas except mercury vapour, for which the ratio has been shown by Kundt and Warbourg to be nearly 1·66.

The spectroscope shows that the molecules of a gas are capable of executing vibrations in various periods. They must therefore be material systems, and cannot have less than six degrees of freedom. The ratio of the specific heats cannot therefore be greater than 1·33, and this is too small for most gases. Every additional degree of freedom possessed by the molecules makes the ratio less, and requires that the specific heat of the gas should be greater than is observed to be the case.

In a paper “On Boltzmann’s Theorem on the Average Distribution of Energy in a System of Material Points,” read before the Cambridge Philosophical Society on May 6, 1878, Maxwell showed that whatever be the forces acting upon or between the molecules, provided they be subject to the principle of conservation of energy, the average kinetic energy of any two given portions must be proportional to the number of degrees of freedom of these portions, and hence the total kinetic energy corresponding to an increment of temperature of 1 Superscript ring Baseline normal upper C is shown to be proportional to the product of the number of degrees of freedom into the absolute temperature.

The actual dimensions of molecules were first estimated by Loschmidt in 1865, then by Stoney in 1868, and by Thomson in 1870. At his lecture “On Molecules,” before the British Association at Bradford, Maxwell gave the following

TABLE OF MOLECULAR DATA,

Divided into three ranks, according to the degree of accuracy with which the Quantities are known.

Maxwell’s investigations in the Kinetic Theory of Gases led him to a conclusion which is of great value in the theory of energy. The principle of the dissipation of energy, sometimes called the second law of Thermodynamics, states that it is impossible by means of inanimate material agency to obtain work at the expense of heat by cooling a body below the temperature of the coldest body in the neighbourhood. This principle was first distinctly given by Sir William Thomson. Maxwell showed that it obtains only in consequence of the coarseness of our faculties not allowing us to grapple with individual molecules. If we could seize upon individual molecules, and bring them to rest in the same way as we can lay hold of a fly-wheel, and compel it to do useful work until it has been deprived of all its motion of rotation, we could convert the whole of the heat of a body into work, and bring it to the absolute zero of temperature. As it is, we are at the mercy of the molecules, and capable of obtaining from them only so much work as they are willing to give in the most favourable circumstances in which we are able to place them. Maxwell imagined a quantity of gas, all initially at the same pressure and temperature, to be divided into two portions, A and B, by a partition full of little trap doors which might be opened or closed without the expenditure of energy. Each trap door he supposed placed in charge of a “demon” that is a creature whose eyes are sharp enough to see the molecules and estimate their velocities, and hands agile enough to open and close the trap doors in time to allow or prevent the passage of any particular molecule which is approaching the partition. The operation depends on the difference of the velocities of the particles in the same mass of gas, and the office of the demon is purely selective, so that any mechanism which could be devised to sort the molecules in the same way would be equally effective. Suppose each demon to open his trap door when a molecule is approaching the partition from A with a velocity above the average, but to keep it closed when the velocity of the molecule approaching from A is below the average; while in the case of a molecule approaching the partition from B the door is opened if the velocity be small, and closed if it be great. In this way all the slowly moving molecules will gradually be sorted into the compartment A, while the rapidly moving particles will be accumulated in B. Thus the temperature of the gas in B will be raised, and that in A lowered without any loss of energy or any work being done by an external agent. A heat engine may now be employed, using B as the source and A as the condenser, and doing work at the expense of part of the heat of A until equilibrium of temperature between B and A has been produced, when the services of the demons may be again utilised, and the process repeated until the whole of the heat of the gas has been converted into work. This is contrary to the principle of dissipation of energy, which has thus been circumvented by intelligence. No corresponding method of overcoming the principle of conservation of energy can be devised, and this principle is thus shown to rest on an entirely different kind of footing from that of the second law of Thermodynamics.

The following extract from Maxwell’s article “Atom” in the ninth edition of the Encyclopœdia Britannica is characteristic. Speaking of the teaching of molecular science respecting the size of atoms, he says:—

A paper “On Stresses in Rarefied Gases arising from Inequalities of Temperature,” by Professor Maxwell, was read before the Royal Society on April 11, 1878, and published in the Phil. Trans. for 1879. The notes and appendix added to the paper in May and June 1879 embodied the results of Maxwell’s last investigations in the kinetics of gases. In this paper Maxwell showed that when inequalities of temperature exist in a gas the pressure at a point is not generally the same in all directions, but the maximum and minimum pressures differ by an amount depending on the rate of change of the increase of temperature per unit length in the direction in which this rate is greatest. The stress thus arising from variation in rate of change of temperature varies inversely as the pressure of the gas, and is therefore most conspicuous in high vacuo. If two small bodies are warmer than the air, the line joining them will be a line of maximum pressure, and they will repel each other, while they will attract one another if they are colder than the air. If, however, a ring be placed so as to have the line joining the bodies for its axis and be sufficiently heated the repulsion may be changed into attraction. In the case of a cup, as noticed by Stokes, the variation of the rate of change of temperature is much greater on the convex side than on the concave, where it is nearly uniform, like the electric potential within a hollow vessel, and hence the normal pressure is greater on the convex surface than on the concave, which will account for the motion of the cup radiometer if tangential stresses are neglected. But when the tangential stresses on any portion of gas are considered, it appears, that they, with the normal forces, form a system which is in equilibrium, so that inequality of temperature has no tendency of itself (i.e. without the action of gravity, etc.), to produce currents in the gas. Maxwell therefore concludes that the above explanation is insufficient, and that the true cause of the motion is to be found in the character of the tangential action between the solid and the gas, allowing the gas to slide over the surface of the solid, and thus diminishing the tangential stresses without affecting the normal stresses. In the appendix, dated May 1879, Maxwell determined the character of the tangential action on certain hypotheses respecting the nature of the surface of the solid, and the character of the collisions, and concluded that the gas may slide over the surface of the solid with a finite velocity, and that inequalities of temperature at the surface “give rise to a force tending to make the gas slide along the surface from colder to hotter places.”

Most of the more elementary theorems respecting the kinetic theory of gases are given in a very concise form by Maxwell in his Theory of Heat, the more recent editions of which also give an account of Professor Willard Gibbs’ Thermodynamic Surface. This surface, in which the co-ordinates represent respectively the energy, entropy, and volume of the substance to which it corresponds, was modelled in clay by Maxwell’s own hands in the Cavendish Laboratory. From the clay model a number of plaster casts were taken. These casts Maxwell placed in the sunshine in particular positions, and drew upon them in water-colours the boundary-lines between the light and shadow, which correspond to constant pressure or constant temperature, on the part of the substance. An account of this surface, and many of the properties which it represents, will be found in the work referred to.

We have now presented to the reader a scanty selection from the results of Maxwell’s scientific work. The mere enumeration of his original papers would occupy several pages of this book, and those who are desirous of forming any approach to a true conception of his contributions to science should consult the memorial edition of Maxwell’s papers, edited by Mr. W. D. Niven, F.R.S., and about to be published by the Cambridge University press.

END OF PART II.