203. So far our molecules have been treated either as elastic spheres, exerting no forces on one another except when in actual collision, or else as point centres of force, attracting or repelling according to comparatively simple laws. The time has now come to discard all such restrictions, and treat the question in a more general way, regarding the molecules as general mechanical structures, which may be as complicated as we please, consisting of any number of parts, capable of any kind of internal motion and exerting upon one another forces of any type.

Degrees of Freedom

204. The total number of independent quantities which are needed to specify the configuration of any mechanical system is called the number of degrees of freedom of the system. This number does not depend on the motions, but on the capacities for motion, of the various parts of the system; it is therefore related to the geometrical or kinematical, and not to the mechanical, properties of the system.

For example, if a point is free to move in space, its position can be specified by three quantities, as for instance x, y, z, the rectangular coordinates of the point, so that a point which is free to move in space has three degrees of freedom. A rigid body which is free to move in space has six degrees of freedom, for the position of the body can only be fully fixed when six quantities are known, as for instance x, y, z the coordinates of the centre of gravity of the body, and three angles to determine the orientation of the body.

I have intended that the present book shall provide such knowledge of the Kinetic Theory as is required by the average serious student of physics and physical chemistry. I hope it will also give the mathematical student the equipment he should have before undertaking the study of specialist monographs, such, for instance, as the recent books of Chapman and Cowling (The Mathematical Theory of Non-uniform Gases) and R. H. Fowler (Statistical Thermodynamics).

Inevitably the book covers a good deal of the same ground as my earlier book, The Dynamical Theory of Gases, but it is covered in a simpler and more physical manner. Primarily I have kept before me the physicist's need for clearness and directness of treatment rather than the mathematician's need for rigorous general proofs. This does not mean that many subjects will not be found treated in the same way—and often in the same words—in the two books; I have tried to retain all that was of physical interest in the old book, while discarding much of which the interest was mainly mathematical.

It is a pleasure to thank Professor E. N. da C. Andrade for reading my proofs, and suggesting many improvements which have greatly enhanced the value of the book. I am also greatly indebted to W. F. Sedgwiek, sometime of Trinity College, Cambridge, for checking all the numerical calculations in the latest edition of my old book, and suggesting many improvements.

ELEMENTARY THEORIES

164. The difficulties in the way of an exact mathematical treatment of diffusion are similar to those which occurred in the problems of viscosity and heat conduction. Following the procedure we adopted in discussing these earlier problems, we shall begin by giving a simple, but mathematically inexact, treatment of the question.

We imagine two gases diffusing through one another in a direction parallel to the axis of z, the motion being the same at all points in a plane perpendicular to the axis of z. The gases are accordingly arranged in layers perpendicular to this axis.

The simplest case arises when the molecules of the two gases are similar in mass and size—like the red and white billiard balls we discussed in § 6. In other cases differences in the mass and size of the molecules tend, as the motion of the molecules proceeds, to set up differences of pressure in the gas. The gas adjusts itself against these by a slow mass-motion, which will of course be along the axis of z at every point.

Let us denote the mass-velocity in the direction of z increasing by w0, and let the molecular densities of the two gases be v1, v2. Then v1, v2 and w0 are functions of z only.

We assume that, to the approximation required in the problem, the mass-velocity of the gas is small compared with its molecular-velocity, and we also assume that the proportions of the mixture do not change appreciably within distances comparable with the average mean free path of a molecule.

132. At a collision between two molecules, energy, momentum and mass are all conserved. Energy, for instance, is neither created nor destroyed; a certain amount is transferred from one of the colliding molecules to the other. Thus the moving molecules may be regarded as transporters of energy, which they may hand on to other molecules when they collide with them. As the result of a long chain of collisions, energy may be transported from a region where the molecules have much energy to one where they have but little energy: studying such a chain of collisions we have in effect been studying the conduction of heat in a gas. If we examine the transport of momentum we shall find that we have been studying the viscosity of a gas—the subject of the present chapter. For viscosity represents a tendency for two contiguous layers of fluid to assume the same velocity, and this is effected by a transport of momentum from one layer to the other. Finally if we examine the transfer of the molecules themselves we study diffusion.

For the moment, we must study the transport of momentum. We think of the traversing of a free path of length λ as the transport of a certain amount of momentum through a distance λ. If the gas were in a steady state, every such transport would be exactly balanced by an equal and opposite transport in the reverse direction, so that the net transport would always be nil.

In June 1847 William Thomson, later Lord Kelvin (1824–1907), met Joule at the Oxford meeting of the British Association for the Advancement of Science, and the encounter led Thomson to study Joule's papers on the mutual convertibility of heat and mechanical work. At the Oxford meeting Joule had read a paper describing his measurement of the temperature change in a fluid agitated by a paddle wheel that was turned by a descending hanging weight; he claimed to have determined the quantitative equivalence between the heat generated by the paddle wheel and the mechanical work required to generate that heat. Thomson found Joule's conclusions astonishing; and he reported Joule's work to his brother James Thomson (1822–92), who confessed that Joule's ‘Views have a slight tendency to unsettle one's mind’. The Thomsons' sense of intellectual disorientation arose from their belief, derived from the work of Sadi Carnot (1796–1832), that heat was conserved in the generation of mechanical work by heat engines. This theory seemed to contradict Joule's claim that heat must be consumed in the generation of work. The unravelling of the apparent contradiction between the theories of Carnot and Joule was to lead to the formulation of the science that in 1854 William Thomson was to term ‘thermo-dynamics’, the theory of the mechanical action of heat.

In his 1900 lecture ‘Nineteenth century clouds over the dynamical theory of heat and light’, William Thomson pointed to two problems facing the mechanical theory of nature: the failure to explain the mechanism of the motion of the earth through the ether, and the difficulty the concept of the equipartition of energy posed for the construction of molecular models. Thomson highlighted two ‘clouds’ that threatened his elaboration of mechanical models of physical phenomena, but there were wider dimensions to the difficulties that physicists perceived in the conceptual rationale of the mechanical theory of nature.

The traditional programme of mechanical explanation elicited diverse responses from physicists in the 1880s and 1890s. Thomson's ether models and Boltzmann's lectures on field theory continued the programme of elaborating detailed mechanical models of phenomena. Boltzmann strove to provide an exhaustive treatment of every detail of the structure and motions of his mechanical models of the electromagnetic field; and Thomson declared that the construction of a mechanical model of a phenomenon was the criterion of the intelligibility of that phenomenon. Nevertheless, the conceptual difficulties associated with the enunciation of mechanical models were well understood. Maxwell had pointed out that such models could not provide unique explanations of phenomena and had drawn attention to the dangers of confusing representation and reality, and though he remained committed to the ultimate aim of formulating a ‘complete’ mechanical theory of the field, in his Treatise he employed an analytical formulation of dynamics, rather than a specific mechanical model.