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.

The 1890s were hardly gay for the majority of Americans. Before the decade was halfway completed an economic depression of unheard-of magnitude had swept the country and threatened the very existence of millions of blue-collar workers, small businessmen, and even bankers. Following the panic of 1893, over 600 banks closed, 16,000 business firms went bankrupt, and 2.5 million laborers were suddenly jobless. The 1,300 strikes by workers in most heavy industries, coupled with the obvious plight of thousands of workers who marched on Washington in “General” Coxey's army, led one senator to fear that the country was “on the verge of revolution.”

In New York, the effect of the Depression was extremely severe. “Times were hard,” recalled Lillian Wald, the famous founder of the Henry Street Settlement. “In the summer the miseries due to unemployment and rising rents and prices began to be apparent, but the pinch came with the cold weather.” The winter of 1893–1894, the first of a depression that was to last through 1897, was most memorable to Wald. She observed firsthand “the extraordinary sufferings and the variety of pain and poverty” of those living in the tenements of New York's lower East Side.

The city's merchants were also severely affected by the Depression, as were the small charity institutions that depended upon their contributions. Coming on the heels of a steady rise in the costs of supplies, fuel, and food, the shock of reduced support was disruptive and, for some facilities, fatal.