time, i.e. the pressure of the gas, is increased by an addition of heat.

Beyond these considerations no further proof is needed of the proposition that the pressure increases as the square of the molecular speed, or, what comes to the same thing, as the energy of the molecular motion. In agreement, therefore, with the general principles of thermodynamics it follows that the mechanical measure of heat and of temperature is the kinetic energy of the molecular motion.

10. Mean Value and Components of the Energy

The closer investigation of the relation between the temperature of a gas and the kinetic energy of it's molecules is rendered difficult by the circumstance that the molecules have not all the same speed, and, therefore, not all the same energy. This consideration is really identical with this other, that the energy of each particle changes on collision. If, however, we can say that the resultant action of the impacts which each particular molecule makes in a fairly long time with its energy ever changing is equal to that which would result if the impacts all occurred with a uniform mean energy, then we must allow that the resultant action of the impacts of all the molecules is the same as if the molecules have all a uniform mean energy of motion.

We gain a further advantage in our calculation by making use of the proposition that, just like a velocity or a force, kinetic energy may be separated into three components, of which each corresponds to a component of the motion in a given direction. The whole energy is equal to the sum of its components, as is easily seen from the known formula √(u2 + v2 + w2)

@=

for a velocity @ in terms of its rectangular components u, v, w; for this gives

mw 2 =

mu2 + 1mv2 + }mw2.

This proposition enables us to substitute a simpler motion for that which really goes on in the gas near the walls of the vessel and produces pressure on it: we divide

the kinetic energy of all the molecules into three parts which in the mean are all of equal magnitude, one of them being the energy of a motion at right angles to the wall, and the others corresponding to motions which are parallel to the wall and at right angles to each other.

Only the first of these components of the energy comes into account in the estimation of the pressure on the vessel, and we therefore find the correct value of the pressure by ascribing to all the molecules a velocity perpendicular to the wall and a kinetic energy equal to one-third of the total mean kinetic energy of a molecule.

This result of our investigation is identical with the assumption with which Joule1 and Krönig' started in their calculations, as they assumed a gas, enclosed in a cube, to press as strongly against the faces as if one-third of the molecules moved parallel to each of the three directions of the edges, so that each face was impinged upon by only one-third of all the molecules.

11. Calculation of the Pressure 3

With this simplified assumption it is easy to calculate the value of the pressure from the resultant action of the impacts which the surface undergoing pressure receives from the molecules that meet it.

This surface, which we will call the stressed surface, may be taken either as a mathematical plane or surface inside the space filled with gas, or as a wall of the containing vessel. The former assumption has the advantage of allowing the calculation to proceed without

1 Mem. of the Manchester Lit. and Phil. Soc. [2] ix. 1851, p. 107; Phil. Mag. [4] xiv. 1857, p. 211.

Berlin 1856; afterwards reprinted in Pogg. Ann. xcix. 1856, p. 315, and in many other periodicals.

Other calculations beside those of Joule and Krönig: Clausius, Pogg. Ann. c. 1857, p. 353; Maxwell, Phil. Mag. [4] xix. 1860, p. 29, xxxv. 1868, p. 195; Stefan, Wiener Sitzungsberichte, xlvii. 1863, p. 91; O. E. Meyer, De Gasorum Theoria, Vratisl. 1866; Pfaundler, Wien. Sitzungsber. Ixiii. 1871, p. 159; v. Lang, ibid. lxiv. 1871, p. 485, Pogg. Ann. cxlv. 1872, p. 290; Saalschütz, Schr. d. phys.-ökon. Ges. zu Königsberg, 19. Jahrg. 1878, Sitzungsber. p. 45.

further hypothesis being necessary; here, however, I will first investigate the value of the pressure exerted on a solid wall by reason of its greater intelligibility.

For this it is necessary to introduce a hypothesis as to the magnitude of the forces exerted by impact against the wall. In choosing this we shall have to be guided by the consideration that a gas suffers no loss of energy through exerting pressure on the solid walls of its enclosure; the gas therefore receives back from the wall the energy it has given to it. If this is true for the gas as a whole we shall have also to assume for each one of its molecules that at every single impact against the wall its stock of kinetic energy remains undiminished. We thus arrive at the hypothesis that each molecule is, like a perfectly elastic ball, thrown back from the wall with the same speed with which it struck it. A molecule that impinges perpendicularly against the wall receives an impulse which is sufficient not only to stop its motion, but also to give it an equal speed in the reverse direction. The magnitude of this impulse is expressed by the product 2mG, wherein m denotes the mass of a molecule and G its speed; and just as great is the impulse exerted on the wall by the molecule during the impact.

To obtain from this the total force exerted on the wall we have to multiply this expression by the number of impacts in the unit of time.

Although this calculation can be made for every possible shape of the enclosure containing the gas, we will for simplicity consider the gas to be in a rectangular parallelopiped, whose edges are a, ß, y in length, so that its volume is aßy. If now there are N molecules per unit volume, there are Naßy molecules altogether. According to Joule's representation of the case, which, as was proved in § 10, may be used in the calculation instead of the real circumstances, one-third part of this number, or Naßy molecules, move in the direction of the edge y perpendicularly against the two faces aß. These faces will be struck alternately by the molecules moving to and fro between them.

The number of impacts on one of these faces in the time-unit is found from the consideration that between every two successive impacts by one and the same molecule there elapses the interval during which the molecule passes to and fro between the faces, that is, the interval in which it traverses the path 2y with the speed G. The number of impacts, therefore, which a single particle makes on the face in a unit of time is the ratio of the path G traversed in the time-unit to the length 2y of the path to and fro, and is thus G/2y; hence the number of impacts on a face aß in the unit of time made by all the particles is

Naßy × G/2y = NGaß.

The product of this number into the impulse 2mG, which is in the mean exerted at each impact, gives for the whole impulse exerted on the face aß in the unit of time, that is, for the total force exerted on it,

p being the pressure; so that the pressure on the surface is given by

This formula confirms what has been deduced before, viz. that the pressure p is directly proportional both to the square of the speed (§ 9) and also to N, the number of molecules in unit of volume, and therefore to the density of the gas; it is consequently inversely proportional to the volume (§ 6).

12. Another Calculation of the Pressure

I do not wish, however, to be content with this one calculation of the pressure, as it suffers from the defect of containing an unproved and unprovable hypothesis which it would have been easy to avoid-I mean the hypothesis that the laws of elastic impact hold for the collisions of molecules, even if only to a limited extent. We do not need this hypothesis if we investigate the pressure in the interior of the gas in place of that on the walls; and this

interior pressure can be calculated by the following method, which is carried out with greater strictness and generality in the Mathematical Appendix No. 1, §§ 1*–7*.

Consider the space occupied by the gas to be divided by a plane into two halves, a right half and a left half, and mark off a bit of this plane of unit area. On this unit area the one half of the gas presses with the same intensity from its side as the other half from the opposite side. For the right half would be moved from left to right by the pressure exerted on it by the left half, if it did not itself exert an equal and opposite pressure. Now, we measure a continuous force by its impulse in a unit of time; in the meaning of our theory, therefore, the pressure is nothing else than the momentum which is transferred in unit of time through the unit area of the plane from one half of the gas to the other, or, rather, as need hardly be specially specified, it is the component of this momentum in the direction of the pressure. To find the value of the pressure we have therefore to calculate the momentum perpendicular to the unit area which is transferred from one half of the gas to the other by the molecules that cross the unit area in a unit of time.

If for simplicity we retain Joule's conception of breaking up the motion into three components, we have to assume that one-third of all the molecules move perpendicularly to the plane. One-half then of this one-third-i.e. one-sixth of the whole-move at any moment from left to right, while an equal number move from right to left.

In a unit of time those molecules only can cross the plane whose distances from it at the beginning of the timeunit are less than the length of path travelled during the time-unit. Hence all the molecules which cross the unit area from left to right in a time-unit come from the cylinder, whose base is the unit area, and whose height is measured by the speed G, and whose volume therefore is numerically equal to G. The number of molecules therefore which cross unit area of the plane in unit time from the left half of the gas to the right is NG, if N, as before, represents the number of molecules in the unit of volume.