exert pressure on each other, as has been proved by manifold observations.1

The meaning of this law is simply that a mixture of two or more gases possesses the same amount of kinetic energy as its components taken together, and the correctness of this fact is proved by the observation that if two gases at the same temperature and pressure are mixed together there is neither generation nor absorption of heat.

18. Heating by Compression

Remembering that the pressure, energy, and temperature of a gas increase together in constant ratios, we have at once an explanation of the fact that the temperature of a gas is raised by compression and lowered by expansion. Even without employing the assumptions of the kinetic theory, it is not difficult to see that a diminution of volume caused by heightened pressure must be bound up with an increase of energy, and that part of this energy may be transformed into heat; expansion, on the other hand, requires an expenditure of mechanical or heat energy to overcome the opposing external pressure. All, however, is not explained by this general consideration. We obtain a deeper insight into the nature of the phenomenon when we investigate more closely the nature of the molecular motion in a gas that is being compressed or expanded.

Let the gas be in a cylinder which is closed by a movable piston. To keep this in equilibrium a pressure must be exerted upon it which will balance the action of the molecules impinging on it. But when an excess of pressure acts on the piston from the outside, the piston is driven into the cylinder; and during this motion of the piston the forces. that come into play in the collisions with the oppositely moving molecules are increased. The molecules are, therefore, thrown back with greater vigour, and consequently

For instance, Lamont, Pogg. Ann. cxviii. 1863, p. 168; Schlömilchs Zeitschrift, viii. 1863, p. 72, ix. 1864, p. 439; Bunsen, Gasometrische Methoden, 1857, p. 209; 2nd ed. 1877, p. 267; O. E. Meyer and F. Springmühl, Pogg. Ann. cxlviii. 1873, p. 540.

D

attain a greater speed on rebound from the moving piston than if they strike it when fixed. But since the kinetic energy of the molecular motion is nothing but heat, it is obvious that the compressing motion of the piston communicates heat to the gas.

The reverse occurs when the pressure on the piston is so small as to be overcome by the impacts of the molecules. The piston then moves in the same direction as the molecules that strike it, which therefore attain a less speed by the impact, as they give part of their former momentum to the piston. The gas consequently cools in doing work by pushing the piston out.

2

In this way the heating of a gas by compression was explained by Krönig1 and Clausius. A mathematical theory of the phenomenon has been given by Woldemar Voigt.3

It has been shown by Clausius that the heat produced by pressure can easily be calculated on the grounds of our theory, and that it is equal to the work done. In a rather later memoir he gives a proof which we here reproduce.

We will, for simplicity, continue to use Joule's procedure, described in § 10, and therefore assume not only that all molecules possess the same mean speed G, but also that only one-third of the molecules are to be taken into account in calculating the impacts on a wall of the containing vessel. This assumption is admissible if the compression takes place so slowly that the disturbance of the equilibrium has always time at once to subside. With this supposition the number of molecules which in unit time meet unit area of the wall of the vessel is NG, by § 12, and the number therefore in unit time which strike the surface F of the piston is

FNG.

1 Grundzüge einer Theorie der Gase, 1856 ; also Pogg. Ann. xcix. 1856, p. 315. Pogg. Ann. c. 1857, p. 365; Abhandl. pt. 2, 1867, p. 242.

Gött. Nachr. 1885, No. 6, p. 228. See Natanson, Wied. Ann. 1889, xxxvii. p. 341.

Mech. Wärmetheorie, iii. 1889-91, § 14, p. 29.

Each of these molecules would, after impact, rebound in the opposite direction with unchanged speed if the piston were at rest. But let the piston which compresses the gas move forward with a speed a, in the direction opposite that of the molecules which strike it with speed G. The strength of the rebound is thereby increased in the ratio in which the relative velocity G+ a exceeds the molecular speed G. A rebounding molecule therefore experiences in the impact an impulse, which is not 2mG as before, but the greater one, 2m(G + a), which results from its losing its initial speed G, and gaining the speed G + 2a in the opposite direction; its kinetic energy therefore increases during the impact by

This we may with sufficient exactness replace by 2maG, since we have assumed that the compression goes on so slowly that every disturbance of the molecular motion at once subsides; for we thereby also assume that the speed a of the piston is negligible in respect of the speed G.

Since each molecule gains this amount of energy at every collision, the whole gain of energy by the gas in unit time due to the impact of FNG molecules, as above, on the piston is given by

NmG2Fa.

=

This product has a very simple meaning; for the pressure of the gas which the piston has to overcome is p NmG2, by § 11, and the diminution which the volume V of the gas experiences in unit time is 8V Fa, as in this time. the piston moves through the length a in the cylinder, the sectional area of which is F, and therefore the expression found for the increment of energy is

=

NmG2. Fa = p dV.

It is thus proved that the kinetic energy gained by the gas. during the compression is equal to p SV, the work which the piston must do to overcome the pressure of the

gas.

19. Cooling by Expansion

In exactly the same manner the reverse phenomenon may be explained by the kinetic theory, viz. that a gas must cool when it does work by expanding, and that it thereby loses an amount of molecular energy equal to the work done.

If by its pressure the gas pushes back the piston with a speed which, as before, we will denote by a, the molecular speed of a molecule which impinges on the piston diminishes from G to G2a, and there passes therefore from the molecule to the piston at each impact the energy 2maG. Thus the molecules that strike the piston in unit time, FNG in all, lose the total energy

FNG.2maG= NmG2. Fa = p dV,

which is the work done by the gas in expanding through the volume V against the pressure p.

The rise of temperature that accompanies the compression of a gas and the fall that results from its expansion can from this be easily calculated if the value of the specific heat of the gas at constant volume is known. We have only to apply to this problem the general theorem of thermodynamics that heat and energy are equivalent to each other. If we represent by A the heat which is equivalent to a unit of work, Ap SV is the heat which is added to the gas during the compression of its volume from V to VSV or which leaves it during the expansion from V to V + SV. We can otherwise express this heat in terms of 89, the change produced in the temperature 9. If c is the specific heat at constant volume, pV the mass of the gas in the cylinder, and therefore pVc its heat-capacity, the relation between the heat produced by compression and the corresponding rise of temperature is

pVc 89 = - Ар бV,

and this holds good too for the case of the gas cooling by expansion. The negative sign has to be introduced into the formula to indicate that an increase of volume corresponds

to a diminution of temperature, and vice versa. The change of temperature which occurs is therefore

Here c denotes the specific heat at constant volume and not that at constant pressure. That this is so we shall easily see by again analysing the procedure. Without transgressing the law of the conservation of energy we can thus picture the transaction; that the work of compression p&V first produces a progressive velocity a of the gas, the volume being diminished without the to-and-fro motions of the molecules being altered, and that then, on the piston ceasing to move forwards, the energy of the progressive motion communicated to the gas is transformed into heat without change of volume by the collisions of the molecules. In the calculation, therefore, it is the specific heat at constant volume that is to be taken into account.

20. Vaporisation

Many gaseous bodies are condensable into liquids by application of pressure only, without the necessity of removing heat from them. Such substances are called vapours, in contradistinction to gases proper.

The cause of liquefaction by pressure alone we can only look for in the forces of cohesion. If the molecules of a gas are brought nearer each other by increase of pressure, those forces are exerted in greater degree; and it may happen that, if the molecules are brought near enough together, their action is so much increased that the molecules cannot separate any more from each other. For this it is necessary that the kinetic energy of the molecules shall be no longer sufficient to overcome the energy of the cohesive forces. If this limit is reached, the vapour begins to change into liquid.

There is now a condition of equilibrium, in which one part of the substance remains liquid and another hovers above the liquid as vapour. In this vapour the molecules