15. Absolute Zero of Temperature

The law just found allows the position of the so-called absolute zero of temperature for gaseous bodies to be determined.

If heat is nothing else than the kinetic energy of molecular motion, the temperature at which a gas possesses no more heat must be identical with that at which its molecular motion has disappeared, and all atoms and molecules remain in a state of perfect rest.

The expression that has been found for the molecular speed G shows that this speed vanishes when

1 + a9 = 0.

If from the measures of Magnus,1 Regnault, Jolly,3 Recknagel, and others, which are all in agreement with each other, we take 0.00367 for the value of a in the case of air when the Centigrade scale is used, and put this in the last equation, we find

9 =

272°.5 C.

If we

for the required temperature of the absolute zero. reckon temperature, not from the melting-point of ice arbitrarily chosen to start from, but from this absolute zero, then we obtain for the absolute temperature

or, in the general case, for all scales in use we have

Θ = a + I,

where the value of the constant a is to be taken as the reciprocal of the coefficient of expansion of air for the scale in question.

Pogg. Ann. lv. 1842, p. 25.

2 Mem. de l'Acad. de Paris, xxi. 1847, p. 73; Pogg. Ann. lv. and lvii.

Pogg. Ann. Jubelband, 1874, p. 82.

Pogg. Ann. cxxiii. 1864, p. 127, table i.

Mendelejeff, Ber. d. deutsch. chem. Ges. x. 1877, p. 81.

[This requires definition; on the Fahrenheit scale, for instance, the coefficient of expansion is usually defined with reference to an initial volume at

Introducing this absolute temperature, we have for the mean molecular speed the formula

G = H√O,
Ꮎ,

where H is a constant, the meaning and value of which are easily decided. The molecular speed, therefore, is a magnitude which increases proportionally to the square root of the absolute temperature.

This determination of the absolute zero and of absolute temperature deserves the more notice, as it is the same, or very nearly the same, for all gases; for, as experiment shows, all gases have very nearly the same coefficient of expansion, and therefore the position of the absolute zero is approximately the same for all gases.

A real meaning, however, is perhaps not to be ascribed to the zero so found in the sense of its really denoting a temperature at which all molecular- and heat-motion ceases so that there is no more heat. For it is evident that if there is no more heat-motion there is also, according to our theory, no more tendency to expand, and therefore no body can remain in the gaseous state. The formula of Gay-Lussac's law cannot strictly be applied at such low temperatures, since possibly--and certainly for such gases as are condensed to liquids at temperatures above - 273° C.this law loses its validity at some higher point, and another takes its place. The absolute temperature that has been introduced has therefore more the signification of an auxiliary mathematical function than of a physical reality.

the freezing-point of water-viz. 32° F.-and in the case of air is taken to be

5

9

× 0·003670 = 0·002039, whereas in the text an initial volume corresponding

to = 0 is required on all scales; so that for air on the F. scale we must take

a = 0-002039 : (1 – 0002039 × 32) = 0002181. Tr.]

1

1 [In fact, H = 485 ÷ √(272·5 × s) = 29·38/ √s metres per second for the Centigrade scale.-TR.]

2 See § 32.

* On other determinations of the absolute zero of temperature see Gehler's Physikalisches Wörterbuch, x. p. 115.

16. Pressure and Energy

The formula found for the pressure of a gas,

brings this magnitude into very close relation with another, namely, the kinetic energy of the molecular motions. Since the density p measures the mass of gas contained in the unit of volume, the magnitude

K = pG2

is nothing else than the amount of kinetic energy possessed by the molecules in unit volume.

The simple relation deduced from these two formulæ, viz.

K = p,

enables us to express the molecular energy by a magnitude which is directly amenable to observation. The pressure and kinetic energy of a gas stand to each other in an invariable relation which is independent of the temperature.

This simple relation is nothing else than an expression of the thought which underlies our theory. Both magnitudes, pressure and energy, have their origin in the molecular motion; they are even completely alike in their nature. Their difference consists only in the difference of the units in which their values are expressed. For the pressure which the walls of a gas-holder support forms also a measure of the kinetic energy of the contained gas. Both magnitudes change proportionally to the absolute temperature, and we have

where Po, K, are the values of the pressure and kinetic energy per unit volume at the temperature of the freezingpoint of water, which on the [Centigrade] scale of absolute temperature is

= 272.5.

On an ordinary scale of temperature wherein a is the coefficient of expansion [see note 6, p. 28] we have

p = P(1 + a9), and K = K(1 + ad).

The latter formula, which is the mathematical expression for the proposition named several times already [§§ 9, 14], that the energy of the molecular motion is the mechanical measure of the temperature, shows that the kinetic energy increases by the same amount for every degree of temperature.

17. Dalton's Law for the Pressure of Mixed

Gases

From this relation between the pressure and the kinetic energy of molecular motion a very important conclusion may be drawn if we extend our consideration to a gaseous medium containing molecules of different kinds, that is, to a gaseous mixture.

For such a mixture the calculation of the pressure exerted would be carried out in exactly the same way as was done in § 11 in the special case of a simple gas. The pressure on a surface is, in the more general case of a mixture of gases, also measured by the sum of the impulses of the molecules on a unit of area in a unit of time; its value is therefore represented by the total energy given up to the surface by all the different kinds of molecules present.

The formula for the value of the pressure exerted by a mixture of different gases therefore takes the slightly modified shape

p = (K1 + K2 + ...),

where the magnitudes denoted by K are the values of the kinetic energy per unit volume of the molecular motions in the single components of the mixture, and are given by

K1 = p,G, K2 = p,G,2,...

2

P1 Pa... being the densities of these components, and G,, G2,... the mean speeds of their molecules.

But the values of the pressures which the components of the mixture would severally exert if separately occupying the volume of the mixture are

or the pressure exerted by the mixture is equal to the sum of the pressures separately exerted by its several components.

Thus the law found by Dalton' for the pressure of mixed gases, and confirmed and defended by Henry' against many attacks, follows as a necessary consequence from the assumptions underlying our theory. Of these one was assumed by Dalton, viz. that molecules of different gases act on each other neither attractively nor repulsively, and it is the most important for our present case; for it is this assumption which entitles us to consider that we have to take into account kinetic, and not also potential, energy.

That this assumption is not absolutely exact, but only approximate, has already several times been pointed out [§§ 4, 8]. Just as Boyle's law, this of Dalton can have only a limited validity."

Dalton's hypothesis, that molecules of different gases neither attract nor repel each other, is often taken to mean that they do not exert pressure on each other. This is quite inadmissible on our theory, for molecules of different kinds moving about in a given volume collide with each other just as much as if they were of the same kind, and, consequently, must exert pressure on each other, as pressure is nothing but the sum of the actions produced by impact.

And this reading of Dalton's hypothesis is also in disagreement with experiment; for different gases do really

Mem. of the Manch. Lit. and Phil. Soc. v. 1802, p. 535; Gilb. Ann. xii. 1802, p. 385.

2 Nicholson's Journal, viii. 1804, p. 297; Gilb. Ann. xxi. 1805, p. 393.

Regnault, Mém. de l'Acad. xxvi. 1862, p. 256; Andrews, Phil. Trans. clxxviii. 1887, p. 45; Cailletet, Journ. de Phys. [1] ix. 1880, p. 192; Galitzine, Ueber das Dalton'sche Gesetz, Inaug. Diss. Strassburg, 1890; Wied. Ann. xli. 1890, pp. 588, 770; Gött. Nachr. 1890, No.1; U.Lala, Comptes Rendus, cxi. 1890, p. 819, cxii. 1891, p. 426; Naturw. Rundschau, vii. 1892, p. 188.