Imágenes de páginas
PDF
EPUB

used the formula for the coefficient of viscosity as given by Stefan, viz. :

n = πNmQL,

which, if we are contented with a somewhat less accuracy, seems quite justified, and is therefore mentioned frequently in memoirs. The reason that we have arrived at different values of the numerical coefficients in our formulæ for D and ŋ, viz. π = 0·39270 and 0·30967, is that in the calculation of a higher degree of approximation has been aimed at and attained.

We have already met with a similar uncertainty of the numerical factors that come into formulæ ; it is in all cases caused by the different way in which the mean is taken of the varying properties and circumstances of the molecules. In an elementary theory it is not, indeed, possible to keep up the consideration of all conceivable particular cases right on to the end through the whole calculation, and then at the conclusion-and not till then if we are to be accurateto form the sum and take the mean. We are obliged, on the contrary, for greater simplicity and clearness of procedure, not only to deduce the mean result for each part of the calculation by itself, but also to employ from the first average values of the magnitudes that come into the calculation instead of the real ones. Since the concluding result of such a calculation can be only approximate, we must not be surprised that the theories given by different investigators have led to different values of the numerical coefficients. All, however, agree in concluding that the conductivity and coefficient of viscosity are connected by the relation

Ε = κη,

where c is the specific heat of the gas at constant volume, and a numerical coefficient which has the same value for all gases. This value lies between 0.5 and 2.5.

In §57* of the Mathematical Appendices to this work I have given a calculation in which I have striven to find

Wien. Sitzungsber. 1872, lxv. Abth. 2, p. 363. Compare § 48* in the Mathematical Appendices.

the value of the coefficient as exactly as possible. This calculation, in which the mean values have been taken with due regard to Maxwell's law of distribution of speeds, has given

t = 1.6027 nc.

If we compare the mathematical calculation which has led to this larger value of the coefficient with that here deduced in an elementary way, the difference will perhaps astonish us, and give rise to the objection that in laying down the mathematical formula we have taken into account the kinetic energy of the molecular motion only, while at the end we have substituted for this kinetic energy simply the specific heat at constant volume c multiplied by the absolute temperature. This procedure seems to contradict the view as to the ratio of the molecular energy to the total energy which is put forward in § 53; for it is there proved that the kinetic energy of the rectilinear motion of the molecules of a gas forms only a part of the whole energy contained in the gas. We might therefore be inclined to assume that the calculation in which account is taken only of the energy of the molecular motion will give a result, the validity of which should be limited to the propagation of that energy only; and that we should therefore consider the value 1.6027 of the coefficient to apply only to the propagation of the molecular energy; and we might consider it possible that the remainder of the energy forming the heat of the gas, and therefore the energy of the motions which the individual particles execute within the molecules and the whole potential energy, may be propagated with a different speed, either smaller or larger, than that of the kinetic energy.

In fact, the assumption has many times been made, especially by Stefan' and Boltzmann,2 that the kinetic energy of the molecular motion is passed on from place to place with greater speed than the remaining energy, which in Chapter V. we have termed the atomic energy.

Wien. Sitzungsber. 1875, lxxii. Abth. 2, pp. 74, 75.

2 Ibid. p. 458; Pogg. Ann. 1876, clvii. p. 457.

We

were at that time obliged to yield to this view, because no other possibility was seen of bringing the theoretical law

Ε = κητ

into a complete agreement with the observations then published. It had, in fact, been apparently established that a smaller value must be assigned to the factor for gases whose atomic energy forms the greater part of the whole energy than for the other gases in which the molecular energy exceeds the atomic energy. These facts would really be very simply explained on the hypothesis named, according to which the gases with preponderating molecular energy must have a better conductivity than the other gases with greater atomic energy and smaller molecular energy.

Theoretical reasons could also be adduced for this hypothesis which at that time seemed to be unavoidably necessary from the results of the experimental measures that had been made. If we consider the processes that occur during the encounter of complicated molecular structures, we may become inclined to the view that the motion of both centres of gravity will chiefly be altered by the collision, while many parts of each complex of atoms may scarcely be disturbed in their own motions. The molecular energy would therefore be carried over from place to place with greater speed or in a higher degree, or, in other words, the conductivity of the gases would be greater for the molecular energy than for the atomic energy.

In the first edition of this book, which was published in 1877, I explained these ideas as possible and admissible because I knew no better explanation of the facts, although even at that time I raised several weighty objections to them. At the same time, however, I pointed to another interpretation, in expressing my belief that all differences between theory and observation might find their explanation in the accumulation of the errors of experiment.

The most important objection that can be urged against this hypothesis, that gases have unequal conductivities for their molecular and atomic energy, is that it contradicts an

important fundamental theorem of the kinetic theory. As has been mentioned in §53, Clausius has established the proposition that in a perfect gas the kinetic energy of the molecules bears an always constant ratio to the whole energy contained in the gas; and this theoretical proposition is experimentally confirmed by the experiments that have been made for the measurement of the specific heats. If now a greater amount of molecular energy were brought to a point in a gas in consequence of the assumed better conductivity for it, the necessary consequence would, according to this proposition of Clausius, be that a compensation would at once result by the atomic energy gaining at the expense of the molecular until the proper ratio was again restored. By this the untenability of that hypothesis might be established, at least for perfect gases.

Now, there are certainly many gases and vapours for which Clausius' proposition cannot hold in all strictness, because their specific heats are not constant, but are highly variable with the temperature. For such cases a different consideration would be in place, which rests on the proposition of the conservation of energy alone. According to this law the discussion of considerations respecting the conditions during an encounter has no bearing on the resolution of our doubt. If a particle has flown from one place to another, it has carried over with it to its new place the whole amount of its energy-not its kinetic energy only, but also the whole of its internal or atomic energy-and it is an entirely unimportant question whether and how this energy is transformed by the collisions that afterwards

occur.

Hence it follows that the conductivity of a gas for every kind of energy is the same, and that if the formula

[blocks in formation]

holds for the conduction of the molecular kinetic energy, it must also hold for the conduction of heat generally.

106. Theoretical Laws of the Conduction of

Heat

The conclusion of the theory which we have found requires the conductivity of a gas for heat to obey the laws which hold for the coefficient of viscosity and for the specific heat.

The kinetic theory of gases has led to the discovery, which has afterwards been confirmed by experiment, that the coefficient of viscosity of a gas is independent of its pressure or density. It also follows from this theory that the same value must be found for the specific heat at constant volume, referred to unit mass of the gas, if the experiment is made with a different volume or at a different pressure; for the number of molecules of a gas which are contained in unit mass of the gas require under all circumstances the same addition of energy when this mass is warmed by 1 degree without expanding, and therefore without doing work. In agreement with this theoretical result the experiments of Regnault' have shown that the specific heats of air, hydrogen, and carbonic acid, measured at constant volume, are independent of the pressure.

Since, then, not only the coefficient of viscosity, but also the specific heat, is independent of the pressure of the gas, the theory leads to the law laid down by Maxwell and by Clausius that the heat-conductivity also of a gas for heat is not variable with its pressure.

E.

Regnault has further found that the specific heat of chemically simple gases is independent also of the temperature. Probably all gases whose molecules are composed of only two atoms have this property, to judge from Wiedemann's2 observations. On the kinetic theory, therefore, the heat-conductivity of a diatomic gas increases with the temperature according to the same law as its coefficient of viscosity.

The laws we have cited for the specific heat do not, however, hold without limitation. The more easily condensible gases and vapours do not obey these laws, at least in all strictness. Hence, also, the theoretical laws of the conPogg. Ann. 1876, clix. p. 1.

1 Mém. de l'Acad. de Paris, 1862, xxvi.

2

« AnteriorContinuar »