limits that the variable values lying between them give the same mean kinetic energy as if all the values are possible. We have merely to take one limit on the ascending and the other on the descending branch of the curve at corresponding places. The probability that the speed of a molecule lies between given limits is represented on the figure by the area included between the curve, the axis of abscissæ, and the ordinates corresponding to the limits. In this way we find, for instance, that the probability of a value between 09 and 11 of the most probable value is given by 0.2 × 0.8 0.16; that is, 16 molecules out of every 100, or 1 out of every 6, have speeds which deviate less than 10 per cent. from the most probable value. There are, on the contrary, as we similarly find, about 9 molecules in every 100 which possess within 10 per cent. of half the most probable value, and about 11 in 100 with a speed equal, within the same limits, to 14 times this value. There are, further, but 3 in 100 with a speed 4 times less than, and scarcely more than this number with a speed twice as great as, the most probable. 27. Mean Value of the Speed These numbers teach us, as indeed does a glance at the curve, whose ascending branch is steeper than its descending, that the number of particles, whose molecular speed is greater than the most probable, surpasses that of the particles which move with a speed less than the most probable. The most probable speed is therefore not also the arithmetical mean of the various speeds, but the mean value of the speed is greater than the most probable. Similarly the mean value of the molecular energy is greater than the energy of a molecule which moves with the most probable speed. The values of the molecular speed, which we have calculated in § 13 by Joule and Clausius' method, from the pressure exerted by gases, are, therefore, not at all the most probable values of the speed of the molecular motion. Indeed, we cannot strictly regard them as correct means of the various speeds; at least, they are not the arithmetic means, but the values of the speed which correspond to the arithmetic means of the energy of the different particles. (See § 10.) By a simple mathematical consideration we may easily see that the method by which Joule and Clausius calculated the mean values of the molecular speed must in all cases give numbers which are greater than the real arithmetic means. Consider n particles moving respectively with the speeds a, b, c...; the mean value of these different speeds is then Q = (a + b + c + ...)/n. Calculating also the mean value of the molecular energy of a particle, E = &mG2, wherein m, as before, represents the molecular mass, and G the mean value of the speed, we obtain E = \m(a + b + c +...), = so that the mean value G of the speed introduced by Joule and Clausius has the signification G2 = (a2 + b2 + c2 + ...)/n. Comparing this expression with Ω Q2 = (a2 + b2 + c2 + ... + 2bc + 2ca + 2ab+...)/n2, which, as we see from the known relation leads to a2 + b2 > 2ab, Q2 < (a2 + b2 + c2 + . . . + b2 + c2 + c2 + a2 + a2 + b2 + . . . )/n2, or, since each square occurs n times in the numerator, to N2 < (a2 + b2 + c2 + . . . ) / n, we find Ω < G; that is, the arithmetic mean value of the speed is less than the mean value G calculated by Joule and Clausius from the mean kinetic energy. If Maxwell's law is true, this relation, which holds in general between the two mean values, takes the following simple form that is equally true for all gases, viz. (see § 19* of the Mathematical Appendices) :— = G√(8/3π) = 0.9213 G, which, with extreme approximation, may be written Ω = 13 G. Joule and Clausius' values are therefore greater than the arithmetic means of the molecular speeds by about a twelfth part. The latter may just as easily as the former be calculated from the value of the pressure; for the formula for the pressure p given by Joule and Clausius (§ 13), viz. p = spG2, where p denotes the density of the gas, may be replaced by ρ p = πρΩΣ, from which the arithmetic mean values of the molecular speed for different gases may be calculated, as has already been done in a Latin dissertation that I published in 1866. Further, for the calculation of the most probable value W of the speed, according to Maxwell's theory, we have the formula W = 'Q=G√(2/3); the value W is, therefore, smaller than both the others, and stands to them in a ratio which is the same for all gases. Closely related to this most probable value is a third. mean value of the speed, which is called the value of mean probability, or, more shortly, the mean probable value. The signification of this value, which I denote by O in § 19* of the mathematical theory, is that there are as many particles with speed less than O as there are particles with speed greater than O. Its value lies between W and 2, and we have 01.09 W = 0·96 N. 1 Inaugural dissertation, De Gasorum Theoria, Vratislavia 1866. 28. Values of the Speeds In order to give a clearer idea of these relations I have calculated a few examples of numbers, and more especially for the two gases which are the most important constituents of atmospheric air, namely, oxygen and nitrogen. densities of these gases, according to Jolly,1 are The when at 0° under the pressure of a mercury column 0·76 m. high at Munich, where the acceleration of gravity is 9.8069 m. per sec. per sec. We therefore obtain for the Joule-Clausius mean values at 0° G = 461.2 m. per sec. for oxygen which completely agree with the values given in §13, as deduced by Clausius from Regnault's observations. From Maxwell's law we obtain for the arithmetic means of the molecular speed at 0° and, finally, the most probable values of the speed at 0° are W = 376·6 m. per sec. for oxygen Lord Rayleigh found hydrogen to be 15.884 times lighter than oxygen, and consequently for hydrogen at 0° The law of the unequal distribution of different speeds is Abh. d. Akad. zu München, xiii. 2. Abth.; Wied. Ann. vi. 1879, p. 520. 2 Proc. Roy. Soc. xxiii. 1888, p. 356. shown by the following numbers. Of 1,000 molecules of oxygen at 0° Since this one example will suffice to give a clear representation of the nature of the law, I shall limit myself to giving only the mean values of the molecular speeds for other gases and vapours, and these I shall tabulate together with the values of the specific gravity which have been used in their calculation. I have in this case not referred the density p of a gas to that of water as unity, but instead of this I have introduced, as in § 13, the specific gravity s referred to atmospheric air. This is a procedure which would not be admissible for exact scientific calculations, since atmospheric air is, as Jolly' has shown, by no means always of the same composition. Still, for the purpose in hand, this inexact procedure is justified by there being a still greater uncertainty in the values by reason of the deviations of the gases from Boyle's law. It is on this account that most observers have referred their numerical values to air; it would have served no purpose to reckon them with respect to water. The references appended to the table relate to the determinations of the specific gravities. 1. Calculated from Lord Rayleigh's observations, Proc. Roy. Soc.' xxiii. 1888, p. 356. 3. Biot and Arago, Mém. de l'Inst.' vii. 1806, p. 320. 4. Gay-Lussac, Ann. Chim. Phys.' [2] i. 1816, p. 218; ii. p. 135. Abh. d. Akad. zu München, xiii.; Wied. Ann. vi. 1879. |