duction of heat deduced from these empirical rules cannot be extended without objection to vapours, nor to condensible gases, nor to gases of complicated chemical structure; for these media the laws have only an approximate applicability.

107. Observations of the Conductivity at Different Pressures

That the heat-conductivity of gases is really independent of the pressure, as the theory requires, has been shown by Stefan's experiments, in which a measurement of the conductivity of gases was given for the first time in absolute

measure.

After he had convinced himself by preliminary experiments that the method followed by Magnus is unsuitable for the purpose, by reason of the simultaneous conduction of heat through the walls of the gasholder,2 Stefan employed Dulong and Petit's method, which had shortly before been also used by Narr3 in comparative experiments on the speed of cooling in different gases. In a cylindrical chamber fitted with the gas under investigation was a similar cylindrical holder, which was filled with air or other gas, and provided with a manometer, whereby it was ready to serve as an air-thermometer. Stefan made observations with this apparatus by immersing it, when initially at the temperature of the room, in melting snow, and measuring the speed of cooling of the air-thermometer by observation of the falling condition of the manometer.

It is by this method, or by similar methods which have been partly improved, that most of the later measures of the heat-conductivity of gases have been made. In addition to a second memoir by Stefan' and two investigations by Josef Plank on gaseous mixtures and a few pure gases which are a continuation of it, several more valuable

Wien. Sitzungsber. 1872, lxv. Abth. 2, p. 45.

2 Buff has confirmed this. Pogg. Ann. 1876, clviii. p. 177.

3 Ibid. 1871, cxlii. p. 123.

Wien. Sitzungsber. 1875, lxxii. Abth. 2, p. 69.

5 Ibid. 1875, lxxii. Abth. 2, p. 269.

Ibid. 1876, lxxiii. Abth. 2, p. 123; Carl's Repert. 1877, xiii. p. 164.

2

3

memoirs have been published by Winkelmann,' Kundt and Warburg, Janssen, Graetz, Eichhorn," and others. Schleiermacher made his observations by Andrews' method.

I first bring forward some of Stefan's results for the experimental proof of the theoretical law that the conductivity of a gas is independent of its pressure. From four observations on air under the ordinary pressure of about 750 mm. of mercury Stefan found the following values, referred to the centimetre and second as units, for 107, viz. 554, 560, 552, 554, and thus the mean value

= 0.0000555;

and when he rarefied the air to the pressure 428 mm. he found in the same way f = 0.0000552.

If, then, there is a diminution of the conductivity with the density, this is certainly so small as to elude observation.

Kundt and Warburg obtained similar results by observing the cooling of a mercury thermometer in a closed space. They found the following values for the interval t which was necessary for the thermometer to cool from 59.3° C. to 19.6° C. when the apparatus was filled with the gases named at the pressures specified.

The constancy of the conduction and radiation of heat

Pogg. Ann. 1875, clvi. p. 497; 1876, clvii. p. 497; 1876, clix. p. 177; Wied. Ann. 1877, i. p. 63; 1880, xi. p. 474; 1883, xix. p. 649; 1886, xxix. p. 68; 1891, xliv. p. 177; 1893, xlviii. p. 180.

2

Pogg. Ann. 1875, clvi. p. 177; 1875, p. 160.

previously in abstract in Berl. Monatsber. 3 Wied. Beibl. 1879, iii. p. 701.

Wied. Ann. 1881, xiv. p. 232; Münch. Habilitationsschrift, 1881. 5 Ibid. 1890, xl. p. 697. Ibid. 1888, xxxiv. p. 623; 1889, xxxvi. p. 346. U

comes out plainly from these numbers, at least within the given limits of the pressure. Under greater pressure the regularity was disturbed by currents of the gas.

In the opposite case also, when the pressure was made very small, Kundt and Warburg observed deviations from constancy; there is here, however, no question of currents in the gas but of a phenomenon of a similar kind to that noticed by them in their experiments on viscosity, which have been described in § 81. Just as the slip of a gas on the wall of the containing vessel becomes the more appreciable the further the rarefaction is carried, so a difference between the temperature of the enclosure and that of the gas touching it becomes the more considerable the less the pressure. If the pressure is high and the gas dense, many particles strike against the wall and cause so complete an interchange of heat that only a small difference between their temperatures can arise. But if the pressure becomes smaller, and therefore the number of particles fewer, the difference of temperature increases and may become so great as to amount to several degrees.

This behaviour is quite analogous to that observed in connection with viscosity; for in viscosity the internal friction alone comes into account in dense gases, while in rarefied gases the phenomenon depends, not only on the internal friction, but also on the external friction as well; and, just in the same way, the conduction of heat in dense gases practically depends only on the internal conductivity of the gas, while in rarefied gases it is conditioned by the external conductivity as well. The internal conductivity, like the internal friction, is independent of the density and pressure; the external conductivity, however, alters, like the external friction, with the pressure, and, indeed, according to the same law. In both cases there is a simple proportionality to the pressure, and for the same reason in both cases; for both the external friction and the external conduction increase in the ratio of the number of the particles which meet the wall. For the clear recognition of this behaviour, first noted by Kundt and Warburg, the later memoirs of Crookes,1 1 Proc. Roy. Soc. 1881, xxxi. p. 239.

Smoluchowski von Smolan1 and Brush have essentially contributed.

Winkelmann similarly tested the constancy of the internal conductivity by determining the logarithmic decrement of the series of observed readings of his air thermometer, which decreased with the time in geometrical progression. The following are some of the values found by him :-

Winkelmann examined other gases, too, especially ethylene, and found a similar confirmation of the law. But the examples we have given will suffice to prove the accuracy of the theoretical law and to show us to what low pressures the conductivity remains constant.

108. Calculation of the Heat-conductivity from the Coefficient of Viscosity

But the theory requires more than that the value of the conductivity should not vary with the density of the gas; it furnishes also the law of the variation of the conductivity with the temperature, and even suggests the possibility of calculating the value of the conductivity itself in absolute units for those gases whose viscosity and

1 Wied. Ann. 1898, lxiv. p. 101; Phil. Mag. [5] xlvi. p. 192; Wien. Akad. Anzeiger, 36. Jahrg. 1899, p. 1, and Sitzungsber. 1899.

2 Phil. Mag. 1898 [5] xlv. p. 31.

specific heat are known. This has been already done by Maxwell and Clausius, the first founders of the theory, long before any measurements of the conductivity of a gas were experimentally made. Clausius concluded from his theory that air conducts heat 1,400 times less well than lead, and Maxwell predicted that the conductivity of air is 3,500 times smaller than that of iron; and both these predictions have been since confirmed in striking fashion by Stefan's experiments.

The formula that leads to these conclusions, viz.

has yet to be proved. We have to calculate the theoretical value of the conductivity of different gases from the observed values of their coefficient of viscosity and of their specific heat at constant volume c, and compare them with the observed values of the conductivity.

η

The earlier attempts, and even that made in the first edition of this book, to carry out this calculation and comparison led to no perfectly satisfactory results; the calculated and observed numbers exhibited no general agreement together. A satisfactory agreement was obtained only in the case of gases which contain not more than two atoms in the molecule, while with all other gases no approach to agreement was found. This result led to the hypothesis of assuming two different kinds of conduction in gases (§ 105). This faulty attempt to account for the contradiction between theory and experiment was refuted by Wüllner, who saw and proved that the failure to obtain agreement was only due to the faultiness of the values assigned in the formula to magnitudes, the variation of which with temperature was at that time not known with sufficient accuracy, the values employed having been determined at quite different temperatures.

Wüllner, whose memoir has already been several times mentioned in §§ 55-58 of Chapter V., determined anew ratio of the two specific heats for a series of gases, and Kundunted the dependence of its value on the temperature.

Wied. Ann. 1878, iv. p. 321.