exactly as represented in § 39. Joule and Lord Kelvin have measured with great care the alterations of temperature that occur in the vessels from and to which the flow takes place, and by these observations have proved that the whole amount of work done by the gas in the receiver, when it flows out, is not to be found in the increased energy of the gas in the outer vessel which has been warmed by compression, and in the heat that has been produced by the overcoming of frictional resistances. This phenomenon has no explanation on our theory so far as it has been developed in the preceding chapters; it proves, therefore, that a secondary circumstance has not been sufficiently taken into account, and scarcely leaves room for doubt that the cause which has been neglected is the cohesion of the gases, to overcome which during their expansion into vacuum a part of the heat energy must be taken up.

If we take in hand a thorough comparison of the two laws which bear Gay-Lussac's name with the results of experiment, we see no less clearly that our theory has not so far led us to absolutely strict laws of nature, but only to rules that hold good approximately, though the approximation is certainly excellent.

According to the first of these, all gases are to expand equally under the action of heat; they ought, therefore, all to have the same coefficient of expansion as air, viz. 0.00367. But the value of this coefficient is, for instance, 0.00366 for hydrogen and 0.00370 for carbonic acid. Similar deviations, which in some cases are even larger, are exhibited by other gases too.

Indeed it cannot be said of any one and the same gas that under all circumstances it has the same coefficient of expansion. Magnus has pointed out that the coefficient of expansion must vary if Boyle's law is not exactly obeyed. For if the pressure and volume of a gas are not strictly inversely proportional to each other, there is no reason to expect that both magnitudes will be increased in exactly the same ratio by a rise of temperature. We have therefore, strictly speaking, two different thermal coefficients.

Pogg. Ann. lv. 1842, p. 5.

to distinguish in a gas, viz. the coefficient of volumeincrease and the coefficient of pressure-increment; the former determines the increase of volume that occurs with rise of temperature when the pressure remains constant, and the latter measures the increase of pressure that is produced by heating without change of volume. By his experiments Regnault proved the difference of these coefficients; he found, indeed, that with most gases the volume-coefficient is somewhat larger than the pressure-coefficient, hydrogen alone showing the reverse property. Regnault1 further observed that the two coefficients are not entirely independent of the pressure and temperature of the gas, but that they increase with the pressure and diminish when the temperature rises.

This behaviour of gases may be also indirectly recognised from the observations that have been made on the specific gravities of gases. These are usually referred to the density of atmospheric air, at the same pressure and temperature, as unity. It is therefore sufficient to determine the specific gravity of a gas at different temperatures in order to learn whether this gas has the same mean coefficient of expansion as air, or a different one that varies with the temperature or the pressure.

An instructive example is given by the experiments made by E. Ludwig 2 on the density of chlorine. For the densities of the gas, compared with air, he found the following values:

The falling-off in these numbers shows that chlorine expands more than air. At 200° chlorine attains the same density as it should have according to Gay-Lussac's law in § 30; presumably from this temperature upwards its specific

1 Mém. de l'Acad. de Paris, xxi. p. 96; xxvi. p. 565.

2 Ber. d. deutsch. chem. Ges. i. 1868, p. 232.

gravity remains constant, so that its expansion-coefficient would have become equal to that of air, while at lower temperatures it must be greater.

41. Vapours

With vapours developed from liquids by heat the deviations from the theoretical laws are much larger than with gases proper. On the whole, vapours comport themselves as gases. They obey Boyle's law approximately, so that in their case, too, the pressure and volume vary very nearly in inverse proportion; and further, both pressure and volume increase with rising temperature in almost equal ratio, just as with gases. Dalton's law also holds good for mixtures of vapours and gases with approximate exactness. We might, therefore, apply the kinetic theory also to vapours by ascribing to their molecules, just as to those of gases, a rapid rectilinear motion, and by assuming that this motion increases with the heat.

Whether and how far these assumptions suit a given case can be most easily determined from the vapour-densities. As vapour-density we denote the value obtained for the specific gravity of a vapour when the density of air at the same pressure and temperature is taken as unity. The vapourdensity must, therefore, be independent of pressure and temperature if the vapour obeys Boyle's and Gay-Lussac's laws as exactly as air. Observation has shown that only within certain ranges of pressure and temperature can a vapour-density be looked on as constant.

In such determinations experiment has in general shown that greater values of the vapour-density are found the lower the temperature at which the measurement is made, and that not till higher temperatures are reached do we observe a constant density of the vapour, i.e. a density independent of the temperature. Further, for greater values of the pressure we likewise find greater values of the density of a vapour, as compared with air under equal pressure; and, correspondingly, the vapour-density approaches constancy in its value as the pressure falls off.

As regards the coefficient of expansion, it follows from the reasons given that it is not at all constant under all circumstances, but that it depends on temperature and pressure in such wise that with rising temperature or falling pressure it decreases, and that at a sufficiently high temperature or a sufficiently low pressure the expansioncoefficient for every vapour attains the same value as that of atmospheric air.

42. Saturated Vapours. Absolute Boiling point

A problem to which the endeavours of experimentalists in this direction have been especially directed consists in the measurement of the highest pressure which a vapour can attain in dependence on the temperature. The theoretical investigators have also occupied themselves very greatly with the condition of vapours at their maximum pressure, or of saturated vapours, as they are called. Interesting as are these researches, and important as their results may be in themselves, this limiting case has less significance for the theory of gases than for a corresponding theory of liquids.

For the state of equilibrium of a saturated vapour which lies above its liquid is characterised by this, that the equilibrium is maintained by vaporisation and condensation at the surface of the liquid. The molecules of the vapour which in their to-and-fro motion strike the liquid surface will not all bounce back, but a part will be retained by the force of cohesion. On the contrary, it will happen just as often that a particle of fluid which possesses sufficient speed tears itself loose from its neighbours, and passes into the vapour above. From this it follows that, in the state of equilibrium of a saturated vapour, the kinetic energy of the vapour is equal to the work which is done by the forces of cohesion during condensation to the liquid state. The measurement of the maximum pressure of the vapour has therefore this significance, that it gives a measure for the energy-value of cohesion in the liquid state, while for the determination of the properties and laws of the vapour

state the observation of unsaturated or superheated vapours is of greater importance.

We shall, therefore, here touch upon only one point of especial interest which has resulted from the observations of Mendelejeff' and Andrews.2 As the observations of Frankenheim and others on the capillary rise of liquids in narrow tubes at different temperatures have shown, the cohesion of a liquid decreases greatly as the temperature rises, whence we should conclude that heat probably produces molecular motions in liquids, just as in gases, by which their cohesion is diminished. If we increase the temperature and this molecular motion more and more by addition of heat, we may imagine a point reached at which the forces of cohesion cease to act and the capillary constant which measures them is zero.

On the other hand, the vaporisation which results from the rise of temperature and the consequent increase of molecular motion is prevented, or at least hindered by pressure. We may imagine the process to be this: that pressure and diminution of volume bring the molecules of the vapour nearer together, and cause their cohesive forces, which increase with diminishing distance between the molecules, to come more strongly into play, so as to overcome the expansive force due to the motion of the molecules. This is, however, possible only so long as the kinetic energy of the molecular motion is not too great; if by the addition of heat this energy should become so great as to exceed the sum total of the potential energy of the forces of cohesion which is lost by two molecules which, from being widely separated, come into contact with each other, it is no longer possible to cause the molecules to join together, and it is, therefore, also no longer possible for the vapour to be changed into liquid by pressure. The temperature necessary for this, which Mendelejeff calls the absolute boiling-point, may, according to our former remarks, be determined also from

1 Ann. Chem. Pharm. cxix. 1861, p. 1; Pogg. Ann. cxli. 1870, p. 618. 2 Report Brit. Ass. 1861, ii. p. 76; Ann. Chem. Pharm. cxxiii. 1861, p. 270; Phil. Trans. clix. pt. 2, 1869, p. 575; Pogg. Ann. Erg.-Bd. v. 1871, p. 64; Proc. Roy. Soc. xxiii. 1875, p. 514.

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