It will be noticed, then, that the two causes tending to bring about deviations from Boyle’s law act in contrary directions. In the greater number of gases the effect due to cohesion at ordinary pressure is greater than the effect due to the actual space occupied by the molecules. In the case of hydrogen at ordinary temperature the contrary is the case; if, however, hydrogen is strongly cooled, it shows variations similar to those exhibited by other gases at ordinary temperatures. By heating these gases the effect due to cohesion—to the mutual attraction of the molecules—becomes less and less; in such circumstances these gases show departures from theory in the same sense that hydrogen does at ordinary temperatures.
The effect of mutual attraction among the molecules is to make the volume of the gas less than the theoretical value; the cohesive force may therefore be regarded as equal in effect to a certain additional pressure; that is, (P + A) V = constant, in which A is the measure of the force of cohesion. A, of course, must have relation to the number of molecules mutually attracted: A is proportional to the square of the number of the molecules. But the number of the molecules in the unit volume is proportional to the density of the gas, and in a given mass of gas the density is inversely proportional to the volume. Hence A is inversely proportional to the square of the volume—A = a/V2, hence (P + a/V2) V = constant. Now let us trace the effect of the second cause of variation from the mathematical exactitude of Boyle’s law. The fact that the molecules are not mathematical points means that V in the foregoing expression is not identical with the space in which the molecules move. That space is V - b, in which b is the measure of the aggregate volume of the molecules. Hence the true expression becomes (P + a/V2) (V - b) = constant.
The law of Dalton (Charles) also receives its simplest explanation by the kinetic theory of gases; and, moreover, the departures from the mathematical truth of the statement follow as a necessary consequence of the facts that the molecules have sensible magnitudes and are mutually attracted. We can measure the effect of heat upon a gas in two ways. We can either keep the pressure of the gas constant, and measure the increase in volume; or we can prevent the gas from expanding, and measure the elastic force or pressure it exerts. If the law of Dalton were mathematically true, it would follow that, if the volume of the gas were maintained constant during the heating, its pressure would increase in the same proportion as the volume would have increased if the gas had been allowed to expand, but maintained at a constant pressure. In other words, the expansion-coefficient and the pressure-coefficient should be the same. Experiment shows, however, that they are not identical.
The following table gives the results of a number of measurements by Regnault:
| Expansion (Pressure Constant). | Pressure (Volume Constant). | |
|---|---|---|
| Hydrogen | .003661 | .003667 |
| Air | .003670 | .003665 |
| Carbon dioxide | .003710 | .003688 |
| Sulphur dioxide | .003903 | .003845 |
Variations in the same sense have since been observed by Jolly and Chappuis. With the exception of hydrogen, and probably also helium, all the gases show greater values for the coefficient of expansion than for the coefficient of pressure, and the differences are greater the greater the coefficient of expansion of the gas.
Since, as we have already stated, the law of Boyle is directly related to the law of Dalton, both being dependent on molecular movements, the same course of reasoning used to account for the variations in the case of Boyle’s law applies equally to the case of Dalton’s law. The “law” of Avogadro follows also as a necessary consequence of this explanation of the laws of gaseous pressure and temperature. If all gases show approximately the same increase in pressure when heated under constant volume, and if the increased pressure is due only to the increased energy with which the molecules strike the sides of the containing vessel, it follows that all gases must contain the same number of molecules in unit volume. But as, from the very nature of the case, the laws of Boyle and Dalton cannot be mathematically true, it follows that the laws of Avogadro and Gay Lussac must be only approximations in the same sense.
The law of Graham, connecting the rate of diffusion of a gas with its density, follows also as a necessary consequence of this explanation of the laws of Boyle, Dalton, Gay Lussac, and Avogadro. If the number of molecules in the unit volume of any gas, whatever be its nature and whatever be their mass, is approximately the same, it follows that the mean velocity of the molecules must be variable; their mean velocities must be in the inverse ratio of the square roots of their densities.
The mean velocity with which the molecules of a gas move can be calculated if we know the pressure it exerts, the weight of a definite volume, and the value of the acceleration due to gravity. The square of this velocity in metres per second of time at 0°C. is given by the expression U2 = 3pg/q in which p = pressure per square metre = 10,333 kilograms; g = the gravitation constant = 9.81; q = weight of a cubic metre of the gas at 0°C. and one atmosphere of pressure.
For hydrogen we have