_Expansion _Pressure (Pressure (Volume Constant)._ 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 = 3_pg_/_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 U² = 3 × 10,333 × 9.81/0.0899 whence U = 1842 metres per second; for oxygen we have U² = 3 × 10,333 × 9.81/1.430, whence U = 461. These numbers accord with those demanded by Graham’s law. The density of H being taken as 1, that of oxygen is 16 and √16 = 4; the numbers 1842 and 461 are in the ratio of 4 to 1.