62 MOLECULAR MOTION AND ITS ENERGY 30 



3O< Gay-Lussac's Law of Gaseous Densities 



As simple consequences of the theorems of molecular 

 mechanics that have been given, we deduce certain laws of 

 theoretical chemistry, of which the law of gaseous densities 

 obtained empirically by Gay-Lussac 1 should first be men- 

 tioned: 



Two gases are at the same temperature when the mean 

 kinetic energies of a molecule of each, 



E l = iwA 2 and E 2 = \rnfi?, 



are equal to each other. By 16 they exert equal pres- 

 sures when their kinetic energies per unit volume, 

 K l = _ Pl G } 2 and K, = ^ 2 2 2 , 



are equal to each other. Hence it follows that, if two gases 

 e not only at the same temperature, but also under the 

 same pressure, their densities must be in the ratio of their 

 molecular masses, or 



Pi '- p2 = m i ' m z- 



This proposition deduced from our theory agrees sub- 

 stantially with Gay-Lussac's law, that the quantities of 

 two gases which can chemically combine with each other 

 occupy volumes which, when the measurements are made 

 at the same temperature and under the same pressure, .are 

 either equal or in the ratio of simple integers. More 

 simply expressed, this theorem runs the densities of two 

 gases are in a simple ratio, expressed by integers, to their 

 stoichiometric quantities. If we denote the latter by Q lt Q 2 , 

 and put n { , n% to represent integers, Gay-Lussac's law 

 gives 



Pi ' p2 = n iQi : ra 2 Q 2 . 



This empirical law agrees exactly with that deduced 

 from theory if the molecular masses m of the gases are to 

 their stoichiometric quantities Q in ratios given by simple 

 integers, or 



m, : 7^ = 7^ : n 2 Q. 2 ; 



a condition which is obviously fulfilled if Dal ton's atomic 



1 M6m. de la Soc. d'Arcueil, ii. 1809, p. 207 ; Gilb. Ann. xxxvi. 1810, p. 6. 



