;74 



NA TURE 



[March ii, 1875 



ON THE DYNAMICAL EVIDENCE OF THE 



MOLECULAR CONSTITUTION OF BODIES* 



II. 



T ET us now return to the case of a highly rarefic 1 s^s in which 

 -L-' the pressure is due entirely to the motion of its particles. 

 It is easy to calculate the mean square of the velocity of the 

 particles from the equation of Clausius, since the volume the 

 pressure, and the mass are all measurable quantities. _ .Sup- 

 posinc the velocity of every particle the same, the velocity of a 

 molecule of oxygen would be 461 metres per second, of nitrogen 

 402, and of hydrogen 1S44, at the temperature o' C. 



The explanation of the pressure of a gas on the vessel which 

 contains it by the impact of its particles on the surface ot the 

 vessel has been suggested at various times by various writers. 

 The fact, however, that gases are not observed to disseminate 

 themselves through the atmosphere with velocities at all ap- 

 proachin<^ those just mentioned, remained unexplained, till 

 Clausius, by a thorough study of the motions of an im- 

 mense number of particles, developed the methods and ideas 

 of modern molecular science. 



To him we are indebted for the conception of the mean length 

 of the path of a molecule of a gas between its successive encounters 

 with other molecules. As soon as it was seen how each 

 molecule, after describing an exceedingly short path, en- 

 counters another, and then describes a new path in a quite 

 different direction, it became evident that the rate of diffusion 

 of gases depends not merely on the velocity of the molecules, 

 but on the distance they travel between each encounter. 



I shall have more to s.ay about the special contributions of 

 Clausius to molecular sc'ence. T!ie main fact, however, is, that 

 he opened up a new field of maihematical physics by showing 

 how to deal mathematically with moving systems of innumer- 

 able molecules. 



Clausius, in his earlier investigations at least, did not attempt 

 to determine whether the velocities of all the molecules of the 

 same gas are equal, or whether, if unequal, there is any law 

 according to which they are distributed. He therefore, as a first 

 hypothesis, seems to have assumed that the velocities are equal. 

 But it is easy to see that if encounters take place among a great 

 number of molecules, their velocit es, even if originally equal, 

 will become unequal, for, except under conditions which can be 

 only rarely satisfied, two molecules having equal velocities before 

 their encounter m ill acquire unequal velocities after the encounter. 

 By distributing the molecules into groups according to their 

 velocities, we may substitute for the impossible task of following 

 every individual molecule throuijh all its encounters, that of 

 registering the increase or decrease of the number of molecules 

 in the different groups. 



By following this method, which is the only one available 

 either experimentally or mathematicahy, we pass fro n the 

 methods of strict dynamics to those of st itistxs and probability. 

 When an encounter takes place between two molecules, they 

 are transferred from one pair of groups to another, but by the 

 time that a great many encounters have taken place, the number 

 which enttr each group is, on an average, neither more nor less 

 than the number which leave it during the same time. 'When the 

 system has reached this state, the numbers in each group muit 

 be distributed according to some definite law. 



As soon as 1 became acquainted with the investigations of 

 Clausius, I endeavoured to ascertain this law. 



The result which I published in i860 has since been subjected 

 to a more strict investigation by Dr. Ludwig Boltzmann, who 

 has also appUed his mettiod to the study of the motion ot com- 

 pound molecules. The mathematical investigation, though, like 

 all paits of the science of probabilities and statistics, it is some- 

 what difficult, does not appear faulty. On the physical side, 

 however, it leads to consequences, some of which, being mani- 

 festly true, seem to indicate that the hypotheses are well chosen, 

 while others seem to be so irreconcilable with known experimental 

 results, that we are compelled to admit that something essential 

 to the complete statement of the physical theory ol molecular 

 encounters must have hitherto escaped us. 



I must now attempt to give you some account of the present 

 state of these investigations, without, however, entering into their 

 mathematical demonstration. 



I must begin by stating the general law of the distribution of 

 velocity among molecules of the same kind. 



* A lecture delivered at the Chemical Society, Feb. iS, by Prof. Clerk- 

 Maxwell, F.R.S. (Coiitiuued from p. 359) 



If we take a fixed point in this diagram and draw from this 

 point a line representing in direction and magnitude the velocity 

 of a molecule, and make a dotal the end of the line, the position 

 of the dot will indicate the state of motion of the molecule. 



If we do the same for all the other molecules, the diagram 

 will be dotted all over, the dots being more numerous in certain 

 places than in others. 



The law of distribution of the dots may be shown to be the 

 same as that which prevails among errors of observation or of 

 adjustment. 



The dots in the diagram before you may be taken to repre- 

 sent the velocities of molecules, the different observations of the 

 position of the same star, or the bullet-holes round the bull's- 

 eye of a target, all of which are distriliuted in the same manner. 



The velocities of the molecides have values ranging from zero 

 to infinity, so that in speaking of the average velocity of the 

 molecules we must define what we mean. 



The most useful quantity for purposes of comparison and cal- 

 culation is called the "velocity of mean square." Ifisthat 



velocity whose square is the average of the squares of the velo- 

 cities of all the molecules. 



This is the velocity given above as calculated from the pro. 

 perties of different gases. A molecule moving with the velo- 

 city of mean square has a kinetic energy equal to the average 

 kinetic energy of all the molecules in the medium, and if a single 

 mass equal to that of the whole quantity of gas were moving 

 with this velocity, it would have the same kinetic energy as the 

 gas actually has, only it would be in a visible form and directly 

 available for doing work. 



If in the same vessel there are different kinds of molecules, 

 some of greater mass than others, it appears from this investi- 

 gation that their velocities will be so distributed that the average 

 kinetic energy of a molecule will be the same, whether its mass 

 be great or small. 



Here we have perhaps the most important application which 

 has yet been made of dynamical methods to chemical science. 

 For, suppose that we have two gases in the same vessel. The 

 ultimate distribution of agitation among the molecules is such 

 that the average kinetic energy of an individual molecule is the 

 same in either gas. This ultimate state is also, as we know, a 

 state of equal temperature, llence the condition that two gases 



