407 



APPENDIX III 

 MOLECULAR FREE PATHS 



26*. Probability of a Molecular Free Path of 

 Given Magnitude 



THE first calculation of the mean length of the free path of a 

 molecule was given by Clausius, 1 in that memoir which we 

 may rightly consider as the foundation-stone of the theory of 

 gases which rests on the calculus of probabilities. In Chapter VI. 

 I have made the same calculation by another method, as I thought 

 it better to avoid the use of transcendental functions and the 

 methods of the higher analysis. I will here, however, complete 

 that elementary demonstration by treating the problem in 

 Clausius' fashion. 



No further assumption respecting the state of motion of the 

 gas and its molecules shall be made than this, that at all places 

 within the gaseous medium the motion goes on in the same way. 

 We may therefore suppose not only that the heat-motion at each 

 point has the same energy, but that at each point it takes place in 

 all directions without distinction, so that every direction of motion 

 has the same probability. Together with this heat-motion we 

 assume a translatory motion of the gas as a whole, but with the 

 limitation that this motion must be regarded as constant within 

 limits of space and time which we shall more closely determine. 

 We may, for example, figure to ourselves the gas as flowing 

 through a pipe. 



Within this gas let us consider an arbitrarily chosen molecule, 

 which moves with a given velocity in a given direction. We wish 

 to find the probability that this particle will traverse a path of 

 length x without a collision. 



1 Pogg. Ann. cv. 1858, p. 239 ; transl. Phil Mag. [4] xvii. 1859, p. 81 ; 

 Abhandl. ilber die mech. Warmetheorie, 2. Abth. 1867, p. 260; 2nd ed. iii. 

 1889-91, p. 46. 



