set up In Molecules hy Collisions. 281 



No^v our linal problem Mill be to find this quantity averaged 

 over all the molecules and collisions in the gas. It is at once 

 obvious that the last term vanishes on the average. Hence 

 a correct result is obtained bv ignoring the free vibration (5), 

 and supposing that the collision simply sets up a vibration of 

 amplitude \/XM^^ and therefore of energy iccp-(JJ -[-Y''). 



§ 5. In evaluating X and Y from equation (8), we shall 

 suppose V divided into two parts, and shall consider at pre- 

 sent onlv the part which is contributed bv the motion of the 

 centre of gravity of molecule B. A second part in Y con- 

 tributed bv the rotation and small vibrations of the molecule 

 B will be discussed later (§ 10). 



If we integrate equation (8) by parts we get, since Y 

 vanishes at both limits, 



X -f /Y= - — , I ' ~ e^P'clL ... (10) 

 a^p-y^ dt 



Xot onlv Y but all its differential coefficients must vanish 

 at both limits^ so that we may repeat the integration by parts 

 indefinitely ; after n integrations we obtain 



x+iY=^^i^ r'^%'>^c/^ . . . (11) 



The value of d'^Y/dt'^ will be comparable with unity pro- 

 vided that the unit of time selected is comparable with the 

 scale of time-variation of Y, and that n is not very great. 

 Hence, if i? is great when measured in these units, we see, 

 from the presence of the factor ^~'''+-^^ in (11). that X and Y 

 will be very small. 



Xow for normal air the probable relative velocity of two 

 molecules is of the order of 10'^ cm. per sec. From exjieri- 

 ments on the viscosity &c. of gases, it is found that the dis- 

 tance apart of the centres of two molecules during an encounter 

 — the distance which is usually described as the '* diameter " 

 of a molecide — is about 10~* cm. The appropriate unit of 

 time for that part of Y which we are now discussing is 

 therefore 10~^^ sec. For p the value varies^ so far as we 

 know, from about 2x10-^^ per second in the case of uitra- 

 red light to 8 x U/'^ per second in the case of ultra-violet 

 light. This gives for the value of p in the present units a 

 range from 200 to 800. Since there is practically no limit 

 to the value of n in (11), the factor ^-(''+i) may be made 

 very small. From this we conclude that practically no vibra- 

 tions are set up in molecule A bv the translational motion of 



