8 The Kinetic Theory conception of Matter [CH. I 



Free Paths. Each collision is the termination of two free paths, hence 

 the number of free paths described in the gas just considered is about 

 4*8 x 10 29 per second. It has already been said that the total distance 

 described i.e., the aggregate of these free paths is 677 x 10 22 cms. Hence 

 on division we see that the mean length of these free paths is 1*4 x 10~ B cms. 



It is obvious that the mean free path, being a pure length, will depend 

 only on the diameter of the molecules, and on the number of molecules 

 per cubic centimetre ; it will not depend on the velocities of motion of 

 the molecules. Thus the values we have obtained for the mean free path 

 are approximately true for all gases so long as the molecules are supposed 

 uniformly to be spheres of radius 10~ 8 cms. The free path is, however, 

 inversely proportional to the number of molecules per cubic centimetre of 

 gas. For instance in a vacuum tube in which the pressure is that of half 

 a millimetre of mercury, the density of gas is only 1 : 1520 of the normal 

 ' density, and therefore the free path is roughly equal to a quarter of a milli- 

 metre. 



It appears from these figures that the mean free path of a molecule is 

 about 700 times its diameter in a gas at normal pressure, and is over 

 a million times its diameter when the pressure is reduced to half a milli- 

 metre of mercury. There is therefore every justification for assuming, as 

 a first approximation, that the linear dimensions of molecules are small in 

 comparison with their free paths. 



Comparing the values obtained for the free path with the values 

 previously given for the velocity of motion, we find that the mean time 

 of describing a free path ranges from about 3 x 10~ 10 seconds in the case 

 of air under normal conditions, to about 1*3 x 10~ 7 seconds in the case of 

 hydrogen at a pressure equal to that of half a millimetre of mercury. 



The principal lesson to be learned from the foregoing figures is that 

 the mechanism of the Kinetic Theory is extremely "fine-grained" when 

 measured by ordinary standards. Molecules are, in fact, not infinitely small, 

 and neither is their motion infinitely rapid, but the units of space and time 

 appropriate for the measurement of the motion of individual molecules are 

 so small in comparison with even the smallest quantities which we can 

 measure experimentally that the phenomena exhibited by a gas constituted 

 in the way described will be indistinguishable, so far as experiment and 

 human observation go, from those of a continuous medium. It is for this 

 reason that the hypothesis upon which the Kinetic Theory rests is, and 

 probably will always remain, an unproved hypothesis. 



