162 PHENOMENA DEPENDENT ON MOLECULAR PATHS 68 



Maxwell l has also calculated the mean value of the 

 free path on the assumption of this law. The calculation 

 cannot be here given ; a deduction of the formula will be 

 found 2 in 29*. 



The result of the calculation agrees almost exactly with 

 that just rmentioned which Clausius obtained on the 

 assumption of equal speeds in the molecules. In this case, 

 too, the formula demonstrated in 65 undergoes no further 

 alteration than the addition of a numerical factor, and there 

 results for the mean free path 



L = X 3 /7rsV2. 



The factor, the value of which will be more closely indicated 

 in 96, is nearly the same as that in Clausius' formula; 

 for the latter is f or O75, and the former l/\/2 or 0-707, so 

 that they are approximately in the ratio of 17 to 16 [or, still 

 more nearly, of 35 to 33]. 



The value of the free path that follows from Maxwell's 

 law is somewhat the smaller; there also results from this 

 law a smaller value of the mean speed than that given by 

 Clausius' theory; both results are explained on the simple 

 ground that a shorter path and a slower speed occur more 

 frequently than a longer path and a higher speed. 



69. Molecular Path. -volume 



The name of molecular path-volume has been given by 

 Loschmidt 3 to the content of the cylindrical space which 

 a molecule describes when it traverses its mean free path. 

 The magnitude of this volume is %TTS?L, since the radius of 

 the sphere of action is equal to the distance apart of the 

 middle points of two molecules during collision, and is, 

 therefore, equal to the diameter of a molecule in the case of 

 actual contact during collision ; hence by the foregoing for- 

 mula it is equal to X 3 /4\/2. If we replace in this expression 

 the size of the elemental cube, or of the space that contains 

 a single molecule only, by the number N of molecules 



1 Phil. Mag. 1860 [4] xix. p. 28 ; Scientific Papers, 1895, i. p. 387. 



2 Compare 97. 



3 Wiener Sitzungsber. 1865, Hi. Abth. 2, p. 397. 



