140 HEAT. 



apply to molecular systems ; that, in fact, the conditions assumed by 

 Boltzmann in the proof of his theorem are not realised in actual 

 molecular systems. 



Joule's Approximate Method of Calculating the Velocity of 



Mean Square. The first calculation of the velocity of the molecules was 

 made by Joule by a method which is obviously only approximately correct, 

 but which is valuable in that it enables us to obtain easily results of the 

 right order and which gives us at any rate insight into the principles of the 

 theory. Considering, say, a cubic centimetre, let us think of the molecules 

 as divided into six streams moving perpendicularly to the six faces of the 

 cube, one stream towards each face, and let us omit all consideration of the 

 collisions between molecules. Then the stream moving towards one face 



at any instant has mass ^ where p is the density of the gas. Let the 



velocity of the stream be Y. Then the total mass moving up to the face in 



pV 



one second is that which would be contained in V c.c., or is . But its 



o 



velocity is changed by impact against the face from + V to - V, so that the 



pV pV 2 



momentum imparted to the face in one second is ~ x 2V = u_, Equating 



b 3 



this to the pressure, we get V 2 = . 



P 

 Effusion or Transpiration through a small Orifice into a 



Vacuum. The phenomena of " transpiration " through a small orifice 

 may be generally explained by the aid of Joule's method, though, of 

 course, the method cannot be expected to give a complete account. If a 



gas of density p is contained in a vessel with a small orifice, area S, we 



P Y 



may think of mass -t- as moving up per second towards the face con- 

 6 



VS 

 taining S, and the mass escaping through S will be p-~- per second. 



The mass escaping when the distribution of velocities according to 

 Maxwell's Law is taken into account can be shown to be 



US 12 P VS 3 

 '-4 131TT-1 



where U is the mean velocity and V that of mean square, so that the 

 approximation in Joule's method gives the numerical coefficient too 

 small in the ratio 13 : 18, and this example may serve to show the kind 

 of error introduced by that approximation. 



If we have two vessels containing different gases at equal pressures 

 escaping through equal orifices, the masses escaping per second will be 

 in the ratio p l V l : P 2 V 2 , and the volumes escaping in the ratio V l : V ; 

 or since V x 2 : V 2 2 = p 2 : p v when the pressures are the same, the volumes 

 are as 



\/P 2 : \/Pr 

 Hence the times of efflux of equal volumes are as 



