he gained during the Mixture of Gases. 423 



from state (1) to state (2) work would have on the whole to 

 be spent. Let then m x be the mass of a molecule of oxygen, 

 m 2 that of hydrogen. And let u 2 be the mean square of the 

 velocity of a molecule for oxygen, a 2 2 for hydrogen. Then if 

 in an element of the tube of volume v, there be n x molecules of 

 oxygen, n 2 of hydrogen, the total pressure, which is uniform 



p ii i • n in>iu 1 2 + n 2 m 2 u 2 2 -n , • T i 



tor all elements, is ~ . -but since, on Lord 



6v 



Payleigh's authority, m 1 u l 2 =m 2 u 2 2 , the total pressure is 

 independent of the ratio n x jn^ so long as r*i + ?2 2 is unchanged. 



From this I think it follows that we might, without doing 

 any work, cause the hydrogen and oxygen "molecules to 

 exchange their positions by pairs in such way as to bring the 

 mixture from state (1) to state (2). For the pressure being 

 uniform and the tube horizontal, no forces act on the mole- 

 cules, and therefore no work is done upon them, by reason of 

 the exchanges. And the pressures are not altered by the 

 exchanges. The process may be effected as slowly as we 

 please. This may be an operation which with our clumsy 

 fingers we cannot perforin, but it is not an operation which 

 essentially requires work to be spent. I then put forward the 

 statement that in a complete cycle of operations at the end of 

 which, as at the beginning, the tube containing the gases is 

 horizontal and in the same position, and the total pressure p 

 uniform,, but at the beginning the gases are uniformly mixed, 

 at the end partially separated, no work is done. 



It is necessary now to consider whether Lord Ravleioh has 

 proved anything which is inconsistent with this statement. I 

 think he has not. 



At p. 314 he supposes the two gases uniformly mixed in a 

 reservoir, the partial pressures being P x for oxygen, P 2 for 

 hydrogen. At the top of this reservoir is a vertical tube, in 

 which the same two gases are in equilibrium in vertical 

 column, the pressure of the combined gases at the base being- 

 Pi +P 2 , as in the reservoir. The partial pressures diminish 

 as we ascend the tube, for oxygen more rapidly than for 

 hydrogen. There is thus a partial separation of the gases, in 

 addition to condensation of both gases towards the base. He 

 then takes an element at the top, and compresses the gases in 

 it till they have the combined pressure of the base, Px + Ps. 

 Then he brings the compressed element to the level of the 

 base. It is proved that the work spent in the compression 

 exceeds that gained from gravity in the descent of the element 

 by an amount which we may call tAV. When the same 

 operation has been performed on every element of the tube, 



