424: The Work gained during the Mixture of Gases. 



the quondam vertical column now occupies a horizontal 

 position, in which the combined pressure of the gases is 

 P x + P 2 at every point, but oxygen is relatively denser at the 

 end of the tube which was the base of the column. There 

 has been effected a partial separation of the gases. And 

 work W has been spent on the whole. I understand Lord 

 Eayleigh to mean that this work W has been spent in the 

 separation of the gases. It is understood that, as he suggests, 

 we may suppose movable pistons interpolated to prevent 

 diffusion for as long as we please. 



This operation o£ Lord Iiayleigh's is the second half of a 

 complete cycle of which I will now supply the first half. The 

 gases to begin with are uniformly mixed in a horizontal tube 

 AB, and their combined pressure is F l + P 2 at every point. 

 Divide the tube into N elements each containing the same 

 number of molecules and numbered consecutively 1 . . . N 

 from A. Set up a model vertical column A'B' containing the 

 same number of molecules of each gas as AB, but in which the 

 gases have attained the state of equilibrium, with P 1 4-P 2 the 

 total pressure at the base. Divide the column into N elements 

 each containing the same total number of molecules and 

 numbered consecutively 1 . . . N from the base A'. To every 

 element of the tube now corresponds an element of the column, 

 and each contains the same number of molecules — but the 

 proportions in which the gases are mixed are not the same in 

 the one element as the other, nor are the volumes the same. 

 Then, Process (1), interchange molecules of oxygen and 

 hydrogen in the horizontal tube until each element contains 

 the same number of molecules of either gas as the corre- 

 sponding element of the column. In that process the pressure 

 is not altered at any point, and (as I say) no work is done. 

 Process (2), let each element of the tube expand until it has 

 the same volume and the same pressure as the corresponding 

 element of the column. The horizontal tube is now a fac- 

 simile of the vertical column, and may be set vertical without 

 further disarrangement of the gases. It is in fact, when so 

 set up, exactly Lord Rayleigh's vertical column, p. 314, and 

 shall be dealt with in the way that he describes. My processes 

 (1) and (2) constitute together the first operation of a 

 cycle of which Lord Rayleigh performs the second. 

 Evidently, since both my expansion and his compression 

 take place after Process (I), the work gained in expansion is 

 exactly equal to that spent in compression. Also since the 

 centre of gravity of the molecules in the tube is at the end 

 on the same level as at the beginning, no work on the whole 

 has been done by or against gravity. Therefore, on the 

 whole no work has been gained or spent unless it is spent in 



