IN THE PRESENCE OF DUST-FREE AIR AND OTHER GASES. 
29.5 
Supersaturation Resulting from a Given Expansion. 
By the supersaturation is here meant the ratio of the actual density of the vapour 
when the expansion has just been completed, and the minimum temperature has 
therefore been reached to the density of vapour which is in equilibrium over a flat 
surface of water at that temperature. 
It is assumed in what follows that the expansion is completed before any 
appreciable amount of water has had time to condense on the walls, or in drops 
throughout the moist gas. 
To find the lowest temperature reached we have the well-known equation for the 
cooling of a gas by adiabatic expression, 
e,~\vj ’ 
where 6^, 9^, are the initial and final absolute temperatures, and y is the ratio of the 
specific heat at constant pressure to that at constant volume. This has been assumed 
below to be the same as in the dry gas, the effect on y of the small quantity of vapour 
present being neglected. The error introduced in this way, as pointed out by 
R. V. Helmoltz,"^ is inappreciable at temperatures below 30° C. 
Knowing 6^ and vjv^, the ratio measured in these experiments, we can, therefore, 
calculate 02 - 
Let TT^, 713 be the pressure of saturated vapour over a flat surface of water at the 
temperature 0 ^, 02 respectively, rr^ is then the initial pressure of the vapour before 
expansion. The volume of the vapour is suddenly changed from to % We cannot, 
however, calculate the resulting change in its pressure, there being no reason to 
suppose that Bovle’s law is even approximately obeyed by the highly supersaturated 
vapour. There is no such uncertainty, however, as to the density of the vapour, 
which must change inversely as the volume. It is for this reason that the super- 
saturation is here defined as the ratio of the actual to the equilibrium density over a 
flat surface, and not in terms of the corresponding pressure. 
The supersaturation, according to the above definition, is equal to 
O _ ' / 
^ = P jp-i^ 
where p is the final density of the vapour before condensation takes place, and p^ is 
the density of the saturated vapour at the temperature 6^. 
But — = — ! therefore, S = — X • 
Pi '^2 P2 ^’2 
Now the actual density of saturated water vapour in the presence of air at ordinary 
atmospheric temperatures, has been shown by ShawI' to agree very closely with the 
* ‘ Wied. Ann.,’ xxvii., p. 508 (1886). 
t ‘ Phil. Trans.,’ 1888, A, p. 83. 
