in the Analogy between Solutions and Gases. 89 



(supposing unit volume to contain unit mass), as regards 

 gas and solution be P and p (which will afterwards be shown 

 to be equal), then, from Boyle's law, the pressure in gas 



P p 



and solution will be — and £ respectively. 



Now let the pistons (1) and (2) be raised so as to liberate" 

 unit weight of gas from the solution, and increase the gaseous 

 volume v by dv, in order to equalize its concentration with 

 that of the gas in the left-hand vessel, and by depressing the 

 pistons (4) and (5) let us redissolve the freshly liberated gas, 

 and then reduce the volume of the solution V + dY by the 

 amount dY in the cylinder with semipermeable walls ; then 

 the cycle is complete. 



AYork has been done in six separate ways ; let us number 

 them (1), (2), (3), (4), (5), and (6). Now (2) and (4) are 

 equal in amount, but opposite in sign, since they refer to 

 change of volume v and v + dv in opposite directions under 

 pressures which are inversely proportional to the volumes. In 

 similar manner the sum of (1) and (5) is zero ; so that the 

 point requiring proof is that (3) + (6) = 0. Here (3) repre- 

 sents work done by the gas in increasing its volume by dv, 



P P 



under a pressure of — , therefore (3)= — dv ; while (6) repre- 

 sents the work done by the solution, while it decreases in 

 volume by dV, under a pressure of ^, therefore (6)= — 4fdY, 

 The statement is therefore 



*dv=£dV; 



v V 



and as v :Y = dv : dY, P must be equal to p, which was to be 

 proved. 



This conclusion, which will receive in the sequel ample 

 confirmation, lends, on the one hand, support to Gay-Lussac's 

 law in its application to liquids : — If gaseous pressure and 

 osmotic pressure are at the same temperature equal to one 

 another, then equal changes of temperature must affect both 

 equally. On the other hand, this relation allows of a con- 

 siderable extension of Avogadro's law, always provided that 

 we may substitute the conception of osmotic pressure for 

 gaseous pressure : — under equal osmotic pressures and at the 

 same temperature, equal volumes of all solutions contain equal 

 numbers of molecules ; and, moreover, the same number of 

 molecules which would be contained in an equal volume of a gas 

 under the same conditions of temperature and pressure. 



