of Mixed Solutions, 285 



the determinant to the left is symmetrical, and the number of 

 thermodynamic quantities reduces to \r{r + l). For a binary 

 solution we get the very simple result 



dfodcx _, .dU 



9ci dx dx 



In the earlier paper* the quantity q^, is found for binary solutions 



/fo 



if, 



= ^o = ^,|p + c.(H-c,)3^}, 



dfa dc^ _1 ^ ^^ ^ ^^dp dU 



P 

 which is the same result as that found in the previous paper. 



and |:.-. = ic,(l+o,)|^^, 



dCi dx p oCi ax 



Part IL — Extension of the conception osmotic pressure 

 to solutions in general. 



I 6. In the previous development we have seen that the 

 thermodynamic potentials are very apt for a general mathematical 

 treatment. The thermodynamic quantities appearing in the 

 equations are mathematically very simple, all of them being 

 derived from a single function by means of a simple mathematical 

 operation. In spite of their mathematical simplicity, the thermo- 

 dynamic potentials have their disadvantages when we are trying 

 to get a clear idea of what they contain physically, and their 

 interpretation is in fact the important thing when we are going to 

 utilise them in any special case. Here it is that the importance 

 of the osmotic pressure comes in ; and even if we consider the 

 thei'modynamic potential to be the basis for the theory of solu- 

 tions, the generalisation of the conception of osmotic pressure will 

 still be of importance for a general treatment of solutions. The 

 equations connecting the osmotic pressure with the thermo- 

 dynamic potentials are from one point of view to be regarded as 

 equations defining the osmotic pressure, but will at the same time 

 give a physical interpretation to the thermodynamic functions. 



The conception of osmotic pressure may be generalised in a 

 number of ways ; it will be our task to find the most usefid. Let 

 the solution, as before, consist of {r + 1) substances, and suppose 

 that we have a membrane permeable to {i) of them but imper- 

 meable to the rest. The solution may be supposed to be con- 

 tained in a cylinder and the membrane to have the form of a 

 piston. We imagine that there is at first no fluid above the piston. 

 If we apply a pressure sufficiently high, some substance of the (^) 



* Phil. Mag. [6] 13, p. 599. 



