Optical Activity of Solutions. 823 



is the density of the solution we shall have (d — c) grams of 

 inactive solvent per unit volume so that 



pi = c, cr 1 = d — c, p 2 = G- 2 — • • • =0. 



Thus in this case 



2(0 . re i-s(d-c) 



n 2 " [1— a{rc + sd— c)] 2 ' 



Although there is no actual theoretical reason for the 

 assumption, yet a sufficiently good approximation is obtained 

 if we put 



d = d + d 1 c, 



d and d 1 being constants of the substances involved. 

 Thus we have 



2co , cfr + sdj — l)-\-sd 



^^ y> (> - . 



n 2 ' [1 — asd — ac(r-\-sd 1 ^ l)] 2 ' 



The quantity usually determined is the specific rotation 

 of the solution, i. e. the rotation produced by a solution 

 containing 1 gram of active substance per c.c. of solution, 



and this is always defined as — . If we denote it by [a>] we 

 have c 



r , co c(r + sd 1 — l) + sd 



[ft)J =- =r' 



c [1 — asd — ac(i*+ sd 1 — l)] 2 



a formula which expresses the specific rotativity of the 

 solution as a function of the concentration : the constants in 

 it are dependent essentially on the nature of the solvent and 

 active solute. This is the general formula, but if the terms 

 in c are small as is usually the case, various approximations 

 may be made by expanding it in different ways. Direct 

 expansions in powers of c would give a result 



[a>]=A + Bc + Cc 2 + ..., (i.) 



or expansion of numerator and denominator separately 

 would give 



(A' + B> + A'C _ B'c 



so that each of the empirical relations quoted may be re- 

 garded as approximations to the more general formula, which 

 we can write in the form 



H=7c^r (m ° 



3K2 



