towards a Dynamical Theory oj Solutions. 27 



and from ^(0-92-^) =0-104 to 0-2116 



= 0-0655 + 0-5445^(0-92 - Pl ) + 0*0040 



— l-5{^ 1 (0-92-p 1 )~0-155} 2 . . (11) 



With these equations the density of mixtures of alcohol 

 and water up to ^> 1 = , 46 is given with a maximum error of 

 the order 2 in 10,000. 



If the process we are studying is really the conversion of 

 the trihydrol of w T ater into dihydrol by alcohol up to 

 ^ = 0-46, then in interpreting ( A + p A 2 /2)/p 1 p 2 p as — {x l lp 2 

 + x 2 /p 1 )/2 we shall have to give a somewhat complicated and 

 arbitrary meaning to x 2 , because, as the trihydrol is com- 

 pletely changed into dihydrol, the constituents of water 

 when they come together in a mixture containing 46 per 

 cent, of alcohol have never time to form water again, the 

 effects of alcoho! continue even when contact with alcohol 

 has ceased. Either we must return to our original equation 

 and treat p 2 as a variable, or we must regard x 2 as including 

 changes caused remotely as well as immediately by contact 

 with alcohol. It is simpler to interpret (9) in the following- 

 direct manner. Let water consist of a trihydrol and 1 — a 

 dihydrol, and let co denote the contraction when a gram of 

 trihydrol changes into dihydrol. Let p 1 of alcohol convert 

 the fraction / of the trihydrol ap 2 in p 2 o£ w T ater, then the 

 whole contraction on mixing is # /ap 2 a) = A, which from (9), 

 neglecting the small term pA 2 /2, is of the form 



PlP2P {b+c Pl (p-92-p 1 -)}. 



For the moment treat 0*92 as 1, then this becomes 



p x p 2 p{b + cp lP2 ), 



that is to say, jap 2 the mass of trihydrol changed into 

 dihydrol by p 1 alcohol consists of two parts, one proportional 

 to p±p 2 due to the direct action of pi alcohol on ap 2 tri- 

 hyclrol, and the other proportional to p ip 2 2 and so due to a 

 conjunction of two molecules of alcohol near two molecules 

 of (H 2 0) 3 in such a position as facilitates the change to 

 3(H 2 0) 2 . We may take the change from 1 to 0*92 to be 

 due to a small term caused by the same conditions as make 

 the potential energy a minimum when alcohol and water are 

 in the proportion 'C 2 H 5 OH to 3H 2 0, that is 0"46 to 0*54 

 rather than 0'5 to 0'5. 



