Grafts et al. 



34 



Water in Plants 



Bancroft and Davis (1928) and Lewis (1908) have derived similar 

 equations. 



By Raoult's law if — = xi = 1 — xo, where xi = mol fraction of 



Po 



the solvent and X2 the mol fraction of the solute, the osmotic pressure 

 equation becomes 



P„V = — RTlnd— xo) (3) 



for an osmotic system in which the vapor of the solvent obeys the gas law. 

 This gives the osmotic pressure law in terms of the mol fraction of solvent 

 in the solution. If one expands the term [ — ln(l — X2)] he obtains 



P„V = RT (X2 + y2 X.J + Vi xo^ . . .) (4) 



For dilute solutions the higher powers can be neglected and the equation 

 becomes 



P„V = RTx2 (5) 



Likewise, for very dilute solutions if n^ = number of mols of solvent in 

 the solution and n2 = number of mols of solute, then 



no — 



xo = — - and P„ V 



RT 



no 



(6) 



Again for dilute solutions m V may be replaced by Vj, the volume of 

 solvent associated with n2 mols of solute in the solution, and 



P„ Vi = na RT (7) 



This is the equation used by Morse (1914). It gives the relation of 

 osmotic pressure to concentration of the soluti'on in weight molality and is 

 accurate to a concentration of about 1 molal. 



By one more assumption, namely that volume and weight molar solu- 

 tions of a solute do not differ (valid only for very dilute solutions) one 

 can substitute V, the volume of the solution, for Vj, the volume of sol- 

 vent, and obtain PV := nRT where n := the number of mols of solute in 

 the solution. This is the van't Hoff law ; as he emphasized, it applies only 

 to very dilute solutions. Its principal value is to show the analogy between 

 the gas law and the osmotic pressure law. 



To indicate the type of results to be obtained by use of these equations, 

 Table 8 reports values obtained by Frazer and Myrick (1916) and pres- 

 sures calculated by several formulae. 



Table 8. — Observed and calculated osmotic pressure values for sucrose solutions at 

 30° C. (R = .08206, V = 18.052. Vi = 1001.9, T = 303°) : — 



* These values were not corrected for the change in V with concentration. 



From these results it is shown that van't Hoff's equation falls far short 

 of providing agreement with the observed values; Raoult's law is some- 



