Chapter III —19— Solutions 



The ideal equation for one mol of gas is usually given in the form 



PgV = RT (2) 



where Pg = the pressure of the gas against the walls of the container, 

 V = the volume of the gas, 



R = a proportionality constant commonly termed the molar gas constant, and 

 T = absolute temperature. 



Though useful for gases at high temperature and low pressure this law, 

 because it neglects attractive forces between molecules and provides no cor- 

 rection for the space occupied by the gas molecules, is invaUd for gases at 

 low temperatures and high pressures. Thermodynamically it is based on the 

 assumption that gas molecules are elastic spheres having no attractive 

 forces and occupying no space. 



To correct the weaknesses of this law van der Waals derived the 

 equation 



(Pg + ^) (V-b) = RT (i) 



in which 



-^is a measure of the attractive forces of the gas molecules or the co- 

 hesive forces as applied to liquids. V-b is the free space unoccupied by 

 molecules. The van der Waals constants a and b have been widely used 

 in dealing with the properties of non-ideal gases and liquids. 



Many other refinements to the gas laws have been offered. Emphasis 

 here is on the fact that just as the ideal gas law finds an analogy in van't 

 Hoff's law of osmotic pressure, the van der Waals equation, and subsequent 

 improved equations have been applied to the relations of solutions. 



Meanwhile Raoult had proposed a law relating the vapor pressure of 

 a solution to the mol fraction of the solvent or solute in it. 



p = p°Xl = p° (1— X2) (4) 



where p = the vapor pressure of the solution, 



p° = the vapor pressure of the pure solvent, 

 XI = the mol fraction of the solvent, and 

 X2 = the mol fraction of the solute. 



Lewis (1908) recognized the validity of Raoult's law as applied to 

 both dilute and concentrated solutions and Hildebrand (1924) has pointed 

 out its wide applicability to studies on solutions. Ideal solutions may be 

 defined as those that obey Raoult's law at all temperatures and pressures. 

 Only liquids having equal changes in pressure with temperature at constant 

 volume will obey Raoult's law. Such liquids mix without heat of dilution 

 or change in volume. 



From the above considerations it is possible to formulate a thermo- 

 dynamic equation of state to cover all liquids. 



(w).-(w)v- 



where P = pressure on the liquid, 



E = total energy of the liquid, 

 V = volume of the liquid, and 

 T = absolute temperature. 



Hildebrand (1936) gives the following empirical expression for the 

 members of this equation : 



