I40 ATOMS, IONS, SALTS, AND SURFACES 



Madelung and Born find that if Coulomb's law is assumed to hold for the attractive 

 forces, it is necessary, in order to accord with the known values of the compressibility 

 of solid salts, to assume that there is in addition a repulsive force between the ions 

 which varies as about the inverse tenth power: 



ylO 



According to this type of theory, solid salts consist entirely of ions, and in this 

 sense the salt is already completely ionized. 



THE IONIZATION OF SALTS IN AQUEOUS SOLUTIONS 



The dielectric constant {D) of ordinary solid salts is about 5, while that of water at 

 ordinary temperatures is about 80. Since the attraction between charged particles 

 (equation |i]) varies inversely as the dielectric constant, the attractive forces between 

 ions of op])osite sign of charge should be very much less in water than in solid salt. If, 

 therefore, it is assumed that the solid salt is completely ionized, it would be unreason- 

 able to assume that the same salts are less than completely ionized when dissolved in 

 water. 



The idea that salts in aqueous solution are completely ionized was suggested by 

 Sutherland in 1907. The theory has been put in more definite form by Bjerrum, by 

 Milner, and by Debye and Hiickel. Their fundamental idea is that the electrical at- 

 tractions between ions of unlike sign, and repulsion between those of like sign, give 

 rise to the following effect: on the average any positive ion will be immediately sur- 

 rounded by more negative than positive ions, while any negative ion will be surrounded 

 by more positive than negative ions. The most important result is that when the solu- 

 tion is diluted the separation of the ions involves the expenditure of energy.^ 



' Debye and Hiickel have calculated this electrical internal energy by assuming that the charges 

 on each ion are concentrated at a point, and that the distribution of these points is determined by 

 probability. They use the probability relation of Boltzmann, developed in connection with the kinetic 

 theory of gases, together with the equation of Poisson derived from the laws of electrostatics, includ- 

 ing Coulomb's law. 



Suppose that we have an ion of valence + n and charge + nE. In any concentric shell of thick- 

 ness dr, the potential is P and the density of the electric charge is p. 



The average kinetic energy of the molecules at a temperature Tis 3/2 kT. The Boltzmann constant 



k is equal to the gas constant 7? divided by the number of molecules in a gram molecule {6.o6Xio^'5). 



According to Boltzmann's principle, if the molecules are distributed in a field of force, such as in 



an electrical field, the distribution will be such that the number of molecules, instead of being equal 



-E 



to N, the number present in the absence of the field of force, will be equal to Ne *^ . Thus the number 



present is modified by a factor in which the base of Naperian logarithms is raised by a power equal 



to the potential energy of the molecule divided by two-thirds of its mean kinetic energ>'. 



The equation of Poisson applies to the variation of the potential P around a point when it is 



distributed with spherical symmetry. It may be expressed as follows: 



r^ dr \ dr ) ~ dr' 

 The electrical density is represented by p. 



2 dP _ 47rp 

 '^~r'dr'~ D' 



