THEORY OF COMPLETE DISSOCIATION The classical theory of Arrhenius may be summarized as having correlated the experiments of Kohlrausch and van't Hoff, postulated a very considerable degree of dissociation in aqueous solutions of salts at ordinary concentrations, and provided a method of calculating the degree of dissociation. The theory ac counted for and correlated three common properties of acids, bases and salts, namely, (I) the practically instantaneous ex change of radicals, as compared to the slow reactions of organic chemistry, (2) the increase in equivalent conductivity with dilu tion to a finite limit set by the mobilities of the ions, (3) van't Hoff's observation that the osmotic properties of this class of sub stances always exceeded that predicted by his gas law equation it = cR T, which he had demonstrated was valid for non-electro lytes in dilute solution.
The net results of recent research have shown that the be haviour of the typical strong electrolytes in solvents of high dielectric constant is best explained by the assumption of a com plete, or at least practically complete, ionization, while the ioniza tion theory retains its original form in all essential details when applied to weak electrolytes (cf. J. N. Bronsted and V. K. La Mer, J. Amer. Chem. Soc., 1924; V. K. La Mer, Trans. Amer. Elec trochem. Soc., 1927). In the Arrhenius theory, the degree of dis sociation a is, as shown above, calculated from either the con ductivity ratio ,u, / or A, / where A, is the equivalent con ductivity at the concentration c, or from van't Hoff's factor i in the osmotic equation for electrolytes for according to Arrhenius i is to be interpreted as the quotient of the number of particles present divided by the number which would be present if no dissociation had occurred. If each indi vidual particle behaved independently of its neighbours, i.e., obeyed the gas laws as given by van't Hoff's law then Arrhenius's interpretation of the meaning of i would be valid. However, as is discussed later, the electric charges upon the ions set up elec trostatic fields, which do not allow the ions to behave independ ently as demanded by the gas laws. The effects of these inter ionic attractions were negl€ cted in the development of the classical theory, whereas, according to the modern point of view, the anomaly of strong electrolytes is to be attributed entirely to their operation. If Arrhenius's assumption in regard to the interpreta tion of i were correct we should obtain for a salt dissociating into v ions that and from Ostwald's dilution law, as shown previously, we should find that In the following tables the first summarizes the results from conductivity measurements upon the typical weak electrolyte acetic acid.
Since an approximately constant value of K is obtained for solutions up to 0.1M, the applicability of the Arrhenius theory must be regarded as established for this class of electrolyte to the concentration indicated. On the other hand the second table shows that large deviations occur in the value of K for the typical strong electrolytes potassium chloride and hydrogen chloride, and the Ostwald dilution law breaks down completely for other strong electrolytes. For instance in the case of high valence salts, like and K varies about one-million-fold for the same range of concentration.
Explanations of Deviations from Ostwald's Law.—At tempts were for many years made to reconcile these deviations by assuming changing viscosity of the solution, solvation of ions, and compound formation between solvent and solute. The diffi culty has, however, only been overcome by adopting a separate theoretical treatment -for strong and weak electrolytes. One rea son for the reluctance in abandoning the classical view of a considerable, yet substantially incomplete, dissociation of strong electrolytes was that, the values of a calculated from the conduc tivity ratio and those calculated from the freezing point measure ments using equation (2) usually agreed very closely. However, more accurate data recently obtained, particularly with the higher valence types of salts, has shown that the agreement is in general not good. The best experimental data indicate that for o• i N HC1 while the electromotive force measurements upon H2 - HC1- AgCI cells would indicate a value of 0.804. G. N. Lewis has shown that very close agreement is obtained between data from freezing point determinations when treated by strict thermodynamic methods, and e.m.f. data but not with that derived from the conductivity ratio.
The main theories which have adopted the assumption first put forward by Sutherland (Phil. Mag., 1902, 1906) that complete dissociation occurs with strong electrolytes are those of Bjerrum, Milner, Ghosh and Debye and Huckel. Bjerrum was led to this point of view from the observation that the optical properties of solutions of the complex chromium salts are practically independ ent of the concentration. Milner (Phil. Mag., 1912, Trans. Fara day Soc., 1919) first put forward the theory that the fall in conductivity with increasing concentration is due to a decrease in the mobility of the ions through the influence of inter-ionic forces which cause an increase in the occurrence of the associa tion of the ions. The term virial is used to denote the electrostatic forces of an ion multiplied by the distance over which it acts. Values for the molecular lowering of the freezing point are ac counted for quantitatively by the theory but no expression has been derived by it to enable the evaluation of conductivity as a function of concentration.
In a theory of Ghosh (Trans. Chem. Soc., 1918), a new depart ture is made in the representation of the properties of electrolytes. Complete ionization of electrolytes is assumed, and from a consid eration of the electrostatic energy of an ionized electrolyte, simple equations are derived to express electrolytic conductivity in aque ous and non-aqueous solutions. On the basis of this hypothesis, the following expressions are derived to express the relation between conductivity and the physical properties of the solution : For binary electrolytes : 2 1,12N = 3RT loge — D'3 %IV where N is Avagadro's number (6.16X e the charge on a univalent ion (4.7 X e.s.u.), D the dielectric constant of the solvent and V its molecular dilution, R the gas constant = 8.315 X I in c.g.s. units. These expressions have been found applicable to a large number of aqueous and non-aqueous solutions.
Further considerations that support the theory of complete dissociation are such general observations as the fact that aqueous solutions of the halogen acids exhibit no appreciable vapour pres sure of the hydrogen halide below a concentration of 3M., and experiments which have shown that the salt is completely extracted by water from benzene in which it is quite soluble. The studies of the Braggs (W. H. Bragg and W. L. Bragg, X-ray and Crystal Structure, 1924) and others on the structure of crystals by means of X-rays furnishes more conclusive evidence for the new point of view. They find that no molecules of NaCl are present in the solid salt ; instead the crystal structure consists of sodium and chlorine ions arranged in a cubic lattice, such as that each sodium ion is surrounded at equal distances by six chlorine ions and similarly each chlorine ion by six sodium ions. That the forces holding a crystal of salt together are due to the electrostatic forces between the charged' ions has since been established by the calculations of Born, Debye and Scherrer, Fajans, Madelung, and others upon the magnitude of the so-called space lattice energy.
The activity of a component in a solution can be defined most simply by the relation where is the activity and F is the partial molal free energy or Gibbs chemical potential. The activity is consequently a thermo dynamic quantity having the dimensions of a concentration, and is determined by the work which is necessary to transport re versibly a gram mol. of the component at constant temperature from some standard reference state, whose partial molal free energy is represented by F°, to the given state under con sideration.
Since van't Hoff demonstrated that the behaviour of any solute, regardless of its specific characteristics, continuously ap proaches and finally obeys the gas laws when its concentration in a given solvent approaches a zero value, the most convenient reference state for solutes is one for which the activity of the solute becomes equal to the concentration at infinite dilution. In the case of strong electrolytes where independent means of determining the true concentration of the undissociated molecule are lacking, the standard state of the undissociated molecule is chosen such that the dissociation constant K becomes unity.
Fig. 2 summarizes the general results of the course of the activity coefficient with concentration f f2, and f 3 refer to the activity coefficient of a univalent, bivalent and trivalent ion or, what amounts to the same thing, the mean (geometric) activity coefficient of the ion comprising a uni-univalent, bi-bivalent or a tri-trivalent salt. A uni-univalent salt is one in which both ions are univalent, e.g., sodium chloride. In this figure the salt or ion is present in very small amount in a solution of a uni-univalent salt of concentration c expressed in gram mols. per litre. When a univalent ion is dissolved in o• 1 M KC1 its activity coefficient is about 70% of its value in an ion-free-medium, while in the case of a trivalent ion the activity coefficient falls to 0.' o at c = o. I M. This means that the solubility of a tri-trivalent salt is ten times as great in 6.IN KC1 as in pure water. We also find as the con centration decreases that the limiting tangent for f assumes an infinite value, just as it did for the osmotic coefficient discussed previously. Obviously, the same objections to the classical theory hold for the activity coefficient of an ion (or salt) as for the osmotic coefficient, since the classical theory demands a finite limiting slope.
Ionic Strength.--When the salt acting as the solvent is of a valency type higher than the uni-univalent or when it is com posed of a mixture of salts of different valence types the problem becomes more complex. From a study of the data available in 1921 Lewis and Randall, however, were able to formulate a re markably simple law which holds in the majority of cases studied thus far; namely that in dilute solutions the activity coefficient of a given strong electrolyte is the same in all solutions of the same ionic strength. The ionic strength µ is defined as one-half the sum of the stoichiometric molar concentration of each ion multiplied by the square of its valency or charge. Thus: where z is the valency and m the molar concentration. That is to say, if were employed as the solvent salt in fig. 2 in stead of NaC1 we should use the ionic strength of the solution instead of the molar concentration, which in this particular case multiplies the concentration scale by the factor 3. If a (2-2) or (3–I) solvent salt had been used the corresponding factors would be 4 or 6 respectively. If the concentration of the ions or the salt whose activity coefficient is measured is present in appreciable amounts, it must be included in the summation in addition to the solvent salts to determine the ionic strength. The generalization that the activity coefficient of an ion is determined in dilute solution solely by its valency and the numbers and valen cies of all the surrounding ions, and not by specific individual properties as is the case in weak electrolytes, constitutes the strongest evidence for the theory that in strong electrolytes the anomalies are due to electrical effects and not to incomplete dis sociation.
A derivation is made of the energy effect due to electrical forces between the ions by means of two general principles. One, the so-called Boltzmann's principle, is borrowed from the kinetic theory, and the other, known as Poisson's equation, is derived from the laws of electrostatics and involves Coulomb's law. The following considerations will become clearer by reference to fig. 3. In this figure the dot at the centre represents an ion of valency +zi and charge and this produces in a shell of volume dv situated between the distances r and r-+dr, a potential ik and a density of electric charge p.
The Boltzmann principle may be stated as follows: When a large number of molecules possessing an average kinetic energy .*kT are distributed throughout a region in which there prevail at different points different fields of force (and therefore, dif ferent electric potentials) whereby any kind of molecule A in any given volume-element dv acquires a potential energy E, the number of such molecules will equal the number n per unit volume in a place where this energy is zero, multiplied by the factor e E/kT and by the volume dv.
The following expression for a uni-univalent electrolyte is ob tained for the density of free electricity p in a unit of volume (1 cu.cm.) of solution; p = E+ekti/kT) = — ane ) (6) k where n is the total number of ions present. The electrical po tential and the density of electricity p are connected by the Poisson equation of electrostatics; viz.: where r is the distance, and D the dielectric constant. This differential equation shows how the potential gradient, or field strength, d/dr varies with the electric density p and the distance r. From this equation Debye and Heckel derive the expression: Substituting this value of p in equation (7) and writing a single constant in place of the resulting coefficient of we have (9) By integration, the following expression is derived for the total potential where a;, is the average apparent diameter, or closest distance of approach of the ions. The first member in (io) represents the potential of an ion in an ion-free medium of dielectric constant D, where it is not influenced by surrounding ions, the second number consequently represents the potential at the surface of the ion, due to the presence of the surrounding atmosphere of ions. When the concentration is small, x is small and equation (io) reduces to From equation (9) it is seen that 1/x has the dimensions of distance, and by its use Debye and Hiickel avoid the introduction of an average distance between ions based upon the cube root of the concentration. They show further that the use of such an average distance is incorrect, since the ions are not restricted to oscillating above fixed points as in a crystal, but are free to move about. i/x which corresponds to the thickness of the Helmholtz double layer is therefore called a characteristic or probability distance, and is more exactly defined as the distance at which the potential difference has fallen to 1/e of its value due to the distribution of the surrounding ions. For aqueous solutions (9) yields for a uni-univalent salt X = Aim (14) where m is the molar concentration and may be replaced by the ionic strength A. In general X = 0•327 X 14/µ (i 5) 1/X is about ten times the diameter of an ion at but of the same order of magnitude as an ion for a i •oµ solution.
Having determined /i the potential at the surface of an ion due to the ionic atmosphere, Debye (Physik Z., 1924), shows that the electrical work of dilution is equal to the work required to build up at a finite concentration the Boltzmann distribution, by charging the ions isothermally and reversibly from zero po tential to i1' . The electrical free energy of dilution then be comes where Es represents the summation of the different types of ions present of . the number per cu. cm., ni • • • • . • n,. By adding the quantity to the classical expression for the free energy of an ideal solution we obtain the free energy F of an ionic solution.
The activity of a component in a solution can be defined most simply by the relation of G. N. Lewis where E is the activity, J. is the partial molal free energy, or Gibbs chemical potential, and is the partial molal free energy of some standard reference state, such as that of a high degree of dilution when the gas laws are obeyed. The activity coefficient for an ion of the ith sort may be defined by the relation where Pe is the partial molal electrical free energy of that ion. From the above and equation (i6) we have When the concentration is expressed as the ionic strength (cf. equation 5) in mol. per litre, and x is replaced by equation (14) the following simple limiting expression is obtained for the ac tivity coefficient of an ion in aqueous solution at C; — logio f (Bait) = 3 a(— ziz2)yµ = o• 5ziz24 in aqueous solution. (20) In place of the activity coefficient, use is frequently made of the osmotic coefficient which may be derived from the relation 4 = where i is defined in equation (I) and v is the number of ions into which the salt dissociates. The osmotic coefficient is thus the factor necessary to convert the ideal (van't Hoff) osmotic pressure to the observed value 7r, or, in terms of freezing point measurements, it is the ratio of the observed molal lowering to the theoretical value for the molal lowering.
From the square root dependence of In f upon the concentra tion the following relation obtains Accordingly from (14) I —4)= a(— VA= o.38(— ziz2) VP, in aqueous solution (21) The above deductions have so far related only to ideal salt solutions in which the dimensions of the ions are negligible in comparison with the characteristic distance 1/X. When the concentration becomes of the order of o•oi i or greater it is no longer permissible to neglect the factor in the ex pression (12) for the potential. In the evaluation of the activity coefficient the factor involving a retains the same form, and instead of (2o) we have As an expression valid for more concentrated solutions, a is an average value for all the individual ai values expressed in A.U. cm.) units and may be considered as the distance of closest approach of two ions. A consideration of these equations shows that if we plot either —logio f or 1— . against the square root of the concentration, the data should fall upon a straight line the slope of which is determined solely by the valencies, the dielectric constant of the solution and the absolute temperature.
The simplest and most useful method of testing the relation expressed in equation (14) in very dilute solutions consists in de terminations of the solubility of very sparingly soluble salts in solutions of other salts of varying concentration. It follows from thermodynamical principles that the activity of a solute in a saturated solution must always equal the activity of the solid solute. If in a saturated solution the ions of the saturating salt have the mobilities and the numbers per molecule of the ions 'I and and the corresponding activity coefficients f 1 and /2, then the equation ni f f = constant holds thermo dynamically at constant temperature. If various foreign sub stances (either electrolytes or non-electrolytes) are added to a solution in equilibrium with a solid phase, the stoichiometric solubility (s) changes but the activity of the dissolved particles saturating the solution remains the same, i.e., where is the activity coefficient and the solubility in what is otherwise pure water, while f and s refer to these values in a sol vent prepared by adding foreign substances to the solution. In the ideal region and for no ion in common, the relation becomes in s/ = — In f = (Alµ — '4 o) (23) where µ and are the ionic strengths in the two media. The limiting laws have been tested at high dilutions by Bronsted and La Mer by determining the increase in solubility of certain diffi cultly soluble colbaltammine salts to o.00005M) on the addition of small amounts (o.0005 to o•oiM) of foreign salts which do not interact chemically with the saturating salt. Fig. 4 where f is plotted against -V µ summarizes the results for very high dilutions.
These data verify the theory in the following respects : (I) The value of a as calculated by the theory is correct within the limits of error for which D is known for pure water since the slope of line (I) equals 0.50. (2) . The valence factor (zit for a single ion) is correct, since the slopes of the lines are in the cor responding ratios I :2 :3 :9. (3) The principle of ionic strength is valid for the simpler types considered, since the values of f when compared for different solvent salts at the same ionic strength agree among themselves for a saturating salt of a given valence type. The theory as developed above deals only with the so-called ideal salt solutions, where the properties of the solution depend simply upon the numbers and valencies of the ions. With rising concentration the activity coefficient depends to an increasing degree upon the chemical character of the ions in solution and particularly on the influence of ionic size. For these solutions the expression given in equation (2 2) is applicable. Further confirma tion of the validity of the Debye theory is obtained from the con cordance in the values of a, the mean effective diameter of the ions, as determined from freezing point and from solubility meas urements.
The simple form of the Debye theory has been extended by }Rickel to more concentrated solutions by introducing the assump tion that the dielectric constant is a linear function of the con centration of ions. The following equation is derived: 4 —login f = B2c, where B is a function depending on the dielectric constant and c is the concentration in molecules per litre. This expression differs from equation (2 2) by the added term B2 c. This equation has been found to give a good representation of the data for a number of solutions.
It seems logical to assume that even with a completely ionized salt such as potassium chloride, occasionally a K', and C1' ion may collide under such favourable circumstances that a rearrangement in electron orbits occurs, whereby a true undissociated molecule possessing a pair of shared electrons is formed. The reciprocal of the dissociation constant of the mass action function is a measure of the number of such molecules. With hydrogen and acetate ions it is very frequent, since K=1.8 X I With the strong electro lytes the probability would be vanishingly small and in an ideal strong electrolyte zero. It has been computed by different meth ods that the true dissociation constant of HC1 in water is of the order of to Io'. This means that the probability of the exist ence of an undissociated molecule of HC1 is only about one in a million when the total concentration is one molar. Knight and Hinshelwood (J. Chem. Soc., 192 7) from the distribution of HC1 between benzene and water find that the transition to the non polar form occurs at about 13 molar in water. In the case of most salts the probability that its ions will share an electron pair is without doubt much less, so small in fact, that it is a question of no immediate practical importance whether we consider potas sium chloride to be completely ionized or whether we say it is only practically completely ionized.