Sept., 1938
IONIZATION CONSTANTS OF INORGANIC OXYGENACIDS [CONTRIBUTION FROM THE JOHNS
2047
HOPKINS UNIVERSITY ]
The Calculation of the Ionization Constants of Inorganic Oxygen Acids from their Structures BY ALEXANDER KOSSIAKOFF~ AND DAVIDHARKER
Introduction I n a given solvent, the free energy of ionization of an acid is directly related to its atomic configuration. The object of this paper is the formulation of a theoretical approach to the evaluation of this relationship. Due to the importance which is generally attached to theoretical calculations correlating physical properties with the structures of molecular systems, it is hoped that the treatment presented here will prove of general interest. The complicated nature of most acid molecules and of highly polar solvents renders a rigorous calculation of the free energy of ionization impracticable, if not entirely impossible. However, if the treatment is limited to non-resonating oxygen acids a number of reasonable assumptions may be made which facilitate a direct calculation of this quantity. The method of evaluation given here depends on the resolution of the free energy of ionization into one part which is essentially constant for all acids in a given solvent, and another part which depends strongly on the structure of the acid. The constant part constitutes a parameter in the equation; the variable part can be explicitly calculated from structural data. Having determined the parameter from the experimental ionization constant of a single acid, the constants of other acids are uniquely determined. The procedure is simple and direct, enabling a calculation of the ionization constant of an inorganic oxygen acid to be made in a short time with an average accuracy of *0.8 in log K . Previous Work.-The wide variation found in the ionization constants of acids has aroused the interest of many investigators. The majority of these have contented themselves with making purely qualitative observations. G. Oddo2 noted that the ratio of oxygen to hydrogen atoms in an oxygen acid was related to its strength. Hantzsch and Langbein3 observed that the strength of an acid increased with the number of oxygen atoms per replaceable hydrogen and with the negativity of the central atom. Similar qualitative correlations (1) American Can Co. Fellow, Johns Hopkins University. (2) G. Oddo, Gaze. ckim. i t d , 62, I, 56 (1922). (3) A . Hantzsch and W. Langbein, Z . anorp. allgem. Chem., 204, 193 (1933).
have been made by a number of other author^.^ The problem has been treated quantitatively by Smallwoods and by van Arkel and Carriere.6 Smallwood calculated the relative strengths of some organic acids on the basis of dipole interaction. Since carboxylic acids are resonating systems, it is impracticable to check Smallwood’s calculations by the method of this paper. Van Arkel and Carriere attempted t o calculate the strengths of acids from the assumption that the charge distribution in the acid molecule is determined by the ions formed by the separate atoms. This assumption is not in agreement with the known structures for most of these molecules, and it is consequently not surprising that the treatment does not lead to results in agreement with experiment. It may be noted that neither Smallwood nor van Arkel and Carriere have considered the properties of the solvent. Mechanism of Ionization.-There is good evidence that the ionization of an acid takes place through the transfer of the proton from the anion to a water molecule followed by the subsequent exchange of protons between water molecules.~ The absence of free protons in solution, together with the abnormally large mobility of hydronium ion, provide ample justification for this picture. As will appear later, the mechanism involving proton exchanges is extremely valuable in simplifying the calculation of acid strength. The process may be symbolized thus A-H
A-
0-H
0-H
0-H
H
H
H
I
H-0-H+
4 (1)
I
H
A-
I
H-0
I
0-H
0-H
H
H
1
.1w
H-0-H+
I
H A-
H-0
H
4 (3) I
H
H-0
4
I
I
H
I
0-H
I
H H-0-H+
I
H
(4) R. Oda, S. Namamoto, J . SOC.Chem. Ind., Japan., 36, Suppl. 543 (1932); H. S. Simms, THISJOURNAL, 68, 1239 (1936); T . M. Lowry, Chemistry &Industry, 42, 1048 (1926). (5) H. M. Smallwood, THISJOURNAL, 64, 3049 (1932). (6) A. E. van Arkel and Carriere, Chcm. Weekblad, 33, 182 (1936). (7) J. D. Bernal and R. H. Fowler,J . Chcm. Phys., 1, 515 (1933).
2048
ALEXANDER KOSSIAKOFF AND DAVIDHARKER
and so forth. This process removes the proton from the anion and reorients the water molecules in the immediate vicinity of that oxygen on the anion from which the proton has been removed. Thus, there is no exteiisive displacement of water, but there is a certain amount of reoriefltation. I t will be seen from the subsequent discussion that a calculation of the free energy change involved in this process is a good approximation to the total free energy of ionization of a n acid. By confining our attention t o non-resonating oxygen acids of the non-metallic elements, we are enabled t o make approximations which permit the calculation of their free energies of ionization. Oxygen acids-those in which the removable proton is attached t o an oxygen atom-comprise a large majority of the inorganic acids, and consequently the limitation of the treatment t o this class causes no great loss in scope. The lack of knowledge concerning the energetics of resonating systems requires the elimination of the acids of boron, carbon, and nitrogen from the present treatment. A careful consideration of the ionization process for the non-resonating oxygen acids makes the following assumptions reasonable : 1. The forces between a proton and an acid ion are due to an electron-pair bond with the nearest oxygen atom and to the electrostatic field of the ion. 2. The hydroxyl electron-pair bond is the same for all acids. Variations in strength from one acid to another are due solely to changes in the electrostatic field and symmetry of the ion. 3. Acid and anion molecules form hydrogen bridges to adjacent water molecules, and orient the more distant water molecules by their electrostatic fields. These effects constitute the hydration of the molecules. The first assumption is based on the fact that exchange forces diminish rapidly with distance and, therefore, will be extremely small between non-bonded atoms. The second assumption is reasonable due to the similarity of the binding of the central atom to the oxygens in the various acids. The third assumption is an expression of the geometry of hydration of anions and acid molecules. Hydroxyl groups rigidly orient the water molecules surrounding them thus : M--O--)Ihydroxyl oxygen thus : M-0- --H-0
--0
/€I
\"
\H
non-
-
Analysis of the Free Energy of Ionization.Consider an elementary ionization process, in
Vol. 60
which a particular proton is transferred from a given acid molecule to a particular water molecule. The reversible work done on the system is the sum of the work done in each of the following processes. 1. Rupture of the 0-H bond in the acid molecule. 2 . Formation of the 0-H bond in hydronium ion, H&+. 3. Removal of the proton from the electrostatic field of the anion. 4. Reorientation of the water molecules. 5. Compression of the water molecules. 6. Change in volume of the system against atmospheric pressure. Let A I to A6 represent the reversible work done on the system in the processes 1 to 6, respectively. The work done on the system, AA, is given by the expression AA
=
LIP - A ( P V )
At constant pressure A(PV) = -4 the work done against atmospheric pressure. Therefore, the free energy change, AF', for the elementary ionization process, defined above, may be written LF' = A i
+ A B + A S + A4 + As
(1)
Assumption 2 implies that A I is constant for all the acids considered, and At does not depend upon the acid involved. A s is the change in the compression work during ionization. T . J. Webb,* in his calculation of the free energy of hydration of ions, has evaluated the compression work, W,, in terms of the radii and charges of monatomic ions. For a univalent ion of radius 1.5 A,, W, is calculated to be 1 kcal. It increases as the square of the charge for a monatomic ion. However, for a polyatomic ion containing widely separated centers of charge, W , will increase approximately with the first power of the total charge. The polyatomic anions of the oxygen acids are cases intermediate between these two extremes. The increase of W, with the charge of the acid anion will be, therefore, considerably less than second power. The deviations from constancy in AS for the large complex anions under consideration should not exceed 0.5 kcal. The error involved in assuming AS to be constant is, consequently, small enough to obviate the necessity of its calculation for each acid. Since AI, A B ,and Ab are essentially constant for ( 8 ) T J Webb, THISJOURNAL, 48, 2589 (1928).
IONIZATION CONSTANTS OF INORGANIC OXYGENACIDS
Sept., 1938
the acids under consideration, we make the approximation that A1
+ A2 + AS = c
(2)
where C depends only on the temperature. sequently AF'
=
+ A4 + C
A3
Con(3)
+ +
The Calculation of As A4.-It is now necesA4 may be calculated sary to show that A3 by computing the work done in removing a proton from the anion by successive transfers between water molecules. It has been noted that this process reorients the water molecules in the immediate vicinity of that oxygen in the anion from which the proton has been removed. It remains to be shown that this work constitutes practically the entire work of reorientation resulting from the ionization process and, therefore, that Ah is indeed included in the calculation. The molecules of oxygen acids are, in general, complex structures, the peripheries of which consist of strongly orienting groups. The assumption has been made that these groups (oxygen and hydroxyl groups) rigidly orient the water molecules surrounding them in a definite way. The consequence is that upon ionization, which changes one hydroxyl group in the molecule into a free oxygen, only the water molecules surrounding this hydroxyl are reoriented. As an example consider HzS04, which changes upon ionization into HSOI-. The molecules in the immediate neighborhood will be arranged thus /H
0 I
H
/
PI/
\o- - -H-o~s-o~-H- - -0 I \
H/
0 2
? 0
\
