R. J. GRABENSTEWER, 0. T. QUIMBY, AND T. J. FLAUTT
4194
The Acid Dissociation Constants of Substituted Methanediphosphonic Acids: A Correlation with P 3 I Magnetic Resonance Chemical Shift and with Taft
U*
by R. J. Grabenstetter, 0. T. Quimby, and T. J. Flautt The Miami Valleg Laboratories, The Proder and Gamble Company, Cincinnati, Ohio &23G (Received December 19, 1966)
A series of substituted methanediphosphonic acids [RIR2C(POaH2)2] was prepared, and the thermodynamic acid-dissociation constants were calculated from data obtained by potentiometric titration with tetramethylammonium hydroxide at different ionic strengths. The dissociation constants were shown to correlate linearly with the electron-withdrawing power of the substituents as measured by the Taft U* and the Pal nmr chemical shift.
Introduction
It has been shown by Martin and Griffin’ that the Taft equation applies satisfactorily to the first and second dissociation constants of a large number of alkylphosphonic acids (RP03H2). The deviations observed with some compounds were attributed to steric inhibition of solvation. However, their attempts to assign a Taft u* value to the P03H2group by the study of a series of alkylidenebisphosphonic acids failed. With arylphosphonic and arylphosphoric acids containing carboxy ~ u b s t i t u e n t sit, ~was ~ ~ possible to obtain Hammett u values for the P03H-, PO3*-, and HP02groups by application of the Hammett equation. However, Martin and Griffin’ felt that in arylphosphonic and arylphosphinic acids there exists a “possibility of da-pn bonding between the phosphoryl group and the benzene ring,” and, hence, the Hammett u values may not be direct measures of the electroneg% tivity of the phosphorus-containing substituent involved. The whole question of the effect of substituents upon the acidity of organic acids was reviewed by Barlin and P e ~ i n . The ~ complications introduced in gem-acids were not discussed. In the course of study of metal chelating agents, acid dissociation constants were determined for a number of gem-diphosphonic acids (substituted methanediphosphonic acids, R I R ~ C ( P O ~ H ~ )The ~ ) . effect of the substituents upon the acidity of these acids was examined. The Journal of Physical Chemistry
Experimental Section Materials. The gem-diphosphonic acids used in this study were synthesized by a variety of methods summarized briefly in the next three paragraphs. Methanediphosphonate (MDP) was made as the ester by a double Arbusov rearrangement between dibromomethane and triisopropyl phosphite.6 Ethane1,l-diphosphonate (EDP) was made by alkylation of the carbanion of MDP ester, propane-2,2-diphosphonate (PDP) by a similar alkylation of EDP.sa Monobromomethanediphosphonate (Br-MDP) and dichloromethanediphosphonate (C12-MDP) were made by action of aqueous hypohalitegb on the tetraisopropyl ester of MDP. All of these esters were converted to the acids by boiling with an excess of concentrated aqueous hydrochloric acid. The Br-MDP was purified by fractional crystallization as the dianiline salt. For C1,MDP a purer product (less C1 deficiency) resulted from anhydrous ester pyrolysis, eliminating propylene.’ (1) D.J. Martin and C. E. Griffin, J . Organometallic Chem., 1, 292 (1964). (2) H.H.Jaff6, L. D. Freedman, and G. 0. Doak, J. Am. Chem. SOC.,7 5 , 2209 (1953). (3) L. D. Quin and M. R. Dysart, J. Org. Chem., 27, 1012 (1962). (4) G. B. Barlin and D. D. Perrin, Quart. Rev. (London), 20, 75 (1966). (5) C. H.Roy, U.5. Patent 3,251,907(1966). (6) (a) See examples I and 111, British Patent 1,026,366 (1966); (b) See examples V and VI, British Patent 1,026,366(1966). (7) A. E. Canavan, B. F. Dowden, and C. Eaborn, J . Chem. SOC., 331 (1962).
DISSOCIATION CONSTANTS OF SUBSTITUTED 3IETHANEDIPHOSPHONIC ACIDS
The carbonyldiphosphonate (CDP) was prepared by action of hot aqueous NaOH on the tetrasodium salt of Cl2-?tlDP8 and purified by recrystallization of the sodium salt. The methanehydroxydiphosphonate (MHDP) was made by the catalytic hydrogenation of tetrasodium CDP in aqueous solution at a pH above 10; after removal of the catalyst the pH was reduced to 5 (HC1) and the sample purified by recrystallization of the N&HZsalt. The ethane-1-hydroxy-1,l-diphosphonate(EHDP) was made by the action of acetic anhydride on phosphorous acid. Analyses follow for the acids or salts used: MDP: Calcd for CHz(P03H2)z: C, 6.8; H, 3.4; P, 35.2. Found: C, 7.4; H, 3.9; P, 35.4. EDP: Calcd for CH3CH(P03H2)z: C, 12.6; H, 4.2; P, 32.6. Found: C, 12.8; H, 4.1; P, 32.7. PDP: Calcd for (CH&C(P03&)2: C, 17.7; H, 4.9; P, 30.4. Found: C, 17.7; H, 5.1; P, 30.3. MHDP: Calcd for HC(0H)(POaHNa)z: (3, 5.2; H, 1.8; P, 26.9. Found: C, 5.1; H, 2.0; P, 26.0. EHDP: Calcd for CH&(OH)(P03Hz)z.H2O: C, 10.7; H, 4.5; P, 27.7; HzO, 8.0. Found: C, 11.2; H, 4.6; P, 27.8; H20, 8.2. CDP: Calcd for 0C(P03N&)2: C, 4.3; P, 22.3; Na, 33.1. Found: C, 4.6; P, 20.9; Na, 35.1. Clz-MDP: Calcd for C12C(P031\'az)z: C, 3.6; P, 18.6; C1, 21.3; Xa, 27.5. Found: C, 4.2; P, 18.3; C1, 21.3; Na, 27.8. Br-MDP: Calcd for BrCH(P03)2H0.rNa~.,:C, 3.6; H, 0.3; P, 18.5; Br, 23.8; Na, 25.3. Found: C, 3.9; H, 0.6; P, 18.0; Br, 24.2; Na, 25.2. Tetramethylammonium chloride (TMA-Cl) and hydroxide (TMA-OH) reagents often contain enough carbonate to influence the titration curves above pH 7. The former was freed of C02 by adjusting the pH of a solution to about 3 with HC1 and bubbling COrfree nitrogen through it for 0.5 hr via a fritted-glass disperser; the pH was restored to 7 with carbonate-free TMA-OH. Pmsage of a 1.0-1.5 M solution of TMAOH under a C02-free nitrogen blanket through an anion-exchange column in OR form reduced the carbonate level so that none was evident in a titration with standard HC1 solution. Other materials were reagent grade, used as purchased. Methods. Titration. Diphosphonate salts were converted to free acids by cation exchange (Dowex 50W, Hf form) and carefully standardized. All titrations were carried out at 25" in covered cell under a nitrogen blanket, the titrant being added under the surface of t,he magnetically stirred solution from the capillary tip of a semimicro syringe buret (5.0 ml capacity, smallest division 0.005 ml). Pure borax (U. S. Bureau of Standards) was made up to 0.01 M with deionized water
4195
giving a solution of pH 9.18 at 25", which was used for calibrating the pH meter (Radiometer, Model PH4) ; glass electrodes were standardized against this buffer before and after each run. If these readings did not agree within 0.02 pH unit, a fresh glass electrode was used. Readings were estimated to the nearest 0.002 pH unit and carried through computation in this form, the results then being rounded to the nearest 0.01 unit. During a titration both titrant and titrand were 0.005 M in tetramethylammonium diphosphonate, both solutions having been prepared by mixing appropriate volumes of standardized tetramethylammonium hydroxide and diphosphonic acid. In addition the titrant contained a known concentration of HCl (in the range 0.1-0.4 N ) . Since both solutions contained identical concentrations of diphosphonate, there was no dilution effect upon the total diphosphonate concentration during titration, and hence relatively small changes in the ionic strength ( p ) of the titrand during a complete titration. Titrations were run at the mean p of the 0.005 M diphosphonate solution (-0.032) and at a nominal ionic strength of 0.10 and 0.25, obtained by adding identical concentrations of tetramethylammonium chloride to both titrant and titrand. The actual range in p (calculated at the midpoint between successive end points) over the course of a titration of the strongest acid at 0.005 M was: unadjusted, 0.028-0.042; adjusted to mean p = 0.1, 0.094-0.109; adjusted to mean p = 0.25, 0.244-0.259. The calculated mean p was used in computations rather than the nominal p. pKi values were calculated from appropriate pHvolume measurements. Each titration was run, at least in duplicate, at each level of p ; in some cases, further replication was carried out. From each titration a series of K i values was computed; the mean value of all titrations is recorded in Table I. Computational methods are shown in Appendix A. Nmr. Aqueous 0.5 M solutions of the acids and their salts at various degrees of neutralization had their P31spectra obtained on an HR-60 Varian nmr spectrometer at a frequency of 24.3 MHz. Calibration was accomplished by interpolation between two side bands. The estimated inaccuracy is *0.05 ppm ( 5 1 Hz).
Results and Discussion Shown in Table I is the average K at each ionic strength expressed as pK values and the extrapolated (8) 0. T. Quimby, et al., unpublished data. (9) B. T. Brooks, J. Am. Chem. Sac., 34,496 (1912); e.g., Henkel and Cie, British Patent 903,816 (1962); or Henkel and Cie, Belgian Patent 619.619 (1962).
Volume 71. Number I d
December 1867
R. J. GRABENSTETTER, 0. T. QUIMBY,AND T. J. FLAUTT
4196
n
value at zero ionic strength (pK,O), together with the confidence limits (at the 95% level) of the extrapolated value. A least-squares regression equation using all mean pKj values at all ionic strengths was used to calculate the extrapolated values and confidence limits. Table I1 shows the chemical shift (6) of the Pal nmr (85y0 H3P04 = 0.0, external standard) of the free acids and various (CHa)4Nsalts of the acids. A slight shift of 6 with the extent of neutralization is evident. Since 6 varies with the extent of neutralization, attempts to correlate 6 with pK? were made with 6 values chosen appropriate to the pK value (Le., pKlo with 6 of (TMA)4 salt, pKS0with 6 of (TMA),H salt, etc.). There is a linear relationsip between 6 and each series of pKo values. The equations of the correlation lines are: pK40 = 8.51 - 0.156, correlation coeff = 0.99; pKaO= 5.78 - 0.086, correlation coeff = 0.96; and pK2O = 2.46 - 0.036, correlation coeff = 0.77. The equations were derived by the least-squares method as described by Brownlee.l0 The correlation coefficient is a measure of the fit of the experimental points to the computed line. The correlation coefficient of the pKz line is considerably smaller than the others because the slope of the line is not far from zero. However, the scatter of points around the line is no greater than for pKa0and pK40. These correlations suggest acid dissociation constants and the phosphorus chemical shifts undergo parallel changes as the substituent on the carbon atom changes. Since dissociation constants for other series of organic acids have shown linear correlation with Taft u* substituent c o n s t a n t ~ , * Jit~ was , ~ ~ not surprising to find a similar correlation valid for these gem-diphosphonic acids. A linear correlation was found between pKf and the sum of the Taft u* values @a*) of the substituents on the a-carbon atom. The Taft u* values usedl1J2 were U*CH, = 0, U*H = 0.49, U*OH = 1.55, U*B* = 2.80, U * ~ I = 2.94. The equations of the least-squares lines and their correlation coefficients are as follows: pKlo = 11.82 - 0.402u*, correlation coefficient = 0.90; pKaO= 7.75 - 0.30Zu*, correlation coefficient = 0.96; and pK20 = 3.32 - 0.322u*, correlation coefficient = 0.94. The slopes of all of the lines (the coefficients of .Zu*) do not differ greatly. Martin and Griffin’ also observed a similarity of slopes of the pK1-u* and pK2~~~~
~
(10) K. A. Brownlee, “Industrial Experimentation,” Her Majesty’s Stationery Office, London, 1949, pp 63-65. (11) R. W. Taft, Jr., in “Steric Effects in Organic Chemistry,” M. 5. Newman, Ed., J. Wiley and Sons, Ino., New York, N. Y . . 1956, pp 556-675. (12) R. W. Taft, Jr., J . Chem. Phya.. 26, 93 (1957).
The Journal of Physical Chemietty
DISSOCIATION CONSTANTS OF SUBSTITUTED METHANEDIPHOGPHONIC ACIDS
~
~
Table 11: Chemical Shift (a) of
P31
~
4197
~
Nmr Spectra" 7-
Compound
(CHa)zC(P03Hz)z CH3CH(POsEIz)z CHaC(OH (PO& )z CHz(POaHz)3 HC(OH)(POaHz)z ClzC(PO3H2)z O=C(POaHzjl'
Abbreviation
PDP EDP EHDP MDP MHDP ClrMDP CDP
Acid
-26.9 -22.6 -19.8 -17.8 -16.1 -7.9 -13.0
I _ _ -
6, ppm----
(TMA~* salt
-25.6 -21.0 -18.8 -15.7 -14.6 -7.7 $3.4
(TMA)a
(TMAh
salt
salt
-25.1 -20.9 -19.0 -15.7 -14.7 -8.4 +1.9
-25.1 -20.5 -18.6 -15.3 -14.5 -9.4 +0.91
'
(external) as zero; diphosphonates 0.5 M. TMA = tetramethylammonium. This comHO pound exists as HO>C(POSH~)zas the acid and as O=C(POSMZ)Zas the salt; an equilibrium mixture is found in solution. (Paper on '6 measured relative to 85%
CDP in preparation.)
correlation lines of the alkyl monophosphonic acids (RPO3H2). This implies that the inductive effect of the substituent is transmitted to the phosphonate group (or groups) to a.pproximately the same extent regardless of the degree of neutralization of the phosphonate group. It was noticed that in the pK40correlation, one group of compounds had pK4values, which were larger than predicted by the equation, the remainder had values lower than predicted. This observation led to an attempt to correlate each group separately, with the following result: (a) pK4O = 12.13 - 0.402:u*, correlation coefficient = 0.99, and (b) pK4O = 11.29 - 0.353u*, correlation coefficient = 0.99. These correlation coefficients indicate extremely good correlation. The compounds whose pK4O values were u*
correlated with 2u* by (a) were:
CH3 >C (P03H2) 2; H
CH3>C(POaH~)? CH3 ; Fi>C(P03H2)2; and::>C(
PO3H2)2.
Those whose pK40 values were correlated by (b) were: Thus, the two groups are correlated by approximately parallel straight lines separated by about 0.8 pK units. The difference is larger than the experimental error. Martin and Griffin' observed that branched substituents (i-C3H7, i-C4H9, sec-C4He, neo-CsHlo) in alkyl phosphonic acids resulted in deviations from the pK2-a* straight-line relationship, but showed no deviation from the pK1-a* line. This deviation (in the direction of a weakening of the second dissociation of the acids, i e . , larger pKz of monophosphonic acids) was attributed to a steric inhibition of solvation of the dianion, and the authors stated that similar steric effects on acidity had been observed in the carboxylic acids,11J3since steric interference with solvation is a common factor in pro-
ducing deviations from linear Taft relationship~.'~J~ It is proposed that a steric factor is involved in the present work in producing the differences demonstrated by eq a and b. Such differences do not appear in the pKaOand pKzocorrelations, analogous to the findings of Martin and Griffin' for the pK1 of monophosphonates. In alkyl monophosphonic acids' the substituent R is bonded directly to the phosphonate group, whereas in the gem-diphosphonates the substituents under consideration are separated from the phosphonate group by a carbon atom. Branch and Calvin16have pointed out that when a substituent is separated from a functional group by a methylene group, the inductive effect of the substituent on that functional group is attenuated by a factor of 2.8, i e . , o*(CH2R) = a*(R)/2.8. If it be assumed that the attenuating- effect of the carbon atom in a gem-diphosphonic acid proximately the same as that of a methylene group in an alkyl chain, then the effective a* at the carbon atom should be approximately u*eff =
(1/2.8) (EO*)
Since the Martin and Griffin equations were derived with apparent (nonthermodynamic) pK values, presumably measured at low ionic strength, probably a more valid comparison of those equations with the Taft equations of the gem-diphosphonic acids can be made if the gem-acid pK values, which were determined at (13) G. 5. Hammond and D. H. Hogle, J . Am. C h a . SOC.. 77, 338 (1955); P. Q. Bartlett, J. Chem. E d w . , 30, 22 (1953). (14) H. K.Hall, Jr., J . Am. Chem. Soc., 79,5441 (1957): M.M.Kreevoy, E. T. Harper, R. E. Duvall, H. 5. Wilgus, 111, and L. T. Ditsch, ibid., 82, 4899 (1960). (15) G. E. K. Branch and M. Calvin, "The Theory of Organic Chemistry," Prentice-Hall, Inc., New York, N. Y., 1941, p 209.
Volume 71. Number 13 December 1967
R. J. GRABENSTETTER, 0. T. QUIMBY,AND T. J. FLAUTT
4198
low ionic st,rength ( b = 4 . 0 3 ; Le., pKp-03), are used in the correlation rather than the pK,O values obtained by extrapolation to zero ionic strength (Table I). The equations correlating pK?.Oa with u*eff thus obtained are shown in Table I11 along with the Martin and Griffin equations. In the equations for the gem-diphosphonic acid, p* (the slopes) for the 2nd, 3rd, and 4th ionizations are quite similar, indicating that the inductive effect of the substituents is transmitted through the carbon atom to the mono-, di-, and trianion to approximately the same extent, and, strikingly, this p* is not far different from that observed for the monophosphonates by Martin and Griffin. These equations were computed for the data shown in Table I at the lowest ionic strength (-0.03). ~
Table 111 : Taft O* Correlation Equations for Monophosphonic Acids and gem-Diphosphonic Acids corMonophosphonic acids"
pK1 = 2.47 pK, = 7.77
relation
gem-Diphosphonic acidsb
- 1.1210* - 1.1770,
pKl0.0a = not available pK2O.O' = 3.20 - 0.84c*sff 0.95 p ~ a= ~ 7-61 . ~ -~0.84a*eff o.96 p ~ p = a 11.61 - i.iiu*eff 0.89
' Martin and Griffin equations, ref 1. R1,Rt = CH8,CH,; CHa,H; CH8,OH; H,H; H,OH; H,Br; C1,Cl.
As indicated previously, the pK4 correlation was better if two lines were computed pK40.03 = 11.98 - l.llU*& correlation coeff = 0.99 (a) pK2.0' = 11.12
- 0.98~*,it correlation coeff = 0.99 (b)
The same compounds are correlated by the equations designated a and b, shown previously in the Results and Discussion. If the P0,H- group can be assigned a Taft u* value, it should be calculable by using the Martin and Griffin equations for the following dissociations of the gem-diphosphonic acids K¶
-HOsPC(Ri) (Rt)POaH, -HO,PC(R~)(R~)POIH- H+ (1)
-
+
KI
-HOaPC(RI) (R2)POaH-HOIPC(R~)(R~)PO~~H+ (2) The Journal of Pha/&al GhmiatTy
+
The computations were made using the appropriate statistical correction factors according to the procedure of Branch and Calvin.16 The results are summarized in Table IV. The computations are described in the Appendix. _
_
_
_
_
~
~
~
~
Table IV: Calculated Taft u* Values for POaHCompound
Substituents
PDP EDP EHDP MDP MHDP Br-MDP Clz-MDP
CHa, CHI CHa,H CHS;OH H, H H,OH H,Br C1, C1
PKi values a t -0.03pKz pK3
-#
3.07 3.06 2.89 2.95 2.66 2.10
...
7.92 7.38 7.16 7.20 6.92 6.40 6.02
--V*PO~H-
From pKP*Oa
-1.20 -1.66 -2.30 -1.87 -2.21 -2.07
.. .
calcdaFrom pK30.0'
-0.64 +0.014 -0.38 -0.095 -0.31 -0.31 -2.0
See Appendix for method of computation.
Table IV shows that the U* for P0,H- group computed from pK2 are usually more negative than those computed from pK3, and even within a series calculated from a given set of pK values there is considerable variation. The U * P O ~ H - values differ considerably from the values shown by Martin and Griffin' derived from alkylidenebiphosphonic acids. On the basis of the traditional view, it appears that the group written formally as P0,H- in both the mono- and dianion of the diphosphonic acid differs considerably in its electronwithdrawing power as a substituent depending upon the nature (ie., extent of dissociation) of the other phosphonate group present in the molecule. This seemingly contradictory situation can be resolved if it is assumed that the phosphonate groups interact strongly, so that the group written P0,H- cannot be considered an unchanging entity. Rather, both gemdiphosphonate groups should be considered as parts of a single acidic group to which all the acidic protons present in a given anion are bound. Evidence for such interaction is the extremely stjrong binding of the last proton to dissociate from the gem-diphosphonates (characterized by pK4) compared to the last (second) proton to dissociate from the monophosphonate. Both situations formally correspond to the dissociation of a proton from the group POaH-. However, in the case of the diphosphonate, the proton is bound about IO3fold more strongly. I n a study of the acidity of a series of polymethylene diphosphonates, Irani and Moedritzer17 found that the binding of the last proton to (16) Cf.ref 15,p 200. (17) R. R.Irani and K. Moedritrer, J. Phals. Chem., 66, 1349 (1962).
DISSOCIATION CONSTANTS OF SUBSTITUTED METHANEDIPHOSPHONIC ACIDS
dissociate became stronger as the number of methylene groups between the phosphonate groups decreased, a circumstance which would be expected to enhance interaction between the phosphonate groups. A possible mode of interaction in the case of the most strongly bound proton is the formation of a six-membered ring through hydrogen bonding, thus
[
-6
I-
‘L
4’
i
-14-
-18-
w
6
-22-26{
.30
Acknowledgment. We wish to thank Dr. H. H. Jaff 6, University of Cincinnati, Cincinnati, Ohio, for
7
J
2 z
0
I n such a situation, the u* value for the inductive effect of one phosphonate group upon the other would have little meaning; both groups are part of a single acidic group. The near constancy of the slopes of the correlation curves (see p*, Table 111) of the gem-diphosphonates is an indication that the transmission of the polar effects of the substituents to the dissociating protons remains rather constant throughout the neutralization.’* If the two diphosphonic acid groups were acting independently and undergoing rather marked changes in structure as the neutralization proceeded, an effect on the polarizability of the connecting carbon atom might be expected, and, thus, a change in the slope of the correlation curves might be expected. If, however, the diphosphonate group is relatively constant in structure (differing mainly in charge as the protons associate with or dissociate from the group), the polarizability of the bonds to the carbon atom should remain constant. It can be shown by consideration of the dissociation constants of monocarboxylic acids and gem-dicarboxylic acids (substitu1,ed malonic acids) and by calculation of the u*values for COOH and COO-, that, as expected,lgJo there is evidence of interaction between the carboxyl groups of gem-dicarboxylic acids; although, it is somewhat less than the interaction between gem-phosphonic acid groups. The carboxylic acid data and results of computations are shown in the Appendix. Chemical Shijts vs. Z u * . A plot of the chemical shift of diphosphonic acids vs. Zu*of the substituents on the P-C-P moiety reveals an increase in the chemical shift as Xu* increases (Figure 1). This correlation is similar to that found by other investigators,21 with smaller chemical shifts and perhaps more regularity in the present work because the substituent is insulated from the phosphorus atom by an intervening carbon atom.
I
I U J
0 0
:>c.1>] 0
4199
0
1 2 3 4 5 6 SUM OF THE TAFT Ut VALUES OF THE SUBSTITUENT GROUPS
Figure 1. Chemical shift of 0.5 M solutions of substituted diphosphoric acids as a function of the sum of the Taft u* values of the substituent groups: 1, PDP; 2, EDP; 3, MDP; 4, EHDP; 5, MHDP; 6, Br-MDP (estimated from data at other concentrations); and 7, ClrMDP.
helpful suggestions and discussions during the preparation of this paper.
Appendix I . Computational Methods. A . Computation of Dissociation Constants. The dissociation constants were computed using a relationship similar to that used by Bjerrum for computation of complex constants. It is shown below in the form used for the tetraprotic diphosphonic acids d =
[(l - ii)(H+)] a4
[(3
+ [(2 - ii)(H+)2] +
- d)(H+)31 az +
a3
[(4- f4(H+)41 ai (AI)
where d = moles of H f bound per mole of diphosphonic acid; a4= l/K4; a3 = 1/K3K4; a2 = l/K&Kz; and a1 = l/K&KzK1; and K1 = constant for the stepwise dissociation of the first H + to dissociate H4A I_ H+
+ H3A-
(H (H3A-I (H4A) The symbols enclosed in parentheses indicate molar concentrations. The other stepwise constants are formulated similarly. The concentrations of free H +, K1 =
+>
(18) Cf.ref 15, p 196. (19) F. H. W-estheimer and 0. T. Benfey, J. Am. Chem. SOC.,7 8 , 5309 (1956). (20) B. L. Silver, Z. Luz, S. Peller, and J. Reuben, J. Phua. Chem., 7 0 , 1434 (1966). (21) J. R. Van Wazer, C. F. Callis, J. N. Shoolery, and R. C. Jones, J. Am. Chem. SOC.,7 8 , 5715 (1956).
Volume 7 1 . Number Id December 1967
4200
R. J. GRABENSTETTER, 0. T. QUIMBY, AND T. J. FLAUTT
(H+) = 10-PH/y~+, where Y H + = activity coefficient of H+. Y H + was assumed to be equal to that of H + in a solution of HC1 at the same ionic strength. The values of YH t used in the computation were taken from a compilation of Conway.22 The values used are: p = 0.032; /.4 = 0.10; p = 0.25; Y H t = 0.857; YHt = 0.796; and Y H + = 0.765. The value of zi is directly calculable from stoichiometry; thus, the H + added as HCI is (H+)Hc~ =
vo + v1
+ HOH
HA-3
PKT = PKobsd f log (n/m)
Monophosphonie Acids. Ki
R P O a H z G RPOaHpK1,obsd
P&,T
VlCl
RPOaH-
-
A
+ H+
n = 2; m = 1
+ log ( 2 / 1 ) = PKl.obsd + 0.30
K,
where VI = volume of HCl added in titrant; C1 = concentration of HCI added in titrant; V o = original volume of the titrand. Some H + is gained from water by hydrolysis; thus A-'
K observed; n = number of equivalent acidic protons in acid; and m = number of equivalent sites to which H + can return in anion.
RP032-
+
+ H+
n = 1; m = 2
PK~,T = PK2,ob~d log ( 1 / 2 ) = PK2,obsd
- 0.30
gem-Diphosphonic Acids. (2nd and 3rd dissociations)
+ OH-
The H+ gained is equal to (HA-3) = (OH-) =
KW (H +I
-
where K, = ionization constant of water at 25". The total concentration of bound hydrogen (H+)b in the titrand is
+
(H+)b = (H+)HcI
- (E+)
(H+)h,d
From the definition of zi, (z = (H+)b/C2, where CZ = total diphosphonate concentration. Rewriting in terms of the above
vo + v1 VlCl
+-Kw (H+)
(H+)]/CZ
From measured values of Vl and (H+) for a given point, the values and the constants were calculated by means of an IBM-1620 FORTRAN program based on eq A l . Both iterative and algebraic methods have been used; b0t.h methods give consistent results. The mean pKc values were correlat~edwith the square roots of the ionic strengths by a least-squares equation, and the pKi at zero ionic strength was calculated, along with its confidence limits at the 95% level, as shown in Table I. The concentration used (0.005 M ) was too dilute to justify computation of pK values less than 2.0, including all pK1 values and certain pKz values (cf. Table I). B. Computation of Taft Substituent Constants. 1. Statistical Correction Factors. (Branch and Calvin) H,A
H,-1A
+ H+
-
Ri C O O H A R1 COO- + H + Rz>~C~cC
