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The Journal of Physical Chemistry, Vol. 83, No.
IS, 1979
In a similar manner, we define the unit vectors, (xz, y2,z2) and {(2), at site b2. Since the screw diad axis is parallel to the b =is (y axis) { (2) = -{ (1) { (2) = { (1) and { (2) = -3;(1). 9 x X ' Y Y ' Since (14)
A. J. Kresge and Y. C. Tang
perpendicular to the molecular plane of urea determined by the X-ray analysisa8From eq 13,15, and 19-21, we can obtain permissible values for the components of the unit vectors, (xl, yl, zl) and (xz,yz, zz). Substituting these values into eq 12, the possible directions of the internal field at site bl can be obtained as = "b(l)
( ::::)
or
-0.57 -{x(1)x2
+ ry(1)y2- 3;(?z2 = *cos
6'
(15)
Here, f signs correspond to the choice of the angles 6' and 7r - 8,which give two possible directions of From eq 11, one obtains (D1))-1@) = (D(2))-1H(2)
(16)
If (D(l))-lis expressed as (D(1))-1=
(3 5
-0.71
where the + and - signs correspond to the cases of 6' equal to 15 and 165', respectively. As for the two-sublattice model, we have also tried to get probable directions for the internal field, but no set yl, zl) and ( x 2 , yz, z2), satisfying eq 13, 15, of vectors, (q, and 19-21 was obtained for site b. Thus the possibility of a two-sublattice model can be excluded.
References and Notes (17)
then (D@))-l can be written by considering the symmetry property of the four-sublattice model as
Since the experimental results require that HCl) = Ht2),one obtains the following equations by substituting eq 12, 14, 17, and 18 into eq 16: A(xi - 9 ~ 2 )+ D(yi + y2) + F(z1- 22) = O (19) D(x1 + x 2 )
?H~(1)(-o.oo6) 0.64 (22)
+ W Y 1 - Yz) + JWl+ 22) = 0 + E(Y1 + Yz) + C(Z1- 22) = 0
(20) F(x1- x2) (21) The numerical components of the tensor were evaluated by summing up contributions from all the Cu(I1) ions in a sphere of 400 8, around a resonant nitrogen nucleus by use of the lattice parameters determined at room temperature.8 The calculation was carried out by means of a FACOM 230-60 computer at Nagoya University. Since the molecular structure of urea is almost planar in crystals,8,21the directions of {(i) are taken to be
(1) R. Kiriyama, H. Ibamoto, and K. Matsuo, Acta Crystallogr., 7, 482 (1954). (2) H. Kobayashi and T. Haseda, J. fbys. Soc. Jpn., 18, 541 (1963). (3) J. Itoh and Y. Kamiya, J. phys. SOC.Jpn., 17, Suppl. B-I, 512 (1962). (4) R. B. Fllppen and S. A. Friedberg, J . Cbem. fbys., 38, 2652 (1963). (5) A. Dupas and J.-P. Renard, Pbys. Left. A., 33, 470 (1970). (6) A. Dupas and J.-P. Renard, C. R. Acad. Sci. Paris, Ser W , 271, 154 (1970). (7) M. S. Seehra and T. G. Castner, Jr., Pbys. Rev. 6 ,1, 2289 (1970). (8) H. Kiriyama and K. Kitahama, Acta Crystallogr., Sect. B , 32, 330 (1976). (9) Y. Yamamoto, M. Matsuura, and T. Haseda, J . fbys. Soc. Jpn., 40, 1300 (1976). (10) M. Kishita, Nippon Kagaku Zassbi, 83, 264 (1962). (11) R. Ikeda, D. Nakamura, and M. Kubo, J. fbys. Chem., 70, 3626 (1966). (12) G. A. Matzkanin, T. N. O'Neal, and T. A. Scott, J . Cbem. Pbys., 44, 4171 (1966). (13) M. Minematsu, J. fbys. SOC.Jpn., 14, 1030 (1959). (14) D. Nakamura, R. Ikeda, and M. Kubo, Coord. Cbem. Rev., 17, 281 (1975). (15) H. Negita, T. Kubo, and M. Maekawa, Bull. Cbem. SOC.Jpn., 50, 2215 (1977). (16) G. W. Leppelmeier and E. L. Hahn, fbys. Rev., 141, 724 (1966). (17) L. C. Brown and P. M. Parker, fbys. Rev., 100, 1764 (1955). (18) C. Kittel, "Introduction to Solid State Physics", 4th ed, Wiley, New York, 1971, p 529. (19) K. Yamagata, Y. Kozuka, E. Masai, M. Taniguchi, T. Sakai, and I. Takata, J . fbys. SOC.Jpn., 44, 139 (1978). (20) M. E. Lines, J . fbys. Cbem. Solids, 31, 101 (1970). (21) A. W. Pryor and P. L. Sanger, Acta Crystallogr., Sect. A, 26, 543 (1970).
Hydrogen Isotope Fractionation between Water and Aqueous Mono- and Dihydrogen Phosphate Ions A. J. Kresge" and Y. C. Tang Department of Chemistry, University of Toronto, Scarborough College, West Hill, Ontario M 1C 1A4, Canada (Received November 7, 1978; Revised Manuscript Received April 23, 1979)
The NMR method of determining H-D isotopic fractionation factors for rapidly exchanging solutes in protic solvents was extended to multicomponent systems and was then applied to aqueous L2P04-/LP042-buffer solutions (L = H or D). The results obtained, 4(L2PO4-)= 1.03 and +(LP042-)= 0.91, give an isotope effect on the ionization of L2P04-,KH/KD= 3.54, which is in excellent agreement with the directly measured value, KH f KD = 3.43.
Isotopic fractionation factors provide a method of analyzing hydrogen isotope effects which has proved to be especially effective in dealing with solvent isotope effects on reactions catalyzed by acids and bases in aqueous 0022-3654/79/2083-2 156$01.OO/O
so1ution.l This application requires knowledge of the fractionation factors for the acid and base species involved, and, since monohydrogen phosphate-dihydrogen phosphate buffers are frequently used in studies of acid-base 0 1979 American Chemical
Society
The Journal of Physical Chemistry, Vol. 83,No. 16, 1979
NMR Determination of Fractionation Factors
catalysis, we have determined fractionation factors for these ions in aqueous solution. The fractionation factor for dihydrogen phosphate ion had been determined before,2 but our measurements provide an improved value. We made our determinations by the NMR m e t h ~ d . ~ This technique makes use of the fact that the proton NMR signal from a solution in which a solute is undergoing rapid hydrogen exchange with the solvent consists of a composite line whose position, 6, depends upon the relative concentrations of protium at each of the exchanging sites. Introduction of deuterium into the system alters these relative concentrations (unless the fractionation factor is unity), and that changes 6. From this change and the stoichiometry of the system, the fractionation factor for the solute may be calculated. This NMR method has been applied so far only to solutions of a single solute, but extension to additional components, as required for the present study, is straightforward. Consider a solution containing a number of solutes with i hydrogenic sites each undergoing rapid hydrogen exchange with the solvent. The observed proton NMR chemical shift of such a system is equal to the sum of the chemical shifts of the individual sites, 6,, each multiplied by a factor expressing the fraction of exchanging protons present at that site, eq 1. The subscript zero in
6=
NoH
-60
NTH
NzH + -N61I H + -yfi2 ... NTH
NT
I ,\
this expression refers to the solvent, and N represents concentration on the mole fraction scale. Thus NiHis the mole fraction concentration of protium at the ith site; it is equal to Ni, the concentration of the ith site, times the atom fraction of protium at that site. Similarly, NP is the mole fraction concentration of deuterium at the ith site. The sum of NiH and NLDis Ni, eq 2, and the sum of all protium concentrations is NTH,eq 3.
+
NiH NiD= Ni
+ C N i H= N
N ~ H
(2)
T ~
(3)
The definition of a fractionation factor, &, eq 4,when combined with eq 2, leads to eq 5, and that, upon insertion into eq 1 gives eq 6. In dilute solution, the hydrogens of
(4)
(5)
NOH 6=---60+C NTH
NoHNisi (NoH+ NoD@JNTH
(6)
the solvent will greatly outnumber those of all solutes, and NTH N NoH.Under these conditions, NoH= 1- x and NoD = x, where x is the atom fraction of deuterium in the solvent, and eq 6 reduces to eq 7. (7)
This expression shows that the position of the proton NMR signal given by a dilute solution of several solutes undergoing rapid hydrogen exchange with the solvent is
2157
determined by the chemical shift of the solvent, a,, plus a series of terms, each representing an independent contribution from one exchanging site. It should be possible, therefore, by varying each solute concentration in turn, while holding all of the others constant, to evaluate each of the terms and thus to determine each of the fractionation factors. This would involve measurements, first in the absence of deuterium ( x = 0), to determine ai, and then in a solvent of known deuterium content, to determine & In practice, however, it may not always be possible to supply each of the exchanging sites independently and thus to vary only one N , at a time. That was in fact the case in the present study, but, fortunately, sufficient other information was available so that this problem could be overcome.
Experimental Section Materials. Potassium dihydrogen phosphate and dipotassium hydrogen phosphate were best available commercial grades (Fisher Scientific Co.) and were dried before use. Solutions were prepared by using doubly distilled, C02-free water or commercial DzO (Merck Sharpe and Dohme) as received. NMR Measurements. Measurements were made with a Varian T 60 NMR spectrometer operating a t ambient temperature; this gave a constant probe temperature of 35 "C. Chemical shifts were measured with respect to 1,1,2,2-tetrachloroethaneas external reference. Bulk susceptibility corrections were not made, inasmuch as absolute values of 6 were not needed; extraction of the desired information from the data used only changes in 6, and a constant correction, such as that for bulk susceptibility, would have dropped out. Spectra were recorded on standard chart paper at a slow sweep rate (sweep time = 250 s) and a sweep width of 100 Hz. Signal separations were measured with a millimeter ruler; distances were estimated to 0.1 mm, and chemical shifts were obtained by multiplying these by the appropriate scale factor (1 mm = 0.4 Hz). Results The NMR chemical shifts of two groups of solutions containing dipotassium hydrogen phosphate and potassium dihydrogen phosphate were measured. In one group, the concentration of the first salt was varied while that of the second was held constant, and in the second group the roles of the two salts were reversed. In both groups, measurements were made in HzO and in 5 % Hz0-95% DzO;for the first group, additional measurements were also made in 50% H20-50% D20.The data so obtained are summarized in Tables I and 11. For all five series of measurements, the observed chemical shift proved to be an accurately linear function of the salt concentration being varied; Figure 1shows some typical examples. The data were therefore subjected to linear least-squares analysis, and the slope and intercept parameters so obtained are listed in Tables I and 11. Discussion The solutions examined in the present study contain three solute components: LP042-,LzPO,, and K+.4 The first two of these solutes each have chemically bound hydrogens which exchange rapidly with the solvent; each will therefore contribute a term to the expression for 6 (eq 7). In addition, K+, though formally without exchangeable hydrogens, is solvated, and its solvating water molecules can be distinguished from those of the bulk solvent in terms of both 66 and 4;Ib this solute species will therefore also contribute a term to eq 7. These considerations lead
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The Journal of Physical Chemistry, Vol. 83, No. 16, 1979
TABLE I: Chemical Shifts of Aqueous Solutions at Constant Potassium Dihydrogen Phosphate Concentration and Varying Dipotassium Hydrogen Phosphate Concentrationa [KzLPO, 1 M
1 0 3 ~ b
6,c
A. J. Kresge and Y. C. Tang
TABLE 11: Chemical Shifts of Aqueous Solutions a t Constant Dipotassium Hydrogen Phosphate Concentration and Varying Potassium Dihydrogen Phosphate Concentrationsa [KL,PO,], M
Hz
6 , c Hz
103Nb
100% H,O 1.79 91.32 3.58 90.64 5.36 89.56 7.13 88.64 8.90 87.64 10.66 86.68 6 = 92.40 i. 0.08 -- (533.4 i. 13.7)N
0.10 0.20 0.30 0.40 0.50 0.60
100% H,O 1.79 91.40 3.58 90.12 5.36 88.88 7.13 87.56 8.90 86.28 10.66 85.04 6 = 92.68 I 0.04 - (719.2 * 10.O)N
0.10 0.20 0.30 0.40 0.50 0.60
50% H,O-50% D,O 0.10 1.81 92.56 0.20 3.59 91.36 0.30 5.38 89.84 0.40 7.16 88.16 0.50 8.93 87.24 0.60 10.69 85.80 6 = 93.96 i 0.16 - (769.2 t 24.8)N
0.10 0.20 0.30 0.40 0.50 0.60
5% H,O-95% D,O 1.80 92.96 3.60 91.76 5.39 91.08 7.17 90.04 8.95 89.16 10.72 88.16 6 = 93.84 i 0.08 -- (526.0 I 14.5)N
a [K,LPO,] = 0.10 M. N = mole fraction concentra6 = observed chemical shift with tion of KL,PO,. respect to external CHCl,CHCl,, no susceptibility correction applied. Values listed are averages of four to six measurements on same solution.
5% H,O-95% D,O 1.80 93.48 3.60 91-80 5.39 90.40 7.17 88.88 8.95 87.48 10.72 85.96 6 = 94.88 i 0.08 - (834.2 I 9.4)N
0.10 0.20 0.30 0.40 0.50 0.60
water at 25 O C is 3.7€i6 As a result of such RGM deviations, fractionation factors defined in terms of total H and D contents of multihydrogenic sites, as in eq 4, are not a [KL,PO,] = 0.10 M. N = mole fraction concentraconstant; they vary with the deuterium content of the 6 = observed chemical shift with tion of K,LPO,, respect t o external CHCl,CHCl, , no susceptibility correcwater in which they are determined. This difficulty may tion applied. Values listed are averages of four t o six be overcome by basing fractionation factors upon indimeasurements on same solution. vidual isotopic species, and there are certain advantages to choosing isotopically homogeneous substances for this g6 7 purpose.lbJ Thus, for a substrate with n equivalent hydrogenic sites, the constant “practical” fractionation factor defined in this way is given by eq 10.
g4L 92
84
0
I
I
I
2
4
6 103 N
8
10
12
Flgure 1. The relationship between the position of the proton NMR signal, 6, and the mole fraction concentration of K,LP04, N, for aqueous solutions containing a constant concentration of KL2P04: lower line, 100% H20; upper line, 5% H20-95% D20.
to eq 8, in which the subscripts 1,2, and 3 refer to LPOZ-, LzPO4-, and Kt,respectively.
This equation and its generalized precursor, eq 7 , were derived by use of a simple theory of solvent isotope effects which does not take into account deviations from the rule of the geometric mean (RGM). The theory assumes that isotopic equilibration between equivalent exchangeable sites is statistically controlled; for example, that the equilibrium constant for the disproportionation of HzO and DzO into HDO, eq 9, is 4.00. But this is not strictly correct;
K = [HD012/(~H~Ol~DzOl) (9) the best value of K determined experimentally for liquid
In the present case, RGM deviations are likely to be significant only for the water species. They will not, of course, apply to LPOZ-, with its single equivalent hydrogen; and they are likely to be negligibly small in L2P0,, where the two hydrogens are fairly remote from one another, separated as they are by three other atoms. On the other hand, RGM deviations should be significant for the water molecules in the solvation shell of Kt.However, their magnitude here is likely to be similar to that for the water molecules in the bulk solvent, and the two RGM deviations will therefore tend to offset one another in their effect on 4; as a result, RGM corrections are likely to be insignificant for the term of eq 8 which refers to K’. RGM deviations were consequently taken into account in the present study only when dealing with the terms of eq 8 which refer to L P O t - and L2P04-,and here only as they applied to the solvent species. The modified version of eq 8 which this produced is given as eq 11. 61N1 ( 1 - ~ ) (+l&[D20]1’2/[H~Ol’/2) 63N3 62N2 (11) (1- x ) ( l + $2[DzO]1/z/[H20]1/2) 1 X + 3Ccb3
+
6=&+
+
Not all of the concentration variables of this equation could be changed independently; since K+ was supplied as the counterion to LPOZ- and L2POi,its concentration was fixed by those of these other two ions, Le., N 3 was
NMR Determination of Fractionation Factors
The Journal of Physical Chemlsfty, Vol. 83, No. 16, 1979
TABLE 111: Chemical Shifts and Fractionation Factors of the Monohydrogen Phosphate and Dihydrogen Phosphate Ions @
ion LP0,2+ L,PO,-
6,a
ppm
-17.3 -11.5
35 "C
25 "C
0.900,b O.90gc 1.028'
0.901 1.029
a At unit concentration o n the mole fraction scale with respect to pure water; the negative sign indicates a downfield shift. Determined in 50% H,O-50% D,O. Determined in 5% H,O-95% D,O.
necessarily equal to 2N, + Nz. Introduction of this relationship into eq 11 gives an expression, eq 12, which 6 = 6 0 + ,
2159
The present value of 4 for LzP04-, 42= 1.03, is significantly greater than 42= 0.80 measured beforee2 In the previous work, however, no allowance was made for the counterion (Na+);the value obtained, moreover, coupled with the known isotope effect on the ionization of L2P04,8 gave, via eq 13 with 1 = 0.69,1bs3a value of 4 for LP042-, L2P04-~t LPOdZ- L+
+
KH/KD= = 3.43 (13) 0.51, which was judged to be improbably It is significant in this respect that the presently determined values of 41and 42lead to an ionization isotope effect, KH/KD= 3.54, which is in excellent agreement with the measured value, KH/KD= 3.43.8 Acknowledgment. We are grateful to the National Science and Engineering Research Council (formerly National Research Council) of Canada and the Connaught Fund of the University of Toronto for their financial support of this work. $1=
References and Notes
shows that the gradient of 6 upon N1 is determined by 63 and 43 in addition to 61 and &, and that the gradient of 6 upon N2 is likewise fixed by J3 and b3 as well as by aZ and 42. Experiments of the kind performed here cannot, therefore, by themselves, yield values of +1 and 42. Fortunately, a3 and 43 are known from other work, and with this additional information and $2 may be determined. The results obtained in this way are listed in Table 111. They are based upon $3 = 0.971b and 63 = 0.047 ppm at 35 "C; the latter is an interpolated value calculated from a3 = 0.050 ppm a t 25 "C and a3 = 0.046 at 40 0C.5 Since the present experiments were done at 35 "C, the results were corrected to 25 "C with the relationship 4 = exp(AGO IRT).
(1) (a) A. J. Kresge, Pure Appl. Chem., 8,243 (1964); V. Gold, Adv. Phys. Org. Chem., 7, 259 (1969); R. L. Schowen, Prog. Phys. Org. Chem., 9, 275 (1972); R. A. More O'Ferrall in "Proton Transfer Reactions", E. F. Caldin and V. Gold, Ed., Chapman and Hall, London, 1975, Chapter 8; (b) W. J. Albery in "Proton Transfer Reactions", E. F. Caldln and V. Gold, Ed., Chapman and Hall, London, 1975, Chapter 9. (2) M. M. Kreevoy, R. A. Landholm, and R. Eliason, J . Phys. Chem., 73, 1088 (1969). (3) A. J. Kresge and A. L. Allred, J. Am. Chem. Soc., 85, 1541 (1963); V. Gold, Proc. Chem. SOC., 141 (1963). (4) The symbol "L" is used to designate hydrogen of unspecified isotopic identity, Le., L = H or D. (5) J. Davies, S. Ormondroyd, and M. C. R. Symons, Trans. Faraday Soc., 67, 3465 (1971). (6) W. A. Van Hook, J . Chem. Soc., Chem. Commun., 479 (1972); J. W. Fyper, R. S. Newburg, and G. W. Barton, Jr., J . Chem. Phys., 48,2253 (1967); L. Friedman and V. J. Shiner, Ibd., 44,4639 (1966). (7) A. J. Kresge, Y. Chiang, G. W. Koeppl, and R. A. More O'Ferrall, J . Am. Chem. Soc., 99, 2245 (1977). (8) R. Gary, R. 0. Bates, and R. A. Robinson, J. Phys. Chem., 68, 3806 ( 1964). (9) The calculation made in ref 2 giving Cp = 0.51 was based upon KdKD on the molality scale; correction to molarity concentration units gives 4 = 0.57. The latter is still considerably short of expectation.
