J. Phys. Chem. 1984, 88, 4394-4397
4394
the shell is also obtained from the second term of the above equation as
respectively. The charge q(8) per unit area on the shell can be obtained from the Gauss theorem, that is q(0) =
The distances rand r'can be expressed in terms of the distances R , x , and y , and the angle 8 shown in Figure 5 as r = [R2 x2 - 2Rx cos
r'=
+ [ R 2 + y 2 - 2Ry cos 8]112
--
B~ substituting eq A-4 and A-6 into eq A-7, we can obtain an expression for q(8) as q(e) =
4 x z - az 41r a
- -[a2+ x2 - 2ax cos 0]-3/2
(A-8)
Fractionation of Deuterium and Protium between Water and Methanolt J. H. Rolston* and K. L. Gale Atomic Energy of Canada Limited Research Company, Chalk River Nuclear Laboratories, Chalk River, Ontario, Canada KOJ 1JO (Received: October 3, 1983; In Final Form: December 28, 1983)
The overall deuterium-protium separation factor, a,between hydrogen gas and aqueous methanol mixtures has been measured over the full composition range at temperatures between 25 and 5 5 OC. At each temperature a increases smoothly with increasing mole fraction of methanol but the values fall significantly below the straight line joining the separation factors for the methanol-hydrogen and water-hydrogen systems. The equilibrium constant, Kl(l), for exchange of a deuterium atom tracer between the hydroxyl groups of methanol and liquid water, calculated from the values of cy for each solution, is independent of composition within experimental error. The value of Kl(l) at 25 OC is 0.54 f 0.02, so that deuterium favors the methanol environment rather than water. The dependence of Kl(l) on absolute temperature, T, is given by the expression In K,(l) = -0.776 + 52.6/T, which corresponds to a reaction enthalpy of -0.43 kJ-mol-'.
Introduction The facile exchange of hydrogen isotopes between water and organic hydroxyl groups is of interest in the study of isotope effects in biological systems where it is generally assumed that a change in the isotopic content of the solvent produces a corresponding change in the isotopic content of the hydroxylic hydrogens in cellular materials.' Extensive deuteration impairs the survival of many higher organisms, whereas some algae and bacteria are able to withstand complete replacement of protium by deuterium. Such replacement solvent isotope effects are manifest in sizable changes on reaction rates when solutes exchange their hydrogens with the solvent or when there are specific solute-solvent interactions present.2 Fractionation factors, introduced by Gold3 and Kresge4 to describe isotopic exchange equilibria between the bulk solvent and hydrogen atoms in specific positions within the solute, have provided a quantitative basis for understanding these effects on simple chemical reactions. Extension of the theory to more complex biological systems or to mixed solvents requires knowledge of the equilibrium partitioning of deuterium between the different types of hydroxylic hydrogens. In addition, liquid phase diagrams and thermodynamic properties, such as solution enthalpies and excess free energies, show effects attributable to exchange reactions typified by those in the methanol-water system. At low deuterium concentrations a random distribution would lead to a liquid-phase equilibrium constant Kl(l) = 0.500 for reaction 1. Early experimentaP6 and theoretical7studies indicated CH,OH(l) + HDO(1) s CH,OD(l) + HZO(1) (1) that deuterium favored the water species with the equilibrium constants at 25 OC significantly less than the statistical value. In contrast Bertrand and Burchfields reported a value of Kl(l) = 0.6 from their thermochemical study of water-methanol mixtures. Combination of independent measurements of deuterium-protium separation factors between hydrogen gas and liquid water9 and liquid methanolI0 has been used to estimate a value of Kl(l) =
0.56 f 0.02 at 25 OC.ll This is in good agreement with the experimental value obtained by weighting the data of ref 6 according to the precision of the spectroscopic method." More recently Khurma and FenbyI2 have recalculated reduced partition functions for methanol by using harmonic frequencies derived from deharmonized values given by Mallin~on.'~At 25 OC a value of K,(l) = 0.54 is reported whereas a value of 0.58 was obtained with the spectral frequencies reported by Schlegel, Wolfe, and Bernardi.I4 The revised values remove the previous discrepancy between calculated and experimental enthalpies of deuterium exchange between methanol and water. These, and the estimate of the deuterium isotope separation factor of water against methanol quoted by Bigeleisen, Hammond, and Tuccio,IS show that the statistical mechanical calculations agree reasonably well with the experimental values derived from (i) thermochemical (1) J. J. Katz and H. L. Crespi, "Isotope Effects in Chemical Reactions", C . J. Collins and N. S. Bowman Eds., Van Nostrand-Reinhold, New York, 1970, ACS Monogr. No. 167, p 286. (2) L. Melander and W. H. Saunders, Jr., "Reaction Rates of Isotopic Molecules", Wiley, New York, 1980, p 202. (3) V. Gold, Trans. Faraday Soc., 56, 255 (1960). (4) A. J. Kresge, Pure Appl. Chem., 8, 243 (1964). 32, 1033 (1936); J. 0. Halford and (5) W. J. C. Orr, Trans. Faraday SOC., B. Pecherer, J . Chem. Phys., 6, 571 (1938). (6) H. Kwart, L. P. Kuhn, and E. L. Bannister, J . Am. Chem. SOC.76, 5998 .... (1954). (7) D. V. Fenby, Aust. J . Chem., 30, 2371 (1977). (8) G. L. Bertrand and T. E. Burchfield, J . Phys. Chem., 79, 1547 (1975). (9) J. H. Rolston, J. den Hartog, and J. P. Butler, J . Phys. Chem., 80, 1064 (1976). (10) J. H. Rolston and K. L. Gale, J. Phys. Chem., 88, 163 (1984). (1 1) D. E. Clegg, and J. H. Rolston, J . Chem. SOC.,Chem. Commun., 1037 (1978). (12) J. R. Khurma and D. V. Fenby, Aust. J . Chem., 32, 465 (1979). (1 3) P. D. Mallinson, J. Mol. Spectrosc., 58, 194 (1 975). (14) H. B. Schlegel, S. Wolfe, and F. Bernardi, J. Chem. Phys., 67, 4181 \ - - -
I .
(\ 1- -977) " I '
(15) J. Bigeleisen, W. B. Hammond, and S. Tuccio, Nucl. Sci. Eng., 83, 473 (1983).
T AECL-8506
0022-3654/84/2088-4394$01 S O / O
Published 1984 American Chemical Society
Fractionation of Deuterium and Protium
The Journal of Physical Chemistry, Vol. 88, No. 19. 1984 4395
data, (ii) the ratios of separation factors, and (iii) the statistically weighted spectroscopic study. All support a value of Kl(l) significantly greater than 0.500. This conclusion is strengthened by the recent thermochemical study of Khurma and FenbyI6 which assigns K,(1) = 0.6 f 0.1 for methanol as well as for ethanol and 1-propanol. Vapor pressure measurements have also been used" to obtain molar excess free energies and a value of Kz(l) = 1.09 f 0.02 at 25 OC.
+ DzO(1) + 2CH,OD(1) + HzO(1) HzO(1) + DzO(1) ~t 2HD0(1)
ZCH,OH(l)
(YL =
(3)
*
Experimental Section Procedure. The flask used to equilibrate the hydrogen and aqueous methanol mixtures was similar to that used in earlier studiesl0 in which catalyst spheres of wetproofed platinized carbonZ0 were suspended in the vapor space above the liquid. Methanol (BDH, ARISTAR) was dried over activated 4A molecular sieves (BDH). Stock solutions of methanol and water containing known initial deuterium atom fractions (FMi= Fwi= 0.030) were prepared by dilution of weighed amounts of heavy water (atom fraction = 0.99969) and dry methanol (CH,OD, Mercke Sharpe and Dohme, atom fraction = 0.9798 by NMR) with light water and methanol, respectively. Intermediate compositions were prepared by dispensing, in a moisture-free glovebag, weighed amounts of these solutions from sealed syringes. The cell used for equilibration with hydrogen was filled and brought to temperature as previously described.lo Analysis. Hydrogen gas standards, prepared for earlier studies on methanol,20were used to calibrate the Micromass 601 mass spectrometer. Equilibrated samples of the pure solvents with hydrogen gas gave overall deuterium-protium separation factors, a , in good agreement with previous studies of methanolI0 and water.21 Calculations. At low deuterium concentrations dideuterated species can be neglected (at 25 O C with FLe = 0.030 the mole fractions of HzO, HDO, and DzO are 0.941,0.058, and 0.00093, respectively) and the atom fraction of hydroxylic hydrogens from methanol, XMe,can be written in terms of the moles (Ni)of species i in the liquid as
The deuterium atom fraction present in the liquid phase a t equilibrium, FLe,at each temperature can be expressed in terms of XMe,Xwe, and the corresponding deuterium atom fractions in the hydroxyl groups of methanol, F M e = NMaD/(NMaD NMaH), and water, Fwe= NHw/(2NHz0 2NHDO), as shown in eq 4. The
+
(16) J. R. Khurma and D. V. Fenby, J . Phys. Chem., 83, 2443 (1979). (17) Z. S. Kooner, R. C. Phutela, and D. V. Fenby, Aust. J . Chem., 33, 9 (1980). (18) V. Gold, Trans. Faraday SOC.,64, 2270 (1968). (19) A. J. Kresge and Y . Chiang, J . Chem. Phys., 49, 1439 (1968). (20) J. P. Butler. J. H. Rolston. and W. J. Stevens. "Seuaration of Hvdrogen Isotopes", H. K. Rae, Ed., American Chemical Society, Washingtoh, DC, 1978, ACS Symp. Ser. No. 68, p 93. (21) J. H. Rolston, J. den Hartog, and J. P. Butler, J. Phys. Chem., 80, 1064 (1976).
FMe(l - Fwe))(Fwe(l - F M e ) )
(5)
It follows that FLe
= [(XMeaL/C'(Fwe))
+ (1 - X M ~ ) ] F W ~
(6)
where C'(Fwe) = 1 + F W y a L
(2)
If we use the relation Kz(l) = [K1(1)]2K3(1)with K3(1) = 3.85,18319 it follows that Kl(l) = 0.53 0.01. The uncertainty in this value is attributed to the variation of K,(1) with the mole fraction over the range 0.4-0.8. While all recent studies indicate that equilibrium 1 lies to the right, there has been little success in determining Kl(l) over a range of solvent composition or temperature. The following summarizes the results of such a study in which Kl(l) has been obtained from direct measurements of the overall deuterium-protium separation factors between hydrogen gas and aqueous methanol mixtures.
+
separation factor between the hydroxyl group of liquid methanol and water in the mixed solvent is given by
-
1)
(7)
The overall liquid separation factor, a , between hydrogen and the mixed solvent is defined in terms of F L e and the deuterium atom fraction in the hydrogen at equilibrium, xe, by the relation CY
= FLe(l - Xe)/(Xe(l - FLe))
(8)
and can be calculated from XMe,Fwe, and aL. Values of Fwefor a given solution were obtained at each temperature with the aid of a deuterium mass balance taken over the gas and liquid phases and the assumption that the separation factor, aL,for partitioning between methanol and water is the same for liquid and vapor phases
where A(Fwe) = CYVW+ Fwe(l - ~ V W )
(10)
+ F M ' ( ~- CYVM)
(1 1)
+ FWi((YL- 1)
(12)
B ( F M ~=)
~ V M
C(F,') = 1
The symbols FMi,Fwi, xi, and xe refer to the initial (i) or equilibrium (e) deuterium atom fractions present in the methanol (M) and water (W) of the liquid and hydrogen (x), respectively. R is the atom ratio of the exchangeable hydrogens in the initial hydrogen to those of the combined liquid phase at equilibrium and S is the atom ratio of exchangeable hydrogens in the vapor to those in the liquid at equilibrium. The separation factors, aVM, between methanol vapor and liquid methanol, and avw, between water vapor and liquid water, are defined respectively in terms of atom fractions by
Results Values of the overall separation factor a between hydrogen and a series of aqueous methanol mixtures, determined at 25, 40, and 55 O C , are listed in Table I. A plot of a as a function of the equilibrium mole fraction of methanol in the liquid is shown in Figure 1. The a values were calculated from the initial composition with corrections for the quantity of methanol and water vapor obtained by interpolation of the vapor composition data reported in ref 17 and 22. At each temperature a increases smoothly with increasing methanol fraction; however, all data points fall significantly below the dotted line joining the separation factors of hydrogen with respect to the pure components. By combining eq 6, 7, and 9 and assigning values of aL one can calculate a range of a values for comparison with the experimentally measured value at each solvent composition. For each solution the value of aLwhich gives agreement between calculated and experimental a values is listed in column 6 of Table I. (22) M. L. McGlashan and A. G. Williamson, J . Chem. Eng. Data, 21, 196 (1976).
4396
The Journal of Physical Chemistry, Vol. 88, No. 19, 1984
Rolston and Gale
TABLE I: Separation Factors, a, between Hydrogen Gas and Aqueous Methanol Solutions D atom fraction separation factor
temp, OC 25.05 25.03 24.96 24.98 24.15 25.44 24.44 26.43
MFM" 0.107 0.207 0.340 0.503 0.644 0.743 0.846 0.897
106xe 7708 6530 5816 10242 7898 7737 7030 7512
106Fwe 28294 24228 21 547 37208 29199 28403 28422 27402
40.06 40.15 39.99 40.17 39.44 40.92 39.52 40.67
0.107 0.207 0.340 0.503 0.644 0.743 0.846 0.897
8362 7171 6385 11193 8673 8428 8484 8259
28308 24254 21 589 37325 29328 28553 28611 27601
55.22 54.89 54.87 54.86 54.12 55.82 54.09 55.43
0.107 0.207 0.340 0.503 0.644 0.743 0.846 0.897
9116 7783 6946 12239 9401 9178 9092 8853
28319 24274 21622 37419 29432 28673 28764 27763
a
ffLb
3.806 3.862 3.891 3.945 4.053 4.082 4.162 4.200
1.036 1.141 1.117 1.115 1.130 1.130 1.130 1.130 1.12 f 0.03 1.137 1.093 1.089 1.112 1.120 1.137 1.118 1.113 1.12 f 0.02 1.021 1.056 1.060 1.078 1.115 1.124 1.130 1.124 1.09 f 0.04
mean 3.509 3.517 3.545 3.613 3.681 3.747 3.792 3.802
mean 3.218 3.240 3.258 3.302 3.406 3.440 3.539 3.535
mean
"Mole fraction of methanol. bThe separation factor aL refers to deuterium fractionation between the hydroxylic hydrogen of methanol
and water.
42
00
01
I
I
I
I
I
I
I
1
02
03
04
05
06
07
08
09
10
ATOM FRACTION METHANOL ( X i )
Figure 2. Comparison of calculated and experimental overall separation factors a with atom fraction of hydroxylic hydrogen atoms in methanol-water mixtures at 25 (top), 40 (middle), and 55 (bottom) O C using eq 15 and selected values of LYL. Solid lines obtained with aL values defined by limiting values aMand aw. Bars indicate 1% error in individual measurements.
,
At XMe= 1 it follows that a = aMwhere aw is the deuteriumprotium separation factor between hydrogen and liquid water and aM is that between hydrogen and liquid methanol.
/'
, ,
40
aw = FW"1
- x e ) / ( x " ( l - F$)]
(16)
At low deuterium atom fractions these separation factors are related to the equilibrium constants of reactions H D HzO(1) F? HDO(1) HZ (18) H D CH3OH(l) s CH,OD(l) Hz (19)
+
+
34-
+
+
measurements.
by the relations aw= KI8(1)and aM= 2KI9(1), respectively. A comparison of the experimental a values with those obtained by using values of aLcalculated directly from the overall separation factors of the pure components at each temperature is shown as the solid lines in Figures 1 and 2. Thus, the curvature of the lines against MFM displayed in Figure 1 is not due to the variation of the aL with composition but arises from the choice of axis and the relationship between the atom fraction of exchangeable hydrogen atoms in methanol and the methanol mole fraction
Discussion Figure 1 shows that the overall separation factor between hydrogen and aqueous methanol solutions increases smoothly with increasing methanol mole fraction (MFM) between the limits imposed by the pure components. An analytical expression for the dependence of a on the atom fraction of exchangeable hydrogens in methanol (XMe)can be obtained by combining the forms of eq 6 and 8 appropriate for the low deuterium atom fractions used in these experiments, Le., when FLeand Fwe
