Richard D. Bardo and
Max Wolfsberg
A Theoretical Calculation of the Equilibrium Constant for the Isotopic Exchange Reaction between H20 and HD Richard D. Bardo and Max Woifsberg* Department of Chemistry, University of California, Iwlne, California 92717 (Received September 22, 1975) Publication costs assisted by the U.S. Energy Research and Development Agency
A theoretical calculation is presented for the equilibrium constant of the isotopic exchange reaction H20 + HD = HDO H2 between 220 and 600 K and compared with experimental data in the temperature range 280-600 K. A correction for the Born-Oppenheimer approximation is included in the calculation. The theoretical curve of the equilibrium constant as a function of temperature differs from the experimental leastsquares curve obtained between 280 and 475 K by a t most 0.7%.
+
Introduction Experimental values of the equilibrium constant K1 for the isotopic exchange reaction HzOtg)
+ HD(g) = HDO(g) + H2(g)
(1)
have been reported in the literature by three groups of worke r ~ . The ~ - ~interest in this equilibrium arises, at least in part, from the possibility of economically producing heavy water from the exchange between water and hydrogen. The latest data are described in the preceding paper3 and were obtained by combining measured data for an equilibrium such as that in eq 1,for liquid water, with values of the vapor pressure ratio H20/HDO deduced from experiment. The experimental values of K1 will be compared here with those obtained theoretically by statistical and quantum mechanics. It should be noted that effects of gas phase nonideality were reasonably considered negligible in the experimental evaluations of K1. Theoretical Calculation Bron, Chang, and Wolfsberg4 (BCW) obtained ideal gas phase partition function ratios QHDOJQH~O and QHD/QH~ a few years ago within the framework of the Born-Oppenheimer (BO) approximation and expressed the logarithms of these ratios in a power series of 1/T (where T is the absolute temperature). The series reproduced the partition function ratios with a maximum error of 0.1% over the temperature range 220-600 K. Within the framework of the BO approximation, the equilibrium constant can be expressed in terms of the partition function ratios as
Ki =
(QHDO/QH~O)/(QHD/QHJ
(2)
With use of the above series expansions, the following expression is then obtained for K1 In Kl(BCW) = -0.17739
+ 0.61704
+ 1.12486
(7)'-
0.36997
(7)3
+ 0.07978 (y)4(3) Equation 3 can be used between 220 and 600 K. Since it forms the basis of the values theoretically calculated here within the framework of the BO approximation, comparison is only made The Journal of Physical Chemistry, Vol. 80,No. 10, 1976
with those experimental data below 600 K (Cerrai et a1.2also reported data a t seven temperatures between 600 and 1015
K). Recently, Kleinman, Wolfsberg, and Bardo516 have considered that calculations such as Kl(BCW) must be corrected for the BO approximation. Within the BO approximation, the molecular quantum mechanics problem can be divided into one of electronic motion (with nuclei fixed) and one of nuclear motion. The potential function for nuclear motion is the electronic energy as a function of internuclear coordinates. The minimum of this potential function will be designated here as the zero of vibrational-rotational-translational energy (ZVRTE) of the molecule. Since the electronic energy of a molecule is independent of the nuclear masses of the substituent atoms, the ZVRTE of a molecule is independent of isotopic substitution. Thus, for an exchange reaction such as eq 1,the energy change between the ZVRTE of reactants and the corresponding ZVRTE of products vanishes. When first-order corrections to the BO approximation are considered, this energy change is no longer equal to zero, but is given by AE = -ABOELE = C(H2) - C(HD) - C(H20) C(HD0) (4)
+
where C is the isotope dependent adiabatic correction evaluated a t the equilibrium BO internuclear configuration of the respective molecule. AE leads to a multiplicative correction factor K(B0ELE) to the equilibrium constant calculated within the BO approximation K(B0ELE) = exp(ABOmE/kT) (5) Kleinman and Wolfsberg5 have shown that K(B0ELE) is expected to be the main correction to the BO approximation for theoretically calculated isotope exchange equilibrium constants among diatomic molecules. They have also shown that one expects correction factors of the order of a few percent in the case of H/D exchange equilibria involving diatomic molecules. The assumption is now made that K(B0ELE) represents the main correction factor to K1 from failure of the BO approximation. C values for Hz and HD have been previously rep0rted.j Evaluations of C for HzO and HDO have just been ~ o m p l e t e d Both . ~ sets of C values were evaluated with electronic wave functions which may be described as being close to the Hartree-Fock limit. It is found that (appropriately rounded-off)
Isotopic Exchange Reaction between H20and HD
1069
TABLE I: Comparison of Xi Calculated with the BO Approximation Kl(BCW) (Eq 3), K1 Calculated with Inclusion of the Adiabatic Correction Kl(BW) (Eq 8), Experimental Values Kl(expt), and the Values from the Least-Squares Fit Experimental Curve Ki(ls expt) (Eq 9)"
A
B
C
279.7 280.0 286.0 288.0 291.2 292.4 295.4 297.0 304.0 305.4 313.1 313.4 314.0 323.5 324.5 330.2 333.9 344.4 354.6 355.3 364.2 367.9
4.01 4.00 3.86 3.82 3.75 3.73 3.66 3.63 3.50 3.48 3.35 3.34 3.33 3.19 3.18 3.10 3.05 2.92 2.81 2.80 2.71 2.68
3.93 3.92 3.79 3.75 3.68 3.66 3.60 3.57 3.44 3.42 3.29 3.29 3.28 3.14 3.12 3.05 3.00 2.88 2.77 2.76 2.67 2.64
3.87 f 0.02c 3.94 3.79 3.73 f 0.04 3.60 3.64 f 0.04 3.61 f 0.01 3.46 f 0.04 3.41 f 0.01 3.42 f 0.02 3.37 3.31 f 0.00 3.28 3.16 3.08 f 0.00 3.09 2.99 f 0.04 2.87 f 0.02 2.74 2.73 2.65 f 0.04 2.62
3.03 2.98 2.86 2.75 2.74 2.66 2.62
324.2 332.2 337.2 347.2 350.7 352.6 362.7 370.2 382.2 391.7 407.2 421.6 441.2 459.7 475.2 492.2 524.2 553.2 588.2
3.18 3.07 3.01 2.87 2.85 2.83 2.73 2.66 2.55 2.48 2.37 2.28 2.17 2.08 2.01 1.95 1.84 1.75 1.67
3.13 3.02 2.96 2.84 2.81 2.79 2.69 2.62 2.52 2.44 2.34 2.25 2.14 2.05 1.99 1.92 1.82 1.74 1.66
3.04 f 0.06 2.84 f 0.06 2.77 f 0.06 2.81 f 0.05 2.79 f 0.05 2.72 f 0.05 2.65 f 0.05 2.54 f 0.05 2.48 f 0.05 2.46 f 0.05 2.30 f 0.05 2.24 f 0.05 2.15 f 0.04 2.08 f 0.04 1.99 f 0.04 1.95 f 0.04 1.85 f 0.04 1.73 f 0.04 1.60 f 0.04
3.11 3.00 2.94 2.83 2.79 2.77 2.67 2.60 2.50 2.43 2.33 2.24 2.14 2.05 1.98 1.92 1.82 1.74 1.66
353.2 373.2 393.2 413.2 433.2 473.2
2.82 2.63 2.47 2.33 2.21 2.02
2.78 2.59 2.43 2.30 2.18 2.00
2.79 2.57 2.42 2.29 2.17 1.98
2.76 2.58 2.42 2.29 2.18 1.99
3.91 3.90 3.77 3.72 3.66 3.63 3.58 3.55 3.42 3.40 3.27 3.27 3.26 3.12
0.6 2 .O
2.5
3.0
3.5
40
IOOO/T
-
Figure 1. Plot of In K1 vs. lOOO/T: (- -) K1 calculated withink0 approximation K1(BCW) eq (3); (-) K7calculated with inclusion of adiabatic correction Kl(6W) (eq 8); (- -) least-squares fit to experimental values Kl(ls expt)(eq 9) in the temperature range T = 280-475 K.
3.11
a The temperatures T were chosen to be those of the experimental values. A refers to experimental values from ref 3, B from ref 2, C from ref 1. Kl(expt) values for ref 3 are obtained by combining the measured values of the equilibrium constant for reaction 10 with the vapor pressure ratio H20/HDO given by eq 10 of ref 3. Whenever error limits are given for Kl(expt) from ref 3 under A, more than one experiment was performed at this temperature and the error limits show the range of the results. For Kl(expt) under B, the error limits are those given by the authors of ref 2.
ABOELE = -3.8 cm-l
(6)
K(B0ELE) = exp(-5,5/T)
(7)
and
When the correction factor K(B0ELE) is applied to K1 evaluated within the BO approximation [K1 = Kl(BCW)K(BOELE)], the following expression is now obtained for K1: In K1(BW) = -0.17739
+ 1.10658
- 0.36997
(y)3 + (F)4 0.07978
(8)
In Table I, Kl(BCW) corresponding to the BO approximation, and K1(BW) corresponding to inclusion of the adiabatic correction are compared with the experimental data, Kl(expt), below 600 K. Comparison is also made with Kl(ls expt) which corresponds to the least-squares curve obtained by Rolston et al? to fit their own data and that of the other two series of measurements between 280 and 475 K In Kl(ls expt) = -0.2735
449 2 + 2380 +T T2
In Figure 1,comparison is made between Kl(BCW), KI(BW), and Kl(ls expt) over the temperature range 270-480 K. Such agreement between theoretical calculations and experimental results, obtained independently, must be termed as excellent. Discussion The numerical values of Kl(BCW), K1(BW), Kl(expt), and Kl(ls expt) in Table I demonstrate the good agreement between theory and experiment and the fact that Kl(BW) leads to better overall agreement with experiment than does Kl(BCW). The numerical values show that the correction factor K(B0ELE) = Kl(BW)/KI(BCW) is about 0.98 in the room temperature region and decreases with increasing temperature. If the numerical values are not rounded off as The JOUrn8l of Physical Chemistry, Vol. 80, No. 10, 1976
1070
Richard D. Bardo and Max Wolfsberg
in Table I, it follows that Kl(BW)/Kl(ls expt) is about 1.003 at 273 K, reaches a maximum of 1.007 around 330 K, and then decreases to 1.001 a t 480 K. While Kl(ls expt) really only applies to the temperature range below 475 K, Kl(ls expt) and Kl(BW) are in almost perfect agreement a t the relevant temperatures above 475 K. The comparison in Figure 1 clearly confirms the good agreement between the theoretical curves Kl(BCW) and K1(BW) and the experimental curve Kl(ls expt). Moreover, the inclusion of the adiabatic correction in K1(BW) leads to better agreement with the experimental curve. As already noted by Rolston et the measurements in ref 2 at 332 and 337 K appear spurious. For most of the other measured values in ref 2, there is agreement with K1(BW) when the stated error limits of the experiments are considered. The measured values of ref 1are in excellent agreement with the K1(BW) values. As noted above, Rolston et al.3 combined data for the equilibrium constant HzO(1)
+ HD(g) = HDO(1) + Hz(g)
(10)
with the vapor pressure ratio HzO/HDO. The analytic expression which they employed for the vapor pressure ratio reproduces the experimental data of Majoube8 mostly to better than 0.1%.Since Majoube’s measurements of the vapor pressure ratio and the measurements of Rolston et al. of the equilibrium constant of eq 10 were both carried out with a very low D/H ratio, any effects due to liquid nonideality in HzOD20 should cancel in the evaluation of K1 from these data. The main error in values of K1 deduced experimentally by Rolston et al. most probably arises from the uncertainty in their measured equilibrium constants for reaction 10. Rolston et al. estimate the overall accuracy of their measurements to be f l % . They also state that this error estimate is reasonable since it approximately equals the standard deviation (f1.2%) of their five measurements at 333.9 K. One might regard this estimate of accuracy somewhat optimistic when one looks a t their experimental values centered around 293.5 K, but, as Rolstong has pointed out, the situation does not look so bad if one experimental point a t 297.0 K is omitted. Kl(ls expt) of Rolston et al. is obtained by combining a least-squares fit expression to equilibrium constant data for reaction 10 with the above mentioned analytic expression for the vapor pressure ratio HzO/HDO. It is to be noted that the equilibrium constant data for reaction 10 included not only their own data but also that of ref 1 and 2 appropriately converted to the liquid phase. Rolston et al. estimate the error in the leastsquares fit expression for the equilibrium constant of reaction 10 to be f1.3%. BCW4 evaluated isotopic partition function ratios within the BO approximation on the basis of isotope independent force fields and equilibrium geometries. The philosophy underlying theoretical isotope effect calculations carried out in this laboratory and by most other workers in this field is that such a procedure is superior to using available spectroscopic data on isotopic molecules directly in the evaluation of isotopic partition function ratios, since small errors in isotope effects on spectroscopic measurements can lead to large errors in the calculations.1° It must also be noted that some of the factors which enter into partition function calculations such as Gall cannot usually be directly deduced from spectroscopic data. In the BCW calculations, the partition function ratio was calculated in the so-called harmonic approximation4 and then various correction factors were applied on the basis of perturbation theory. It was shown that the only major The Journal of Physical Chemistry, Voi. 60, No. IO, 1976
correction factor is the anharmonicity correction ANHC except in the case of the molecular hydrogen molecules where the correction for quantum rotation QMRC is also important. It was pointed out by BCW that the choice of molecular potential functions could introduce uncertainties into the partition function ratios. The potential functions are based on spectroscopic data (interpreted within the BO approximation5). It is not felt that there is much problem with the partition function ratios of molecular hydrogen. For the water partition function ratios, a potential function deduced by Kuchitsu and Morino12 (KM) was employed. This potential function is based on an evaluation of spectroscopic data by Benedict, Gailar, and Plyler13 (BGP). The partition function ratio QHDO/QH~O has also bee: evaluated from the MUBFF force field of Papougek and P11va.l~This force field was also chosen to yield agreement with the BGP data. QHDO/&H~O calculated in the harmonic approximation with the MUBFF force field differs from that calculated with the KM force field by less than 0.1% over the temperature range considered here. If the anharmonicity correction is also considered, a maximum deviation of 0.2% occurs a t the lowest temperatures of the experiments here and the deviation decreases with increasing temperature.15 In view of the above, the KI(BCW) values are expected to be quite precise values of the equilibrium constants within the framework of the BO approximation. It is suspected that the main uncertainty in the K(B0ELE) correction factor, which takes into account the deviation from the BO approximation, is in the use of Hartree-Fock limiting wave functions rather than configuration interaction wave functions; at this time no uncertainty estimate can be given. Overall, considering the accuracy of both the theoretical calculations and the experiments, the agreement of Kl(BW) and Kl(ls expt) should not be really surprising. Nevertheless, the agreement is very gratifying. K(B0ELE) has been calculated for very few e q ~ i l i b r i a .The ~ ? ~present instance represents the first time where a theoretical calculation of an isotopic exchange equilibrium constant, including a nonnegligiblel6 adiabatic correction, has been compared with experiment.
Acknowledgment. It is a pleasure to acknowledge discussions with Dr. R. A. Rolston, and also the technical help of L. A. Laing in the preparation of this manuscript. This research was supported by the U S . Energy Research and Development Agency under Contract No. AT(04-3)-34, Project Agreement No. 188. References and Notes H. Suess, Z.Naturforsch., 42, 328 (1949). E. Cerrai, C. Marchetti, R. Renzoni, L. Roseo, M. Silvestri, and S.Villari, Chem. Eng. Prog. Symp. Ser., 50, No. 11, 271 (1954). J. H. Rolston, J. den Hartog, and J. P. Butler, J. Phys. Chem., preceding paper in this issue. (4) J. Bron, C. F. Chang. and M. Wolfsberg, 2.Naturforsch. A, 28, 129 (1973). it may be noted that atomic masses, rather than nuclear masses, were used in evaluating the partition function ratios. This choice will be the subject of a future discussion (see also ref 5). (5) L. Kleinman and M. Wolfsberg, J. Chem. Phys., 59,2043 (1973); 60,4740, 4749 (1974); M. Wolfsberg and L. Kleinrnan, Am. Chem. Soc.,Symp. Ser., 11. 64 (19751. (6)R . D . Bardo and M. Woifsberg, J. Chem. Phys., 82,4555 (1975). (7) R. D. Bardo and M. Wolfsberg, to be submitted for publication. (8) M. Majoube, J. Chim. Phys., Physiochim., Biol., 68, 1423 (1971). (9) J. H. Rolston, private communication. (10) Such a procedure was employed, however, by Y. Bottinga, J. Phys. Chem., 72, 4338 (1968), in his calculation of KI. (11) M. Wolfsberg, A. A. Massa, and J. W. Pyper, J. Chem. Phys., 53, 3138 (1970).
Indefinitely Self-Associating Systems (12) (13) (14) (15)
1071
K. Kuchitsu and Y. Morino, Bull. Chem. SOC.Jpn., 38, 814 (1965). W. S. Benedict, N. Gailar, and E. K. Plyler, J. Chem. Phys., 24, 1139 (1956). D. PapoJek and J. Pliva, Collec.Czech. Chem. Commun., 29, 1973 (1964). M. Wolfsberg, J. Chem. Phys., 50, 1484 (1969), has previously calculated K1 within the BO approximation with use of the MUBFF force field for water. This calculation differed slightly from the BCW calculation since in the BCW
calculation all constants were calculated directly from the force field while in the earlier calculation the values calculated by Papousek and Plivai4 from their force field were used (see ref 11). (16) Bardo and Wolfsbergs have shown that, for self-exchange equilibria such as H20 -t D20 = 2HD0, the adiabatic correction makes no contribution to the equilibrlum constant.
Analysis of Sedimentation Equilibrium Results Obtained with Indefinitely Self-Associating Systems Using a Procedure Based on Laplace Transformation L. W. NIchol,* P. D. Jeffrey, and B. K. Milthorpe Department of Physical Biochemistry, John Curtin School of Medical Research, Australian National University, Canberra, A.C. T. 260 1, Australia (Received December 22, 1975)
Equations are developed in closed form which permit the simulation of the distribution of total concentration vs. radial distance obtainable in the sedimentation equilibrium of a system of specified initial concentration undergoing indefinite self-association. Simulated distributions obtained with a variety of systems involving one or more equilibrium constants are used to test an analysis procedure which fits the distribution to a function capable of inverse Laplace transformation and leads to a specification of the relative amounts of the species in the cell as a function of their molecular weights. It is shown that such results may be related to the equilibrium concentrations of oligomeric forms at each radial distance, thereby permitting successive equilibrium constants appropriate to the indefinite self-association to be estimated. Several systems have been shown by analysis of sedimentation equilibrium results to be indefinitely self-associating.14 In general these may be specified by A,-l+ A1 Ajr one or more equilibrium constants governing the successive addition of monomer to higher polymeric forms. Major emphasis has been placed on cases (isodesmic) where only one equilibrium constant pertain~.l-~ The interpretation of more complicated systems is undeniably more difficult, since attempts to define successive equilibrium constants on the basis of the weight fraction of monomer as a function of total concentration7 are known8 to be subject to considerable uncertainty when the number of oligomeric states is large. Concurrent with the interest in self-associating systems has been the development of an analysis procedure based on the use of Laplace transformation to define the composition of a solution containing several noninteracting solutes?-ll It is the aim of the present work to explore the possibility that the latter method may be applied to chemically interacting systems undergoing indefinite self-association of monomer. Such an approach has the potential of not only elucidating isodesmic self-associations, but also of providing estimates of the successive equilibrium constants appropriate to more complicated systems.
Basic Theory The sedimentation equilibrium distribution of each species i (i = 1 , 2 , . . . , a) in a thermodynamically ideal system comprising monomer (i = 1)in chemical equilibrium with a series of polymeric states (Aj-1 + A1 * Aj) is described by ci ( p ) = C i ( ~ , ) e ~ M ~ ( ~ 2 - ~ m 2 ) (la)
4i = (1- uip)w2/2RT
(lb)
where ci(r) is the weight concentration at radial distance r of species i whose molecular weight and partial specific volume are Mi and ui, respectively; r , is the radial distance of the meniscus, w the angular velocity, p the solution density, and T the temperature. It will be assumed that all 4i = 4, implying through the equality of all fi; that no volume changes are inherent in the formation of successive polymers: in turn, this means that successive equilibrium constants are independent of pressure and hence of radial distance. The amount of species i in the cell does not equal the corresponding amount in the original solution,12but is given by13
where 6 and h are the cell sector angle and thickness and b = (rb2 - rm2),rb being the radial distance of the base. Combination of eq l a and 2 yields
ti = 4Mib and t = (rb2- r2)/b (3b) Equation 3 may be divided by the total amount of solute in the cell, which according to mass conservation is given by QT
= @oh(b/2) = C (QiLii
(4)
1
where c* is the original loading concentration. Thus
-
ci(E) = c?’tifietL(l-E)/(eti 1)
(5)
where f ; = (Qi)cell/QT.It follows that the total concentration at each radial distance is The Journal of Physical Chemistry, Vol. 80, No. IO, 1976
