J. Phys. Chem. 1883, 87,4715-4720
coefficient. The two regions described above are representative of a membrane diffusion controlled and an aqueous diffusion film controlled permeation process. Acknowledgment. We thank D. Kalina and C. Cianetti for performing the Kd measurements of Figure 5A and the 7 measurements of Figure 4. This work was performed
4715
under the auspices of the Office of Basic Energy Sciences, Division of Chemical Sciences, U.S. Department of Energy, under Contract W-31-109-Eng-38. Registry No. Americium, 7440-35-9; formic aid, 64-18-6; noctylphenyl(N,N-diisobutylcarbamoylmethyl)phospineoxide, 83242-95-9.
A Quasiclassical Trajectory Calculation of the H Constant
+ C2H4
-
C2H, Bimolecular Rate
Kandadal N. Swamy and Wllllam L. Hase" Department of Chemlsfry, Wayne State Universlfy, Defroif, Michigan 48202 (Received: April 18, 1983)
-
The dynamics of the H + C2H4 C2H5 recombination reaction are studied with quasiclassical trajectories. For the 200-500 K temperature range the quasiclassical trajectory bimolecular rate constant agrees with that calculated from activated complex theory with a tunneling correction.
Introduction For modeling the role of elementary bimolecular reactions in chemical reaction mechanisms it is important to have accurate methods for calculating their rate constants. A converged three-dimensional quantum reactive scattering calculation on a quantitative potential energy surface is the most accurate theoretical way to determine a rate c0nstant.l However, basis set limitations preclude the general applicability of such calculations. To date, only for the H + Hz exchange reaction has a converged threedimensional quantum reactive scattering calculation been c~mpleted.~~~ Transition-state theory is the most widely used method for calculating bimolecular rate constants. Important assumptions concerning the reaction dynamics are made to derive the theoretical expression for the rate ~ o n s t a n t . ~ Two corrections are emphasized to make the calculation of the rate constant more quantitative. By using the variational criterion to choose the transition state, the "best" bottleneck (i.e., dividing surface) is found along the reaction path and an approximate classical upper bound to the rate constant is reali~ed.~ An exact classical upper bound results if a dynamical prescription is used for selecting the dividing surface.6 In transition-state theory the threshold for reaction is the vibrationally adiabatic barrier height. In a three-dimensional quantum scattering calculation reaction occurs at energies below the vibrationally adiabatic barrier as a result of effects referred to as tunneling and vibrational nonadiabaticity. These effects are accounted for in the transition-state theory bimolecular rate constant by including a one-dimensional tunneling correction?-l0 Recent (1)R. B. Walker and J. C. Light, Annu. Reu. Phys. Chem., 31, 401 (1980). (2)A. B. Elkowitz and R. E.Wyatt, J.Chem. Phys., 62,2504 (1975). (3)G. C. Schatz and A. Kupperman, J.Chem. Phys., 65,4668(1976). (4)W. H. Miller, Acc. Chem. Res., 9,306 (1976). (5)D. G. Truhlar and B. C. Garrett, Acc. Chem. Res., 13,440(1980). (6)P. Pechukas and E.Pollak, J. Chem. Phys., 71,2062 (1979). (7)R.A. Marcus and M. E. Coltrin, J. Chem. Phys., 67,2609(1977). (8)W. H. Miller, N. C. Handy, and J. E. Adams, J . Chem. Phys., 72, 99 (1980). (9)J. M. Bowman, G.-Z. Ju, and K. T. Lee, J.Phys. Chem., 86,2232 (1982). 0022-3654/83/2087-47 15$01 SO10
work has focused on the best procedure for making this Quasiclassical trajectories provide another means for calculating bimolecular rate constants. For reactions with an early transition state, so that the transition state more closely resembles the reactants than products, quasiclassical trajectories often give accurate rate constants."J5 This results from the finding that the vibrational nonadiabaticity of the quasiclassical trajectories can mimic the occurrence in a quantal calculation of reaction at energies less than the vibrationally adiabatic barrier.3J1J6J7 Some reactions for which quasiclassicaltrajectories appear to give accurate bimolecular rate constants are H + H2,11J8F HZ,l1J8C1 + H2, 11719 0 H2,11J5and OH Hz.20 Ab initio calculations21-26show that the H + C2H4 C2H5 reaction has an early saddle point, which has a structure more like that of the reactants than products.
+
+
+
-
(10)D. G.Truhlar, A. D. Isaacson, R. T. Skodje, and B. C. Garrett, J. Phys. Chem., 86,2252 (1982). (11)B. C. Garrett, D. G. Truhlar, R. S. Grev, and A. W. Magnuson, J.Phys. Chem., 84,1730 (1980). (12)R. T. Skodje, D. G. Truhlar, and B. C. Garrett, J . Phys. Chem.. 85. 3019 (1981). '(13)R:T. Skcdje, D. G. Truhlar, and B. C. Garrett, J. Chem. Phys., 77,5955 (1982). (14)A. D. Isaacson and D. G. Truhlar, J. Chem. Phys., 76,1380(1982). (15)G.C. Schatz. A. F. Waener. S. P. Walch. and J. M. Bowman. J . Chem. Phys., 74,4984(1981);k. T.Lee, J. M. Bowman, A. F. Wagner, and G. C. Schatz, ibid., 76,3563 (1982). (16)R.A. Marcus, J . Phys. Chem., 83,205 (1979). (17)J. S. Hutchinson and R. E. Wyatt, J. Chem. Phys., 70, 3509 (1979). (18)V. Khare, D.J. Kouri, J. Jellinek, and M. Baer in "Potential Energy Surfaces and Dynamics Calculations", D. G. Truhlar, Ed., Plenum, New York, 1981,p 475. (19)B. C. Garrett, D. G. Truhlar, and A. W. Magnuson, J . Chem. Phys., 74,1029 (1981). (20)G. C. Schatz and S. P. Walch, J. Chem. Phys., 72,776 (1980);G. C. Schatz, ibid., 74,1133 (1981). (21)C. S. Sloane and W. L. Hase, Discuss. Faraday Soc., 62,210 (1977). (22)W. L. Hase, G. Mrowka, R. J. Brudzynski, and C. S. Sloane, J . Chem. Phys., 69,3548 (1978);erratum, 72,6321 (1980). (23)S. Nagase and C. W. Kern, J.Am. Chem. SOC.,102,4513 (1980). (24)S. Kat0 and K. Morokuma, J . Chem. Phys., 72,206 (1980). (25)H. B. Schlegel, J. Phys. Chem., 86,4878 (1982). (26)H. B. Schlegel, K. C. Bhalla, and W. L. Hase,J . Phys. Chem., 86, 4883 (1982).
0 1983 Amerlcan Chemical Society
4716
Swamy and Hase
The Journal of Physical Chemistry, Vol. 87, No. 23, 1983
TABLE I: Equilibrium Geometriesa ethylene
saddle point
coordinate
this surface
rCH
1.080
1.086
1.085
1.081 a 0.002
r* R
1.330
1.341
1.335
1.334 i 0.002
116.0 180.0 116.0 180.0
116.2 180.0 116.2 180.0
116.4 180.0 116.4 180.0
117.4 i. 0.3
81 AI
a1 A2 a2
MP2/3-21Gb MP2/6-31G*
experimentC
117.4 + 0.3
this surface
1.080 2.000 1.379 104.1 115.9 194.2 116.8 184.8
MP2/3-21G 1.085, 1.084d 1.845 1.346 107.4 115.7 192.6 116.3 177.2
a Bond lengths are given in angstroms and angles in degrees. The coordinates are defined in Figure 1 o f ref 27. Ab initio results from ref 25. J. L. Duncan, M o l Phys., 28, 1177 (1974). The first value is for the carbon to which the hydrogen is adding.
TABLE 11: Harmonic Vibrational Frequenciesa ethylene frequencies type o f mode
symmetry
CH stretch CC stretch CH, scission
this surface
exptb
HF/3-21GC
2955 1669 1313 938 3037 1155 775 902 3025 90 9 2926 1440
3152 1655 1370 1044 3231 1245 969 959 3234 843 3147 1473
3238 1842 1522 1165 3371 1387 1115 1157 3404 944 3305 1640
A, A, A, AU
torsion
CH stretch CH, rock CH2 wag CH, wag CH stretch CH, rock CH stretch CH, scission
saddle point frequencies
Bl, B,, BIU
B2, B2U B2u
B,U B3U
reaction coordinate CH* twistd CH* scission
symmetry
this surface
HF/3-21GC
A'
2946 1579 1290 63 5 3037 1138 80 6 90 3 3026 91 1 2925 1450 630i 277 26 1
33 20 1625 1332 1054 3368 1368 976 1061 3411 922 3308 1690 614i 448 415
A' A' A' ' A' A' '
A' A' A' A'
I
A' A' A' A" A'
a Frequencies are given in c m - ' . C. W. Bock, P. George, and M. Trachtman, J. Mol. Spectrosc., 76, 191 (1979). H* represents the H atom adding t o ethylene. initio results from ref 25.
In addition the reaction has a low vibrationally adiabatic barrier. From these properties one might expect quasiclassical trajectories to yield an accurate bimolecular rate constant. This proposal is examined in the work presented here.
Method of Calculation Potential Energy Surface. The analytic potential energy surface used in this study has been described previously.n A convenient name for the analytic surface is MAPS/ C2H,-I, where MAPS denotes "molecular anharmonic potential written with switching functions" and I indicates version I for the analytic surface. The surface was derived in part from ab initio calculations. The surface satisfactorily reproduces the geometry and harmonic vibrational frequencies determined for ethylene and the saddle point from experiment and ab initio calculations, respectively. Equilibrium geometry comparisons are given in Table I. The coordinates are defined in Figure 1 of ref 27. The harmonic vibrational frequencies for ethylene and the saddle point are compared in Table 11. One minor deficiency in the analytic surface is that it has saddle point frequencies which are too low for the torsion, CH* twist, and CH* scission vibrations. The zero-point vibrationally adiabatic potential along the reaction path is
Ab
where V&) is the classical potential energy along the reaction path, the vi(s) are the frequencies for the 14 vibrations orthogonal to the reaction path, and E , is the zero-point energy for the reactant molecule ethylene. Since the zero-point energies for ethylene and the saddle point are nearly the same as are the frequencies, harmonic frequencies can be used to determine the VaG(s)curve in the vicinity of the saddle point.% The classical threshold for the analytic potential energy surface is 3.50 kcal/mol, and the value of VaC(s)at the saddle point is 3.70 kcal/mol larger than for the reactants. The vibrationally adiabatic threshold is 3.74 kcal/mol and is located slightly nearer the products than is the saddle point.29 Analyses of the temperature dependence of the experimental H + C2H4 bimolecular rate constant suggest a vibrationally adiabatic barrier of 2.0-2.5 kcal/mo1,26p30which is 1.0-1.5 kcal/mol smaller than that for the surface used here. The principal moments of inertia obtained from the analytic surface are 3.40, 16.8, and 20.2 amu A2 for ethylene, and 6.89, 22.7, and 22.8 amu A2 for the saddle point. Selection of Initial Conditions. In the quasiclassical trajectory calculations ethylene is treated as a symmetric top with Z, = Zy = 18.4 amu A2 and I, = 3.40 amu A2.For a collision between an atom and a symmetric top polyatomic molecule the reaction cross section averaged over
14
VaG(s) = V,..(s)
+ iChvi(s)/2 -E, =l
(1)
(27) W. L. Hase, D. M. Ludlow, R. J. Wolf, and T. Schlick, J. Phys. Chem., 85, 958 (1981).
(28) For potential energy surfaces with significant frequency and zero-point energy differences between the reactant and saddle point, anharmonic frequencies are essential in calculating an accurate VaG(s) curve; see, for example, ref 14. (29) W. L. Hase, Acc. Chem. Res., 16, 258 (1983). (30)W. L. Hase and H. B. Schlegel, J.Phys. Chem., 86, 3901 (1982).
Calculation of the H
+ CzH,
-
CH ,,
Bimolecular Rate Constant
vibrational and rotational energies is Sr(EreJ =
C C f'(ni) f'(J,K) ar(Erei,J&,ni) n8JX
(2)
where a, is the cross section for a specific initial state with vibrational energy specified by the quantum numbers ni, rotational energy specified by the quantum numbers J and K , and the relative translational energy ErepP(nJ and P(J,K) are normalized Boltzmann probabilities. The bimolecular rate constant is a thermal average of S,(E,J and is given by
All the quasiclassical trajectory calculations reported here were performed with ethylene in its ground vibrational state; i.e., each mode contains zero-point energy. This restriction is valid for temperatures less than 500 K, where the excitation of higher vibrational levels for any mode is less than 10%. In the previous quasiclassical trajectory excitation of ethylene vibrations was found to cause no significant change in the reactive cross section, and as discussed in a following section accurate rate constants can also be computed with zero-point energy in ethylene for temperatures much higher than 500 K. The principal moments of inertia for ethylene are sufficiently large that for temperatures 200 K and above the rotational quantum numbers J and K can be treated as continuous variable^.^^ The angular momentum along the symmetry axis is L, = K h and the total angular momentum is L2 = J(J + l ) h 2 = (Lx2+ L,2 + LZ2).As shown by Bunker,32L, and L are sampled from the probability distributions P(L,) = exp(-LzZ/2I,kbT)
O IL, I00
(4)
P(L) = L exp(-L2/21,kbT)
L, I L I
(5)
m
Equation 4 is sampled by the Von Neumann rejection method and eq 5 by the cumulative distribution function formula L = [ L , ~- 21,kbT In (1 - R)]1/2
(6)
where R is a random number uniformly distributed in the interval (0-1). Negative and positive values for L, have equal probabilities. The components L, and Ly are found from L, = L sin a sin p L, = L sin a cos p (7) where cos (Y = L,/L and p is chosen randomly between 0 and 2 ~ . The impact parameters for the H + C2H4collisions are chosen from b = bmJ21/2. Ethylene is randomly oriented by rotation through Euler's angles, and the phases for the normal modes are chosen at random. Specific equations have been given p r e v i ~ u s l y for ~ " ~transforming ~~ the normal-mode momenta and coordinates, the rotational angular momenta, the impact parameter, and relative velocity to center-of-massCartesian coordinates which are used in the trajectory calculations. Computational Details. The trajectories were numerically integrated with combined fourth-order Runge-Kutta and sixth-order Adams-Moulton algorithms. A fixed time step of 2.00 X 10-l6s, chosen by backward integration tests, was used for all the trajectories. The trajectories were (31) N. Davidson, "Statistical Mechanics", McGraw-Hill, New York, 1962. (32) D. L. Bunker and E. A. GoringSimpson, Faraday Discuss.Chem. SOC.,55, 93 (1973). (33) C . S. Sloane and W. L. Hase, J. Chem. Phys., 66, 1523 (1977).
The Journal of Physical Chemistry, Vol. 87, No. 23, 1983
4717
computed on the Amdahl47OV/6 computer at the Wayne State University Computing Services Center and on a VAX 11/780 computer in the Wayne State University Biochemistry Department. In order to compute statistically meaningful reactive cross sections, the number of trajectories integrated varies from lo00 to 250 at the lowest and highest relative translational energies, respectively. Values range from 2.5 A (the highest relative chosen for b, translational energies) to 1.4 8, (lowest relative translational energies). Reactive Cross Section Individual trajectories were classified by following bond angles and lengths vs. time. Particular emphasis is placed on the distance (r*) between the attacking H atom and the nearest carbon atom. A vibrationally-rotationally excited ethyl radical CzH5* was considered formed if two or more inner turning points occurred in r* at distances less than the CH equilibrium bond length of 1.08 A. The cross section for formation of these radicals is called the primary reactive cross section. Its experimental analogue is the reactive cross section in the high-pressure limit where the CzH5* radicals are collisionally stabilized. Trajectories which have only one inner turning point in r* at a distance less than 1.08 8, are unimportant for relative translational energies less than 20.0 kcal/mol; i.e., the cross section for this type of event is more than an order of magnitude smaller than that for forming CzH5*. However, this type of trajectory does become important at higher relative translational energies.27 The internal motion of the C2H5* radicals was followed for -3.5 X s (-35 CH vibrational periods) to ascertain the extent of decomposition in this time period. For initial relative translational energies in the 2.5-5.0 kcal/mol interval approximately 5% decomposed, while approximately 10% decomposition ensued for relative translational energies in the range of 6.0-20.0 kcal/mol. Detailed H + C2H4 CzH6*trajectory calculations at a relative translational energy of 30.0 kcal/moP4 show that C2H5* decomposition becomes RRKM after only 2.0 X 10-13-4.0 X s of internal motion. Significant nonRRKM decomposition is unexpected for the C2H5*radicals. Values for the reactive cross section Sr(Erel) are shown in Figure 1 for rotational temperatures of 0 and 300 K, and for ErelI 4.0 kcal/mol. For calculations at Erel= 2.75 kcal/mol and Trot= 0 K, and at Erel= 2.25 kcal/mol and Tmt= 300 K no reactions occurred out of lo00 trajectories. If one reaction had occurred, the reactive cross section would be 0.006 A2 and would make a negligible contribution to the rate constant for temperatures in excess of 200 K. Reaction occurs below the vibrationally adiabatic threshold of 3.74 kcal/mol as a result of vibrationally nonadiabatic effects in the classical dynamics. For Trot= 300 K Sr(EreI) extends to lower values of Erelthan at Trot = 0 K. Each set of points in Figure 1 is fit to the equation
-
SrWreJ
= 4 1 - expl-b(Erei - 411'
Sr(EreJ = 0.0
Ere1
Ere1
2d
(8)
at Erelequal to 6.0 and 10.0 kcal/mol exciting CzH4vi4.0 kcal/mol are fit to eq 8 with parameters a = 3.6671, brations causes only minor variations in the reactive cross b = 0.17243, c = 1.0088, and d = 3.7000. section. Vibrational enhancement of the reactive cross section may occur at E,, less than the vibrationally adiaBimolecular Rate Constants batic barrier. However, as shown in Table I11 these enQuasiclassical Trajectory. Bimolecular rate constants ergies only make a small contribution to the rate constant vs. temperature were calculated from the trajectory S,(E,,) for temperatures greater than 500 K. Thus, no significant
Calculation of the
H
+ C2H,
-
C,H, Bimolecular Rate Constant
displayed in Figure 3 over the range of temperature for which experimental measurement^^^"^ of the rate constants have been made. As the temperature is increased the rate constant calculated from activated complex theory with a tunneling correction becomes larger than that from the quasiclassical trajectories. Such a result is expected on dynamical grounds, since the probability for trajectories to recross the dividing surface increases as the temperature is inThis recrossing violates the postulate of activated complex theory5 and as a result the trajectory rate constant is less than that calculated from activated complex theory.
1I
-’lo
-14.0
The Journal of Physical Chemistry, Vol. 87,No. 23, 1983 4719
I
I
I
I
1000/ T
+
Figure 3. Logarithm of the H C,H, C,H, bimolecular rate constants in cm3/(mdecules) vs. inverse temperature In units of K-’. The sdid line is for activated complex theory with the Wigner tunneling correction. Open and solid circles are the quasiclassical trajectory results with T, = 0 and 300 K, respectively. +
difference is expected in the bimolecular rate constant if a vibrational temperature equal to the translational temperature is used instead of 0 K. Activated Complex Theory. In previous work2eit has been shown that the variational criterion is unimportant in calculating the activated complex theory bimolecular rate constant for the analytic potential energy surface used in the trajectory calculations. Also, the molecular geometry at the vibrationally adiabatic barrier and at the saddle point are nearly identical so that placing the activated complex at the saddle point gives a rate constant similar to that found by placing the activated complex at the vibrationally adiabatic barrier. Thus, to simplify the activated complex was placed at the saddle point. As Truhlar1+l4 has pointed out, this is in general not a good assumption and it only is used here because the geometries, vibrational frequencies, and vibrationally adiabatic potential energies at the saddle point and vibrationally adiabatic barrier are similar; and because the variational criterion is unimportant. Activated complex theory rate constants calculated without a tunneling correction, and with the Wigner,35 E ~ k a r tand , ~ parabolic37tunneling corrections are given in Table 111. In each calculation the activated complex is placed at the saddle point and the imaginary frequency used in the tunneling corrections is that for the saddle point. The energies used in the tunneling corrections are the zero-point energy differences between the reactants, saddle point, and product. At 200 K the different tunneling corrections increase the activated complex theory rate constant by a factor that varies from 1.8 to 2.9. At the higher temperatures tunneling becomes less important and the Wigner, &kart, and parabolic tunneling corrections are similar. Including a tunneling correction in the activated complex theory calculations results in rate constants at the lower temperatures which agree with those from the quasiclassical trajectory calculations. A comparison of the quasiclassical trajectory rate constants with those calculated from activated complex theory with the Wigner correction is (35)H.S.Johnston, ‘Gas Phase Reaction Rate Theory”, Ronald Presa, New York, 1966,p 134. (36)Reference 35,p 40. Also see B. C. Garrett and D. G. Truhlar, J. Phys. Chem., 83,2921(1979),where important corrections to the &kart tunneling equations in H. S. Johnston’s text are given. (37)R. T. Skodje and D. G. Truhlar, J . Phys. Chem., 85,624(1981).
Discussion The important finding from this study is that quasiclassical trajectories and activated complex theory with a tunneling correction give values for the H + C2H4 C2H5 bimolecular rate constant which agree over the temperature range the reaction has been studied experimentally. Though approximations are introduced in the activated complex theory calculations, they are minor for this reactionm and removing them is not expected to negate the agreement found between the two sets of rate constants. Using the variational criterion to choose the transition state instead of placing it at the saddle point will decrease the activated complex theory rate constant, and including curvature12 iri the tunneling correction will increase it. Both of these corrections are expected to be small and will to some degree Since the potential energy surface used in this work differs from the best deduction of the actual surface,25p26,30 the calculated rate constants do not agree with the experimental ones. Specifically, the potential energy surface used here is estimated to have a vibrationally adiabatic threshold 1.0-1.5 kcal/mol too large and the frequencies at the saddle point for the torsion and the CH* bending frequencies are approximately 50% too small. However, these are not major disparities and a surface for which they are removed is expected to also yield a quasiclassical trajectory recombination rate constant which agrees with that calculated from activated complex theory with a tunneling correction. Reaction occurs quantum mechanically at energies below the vibrationally adiabatic threshold from tunneling and vibrational nonadiabatic effects. For the H + C2H4 C2H5 reaction classical mechanics seems to give an appropriate amount of vibrational nonadiabaticity so that the quasiclassical trajectory recombination rate constant agrees with that of activated complex theory with a tunneling correction. This finding is consistent with the work of others, where quasiclassical trajectories give accurate rate constants for surfaces with an early saddle point if tunneling is not too i m p ~ r t a n t . ~ This ~ J ~isJ the ~ ~ situation for the H + C2H4 C2H5 reaction. However, additional work is certainly necessary to determine the applicability
-
-
-
(38)J. H.Lee, J. V. Michael, W. A. Payne, and L. J. Stief, J . Chem. Phys., 68,1817 (1978). (39)R.Ellul. P.Potzineer. B. Reimann. and P. Camilleri. Ber. Bunsenges. Phys. Chem., 85,i07’(1981). (40)KO-ichi Sugawara, K.Okazaki, and S. Sato, Chem. Phys. Lett., 78,259 (1981);Bull. Chem. SOC.Jpn., 54, 2872 (1981). (41)B. C. Garrett and D. G. Truhlar, J. Am. Chem. Soc., 101,4534 (1979). (42)For our trajectories the recrossing occurs without r* attaining the CH equilibrium bond length of 1.08 A; see ref 27. (43)Placing the transition state at the saddle point and making a tunneling correction which neglects curvature is often found to have an empirical accuracy. For example, see B. C. Garrett and D. G. Truhlar, J. Phys. Chem., 83,200 (1979);B. C.Garrett and D. G. Truhlar, ibid., 83, 1079 (1979),where the Wigner tunneling correction is used.
4720
J. Phys. Chem. 1983, 87, 4720-4724
of quasiclassical trajectories to polyatomic bimolecular reactions in general and recombination reactions in particular. In a future study we plan to repeat the quasiclassical trajectory calculations on a more accurate POtential energy surface so that a direct comparison can be made with the experimental rate constants.
Acknowledgment. This research was supported by the National Science Foundation. The authors thank Professors William H. Miller and Donald G . Truhlar for very helpful discussions. Registry No. Atomic hydrogen, 12385-13-6; ethylene, 74-85-1.
Nonlinear Regression Models of Multicomponent Interactions of Anhydropolyols with Aqueous Ammonium Ion by Carbon-13 Nuclear Magnetic Resonance DavM G. Naugler and Robert J. Cushky' Department of Chemistry, Simon Fraser Unlverslty, Bwnaby, Brnish Columbia, Canada V5A IS6 (Received: May 19, 1981, In Final Form: February 23, 1983)
The anydropolyhydric alcohols 1,4-anhydroerythritol and 1,4-anhydrothreitol were synthesized from the correspondingtetratols via acid-catalyzeddehydration, under vacuum, using H+ion exchange resin. Comparison of temperature-dependent 13C NMR chemical shifts in aqueous and nonaqueous solvents demonstrated a significant molecular association interaction between 1,4-anhydroerythritoland water. The chemical shifts of C(1) and C(2) of 1,4-anhydrothreitoland C(1) of 1,4-anhydroerythritolshowed a linear increase with temperature but C(2) of 1,4-anhydroerythritolshowed a decrease between -0 and -55 "C followed by an increase from -55 to -100 "C. A multiligand complex was shown between 1,4-anhydroerythritol,water, and ammonium ion. The stoichiometry of the interaction between 1,4-anhydroerythritol,water, and aqueous ammonium ion was determined by nonlinear regression analysis applied to data obtained from 13CNMR chemical shifts through A general model was variation of temperature, solvent activity, aH,o(c,T), and solute activity, UNH,~,(C,T). constructed for the dependence of the 13C NMR chemical shifts upon temperature and the activities of the two ligand species.
Introduction The oxalane structures 1,4-anhydroerythritol (oxalane3,4-cis-diol), I, and 1,4-anhydrothreitol (oxalane-3,4tram-diol), 11, form the central moiety of the biochemically important molecules adenosine triphosphate, ribonucleic acid, adenosine 3',5'-cyclic phosphoric acid, cyclic adenosine monophosphate, and D-fructose 6-phosphate., The latter molecule is an entry point in the glycolytic pathway. The interactions of I and I1 with other species are governed by the lone pairs of electrons on the oxygen atoms, which may participate in the interactions one, two, or three at a time. The symmetries of I and I1 are C,Jm and C2,, respectively, and, in any interaction with a species of higher symmetry in solution, this symmetry will be preserved on the average. To data a considerable amount of work has been done on the interaction of various metal ions with sugars and related compounds in different solvent systems. For instance, 23NaNMR has been usedl for the study of sugarsodium interactions in pyridine solution. Since it has been shown that the sodium cation coordinates on one face to three or more oxygen atoms from the sugar, and on the other face with nitrogen atoms from the solvent molecules, there is a substantial electrostatic field gradient at the quadrupolar nucleus. Line broadening can be analyzed to reveal details of this complexation. Complexations of lanthanide ions with alditols and polyhydric alcohols have been used as model studies for the complexation of sugars with isosteric Ca(II).2+3 I t was
shown that a lanthanideaorbitol complex in D20solution, for example, was formed with the sorbitol adopting a normal zig-zag conformation while using 0-2,0-3, and 0-4 as ligands. Haines et ale4studied the interactions of carbohydrates with sodium ions in acetone-d6 solution by 'H NMR. In the case of 1,4-anhydroerythritol, the chemical shift data for the three observed protons was analyzed by the Scott modification of the Benesi-Hildebrand method and yielded equilibrium constants for complex formation of KH(1) 6 M-', KH(2) 1M-', and KH(n 2.5 M-le4 The variation in the values obtained by Haines et al.* is clearly the result of the nonvalidity of assumptions inherent in their treatment. This point will be discussed at greater length subsequently. In the present study, we derive the chemical shift expression for multicomponent equilibria based on activities. We also show that, using a canonical approach, the chemical shift equation can also be derived from the partition function which emphasizes the central role played by temperature. Based on nonlinear regression analysis of 13Cchemical shift data through variation in temperature, solvent activity, and solute activity, a model for the multicomponent equilibria of 1,4-anhydroerythrito1 and aqueous ammonium ion is presented.
(1) Detellier, C.; Grandjean, J.; Laszlo, P. J.Am. Chem. SOC.1976,98, 3375. (2) Kieboom, A. P. G.; Sinnema, A.; van der Toom, J. M.; van Bekkan, H. J.R. Neth Chem. SOC.1977, 96,35.
(3) Spoonmaker, T.; Kieboom, A. P. G.; Sinnema, A.; van der Toorn, J. M.; van Bekkan, H. Tetrahedron Lett. 1974, 3713. (4) Haines, A. H.; Synes, K. C.; Wells, A. G. Carbohydr. Res. 1975,41,
0022-3654/83/2087-4720$0 1.50f0
-
-
-
Theory Derivation of Chemical Shift Expression for Species Undergoing Competitive Multiequilibrium (Fast-Ex-
85.
0 1983 American Chemical Society
