Maurice M. Kreevoy' University of Minnesota Minneapolis
The Exposition of Isotope Effects on Rates and Equilibria
The advanced undergraduate and the beginning graduate student are frequently prese~lted wit,h two formulations of the theory of isotope effects: the "physical" formulation, in which the impression is given that nothing can be done without a very large number of vibrational frequencies, all known with i n possible precision; and the "organic" formulation, which often leaves the incorrect impression that potential energy surfaces change under isotopic substitution. The purpose of the present paper is to present an approach to this subject which the author feels is stimulating and without conceptual error; in many cases it, permits the student to reach correct qualitative conclusions from available data. The difference between the ionization constant of acetic acid in water and that in heavy wat,er is a manifestation of the Heisenberg uncertainty principle which can be detected with a pH meter (1, t). Classically there is no effect predicted (9); in fact K H / K Dis 3.3 (4, 5 ) . The origin of the effect is illustrated in Figure 1, in which a one dimensional version of t,he real multidimensional explanation is given. The Heisenberg uncert,ainty principle requires that the product of the uncertainty in the position and momentum of a particle (in this case, several particles, protons) be constant. The more closely the particle is confined by its potential well, the higher is its lowest permitted energy level above the bottom of that well (zero-point energy) in order that, the uncertainty in the momentum will increase as the uncertainty in position is decreased (6). Substitution of D,O for water changes all the mobile protons to deuterons. This reduces the energy required to maintain the same uncertainty product by a factor of 1/.\/2, on both sides of the equilibrium. The net result, however, is to lower the left hand side more than the right, because the factor, I/&, is multiplying a larger value to start wit,h. Partition functions also enter an exact quantitative formulation of these ideas (S), but their influence on hydrogen isotope effects is usually fairly small. The reason for the lower eeropoint energy in the dissociated form is the strong hydrogen bonds in the hydrated proton (7, 8),which permit the protons to move quite a distance along the line of the oxygen atoms before their potential energy begins to rise steeply. A simple way to generalize this is to say that, in a system where protons and deuterons are allowed to a of different deuterons tend to accumulate i n those positions where they are most closely confined by potential barriers. These potential barriers may originate either in the bonds 1
Alfred P. Slom Foundation Fellow, 196G54.
Journal of Chemical Educafion
mhich hold the hydrogen to the rest of the molecule or in nan-bonding interactions (steric repulsions). The purpose of this paper is to present a format for the presentation of these ideas to students, which is believed to tie in better with the established frame of reference of the senior or first year graduate student than those curreutly in wide use. To a reasonable level of approximation the hydrogen isotope effect on an equilibrium constant, K H / K D(9, Q), is given by equation (1) and that on a rate constant, k n / k ~(9, 10) by equation (2).
where o = a svmmetrv number and v = a vibrational frequency wherem* = theeffectiven~assforn~otionalongthe reaction coordinate in the transition state. Symmetry numbers take account of the number of equivalent sites a t which a reaction may take place (11). I n the example given in Figure 1, oESis 2 and onPis 3 but the whole ratio is unity because of cancellation with ooS and oDP. This sort of cancellation occurs when all of a group of equivalent hydrogens are sin~ultaneouslydeuterated, but may fail if only part of a group of equivalent hydrogens are deuterated. The summation is taken over all the vibrational frequencies of the molecules (and transition states) involved. The exponential tenn represents the effect of zero-point vibrational energy.
Figure 1. A one-dimensionmi representation of the dirrociotian of .,.tic in HIO and D ~ O . AII energies are by. common foctor on the rubrtitution of deuterium for hydrogen, but the left. hond lundirrocioted) ride is favored became in rero-point energy is lorger to start with, and so is reduced by a lorger obsolute omount. The harmonic otdllator ~ o l e n t i denemies shown are conaideroble over~implificationr of true potential energy;urver; porticuloriy that for H30+, which probably has o double minimum.
Equation (1) is customarily arrived a t by a consideration of the equilibria shown in equations (3) and (4), where S and P are a starting state and a product, reSa
spectively. Equation 2 can be obtained in a similar way, by replacing the product with a transition state (20,12). These equations serve very well to point up the inlportance of zero-point vibrational energy in determining isotope effects, and, in cases where the necessary vibrational frequencies are known, they can be used for the quantitative estimation of such effects. However, they present serious difficulty to the student (particularly the organic chemist) trying to place isotope effects in his customary frame of reference; inductive effects, steric effects, etc. These effects are all the direct or indirect result of changes in potential energy (IS), while the equilibria shown in equations (3) and (4) are governed by identical potential energy surfaces (14). This dilemma can be avoided by the consideration of t,he equilibria shown in equations (5) and (6). These equilibrium constants cannot be measured, but the influences operating on the ratio K5/K8can be readily Sa
interpreted in conventional terms, since, relative to protium, deuterium tends to concentrate in structures where it is more closely confined. Hence, K g will be larger if the forces, both bonding and non-bonding, confining the hydrogen, are larger in S than in P; if the inverse is true, K6 will be larger than K g ; and K 6 / K 8is identically equal to K H / K D . In the example given above, K K , for the reaction shown in equation (5a), clearly must be substantially CHsCOOH
+ HIO + 3D
+ DzO + 3H
K H / K D is ) less than unity, in accord with the observations. It is not possible to say which of the two effects is the more important. Again, formic acid is a stronger acid by 8% than deutero formic acid, both measured in water (18). The corresponding specific versions of the equilibria shown in equations (5) and (6) are given in equations (5c) and (6c). At present it is not possible to say with confidence, whether KEor K Kshould be the larger, but one can say that the observed effect is very likely of inducHCOOH
tive origin, and due to the change in charge type. Since it is known that inductive effects are generally prop* gated away from their point of origin in an orderly fashion (IS), it can be predicted that other deutero-substituted carboxylic acids will also be weaker than their proton analogues, Kn/Kn, varying with the number of substituted hydrogens and the number of bonds between them and the carboxylic acid group. This prediction is in accord with the facts (18). Primary kinetic isotope effects are very easily explained on this basis. The hydrogen atom is firmly bound in the starting state, and is in the process of being transferred in the transition state. Identifying the transition state with P, K , > Kg,so that K H / K Dcan be expected to be substantially greater than unity, as it usually is in such cases (19). Secondary kinetic isotope effects can be formulated in much the same way as isotope effects on equilibria. A particularly nice example is furnished by the recent work of Melander and Carter (20). They found that the racen~ization of 2,2'-dibromo4,4'-dicarboxy-6,6'dideuterobiphenyl is faster than that of the undeuterated analogue. The pertinent versions of equations (5) and (6) are (5d) and (6d).
larger than K6,because of the freedom for the hydrogens to move along the hydrogen bond axes in HsO+. Thus Kn/KD is predicted to be substantially larger than unity, in accord with observation. The point will be further illustrated. The equilibrium constants for the reactions shown in equations (3b) and (4b) have been measured and K A / K Dis 0.80 (CHM
Ka + (CHdrN *
(15). The alternative formulation is shown in equations (5b) and (6b).
Both steric and Baker-Nathan forces (15, 17) effects would tend to make K g smaller than Ke. Steric compressions tend to confine the hydrogens in the adduct (P)and Baker-Nathan forces tend to make them more mobile in the trimethylborane. Thus K S / K 6(equal to
The two transition states are flat, or very nearly so, and there is considerable compression of the volume available to the hydrogen atoms (21). Accordingly it is readily (and correctly) predicted that Ke > KK, and ka/ko < 1. The effect is quite plainly of steric origin. This approach is readily generalied to isotope effects Volume 41, Number 12, December 1964
involving atoms other than hydrogen. Its qualitative conclusions generally remain unchanged if the partition function and transmission coefficientterms are included in equations (1) and (2) (9, 10). It should be noted that this approach adds nothing fundamentally new to the study of isotope effects. To calculate these rigorously one would still need a complete vibrational analysis for the four species involved. I t does clarify the area within which one can discuss inductive, resonance, and steric contributions to isotope effects without doing violence to the Bigeleisen forrnalism or the Born-Oppenheirner approximation. It points out that no special effects, like anharrnonicity, need he considered to reach semiquantitative couclusions. It serves to emphasize the point that the effects actually under consideration are those of the funetinal group on the molecular vibrations. Literature Cited (1) MEKKELSEN, K., AND NIELSEN,S. O., J. Phys. Chem., 64, 632 (1960). (2) GLASOE,P. K., AND LONG,F. A., J . Phw. Chem., 64, 188 (1960). J., AND MAYER,M. G., J. Chem. Phys., 15,261 (3) BIGELEISEN, (1947).
Journal of Chemical Educofion
LA MAR,V. K., AND CHITTUM,J. P., J. Am. Chem. Soc.. 58, 1642 (1936). KORMAN, S., AND LA MAR, V. K., J. Am. Chem. Soc., 58, 1396 (1936). EYRING, R., WALTER,J., AND KIMBALL,G. E., "Quantum Chemistry," John Wiley and Sons, Inc., New York. N. Y., 1944, p. 77. BUNTON. C. A,. AND SHINER.V. J.. J. Am. Chem. Soc... 83.42 . (i96i). ' ACKERMANN, T.,Zeit. fur Physik. Chem., 27, 34 (1961). MELANDER, L., "Isotope Effects on Reaction Rates," The Ronald Press, Co., New York, 1960, chap. 1. BIOELEISEN, J., J. Chem. Phys., 17, 675 (1949). BENSON, S. W., J . Am. C h a . Sac., 80, 5151 (1958). FROST,A. A,, AND PEARSON, R. G., "Kinetics and Mechh nism," John Wiley and Sons, Inc., New York, 1961, p. 98. Tam, R. W., JR., in "Steric EBkcts in Organic Chemistry," Newmsn, M. S., ed., John Wiley and Sons, Ine., New York, chap. 13. WESTON,R. E., JR., Tetmhedra, 6, 31 (1959). LOVE,P., TAFT, R. W., JR., AND WARTIK, T., Tetrahedron, 5,116 (1959). BARTELL,L. S., J. Am. Chem. Soe., 83, 3567 (1961). SHINER,V. J., Tetrahedra, 5, 159 (1959). STREITWIESER. A,. JR.. AND KLEIN. H. S.. J. Am. Chem. SOC., 85,2759 (i963): MELANDER, o p . eit., chap. 2. Melander, L., AND C ~ RR. , E., J. A n . Chem. Sac., 86, 295 (1964). WESTHEIMER, F. H., J . Chem. Phys., 15, 252 (1947).