II I IV l I IIWIIIIIH 124 170 HTHS ‘E‘HEGKT MAL [E‘W‘Efi é GATE GEN CF CARfiQN-fl [SOTGPE EFFECTS EN THE THERMAL imMEREZA'EEGN OFF CYC LOPfiGPANE Tim“ 5‘9? {he Degree of M. 5. WCEEGEH STATE UNIVERSITY Hams Paul Edward Sachse 3.967 TH ESE: [J L IR R A R Y Pv'liciiigzin State .1 University L ‘—_- ABSTRACT THEORETICAL INVESTIGATION OF CARBON-15 ISOTOPE EFFECTS IN THE THERMAL ISOMERIZATION OF CYCLOPROPANE by Hans P. E. Sachse The unimolecular reaction rate theories of N. B. Slater and of R. A. Marcus were applied to existing rate data for the unimolecular isomerization of cyclOpropane. Complete vibra- tional calculations were carried out for cyclOprOpane, cyclo- prOpane-de, cyclo-Cé2Cl5H6 and cyclo-C%3H6. In the course of the vibrational analysis of cyclopropane, an investigation of the ring modes of species E' resulted in the assignment of the 884(2) cm.l frequencies to the ring vibrations rather than the 1050(2) cm.l frequencies as some other workers have done. The unimolecular reaction rate constants were calcula- ted in the high-pressure limit and also as a function of pres- sure, using both the Slater and the RRKM theories. Then cal- culations were carried out to compute the kinetic isotOpe effect for carbon-15 substituted cyclOprOpane molecules at the high-pressure limit and as a function of pressure. The vibrational analysis of the molecule allowed the necessary amplitude factors for the Slater theory to be cal— culated. The shapes of the curves for the pressure dependence 1 Hans P. E. Sachse of the molecules were in agreement for high pressures; however, the shape of the fall—off of the carbon-15 isotope effect with pressure and the low-pressure limit could not be reconciled with the observed values, reflecting the incorrectness of the underlying assumptions of the Slater theory. For the calculations of the kinetic isotOpe effect using the RRKM theory, a frequency pattern had to be deter- mined for both unlabeled and isotOpically substituted activa- ted complexes. An empirical method was employed to fix the vibrational frequency pattern of the complexes and the complex geometry of Setser was used. The prescription used in assign— ing the complex frequencies involved: the removal of a C-—H bond stretch as corresponding to the internal translational motion along the reaction coordinate; the changing of the three ring modes into a C——C and a 03LC bond stretch, and a C-C-C angle bend in the complex; the shift of the twisting frequencies to simulate hindered rotation of the end CH2 groups in the complex; and, finally, other frequencies were unchanged or changed toward their corresponding prOpylene frequencies, the magnitudes being closer to the prOpylene frequencies as indicated by the large entropy of activation. The resulting icomplex frequencies were adjusted slightly so that the isotOpe effect would have the correct temperature dependence. The isotOpe effect is a ratio of the isotOpic rate con- stants at any pressure and thus small changes in the frequency pattern of the isotOpically substituted complex relative to Hans P. E. Sachse the unlabeled complex will change the value of k/k' at high pressures more than those at low pressures, which are essen- tially independent of the complexes. This has the effect of markedly altering the shape of the fall-off of the isotOpe effect as a function of pressure. It was not possible to construct a physically reasonable vibrational frequency pat- tern for the complexes by the above empirical method that would give reasonable agreement with experiment. The sensi- tivity of the calculated results to small changes in the com- plex vibrational frequency patterns indicate both that the isotOpe effect is a more sensitive test of the details of the theory than the isomerization rate constant results and that the empirical method which has been successful in analyzing the isomerization rate constant results and the deuterium isotope results cannot be used in the case of the relatively small carbon-15 kinetic isotOpe effect. Thus the more direct, but more complicated, approach of performing complete vibra- tional analyses of the isotOpic complexes seems necessary for carrying out RRKM calculations of the carbon-15 kinetic iso- tOpe effect. THEORETICAL INVESTIGATION OF CARBON-15 ISOTOPE EFFECTS IN THE THERMAL ISOMERIZATION OF CYCLOPROPANE By Hans Paul Edward Sachse A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Chemistry 1967 ’i C ‘x :3 9",. IN)" \ \ ” {‘7'} To my parents ii ACKNOWLEDGMENT The author wishes to express his ape preciation to Dr. L. B. Sims for his guid~ ance and encouragement during the course of this investigation. iii Chapter +4 +4 1-4 III. IV. TABLE OF CONTENTS INTRODUCTION . . . . . GENERAL A. B. C) D? APPLICATION OF THE SLATER AND RRKM THEORIES TO THE ISOMERIZATION OF ISOTOPIC CYCLOPROPANES Introduction . The Slater Theory The Marcus Theory DEVELOPMENT OF THE THEORIE (I) Effect of IsotOpic Substitution . Introduction . Th (D The Slater Theory The RRKM Theory . I. Introduction 2. Calculation of the Rate Constant 5. Construction of the Complex Models DISCUSSION . . . . . . «A 0 Nature of the Comparison Between Theory Experiment . . . Slater Calculations RRKM Calculation Results and Comparison with Experiment I. Slater Theory 2. RRKM Theory iv Vibrational Problem 0 47 47 48 TABLE OF CONTENTS - Continued Chapter BIBLIOGRAPHY . . . . . . . . Page 53 TABLE LIST OF TABLES Page Observed and Calculated Frequencies and Principal Moments of Inertia of Isotopic Cyc10pr0pane MOleculeS O O I O O O O 0 I I 0 O O O O O O O O 0 0 51 Symmetrized Eigenvectors for Cyc10pr0pane . . . . . 52 The Calculated Carbon-15 IsotOpe Effect, k/k', as Compared with Experiment a) At the High-Pressure Limit b) As a Function of Pressure . . . . . . . . . . . 50 vi I. INTRODUCTION The thermal decomposition of a uniform gas is termed "unimolecular" if, at a fixed temperature, the rate expres- sion is first-order in reactant concentration at high concen- trations and becomes second-order at sufficiently low concen- trations. The decrease of the so-called first-order rate constant with concentration from a limiting value at high concentrations to a concentration dependent second-order rate constant is referred to as the "fall—off." Early experiments revealed the existence of first-order reactions which appar- ently could not be explained by a second-order collisional activation mechanism. However, the strong temperature de- pendence shown by these first-order rate constants, making them expressible in the Arrhenius form, indicated that the dissociation of a molecule required it to have high energy. And yet, the apparent independence of the rate constant from concentration seemed to indicate that this energy was not ac- quired by collision. It remained for Lindemanni6 to prOpose his mechanism whereby attention was focused on the internal motions of the molecule. At the time little was known about these motions except that they would probably prove to be complicated. Lindemann prOposed that a collisional activation mechanism 1 2 for the attainment of high energies would be possible if the time between collisions were significantly shorter than the lifetime of the high energy or activated molecule. In this manner, a steady-state concentration of active molecules could be maintained and their rate of dissociation would be limited by the concentration of reactant molecules; and the reaction is first-order. On the other hand, as the concen- tration is decreased, the situation would be encountered where the lifetime of the active molecules would be shorter than the time between collisions, such that the rate of re- action would be limited by the second-order collisional ac— tivation process. It should be noted that, in order for a molecule to exhibit unimolecular behavior, it must contain a sufficient number of energy "sinks" (i.e., it must be suffi- ciently polyatomic) to give rise to a time lag in the forma- tion of the dissociation configuration that is long in com- parison with the time between collisions. All of the theories of unimolecular reactions assume that an active molecule will be deactivated on every colli- sion (the strong collision assumption)? so that the rate of deactivation is the same as the collision frequency. The rate of spontaneous decomposition, on the other hand, is de- pendent on both the model for the active molecule and the criterion for the reaction. The difference in the various theories of unimolecular reactions arises then from their treatment of these factors. Thus, assumptions made concern- ing inter- and intramolecular energy transfer, anharmonicity 5 of vibrations, and the randomness of the distribution of en- ergy, which are of fundamental importance in the theory of reactions, can be tested by comparison of experimental data with theory. With the focus on the internal motions of the molecule as significant in the description of the reaction, it can be seen that a study of the effects of isotOpic substitution on the characteristics of the reaction would be a further aid in establishing the validity of the assumptions made in connec- tion with the various theories of unimolecular reactions. Both of the theories under consideration, that of N. B. a me . and that of R. A. Marcusi attempt to establish a Slater3 definite description of the activated complex in terms of its internal motions. The Slater theory assumes that a molecule may be represented as a collection of uncoupled classical harmonic oscillators which are described as the normal mode oscillators. The critical energy for reaction can, in the Slater theory, be equated to the high-pressure experimental activation energy. The theory of R. A. Marcus, referred to as the RRKM theory, is also dependent on the detailed vibra- tional characteristics of the molecule. However, the mole— cule is described as a collection of lightly-coupled quantum harmonic oscillators. The rate of reaction of an active molecule at a particular energy is viewed as prOportional to the ratio of the density of states of the active molecule to that of the activated complex at the given energy. Thus, both theories reflect a dependence on molecular structure through 4 the vibrational modes of the active molecule and of the com- plex. Of the reactions known to exhibit gas phase unimolecu- lar behavior, perhaps the most studied example is that of the thermal structural isomerization of cyc10propane to pro- pyleneilo Other examples include the isomerization of methyl isocyanide to acetonitriléawand the decomposition of nitrogen pentoxideiw4 The only mechanisms prOposed for the isomeriza- tion of cyclOprOpane which have withstood the test of experi- ment and theoretical arguments are the two originally pro- posed by Chambers and Kistiakowsky.6 One involves the migra- tion of a hydrogen in the activated complex with the carbon skeleton essentially undeformed, and the other involves the formation of a tri-methylene-like activated complex accom- panying an Opening of the carbon ring. The first was used by Slater in his theoretical calculations, while the second is indicated by studies of the isomerization of deuterium— substituted cyclOpropaneEEniHInvestigations of the effect of changes in the complex structure on the calculated rate car- ried out by SchlagZ7 and Setseryiresulted in a complex struc- ture involving both the hydrogen migration and the opening of the carbon ring. The frequencies of the complex which are sensitive to deuterium substitution were adequately defined by this comparison. Work by Weston38 revealed a carbon-15 isotOpe effect in the thermal isomerization of cyCIOprOpane. He felt the results could be explained on the basis of either mechanism. 5 In order to assess the importance of ring motions for the isomerizations, Sims and Yankwichg'determined the carbon- 13 isotOpe effect as a funotion of pressure at several tem- peratures. Their results indicate that the reaction coordi- nate is more complex than the hydrogen bridging coordinate originally prOposedf and they conclude, on the basis of the isotopic sensitivity of the magnitude of the isotOpe effect to the ring deformation frequencies, that the reaction co- ordinate includes considerable ring relaxation. However, the temperature and pressure dependence of the carbon-15 isotOpe effect have not been adequately interpreted, even though con- siderable information concerning the carbon~15 isotOpe sensi- tive vibrations of the complex could be obtained; this infor- mation is requisite to a vibrational analysis of the complex for a more complete understanding of the reaction. The purpose of this study was to make comparisons be- tween the experimental carbon-15 isotOpe effect and that cal— culated by the Slater and RRKM theories of unimolecular re- actions. II. GENERAL DEVELOPMENT OF THE THEORIES A. Introduction The temperature dependence of the rate constant of an elementary reaction can be described in terms of the Arrhenius expression k = A expC—Ea/RT) (1) where A is the pre—exponential factor and Ea is the experi- mental activation energy. The usual interpretation of this strong dependence on temperature is that only molecules with high energy are capable of reaction and that there is a mini— mum energy, called the critical energy, necessary for reac- tion. Molecules which have a total energy greater than this critical energy are termed "active" molecules. Experimental- ly one observes an average rate constant over all molecules with energy greater than the critical energy. If ci is defined as the probability of reaction per unit time from state i, where i is an enumeration of the ac- tive energy levels (be they continuous or quantized) and fi is the fraction of molecules in state i, the observed rate constant for the first-order reaction is k chifi (2) i 7 the sum being over all states 1 of the active molecule. The distribution of active molecules, fi, is dependent on the ex— perimental conditions (i.e., the method of excitation) and the specific decomposition rate, 0 is dependent on the 1’ model used to describe the active molecule. B. The Slater Theory Slater prOposedfifithat a polyatomic molecule may be de- scribed as a collection of non-interacting classical harmonic oscillators. The molecular motion is described by a set of internal (or symmetry) coordinates, qr, which may, since the oscillators are harmonic and the potential energy expression is quadratic in the coordinates, be resolved into n normal modes of vibration with frequencies vl,‘v2, ...,‘Vn, energies and phases ¢1, ¢2, ...,CP . Each normal D. 61, 62, .00, en mode is associated with a normal coordinate Qi’ which is a linear combination of the internal coordinates. The frequen- cies are assumed independent; consequently, the energies and phases of the different modes of vibration will be constant in the free molecules and will change only on collision. Thus, there is no energy flow between oscillators. The reac- tion is said to occur when a specified coordinate qr attains a critical value, qro' The internal coordinates are related to the normal co- ordinates by the transformation n qI‘ = Z O(ri Q'i (5) i=1 8 where the afi are the elements of the eigenvector matrix as- sociated with the vibrational secular equation, |<13(F->\E=o, <4) . . . . 4 . where, in the notation of Wilson, DeCius and Crossf G} is the inverse kinetic energy matrix, F is the force constant matrix, A is the eigenvalue and EL the identity matrix. For an harmonic oscillator, the normal coordinates are of the form Qi = Kicos 217(Vit + 4>i). (099514 1), (5) as Where the Qi satisfy the energy relations n _. ' 2 T_ Z:Ql/Al i=1 (6.) n _ 2 and V - Z Qi . i=1 The Xi are related to the normal frequencies, vi, by _ 2 2 )‘i - 417 vi ('7) These restrictions lead to interpretation of the coefficients Ki as K. = V? (8) th where 61 is the energy in the i normal mode. Finally, the expression for the internal coordinates becomes n qr = Zarivei cos 211(vit + gbi) (9) i=1 Now a molecule cannot react unless n qr = Z larii V61 2:qu (10) i=1 where the cosine term is taken to have the maximum value of unity. The values of 6i that satisfy this requirement and at the same time keep the total energy Ezzei content a min- i imum are _ 2 2 4 6io " q~r ari /OL (11) n where 0C2 = 20912 (O: > O) (12) i=1 E = q 2/o<2 (15) Note that it is not necessary that the reaction coordinate be one of the qr. It may be a linear combination of the form n qo “‘EE:3}qr (14) r=l 10 The amplitude factors of equation 5 then would be n I aoi = Z 7rari (l5) r=l Having established the minimum conditions for reaction, the distribution function and the specific rate constants of equation 2 must be determined. If it is assumed that the system is in effective equilibrium, then the population in the energized ranges (ei, 61+ déi), for i = l,2,...,n , where the 61 satisfy equation 16, will be D :16 i C eXp(-E/ka)TT Rig-T (16) i=1 where C is the total number of molecules in the system. The fraction of energized molecules with energies €1,62,...,én reacting per second is L, the average frequency with which qr attains the critical value qro . The collision frequency per molecule and hence the rate of deactivation is again assumed to have the classical form (strong collision assumption) w: zo (17) where H «k T Z = 402 —EE_ ‘fi%) sec mm Hence the number of molecules raised in unit time into this energy range is 11 n dei a)C exp(-E/ka) “'T (18) b i=l However, the pOpulation of activated molecules in any energy range will not correspond to the Boltzman distribution be- cause of the depletion by reaction; it will be some smaller number, such as Cg(el,e2,...,en) del d€2 ... den (19) where g is to be determined. Then the number of molecules reacting per unit time is Cg L del de2 ... den (20) and the number deactivated is d6 de (21) (UCg d6 2 ... n I In the steady state, the number of molecules activated by collision is equal to the number deactivated plus the number which react; hence (Iw+a0s =<~>(kb'l‘)’n eXp(-E/ka) (22) which gives an expression for g. Now by integrating equa- tion 20 over internal energies, l kz—C%%=f...fgl.ldél de2 ooo den 3 (23) S1 en and substituting g, L exp(-E/ka) 1]— dei 1+L/w i=1 m (24) where the integral is over the energy range which corresponds to activation. By letting b . Eo/(ka) , 2.. 'i%i/“ (ZPi'l - n and v3 :vaiz 9 ' 121 the integral may be reduced to k = v exp(-b) IDCO) or I a; 1 x ...,r :4: 1;: where x = (E - EO)/(ka) ° 0 = %} bm‘l rn n fn = (4”)m-l ka) i—iili . i=1 and n is the number of effective or contributing modes. If there exists more than one equivalent reaction (25) (26) (27) coordinate, the rate constant must be multiplied by a statis- tical factor d;corre8ponding to the number of equivalent coordinates, since the theoretical high-pressure rate con- stant is evaluated on the assumption that the reaction will 15 occur whenever any one of the reaction coordinates exceeds 35 its critical value. Thus, as Slater has shown, km: E kmr (2a) where kwr is the rate constant evaluated for one of the reaction coordinates. Note that at high pressures the integral approaches the value of r(m) and k/kon approaches unity. At very low pressures, the value of the integral approaches 9 and the rate constant becomes n ~ 0(1/6 -b -l q k0 = F::I)3( ) = o5) Q, At low pressures the limiting form of the rate constant is proportional to the pressure and is given by 00 coexp(-E /k T) . * - k0 = .0 b N (E ) exp(-E+/ka) dE+ (56) Q, 0 oo = “: N*(E’) exp(-E*/ka) dE* QV 0 If, however, there is more than one isomerically equivalent coordinate, then kE must be multiplied by a sta- tistical factor, the quantity(X, to compensate for the addi— tional number of equivalent reaction paths. It should also be noted here that this development is fOr reactions which require a rigid activated complex, since it does not allow for the presence of any internal rotation in the activated complex. The equations, however, comprise a 22 limiting case of the general expression deve10ped by Marcus and Wieder.‘1L1 The critical energy, E is evaluated by requiring km 0, to have an Arrhenius temperature dependence. Comparing equa- tion 55 to equation 1 and performing a logarithmic differen— tiation with respect to (l/T), one obtains E0 = Ea + (12;) - - RT (57) where (1 ) h . ex (—h ./k T) (E > = ka2 (1.3.91. = 2 U1 p V]. b (58) V (“)T . l - exp(-hvi/ka) l D. Effect of IsotOpic Substitution Kinetic isotOpe effects arise from the fact that iso- tOpically substituted molecules react at different rates. This arises mainly as a result of the effect of the differing masses on the vibrational frequency pattern in the various isotOpic molecules. From transition state theory, the high pressure rate constant can be shown to be prOportional to the mean velocity v in the reaction coordinate and hence inverse- ly prOportional to the square root of the effective mass m, since v z (agar/11111)?é (59) Thus, the ratio of rate constants of isotopic molecules iSiflr Versely prOportional to the effective masses of the atoms 25 related by the reaction coordinate: (6O) where the prime refers to the heavier isotOpic molecule, and ra,will always be greater than unity (termed a "normal" iso- tOpic effect) if the coordinate involves the substituted atom. If the atom substituted is not directly involved in the reaction coordinate, there will still be a small effect, since the reaction coordinate involves a linear combination of all the normal coordinates. At low pressures, where col- lisional activation is the rate-determining step, only a very small, but normal, isotOpe effect is expected because of the relative insensitivity of the collision process to isotOpic substitution. Thus classical theory predicts a decrease in the isotope effect with decreasing pressure. The non-classical RRKM theory, on the other hand, also allows for quantum statistical effects of isotOpic substitu- tion. Considering equation 55, in which the integral term has been equated with Qéfi the vibrational partition function of the activated complex, the ratio of the high—pressure lim- iting rate constants becomes k I H + £7 = Ir. Q". +—, eXp((EO' - EO)/ka) . (61) no :- Q'V up The IrQ Q+ product term is usually slightly less than unity, but the exponential term is large enough to produce a large normal isotope effect at ordinary temperatures. At the low 24 pressure limit, the ratio of rate constants becomes, re- calling equation 56, * I r0 = k . = 27 3x:— exp</kbr> .S . <62) Note that all dependence on properties of the activated com- plex have disappeared in the low-pressure region. S is the F at 1- 5 ratio of integrals of N (E ). S may be very small because at [ any given energy the density of energy levels in the active . molecules is greater for the heavier isotOpic molecules, the effect increasing with energy. Thus, the theory predicts that the isotOpe effect will decrease with decreasing pres- sure and invert at very low pressures. It should be noted that this effect arises from the quantized nature of the oscillators, and should be observed for all unimolecular re- actions at sufficiently low pressures. III. APPLICATION OF THE SLATER AND RRKM THEORIES TO THE ISOMERIZATION OF ISOTOPIC CYCLOPROPANES A. Introduction The theoretical formulations of the reaction rate con- stants for thermal unimolecular gaseous reactions given by Slater and by Marcus are based on a model in which the vibra— tions of the molecule are described by a set of harmonic oscillators. The application of the theories requires a vi- brational analysis of the reacting molecule; the high— pressure experimental activation energy is used to fix the temperature dependence of the high pressure rate constant; the kinetic theory collision diameter is used in calculation of the rate of collisional deactivation. B. The Vibrational Problem The Wilson GF matrix techniquefi§was employed in the vibrational analyses of the molecules. The vibrational prob— lem is expressed in the internal valence coordinates of Deciusw who has shown that a set of four types--namely bond Stretching, valence angle bending, out-of-plane wag and torsion--is sufficient to describe the most general vibra- tional displacement of any molecule. 25 . .. “Tm 26 For a set of Bi internal valence coordinates, the po- tential energy, V, and the kinetic energy, T, for a vibrating molecule may be written as ~ REE mm TREE where f-is the force constant matrix,(G is the inverse kinet- 2V and 2T ic energy matrix, the R are the time derivatives of the in- ternal coordinates and’V implies a transpose matrix. Through the equations of Lagrange, the vibrational problem leads to a secular equation of the form (<60: -X[E)[L = (O (64) where X = 4W2U2 and L.is the eigenvector matrix (i.e., the normalized amplitude matrix) which also describes the trans- formation from internal coordinates, Ri’ to the normal co- ordinates, Qi: lR= HQ (65) As was noted earlier, it is often more convenient to use some linear combination S of the internal valence co- ordinates Ri’ This simplifies the secular equation by re- ducing many off-diagonal elements to zero on the basis of the symmetry of the molecule. An orthogonal matrix klis con- structed such that S‘UIR , (66) 27 from which the expression for S in terms of the normal co- ordinates Q follows: S=UUI=£Q - (67) A "symmetrized"k% and 3;may be defined by 23 = mil (.8) and \}= UIFU (69) which lead to a symmetrized vibrational secular equation (Ml-A )1, = o <70) where A = XE A useful description of the normal coordinates, Q, is the transformation if from normal coordinate Space to Car- tesian space: X=T© . on .F is a 5N by 5N-6 matrix, N being the number of atoms in the molecule, correSponding to 5N—6 normal coordinates and 5N Cartesian displacement coordinates. If the Cartesian co- ordinates are taken three at a time, corresponding to the th and z. displacements of atom a in the i normal xia’ yia la mode, these triples define a set of s—vectorsf3 which can be used to construct the transformation,fi3, from internal co- ordinates to Cartesian displacement coordinates: IR: BX - _ (72) 28 The reverse transformation is given by X‘AR <75) where A is not simply the inverse of B, since B is not square, but is defined by7fl BA=E - W4> 42 The inverse kinetic energy matrix is related simply to Ebe M1438 = G , (75) where w1is the matrix of reciprocal masses. It follows that Jim-1556* . (76) Then, since R= LQ , the transformation from Xto Q is given by X= AU.) = [M'lE (G‘lLQ ; (77) or, recalling equation 64 and comparing equation 77 with equation 71, it follows that T= M’HBJIFM'I = AL (78) which removes the necessity for calculating the inverse of‘G. At this point it will be of interest to note that once« the transformation T has been found, the calculation of the Slater oc’s can follow immediately. Recall that the critical coordinate, qo, could be expressed as a linear combination of the internal coordinates and was related to the normal co- 29 ordinates by equation 54. It is necessary to obtain an ex- pression for ”1. Since the transformation H3, relating inter— nal coordinates to Cartesian displacement coordinates, is known, it follows that any internal valence coordinate chosen as the critical coordinate may initially be described by Q0 = BX (79) where B' is a row vector for the coordinate qO ; and, further, if go is to be a linear combination of contributing reaction coordinates, then B' is a row vector containing the algebraic column by column sum of the B' vectors of the contributing co- ordinates (equation 79 is formally the same for both cases). In light of equations 73 and 65 one can relate qO first to the complete set of internal valence coordinates R by <10 = JB'AR (80) and then to the normal coordinates, Q, by qo = B'AUKQ (El) and, finally, I may be identified as F= [B'A = [B'CIM‘IIBG‘H . (82-2) Finally, the Slater expression for the 038 may be obtained by comparing equation 81 with equations 54 and 55, and 0 QZCE') 05 Q, E’ 81(E') 1.01589 0 O 0 82(E') 0 -0.06155 0.12475 0.52168 . 85(E') 0 -1.44867 -0.06588 -0.00585 S4CE') 0 0.05344 -l.00029 0.02264 Q Ql 02(A2~> A1" : S(Al") 0.90507 A2" : Sl(A2") 1.05980 . 0 82(A2") 0 1.41142 01 02 Q5(E") E" : Sl(E") 1.05980 0 0 S2(E") 0 0.79724 -0.576855 S3(E") 0 1.27625 1.06888 55 calculated cyclopr0pane on cyclo-05H6 frequencies shows less 1 frequency (to 1044 than one percent lowering of the 1051 cm- cm7l), whereas the 884 cm-lfrequency is shifted by 5.5 per~ cent upon substitution of carbon-15. Further confirmation of the present assignment comes from a simple three-center, equilateral triangular, molecular mode1.used for calculation of the ring vibration modes. The "atoms" at the corners of the model were given an effective mass equal to the CH2 sub- group, 14 a.m.u. Force constants for stretching were esti- mated by Badgers' Rule and modified slightly to account for the ring structure. The calculated frequencies, 1194, 890 (2), were in good agreement with the experimental fre- quencies, 1186, 884(2). This identification of the ring modes will have an important effect on the application of the RRKM theory, since the frequencies of the vibrational modes active in the reaction coordinate in the activated complex must be shifted toward the pr0pylene frequencies to simulate the movement along the reaction coordinate. The ring defor- mation modes of cyc10propane become a C——C stretch, a 0340 stretch and a C-—C—-C angle bend in the complex, and, thus, considerable changes in the ring frequencies occur in going to the complex. C. The Slater Theory The kinetic isotope effect observed in the isomeriza- tion of cyclopr0pane and cyclOprOpane-d6 was used as a test 54 of the ability of the Slater theory to predict the character- istics of deuterium isotOpe effect. The evaluation of the Slater integral was fairly straightforward. It was necessary to be able to evaluate the Gamma function for non-integral, specifically half-integral, arguments. An expression developed by Artinz giving twelve decimal-place accuracy was used. The integral itself was evaluated by 52-point Gaussian quadrature using Laguerre polynomials which are orthogonal on the semi-infinite inter- val ( 0,00)?; The integration routine was checked by evaluat- ing the integral for given values of log10 6 and comparing them with those given by Slater?6 The method was also used to evaluate the Gamma function and was found to have an error of one part in one hundred thousand for non-integral values of the argument greater than two and no error for integral values of the argument. The initial single isotOpe calculations of the rate constant in the high-pressure limit and in the fall-off re- gion yielded frequency factors that were too low by a factor of three or four. For the normal cyclOprOpane molecule it is 15 found experimentally to be léiiclo sec-1, whereas a value of 5x1014 seen1 was calculated. This is a definite failing of the theory, since the Slater frequency factor is a weighted average of the molecule frequencies, and the maximum limiting case is then always less than the highest molecular frequency. The calculations revealed that the lepe of the fall- am off curve agrees fairly well with experiment, although the 55 absolute magnitudes of the rate constants were not correct, and the curves were shifted to higher pressures (0.77 and 0.22 log units for cyclo-CBH6 and cyclo-C5D6, respectively). Al— though this has been attributed in part to inefficient colli- sionsf5 it would seem unlikely that a molecule as complex as cyclOprOpane would exhibit the low collision efficiency neces- sary to account for such large shifts. Also, the large dif— ference in the shift for the two isotOpes was not expected. E1 "WW Finally, where full Slater calculations show a decrease in v for a heavier isotOpe and thus a high-pressure frequency fac- tor for k/k' that is greater than unity, Blades5 has reported an experimental ratio of 0.82 for the k/k' frequency factor for deuterium substitution. Further, the fall-off of the iso- topic k/k' with pressure does not have the correct shape. D. The RRKM Theory 1. Introduction The application of the RRKM theory requires that care- ful consideration be given the construction of a complex model, since the activated complex is critical to the descrip— tion of the reaction. The high frequency factor for the isomerization of 15 cyclopr0pane (ca. 10 sec-l) implies a large positive en- tr0py of activation. The observed entr0py of activation ([15; == 7.55 e.uf at 515.80C, as compared to the standard entrOpy of reaction,lfld§)= 8.0 e.u.) indicates that the acti- vated complex has a structure rather close to pr0pylene, i.e., 56 the carbon ring has been deformed to a large extent and one hydrogen has undergone at least partial transfer to an ad- jacent carbon. Several models were considered and the iso- tOpe effects calculated according to this theory. 2. Calculation of the Rate Constant The equation for the rate constant as a function of pressure has been described and the various terms defined in Chapter II, section C. Details of the calculation will be ! _1 ”it considered below: The inertial ratio, Ir’ is the ratio of the rotational partition function of the activated complex to that of the active molecule and is given by Qrot. = IAIBIC II I * IB*IC* Ir = Qrot. A where IA’ IB and IC are the principal moments of inertia. The vibrational partition function, Qv’ has already been described in equation 52. The critical energy, E is related to the high- 0’ pressure experimental activation energy, Ea’ and the average vibrational energies of the complex (+) and active molecule (*) by equation 57. E0 has the significance of being the difference between the lowest energy atate of the complex and activate molecule. If equation 57 is used to calculate E0, a small temperature dependence to E0 is introduced, contrary to the physical significance of E0. This small error can be very 37 important in isotOpe effects, where AEO -.- EO' — E0 (85) appears in the exponential of equation 61 for the high pres- sure isotOpe effect, and considerable error inAEo can result from small errors in E0 and E5 . To insure that the isotOpic rate constant ratio k/k' and the individual rate constants had an Arrhenius temperature dependence, E0 was evaluated from equation 57 for the lighter isotOpic molecule at an in- termediate temperature of the experimental range, and the quantity AQEO evaluated from the zero point energies of the active molecule and the activated complex of both isotOpic species by )+ . (86) ')*- (E -E ... :_ = : AE 7 E E (E zpt. 2pt. o o o Zpt. - Ezpt. Ad130alculated in this way is independent of temperature and has the correct physical significance. The critical energy of the heavier isotopic molecule was then determined as The collision frequency, a): ZP, is necessary, since the strong collision assumption was used in all calculations. 35 Using a collision diameter, 03 of SR, the kinetic theory 001— lision frequency, Z, was calculated from 35 wk T z = 462 b 1%- sec‘1 mm-1 (87) 58 The quantity E: F(EV) is the number of vibrational EifE energy states for a molecule with a total energy, E, since it is assumed that the total energy can be distributed among the various degrees of freedom (vibrational, rotational, transla- tional) in all possible combinations. If F(Ev) is the number of ways of distributing the vibrational energy, Ev, among the vibrational degrees of freedom, then the sum of F(EV) over all EVEEE will be the total number of vibrational energy states at a total energy E. A systematic counting procedure has been deve10pedM)and can be used to evaluate ):P(Ev). The direct count method for a molecule such as cyclopr0pane, how- ever, is extremely time-consuming, even on a computer, so that good approximate expressions are desirable. The approximation to ):P(Ev) developed by Whitten and Rabinovitch:39 was used exclusively for evaluation of both ):P(E;) and N*(E*), since the expression is simple and accu- rate. The expression is s s BEEPOEV) == [EZ(E'+1-(5w) / ( Filjl hvi) (88) where s is the number of active vibrational modes, E2 is the zero point energy, E' E/E Z Tb I 2 .. Lei—1.2 213% , a frequency dispersion parameter, . s V 59 '— exp[-l.0506(E')O'25] (1.05E'58.0) 8 u and E II [5.0(E') + 2.75(E')O°5 + 5.51]-1 . (0.15E'51.0) The approximation is good to about one percent for E220.5Ez and is valid at energies as low as 0.1Ez. Since the complex for the cyc10propane isomerization is assumed to be "rigid," only the vibrational degrees of freedom contribute to the reaction, but the total energy of the complex, E+, can be distributed in any manner between the vibrational degrees of freedom and the internal translational degree of freedom. Equation 88 was then used for evaluating ZP(EV+) for each complex. The density of energy states of the active molecule at It It I! t energy E , N (E ), is related to ZP(EV ) by 82 . KEV") EVEEE e 8 8E ( 9) NYE') = Equation 88 can be differentiated directly with respect to E l l to obtain an expression for N (E ), N*(E*) = s[EZ(E' + l -@w)]S-l/ (F(sfl) E hvi) (90) The reaction path multiplicity, Ch is defined as the number of equivalent but distinguishable reaction paths from 40 a given molecule to products. For normal cyclOpropane this number is twelve, since there are twelve equivalent non— bonded carbon-hydrogen distances that can be considered as reaction coordinates. 52 Construction of the Complex Models The vibration frequencies of the molecule were calcu- A lated as described above, using the Wilson GF matrix method. The vibration frequencies of the activated complex were deter- mined by an empirical method, although one detailed calcula- tion was attempted. The complex was treated as "rigid", ad- mitting no internal rotations. For the calculation, the com- plex was treated as a normal molecule, except that one fre- quency was considered an internal translation along the re- actant coordinate, and hence one molecule frequency was de- leted in going to the complex. For this calculation, the complex was constructed such that it had C2v symmetry, in order to simplify the assignment of force constants. It was assumed that the ring was opened and that no interaction ex- isted between the end CH2 groups. The CH2 groups were con- sidered to be almost in the CCC plane, and the CH2 group on the central atom was taken as perpendicular to the carbon plane. Force constants were assigned from those for cyclo— pr0pane and pr0pylene, adjusting the values to correspond with any loosening or tightening of bonding in going from cyc10propane to the complex: the torsional force constants were reduced for the hindered-rotation in the complex, the 41 C-—C and C==C bond force constants were altered to fit the partial bond characteristics of the complex, and the central carbon CH2 group was treated as in cyclopr0pane, except that the force constants were made to reflect the relaxation of the constrained ring. Calculations made with this model re- turned frequencies which were not in good agreement with the expected shifts from cyclOprOpane. This is attributed to the absence of important interaction elements from the force con- stant matrix for the complex which are difficult to estimate. Further attempts to calculate the complex frequencies were not made. For the isotOpe effect calculations, the models were constructed using an empirical method of fixing the vibra— tional frequency pattern of the complex, and the final fre- quencies were groupedfito facilitate calculation. The complex geometry of Setserx)was used in all of the calculations. This model consists of a C00 angle of 1090, with one hydrogen bridged in the CCC plane. IV. DISCUSSION A. Nature of the Comparison Between Theory and Experiment The kinetic isotope effect in the structural isomeri- El zation of carbon-15 labelled cyclOprOpane (natural abundance), cyclo-Cg2ClEH6, to prOpylene-C15 was calculated using both I ”Hanna‘s“ the Slater and RRKM theories of unimolecular reactions. The isomerization can be written: cyclo—C%2H6————4> 0H50H=0H2 12kl cyclo-CECBH6 ————e> 015H50H=CH2 4k2 ————,> 13’: CHBC H CH2 4k5 _ l5 ————4> CHBCH—C H2 4k4 where k1, k2, k3 and k4 are the respective unimolecular first—order rate constants, and the factors 12 and 4 arise from the reaction path multiplicity for the reactions. The kinetic isotOpe effect has been investigated in detail by Sims and Yankwichaz, but the analysis used by these workers allowed determination only of the intermolecular isotope effect k/k' related to the individual isotOpic rate con- stants ki (i=l,...,4) by36 k 12k ET 3 4R2 + 4k l . 5 + 4k4 (91) 42 45 Comparison of theory with experiment should pr0perly be made between the calculated rate constants ki (i=1,...,4) and the experimental average isotOpe effect k/k' by means of equation 91. This was done in the case of the Slater theory, and it was found that within the calculation error k2=k3=k4. For calculations using the RRKM theory, a specification of the vibrational frequency pattern of the complex is required; the empirical method described below for fixing the complex frequencies and the numerical integration necessary for eval- uation of the RRKM rate integral prevent any real distinc- 4 pattern representing the "average" effect of carbon-l5 label- tion between k2, k3 and k . Therefore, a complex frequency lingwas employed in all the RRKM calculations, and the cal- culated results compared directly with k/k'. For all of the complex models considered, the isomer- ization rate constant k (kl above) was calculated in both the high pressure (km) and fall-off regions (k/km), and the deuterium isotOpe effect kH/kD for the structural isomeri- zation of cyclo-C5D6 was also calculated as a function of pressure, and comparisons made between the calculated and corresponding experimental quantities. B. Slater Calculations The rate constants for isomerization of unlabelled cyclopr0pane (C5H6)’ perdeuterated cyclOprOpane (C5D6) and carbon-l5 labelled cyclOprOpane (Cé2ClBH6) were calculated from the amplitude factors resulting from a vibrational 44 analysis of each molecule (see Chapter III). The critical coordinate was chosen as a non-bonded C-—H distance in each case, which was found by Slater35 to give the best agreement with the structural isomerization rate constant at all pres— sures. Some calculations involving a C-—C critical coordi- nate were also carried out.~ However, only the trends of the calculations using Slater theory were of interest, and a thor- ough investigation was not undertaken, since the main inter- est in this work centered on the RRKM calculations. Other workersmvwhave shown that the basic assumptions of the Slater theory are incorrect. C. RRKM Calculation The calculation of the various quantities entering the RRKM rate expression are discussed in detail in chapter III. The evaluation of the rate integral must be done by numerical integration at all pressures except the high-pres- sure limit, for which the RRKM integral reduces to the vibra- tional partition function of the complex and hence an analyt- ical formula can be used. It follows that in the high-pres- sure limit, errors due to integration disappear, and the calculated results reflect only errors in the theory or in the characterization of the complex. In the high-pressure limit, the prOperties of the complex determine the calcu- lated result to a large degree; as the pressure is decreased, the properties of the complex become less important, and in the low-pressure limit, the rate expression becomes indepen- 45 dent of the complex and depends only upon the properties of the active molecules, which are known quite acurately and cannot be adjusted. It follows that the calculated isotOpe effect will be dependent on the pr0perties of the complex only at high pressures, and because of the reasons given above, comparison of the high-pressure isotope effect (k/k')OD will be considered separately from the isotOpe fall-off curve, k/k' versus pressure. For all of the calculations, N the numerical integration was chosen so that at high pres- ? sures the RRKM integral evaluated by integration agreed with the complex vibrational partition function to better than 0.1 percent. The complex vibrational frequency pattern was deter- mined by changing frequencies of the molecule systematically in going to the complex so that: (1) one frequency was re- moved as representing the reaction coordinate motion (In all cases, a C-—H stretching frequency was chosen, as indicated by the deuterium isotope effect resultszt even though the carbon-l5 kinetic isotOpe effect results indicate that the reaction coordinate must contain appreciable loosening of the ring and C-—C motion in addition?); (2) the twisting frequen- cies of cyc10propane were lowered to simulate torsion of the and CH2 groups in a trimethylene-like complex structure, as suggested by the cis-trans geometrical isomerization of 1,2- cyclopr0pane-d2; (5) the ring frequencies of cyclOpropane (now assigned as 1186(1) Ai and 884(2) E' rather than 1050(2) E' in earlier work) were changed to represent C-—C stretch- ing, CflvC stretching and COO bending of the complex; (4) oth— 46 er frequencies were either unchanged or changed so as to cor- relate closely with corresponding frequencies in pr0py1ene (these other frequency changes had little effect upon the cal- culated isotOpe effect since they are not C-l5 sensitive fre- quencies.); (5) the magnitude of the changes were such that the final complex frequencies were closer to the correspond— ing propylene frequencies than to the cyclopr0pane frequen- cies, since the large entrOpy of activation4 (i.e., large pre- exponential factor) for the structural isomerization suggests considerable loosening of the cyc10pr0pane ring structure in the complex; (6) the resulting complex frequencies of the isotOpic molecule were in each case slightly adjusted in or- der that the Teller-Redlich product rule was closely satis- fied (within 1% generally), and that the high-pressure iso- tOpe effect had the correct experimental temperature depen- dence. The RRKM high-pressure isotOpe effect is given by 4'1 0 + ) exp(AEO/RT) (61), WIW 13?? where * and + refer, as before, to active molecule and acti- I‘ vated complex, respectively. The temperature dependence is determined almost exclusively by the exponential term, so that the requirement on the complex frequency patterns of the isotopic molecules was that _ * + _ AEO = (EZ - EZ') - (EZ EZ) -, AEa (92) where Ez refers to zero-point energy, andAEa to the experi- mental Arrhenius exponential term for the high-pressure isotope effect. "éw‘fg 1.11er 3’ 47 D. Results and Comparison with Experiment 1. Slater Theory The Slater theory results indicated that fairly good agreement can be obtained with the isomerization rate constant (ka,and k/kw vs. P) at all pressures using a C—-H critical coordinate, the only discrepancy with experiment being a small shift (r0.5 log P units) along the pressure axis, which is not If considered serious. The assumption of strong collisions and lfi the harmonic approximation are expected to affect the results in this direction. The same trends were noted by Slater85 using a slightly different vibrational analysis. Poorer agreement is obtained using a C——C critical coordinate. The Slater theory has not been used to calculate iso- tope effects as a function of temperature and pressure, even though some simple comparisons have been madea? The calcu- lated Slater deuterium isotOpe effect was found to be much too large at high pressures, and to encompass a much larger range of values of kH/kD over the experimental interval than observed experimentally; the same general trend was found for the carbon-15 isotOpe effect. Furthermore, the low—pressure limit of both the calculated deuterium and carbon-l5 kinetic isotope effects was slightly greater than unity, whereas the deuterium isotOpe effect has been observed to invert at low pressuresa; the inversion has been shown to be a consequence of the quantized nature of molecular vibrationsfl, and thus should be of general occurrence, and the C-15 isotOpe effect should also invert at low pressures. 48 It appears that the Slater theory of unimolecular re- actions fails to give even qualitative agreement for isotOpe effects in unimolecular reactions, even in cases where the agreement with the overall rate constant is very good. Iso- t0pe effects seem to be a better test of the details of the theory than the overall kinetic rate constant. 2. RRKM Theory Many variations of the complex model were considered for both assignments fo the degenerate ring modes: 1050(2) cm- and 884(2) cm-l. In every case it was found that the high- pressure calculated carbon-l5 isotope effect was too large if the 1050(2) cm.l frequencies were assigned to the ring modes. If, on the other hand, the 884(2) cm.l modes were assigned as ring modes, the above prescription for frequency changes in going to the conplex led to a calculated high-pressure iso- tOpe effect in good agreement with experiment ifAEO were fixed equal toAEa as discussed above. Actually, the high- pressure isotOpe effect may be varied over a rather wide range of values (1.004 to 1.012 for example) by small changes of one or two chl in one or two of the frequencies of the iso- t0pic molecule. The empirical method of fixing the complex frequencies is certainly no better than a few cm.1 at best, so that a difference of one or two cm‘l between the isotOpic molecules is allowed. The real test of the complex frequency model is the pressure dependence of the isotOpe effect. None of the models tested gave good results for the pressure fall- off of the rate constant. The 1050(2) cm"l models all pro— 49 duced isotOpe effects which fell off at much lower pressures than observed, and spanned a range of values of k/k' much '1arger than observed over the experimental pressure interval. The same general trend was found for those models in which Ithe 884(2) cm.l frequencies were assigned as ring modes, but in these cases the effects Were less pronounced. ’ In order to assess the sensitivity of the calculated results to changes in the frequency patterns of the isotOpic “1 complexes,AEO was fixed in a systematic manner to a given 3‘ value using different relative changes for the complex fre- quencies of the isotopic molecule relative to the normal (unlabeled) molecule. In no case did these changes amount to more than 1 or 2 cm-1 in any frequency, and two or three frequencies at most were affected. Changes of this order of magnitude are entirely within reason and possibility, so that the method seems to test the isotope-sensitivity of the empirical method of fixing the complex frequency pattern. The results showed that even these small changes affected very drastically the calculated isotOpe effect fall-off curve, even though the shapes of the fall-off curves for either molecule were but little affected. The isotope eff feet is simply a ratio of the rate constants at any pressure, so that changing even slightly the prescription for the com- plex frequency pattern of the isotopic complex relative to the unlabeled complex will change the value of k' slightly at high pressures, but will have substantially less effect at low pressures; hence, the shape of the fall-off of the 50 Table 5. The Calculated Carbon-l5 IsotOpe Effect, k/k', as Compared with Experiment a) At the High-Pressure Limit W Model (k/k' )0, Exp . 1.008 1 1.0072 1 2 1.0081 LET 5 1.0089 1... 4 1.0070 b) As a Function of Pressure 0.9945 —1.98 0.18 anS.:, 10g Pcalc log Pexp logéiiif 1.0000 CXD CK) —- 0.9995 0.25 2.54 -2.29 0.9990 -0.24 2.18 -2.42 0.9985 -0.62 1.87 -2.49 0.9980 -0.87 1.61 -2.48 0.9975 -1.05 1.48 -2.55 0.9970 -1.24 1.16 -2.40 0.9965 -1.41 0.97 -2.58 0.9960 -1.56 0.75 -2.51 0.9955 -1.70 0.54 -2.24 0.9950 -1.84 0.56 -2.20 51 isotOpe effect versus pressure curve can be quite seriously affected by such small changes. It was not possible for any of the models to make changes in the frequencies of the iso- topic complex sufficient to obtain agreement with the experi- mental fall-off behavior of the isotOpe effect, and still re- tain a physically reasonable vibration pattern. (See Table 5. The extreme sensitivity of the calculated results to such small changes in frequency patterns indicates that the empirical method which has been used very successfully for fixing the complex frequencies in analyzing the isomerization rate constant results“31 and the deuterium isotOpe effect re- sultsy}cannot be used for such small effects as the carbon-l5 kinetic isotOpe effect. Rather, a more direct approach of calculating the complex frequencies by a vibrational analysis of the pr0posed complex structure seems a better method of proceeding. A few attempts at such a vibrational analysis were made, but the problem is onerous because the complex structure (the geometrical parameters of the complex were taken from Setserw; a slightly different choice of complex structure would not seriously affect the results since the calculated quantities are fairly insensitive to the geometry but are more sensitive to the force constants) lacks symmetry and because a large number of interaction potential constants are expected in the force field. Using only a few interac- tion constants taken over from cyCIOpropane, the complex fre- quency patterns calculated were not reasonable; in particular, frequencies very much higher than those for either cyc10pro- 52 pane or propylene resulted for some of the CH2 group motions, which is unreasonable. 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