1532
PETERE. YANKWICH AND RICHARD M. IKEDA [CONTRIBUTION FROM THE
NOYESLABORATORY OF CHEMISTRY,
Vol. 81
UNIVERSITY OF ILLINOIS]
Temperature Independent Factor in the Rate Constant Ratio for Isotopic Reactions : Four Center Systems BY PETERE. YANKWICH AND RICHARD M. IKEDA RECEIVED JUNE 30, 1958 The Rigeleisen-Wolfsberg-Slater treatment of reaction coordinate motion is extended to the calculation for planar fourcenter systems of the temperature independent part of isotopic rate constant ratios. The results for several configuration types are applied to the acidic hydrolysis of urea. Fair agreement with experiment is found for the intramolecular hydrogen transfer mechanisni suggested by Shaw and Bordeaux.
Introduction A ratio of isotopic rate constants is the product of two factors, one temperature independent, one temperature Assuming that the transmission coefficient is unaffected by isotopic substitution
reaction ; in the work on nucleophilic substitutions each reaction was studied at one temperature only and the use of molecular fragment masses seems to do some violence to our notions concerning the factorizability of secular equations. In a study of the carbon isotope fractionation in the hydrolysis of urea,Q Yankwich and Veazie kl/ka (TIF)(TDF) = ( ~ 2 * / ~ l * ) ’ / ~ ( f ’ / f * ) (1) obtained an “experimental” value of TIF by a where thef’s are ratios of isotopic partition func- methodlo which assumed: (i) the applicability of tions for the normal and activated species, and the the Redlich-Teller product rule to the activated term involving the m*’s is the contribution to the complexes as well as to the normal molecules,l’ isotope effect of the reaction coordinate motion. and (ii) the hyperbolic sine representation of viThe identity of the m*’s depends upon the character brational partition functions. Calculated values of the theory applied to the reaction: in the tran- for TIF were obtained for a three-center representasition state theory, m* is the effective mass of the tion of the urea molecule, as follows: (i) by assumactivated complex in the direction of the reaction ing the reaction coordinate motion to be a normal coordinate; where a Pelzer-Slater potential is vibrational mode with an associated force constant employed, m* is the mass coefficient associated of zero magnitude; (ii) by assuming motion in the with the reaction coordinate according to the ex- reaction coordinate describable by a Bigeleisenpressiod Wolfsberg critical coordinate; (iii) by assuming any of several three-center approximations to the two-center method.2 In none of these calculations was satisfactory agreement obtained between calwhere q1 is the reaction coordinate (expressed as an culated and “experimental” values of TIF. In this paper we report the results of an exteninternal coordinate), and mi and ri are the mass and position vector, respectively, of the ith par- sion of the Bigeleisen-Wolfsberg-Slater approach to planar four-center representations of reacting ticle (in the molecule). Bigeleisen and W o l f ~ b e r g ~have , ~ applied eq. molecules, with application to urea hydrolysis. 2 to the computation of the T I F in three-center systems involving asymmetric motion with respect Planar Four-center Models and Reaction Coiirdinates to the central atom; Yankwich and Copeland’ extended the method to include symmetric motion I. Un-branched Chain, Non-cyclic Coiirdinate. about the central atom, to permit application to -Consider the four-center system isotope fractionation in lead oxalate pyrolysis; Bender and Hoeg’ applied the method to several nucleophilic substitution reactions of methyl io1 3 dide but used the masses of molecular fragments inI f the 2-3 bond is being broken and the 1-2 and 3-4 stead of atomic masses.8 In each of these applications one or more factors operated to complicate bonds formed, the reaction coordinate can be the test of the method: in the first the expressed as discussion of examples was not repeated after a @ lrs - rd - Y h - r.1 q = --a Irl - ral sign error was detected; in the lead oxalate system where u, j3 and y are defined following Bigeleisen the stoichiometry of the reaction was a function of temperature and the calculations on the ob- and Wolfsberg,’ and their ratios determine the relaserved isotope effect had to be based on assumptions tive amounts of bond formation and bond rupture concerning the size of another isotope effect in the between the atom pairs. Application of eq. 2 and division by p2 yields
+
(1) J. Blgeletsen, J . Chcm. Phys., 17, 676 (1849). (2) J. Bigeleisen, J . Phys. Chcm., 66, 833 1862). (3) J. Bigelelsen and M. Wolfsberg, J . Chcm. Phys., ai, 1073
(m*fl*)-I
= gin-’A*
(1053).
(4) J. Blgelefsen and M. Wolfsberg, ibid., SS, 1264 (1864). (5) N. B. Slater, Proc. Lccds Philosofihical Soc., 6 , 76 (1848). (6)P. E. Yankwlch and J. L. Copeland, Tars JOIJ~NAL, 79, 2081 (1957). (7) M. L. Bender and D. F. Hoeg, ibid., 79, 6660 (1067). (8) J. Bigeleisen and M. Wolfsberg, footnote 28 in rel. 7.
+
pa-1
+ pv-’B’ -
23 2 cos t#a - -cos ma ms
(3)
(9)P. E. Yankwhh and A. E. Veazie, TEXSJOURN~A, 80. 1836 (1868). (10) P. E. Yankwich and H. S. Weber, ibid., 78, 664 (1866). (11) H. S.Johnston, W.A. Bonner and D. J. Wilson, 1.Chcm. Phys.. P6, 1002 (1857).
RATECONSTANT RATIOFOR ISOTOPE REACTIONS
April 5, 1950
where A = a/& B = r/B, and pi3-I = 3A and €3 represent the amounts of bond formation relative to bond rupture. In cases where the 3-2 and 3-4 bonds are being formed while the middle bond is broken, eq. 3 is obtained, but A and B then measure the amount of bond rupture relative to bond formation. Where a pair of adjacent bonds suffers like fate, the sign of one of the trigonometric terms is reversed and the A'S and B's have to be redefined. The TIF for a pair of isotopic reactions is the square root of the quotient of a pair of equations like (3). II. Un-branched Chain, Cyclic Co5rdinate.In metatheses, cyclizations and atom transfer reactions, a convenient general representation is t)zj-I;
1533
The "experimental" value of TIF for the acidic 0.002.0 In Fig. 1 hydrolysis of urea is 1.016 and Fig. 2,isotifs are shown for the branched chain
*
3
Unfortunately, introduction of a third parameter like A and B would be required. For simplicity, we show as an example the situation where bonds 2-3 and 4-1 are being ruptured and bonds 1-2 and 3-4 are being formed; further, we set y = B. This coijrdinate describes a pseudo intramolecular transfer of atom 3 in a situation where the initial configuration is 4-1-2-3 and the final one is 1-2+3-4. Here p -Q lrl - rd 0 Ira - r21 0 lr, rd 8 Irl - rd which yields
-
+
(m*b')-'
pn-*A'
+
-
(pa-'
AB
B
Fig. 1.-Isotifs, model 111: 14. 12, m' = 13; 41 116'.
-
0
= 10, ms 14, ma = 122.5', 48 = 122.5', 44
fill
- +
+ pw-')B* + B'
O
p14-I
hi
B
where A = a/&and B = B/S. III. Branched Chain.-In this case a central atom is attached to the other three 4
A As an example, the rupture of the 2-3 bond while the other two are being formed can be represented by q -U It, rtl iB h - tal Y Ir4 - rd which yields
-
-
(m*@')-'
+ pz4-'B*
+ 41 - ACOS68 - B
pn-ld'
prc-'
0
0
I
1/8
-
0
Fig. 2.-Xsotifs, model 111: m~ = 16, ma 14, ma = 14= 12, mt' = 13; 1$1 = 150', 6 = 105', #4 = 105'.
mz
four-center model (111) comprising the heavy atom skeleton of the urea molecule, carbon central. + ma - [ A B COS COS C I For Fig. 1, the calculations are based on the normal urea bond an les; for Fig. 2, angle OCN is inwhere A fila, and B = 7/19, creased to 1508 to test the effect of anticipating the Application to Urea Hydrolysis cyanic acid structure. In both cases, a should be The square root of the quotient of an isotopic quite small with respect to 0 because the relative pair of relations such as those in eq. 3, 4, and 5 is change in the carbonyl bonding is small in the rethe TIF for a pair of isotopic reactions in terms of action; similarly, y should be somewhat smaller molecular constants and the parameters A and than B but of the same magnitude.12 Chemical B. A convenient representation of such a function inference, then, suggests that acceptable values of is a plot of lines of equal TIF (isotijs) on an A-B (12) For the case of Fig. 2 , 7 shoiilrl b e more nearly equal t o 6 plane, vide infra. than for Fig. 1 because o I the increase in angle DCN. 2
5:
(5)
1534
PETER
E.
Y A N K W I C H AND
RICHARD M. IKEDA
Vol. 81
A lie near zero, while those expected for B lie former, the bond angles are approximately those of
near 1. The values of TIF in the regions so de- the normal urea molecule, but the results depend scribed are in the neighborhood of 1.023-1.025; but little upon the angle CNH; in the latter, it has the experimental value of 1.016 requires that A and been assumed that deformation accompanies B both be large, 1.5-10. Parameter values this activation, and all bond angles have been taken as large correspond, in the carbon central model, to 90". In both cases, 01 should be somewhat smaller greater C-0 and C-N bond formation, respectively, than 6 but of the same magnitude; similarly, @ than C-N bond rupture-as if upon activation and 6 should be approximately equal. Chemical the molecule became very compact before ex- inference, then, suggests that the acceptable values ploding into products; this is not an attractive of A and B lie near 1 in an elliptical area near the picture of the hydrolysis. center of each figure. Where the bond angles are Isotifs for two versions of the cyclic reaction normal, Fig. 3, the values of TIF in this region are coordinate similar to that suggested by Shaw and 1.003-1.008; where deformation is assumed, Fig. Bordeauxla are shown in Figs. 3 and 4. In the 4, the TIF values lie in the range 1.007-1.016, and agreemcnt with experiment is approached. Probably, it would not be difficult to achieve agreement in terms of this model between experimental and calculated temperature dependent factors in the isotopic rate constant ratio. VA
I '
I
I
0
3.0-
C
O0
i
< W
A
2
2.5-
1 '. v
@ e LL
0 I-
Fig. 3.-Isotifs, model 11: mt = 12, ml' = 13, mz = 14, = 1, m4 = 14; +I = 120". +a = 110", +a = 70°, #, = 60". The dashed line is the value (1.0214) expected for simple C-N bond rupture.
U
J
2.0-
~8
C
'/a I 1
0.2
I A
(13) W.H. R . Shaw and J. J. Bordeaux, THISJOURNAL, 77, 4729 (10661.
I
0.4
A L (TDF), Fig. 5.-L(TDF) at mean (1/T) ueisus AL(TDF) for the temperature interval ( l W / T ) = 2.75-3.00, carbon 13 isotope effects: (A) 9-particle model for malonic acid and 8particle model for mono-anion of malonic acid decarboxylation; all bond angles 120' ; bond distances-Pauling's rule, force constants-Badger's rule; several different reaction coordinates assumed. (The circles represent the results of individual calculations; the scatter of the results is about average.) (B) - - -, 3-particle and 4-particle models for urea hydrolysis; as in A, except that bond angles were variable and the reaction coordinate was closely simple bond rupture or a n asymmetric stretching vibration. (C) , 3-particle model (carbon skeleton) for malonic acid decarboxylation, intramolecular isotope effect; parameter variation as in B. ( D ) --------- , similar to C, but for intermolecular isotope effect.
-
Fig. 4.-Isotifs, model 11: ml = 12, ml' = 13, ma = 14, 90". ma = 1, wad = 14; 61 = +a = +a
I
0.3
April 5, 1959
INFRARED ANISOTROPYAND STRUCTURE
OF
CRYSTALLINE FORM c STEARIC ACID
15%
two temperatures, well-separated, but within the When the groupings 0-C-N-H and N-C-N-H with normal bond angles are treated as in model range of the experiments. Values of L(TDF), I, the results are virtually identical with those L = 100 In, a t the mean (1/T) are plotted as shown in Fig. 3. For the N-C-N-H model, the ordinates versus the difference in L(TDF) between predicted TIF, 1.022-1.025, is in the neighborhood the two temperatures. The results of several of that for the C-N two-center model (1.0214) such families of calculations are shown in Fig. 6. and high in comparison with experiment. For the Use of these graphs depends upon the fact (eq. 1) L(TDF). From the O-C-N-H model, the predicted TIF is quite sensi- that L(kl/k2) = L(T1F) tive to the value assumed reasonable for y. It observed value of AL(TDF) a figure for L(TDF) seems likely that the change in the N-H bond upon a t the mean value of (l/T) for the calculation activation is small compared with that in the C-N interval is obtained; combination of this figure bond which is being ruptured; however, the with L(kl/ka)a t that temperature yields an “expredicted TIF could correspond well with the ex- perimental” value for L(T1F). Since TIF is a perimental value if B ( = y / B ) were in the range property of the reaction coordinate motion only, contributions of equilibria to TDF do not affect 0.2-0.4. Acknowledgments.-We are indebted to the the applicability of the method to a large degree. For all u (= hcw/kT) 3, there is but small error staff of the University of Illinois Digital Computer Laboratory for cooperation in carrying out numer- in the approximation sinh(ul2) = [exp(u/2)]/2. ous calculations. This research was supported by Using this approximation, i t can be shown that a t the temperature corresponding to the mean value the U. S.Atomic Energy Commission. of (1/T) in the calculation interval (T1,Ts): Appendix L(TDF) = clAL(TDF) c2; c1 is a function of “Experimental” Value of TIF.14-The values of T1and Tz alone, while t2 is a relatively insensitive the temperature independent factor referred to function of the models assumed for the normal and above as “experimental” are obtained by the fol- activated molecules. For several models to yield lowing semi-empirical method. similar graphs as in Fig. 5, it is necessary that they The system under study is described in terms of have similar complexity, similar partial compensamodels for the normal and activated molecules. tion for the effects of low frequencies and similar The various input parameters (particle masses, distributions of high and low frequencies. These bond lengths, bond angles, force constants, label are stringent criteria, and it is remarkable that the positions, nature of the reaction coordinate mo- various calculations described in the caption to tion, etc.) are subject to systematic variation, Fig. 5 yield plots no more divergent. each of many groupings of the variants leading set of input parameters happened to yield ~ o o dcorrespondence with the finally to a calculation of TIFl6 and of T D F a t results of the experiments, the problem at hand would be solved.
+
(a)
>
+
(14) I t is important to note that TIF is nof the ratio AI/As of isotopic Arrhenius A-factors. (16) In general, these TIF values are abandoned for the present purpose. Of course, if a particular calculation based on a reasonable
[CONTRIBUTION FROM THE
This is not often the case, and, since the temperature dependence and magnitude of the isotopic rate constant ratio are the quantittu derived from experiment, our concentration on T D F is required.
URBANA, ILLINOIS
EASTERN REGIONAL RESEARCH LABORATORY’]
Infrared Anisotropy and Structure of Crystalline Form C Stearic Acid and Vaccenic Acid’ BY H. SUSI RECEIVED OCTOBER3, 1958 The infrared spectra of highly oriented crystalline films of form C stearic acid and of vaccenic acid have been obtained using polarized radiation with the plane of polarization perpendicular and at 45’ angles to the principal crystal planes. The anisotropy of stearic acid is discussed in relation to its known structure. The data obtained on vaccenic acid indicate that the substance crystallizes in the orthorhombic system. The factor group of the space group is probably isomorphous with Dah. The main portions of the hydrocarbon chains are packed into an orthorhombic substructure similar to polyethylene and form C saturated acids, but the chains seem twisted near the double bonds, which appear to be nearly parallel in the projection along the C-axis.
Introduction The structure of saturated long chain monocarboxylic acids in their various crystallographic forms is known from single crystal X-ray measurements.* Very few data are available on the structure of the corresponding mono- and polyunsatu(1) Baatern Utilization Research and Development Division, Agticultural Ruearch Service, U. S. Department of Agriculture. (2) Presented in part at the Symposium on Molecular Structure and Spectroscopy, Columbus, Ohio, June 1958. (8) E. v. Sydow, Arkia. Kcmi, 0 , 231 (105b).
rated compounds. Lutton and K0lp4 have concluded from long spacing measurements obtained from powder diffraction data that in the trans-6through trans-12-octadecenoic acids the hydrocarbon chains of the odd compounds are roughly perpendicular to the planes containing the carboxyl groups, whereas the chains of the even compounds are tilted. Odd and even refer to the number of carbons from the carboxyl group to the double bond. (4) E. S. Lutton and D. G. Kolp, Turs JOURNAL, 78, 2733 (1861).
