2756
K.
M.MALONEY, S. P. PAVLOU, AND B. S. RABINOVITCH
Kinetic Isotope Effects in Nonequilibrium Thermal Unimolecular Systems.
Ethyl Isocyanide-d51& by Kenneth M. Maloney,lb S. P. Pavlou, and
B. S. Rabinovitch
Department of Chemistry, University of Washington, Seattle, Waehington 98106
(Received February 10, 1969)
The thermal isomerization of CzDsNC has been studied at 230.9’ over a range of pressures from lo-* to IO2 mm. The fall-offcorresponds to an enhanced value of the “effective number of oscillators” relative to C21SsNC. The difference in the observed high-pressure activation energy of the two species is very small [ ( E B-~E e E ) m = 0.15 kea1 mol-’] or zero. The data were treated on an RRKM quantum statistical basis. Calculations were made for several vibration-rotation models. The calculations are in good agreement with experiment. Ethyl isocyanide provides another reaction system for which the large inverse intermolecular secondary isotope effect predicted earlier is found; I C E / ~ D declines from 1.10 at p = to 0.21 at p N mm; the agreement between theory and experiment supports a value of around 0.2 at p = 0. Introduction The differential quantal effects that occur in unimolecular reactions as a consequence of the change in frequency pattern of the molecule upon isotopic substitution, e.g., of H by D, were originally described quantitatively by Rabinovitch, Setser, and Schneider. An inverse secondary intermolecular isotope effect in nonequilibrium thermal activation systems was predicted and has now been verified for methyl-methyld33s7b and methyl-methyl-d~~~ isocyanide systems, and for the cyclopropane-cyclopropane-& pair.4 This is pure secondary isotope effect of a statistical-weight nature, unlike conventional secondary isotope effects studied in equilibrium systems whose origin is basically mechanistic and which involve a change in critical reaction energy EOupon isotopic substitution; the present effect increases with the increasing degree of isotopic substitution, even at positions remote from reaction site such that AEO = 0, and is a maximum at the low pressure limit. I n continuation of the study of the isocyanide reaction system, the ethyl-& isocyanide has now been examined. Relative to the effect caused by the substitution of three H atoms in perdeuterated methyl-d3 isocyanide, the pentadeuteration in the ethyl-& isocyanide system is expected to lead to a large increase in isomerization rate relative to the light molecule. The present paper reports the temperature and pressure dependence of the C2H5NC-C2D5NC kinetic isotope effect. As in the earlier detailed study of ethyl isocyanidelsa chain lengthening is accompanied by a decrease in the ease of experimental accessibility of the low concentration limit and makes impossible the extension of these studies to the lower limit for the ethyl system or for higher isocyanide homologs.
in the molar ratio 1 :2 by the modified Gautier It was purified by glpc. The isotopic purity was determined by parent peak analyses with an AEI MS/9 mass spectrometer and was found to contain 3.3% C2HD4NC. Apparatus and Procedure. An internal comparison method and procedure similar to that described prev i o ~ s l y *was ~ ~ employed. ~~~ A reactant mixture of C Z H ~ N Cand C2DsNC of composition 1.82:l.OO was used. Different reactors were used in various overlapping pressure regions. Reaction vessels of different sizes, ranging from 1.07 ml a t the highest pressures up to 12 1. at the lower pressures, were used; there were no apparent systematic disparities in the overlap regions. Runs were carried out a t a temperature of 230.9’. The conversion of reactants was kept as low (average -30%) as was convenient for analysis. Analysis. The products were separated from the reactants by means of a short column of AgCN which quantitatively removed unreacted isocyanide. The residual nitrile products were analyzed on the MS/9 spectrometer with the use of parent M and M- 1 (M -2 for CzDsCN) peak intensities at 70-V electron energies. Repeated calibrations were made with standard
Experimental Section Reactant. CzDsNC was prepared by allowing CzDJ (Merck Sharp and Dohme, Ltd.) to react with AgCN
2994 (1964). ( 5 ) (a) K. M. Maloney and B. S. Rabinovitch, J. Phys. Chem. (called part I ) , 73, 1652 (1969). (b) K. M. Maloney and B. 6. Rabinovitch, ibid., 72,4483 (1968).
The Journal of Physical Chemistry
(1) (a) This work was supported by the National Science Foundation. (b) Abstracted from the Ph.D. Thesis, University of Washington, 1968,of K. M. Maloney. (2) B. S. Rabinovitch, D. W. Setser, and F. W. Schneider, Can. J . Chem., 39, 2609 (1961)(called RSS) ; see also B. S. Rabinovitch and J. H. Current, ibid., 40, 557 (1962). (3) (a) F. W. Schneider and B. S. Rabinovitch, J . Amer. Chem. Soc., 84, 4215 (1962); (b) F. W.Schneider and B. S. Rabinovitch, ibid., 85,2365 (1963); (c) B. S. Rabinovitch, .’l W. Gilderson, and F. W. Schneider, ibid., 87, 158 (1965). (4) B. S. Rabinovitch, P. W. Gilderson, and A. T. Blades, ibid.,86,
2757
KINETICISOTOPE EFFECTS IN C2HSNC-C2D5NC Table I : Pressure Dependence and Relative Isotopic Magnitudes of Experimental Isomerization Rate Constants for CtDbNC Determined by Internal Comparison (T = 230.9')
0.00075 0.00075 0.00085 0.0018 0.0021 0.0023 0.0024 0.0079 0.0088 0.0099 0.0388 0.092 0.267 1.93 2.14 2.15 13.31 16.9 22.5 87 87 90
0.249 0.243 0.285 0.328 0.342 0.340 0.353 0.52 0.58 0.64 1.85 3.11 5.8 12.0 12.0 12.4 15.6 16.0 15.8 15.8 15.6 16.0
(0.277) (0.270) (0.309) (0.345) (0.359) (0.357) (0.370) (0.52) (0.58)
OD
1.32 1.27 1.36 1.57 1.53 1.66 1.57 1.68 1.79 1.87 3.56 4.6 7.6 12.7 12.2 13.0 14.4 15.1 14.8 14.1 14.0 14.5 14.5'
0.0178 0.0173 0.0198 0.0221 0.0230 0.0229 0.0237 0.0334 0.0371 0.041 0.119 0.199 0.371 0.77 0.77 0.79 0.99 1'.01 1.00 1.00 0.99 1.01
0.090 0.087 0.093 0.108 0.105 0.114 0.108 0.115 0.123 0.128 0.244 0.318 0.52 0.87 0.84 0.89 0.99 1.03 1.01 0.97 0.96 0.99
0.210 0.212 0.227 0.220 0.235 0.215 0.235 0.310 0.323 0.344 0.52 0.67 0.76 0.94 0.98 0.96 1.08 1.06 1.07 1.12 1.11. 1.10 1.10
(0.210) (0.211) (0.227) (0.220) (0 235) (0.215) (0.235) (0.310) (0.323) (0.344) (0.52) (0.68) (0.76) (0.94) (0.98) (0.96) (1.06) (1.08) (1.08) (1408) (1.10) (1.10) (1.09) I
a Constants in parentheses are uncorrected for heterogeneity. * Isotopic ratios in parentheses are determined from the M 2 peak intensity of CZD~CN. ' Based on the data of part I becomes 14.2; see text. intensity of CpH6CN and the M
-
GH6CN-C2D6GN mixtures for each series of analyses. The results were corrected for C13 and NIs contributions.
Results Corrections to the Data. The experimental corrections described in part I were applied to the present data. The presence of the small percentage of d4 isomer was allowed for in the analytical calculations.
Isotopic Rate Ratio. The relative rates of isomerization of CzHsNC and C2D5NC a t 230.9' are tabulated in Table I and the individual rate dependence on pressure is shown in Figure 1. The observed limiting high pressure ratio ( k H / k D ) , is 1.10 and has declined at the lowest pressure to 0.21 (Figure 2). Temperature Dependence of Rate Ratio. MeasureI
..
0.I
Figure 1. Plot of k/km us. pressure for C2D6NC, 0,and CZHKNC,0, fall-off at 230.9". The solid line in each case represents the prediction of the E-300 models (after pressure factor correction). The data presented for CZHKNC were determined within this study, all by mass spectroscopic analysis; these data accord well with earlier fall-off measurements (see Figure 1, part I). The effect of heterogeneity at the lowest prassurw in a 12-1. reactor is evident (see ref 5b).
- 1 peak
I
IO
1
100
p(mm)
Figure 2. Plot of kH/kn vs. pressure for the CzHsNC-CzD6NC system at 230.9". Experimental data, 0 ; calculated behavior on the E-300 model, , shifted slightly to coincide a t p = m but not corrected for pressure misfits of the separate fall-off curves. It is evident from Figure 1 that, with pressure correction, the calculated E-300 ratios and the observed ratios curves would coincide very closely down to 10-8 mm. Volume 78, Number 8 August 1969
K. n/x. MALONEY, S. P. PAVLOU,
2758
AND
B. S.RABINOVITCH
Table 111: Vibrat.ion Frequencies (cm-1) for the GzDsNC Molecule and Activated Complex E-300 ----Molecul~----
Grouping
complex
2146 (6)
2143 (5)
C-D str 2.15
1.85
Figure 3. Arrhenius plots of log ( k ~ / k ~US. ) , 1/T.
N=C str CDs bend
ments of the ratio ( k ~ / k ~were ) , made over a range of temperatures from 190.0 to 259.6' at) 100 mm (Figure 3). This is in the high-pressure region for both of the reactions. The M peak intensities gave a value of AEam= -0.2 kcal mol-'; the M - 1 and R!I - 2 peak intensities gave a value of AE, = -0.1 kcal mol-' which is intrinsically slightly less reliable. The true value is in any case close t o zero. Measurements of the temperature dependence in the low-pressure region were not pursued. C&NC High-pressure Rate and Fall-of Behavior. The absolute value of k D , measured at 230.9' is 14.5 X sec-I. However, a t this temperature k H m is 15.6 X sec-' from part I, and since ( k ~ / k ~ =) , 1.10 the value of kD, on this basis is 14.2 X sec-'. Because of the large number of measurements made in , the greater accuracy of relative part I for k ~ and internal comparison measurements, the latter value is adopted. The experimental fall-off curve (Figure 1) corresponds to a curvature and shape characterized by a Slater n value of 6.9.
Discussion High Pressures. At the high-pressure equilibrium limit the reaction coordinate is the rearrangement of the activated complex. The secondary isotope ratio is ( J ~ H / ~ D= ) ~
AEo/R T
(IrH&+H&n/IrD&H&+D)e-
( 1)
where I, = u(I+AI+BI+~)'/'/u+(IAIBIc)~'~ is an inertial moment ratio of the activated complex and molecule, and Q+ and Q represent the total partition function of the Table 11: Moments of Inertia (amu, lz) C2HSNCand CzDjNC Molecules and Activated Complexes and Vibrational Partition Functions
CaD6NC E-300 complex CzHsNC E-300 complex
IA
IB
IC
I,
15.36 20.33 12.55 14.64
110.0 129.3 97.81 110.5
126.3 133.3 110.4 117.0
1.281
C&NC (230.9') C2DsNC (230.9') The Journal of Phusical Chemistry
16.65 37.76
CDa torsion
'150 sot50 1185
910 877 766 670 420 249 205 165
}
Ring dof. 1990 1105(5)
j 928 (4)
945 (3)
Ring def. 562 324
product for the internal degrees of freedom of the activated complex and molecule, respectively. Because AEo is close to zero, and because thel, (Table 11)and (Q+/&) ratio of ratios in eq 1 are almost inevitably close to unity, only a small pure secondary isotope effect occurs under equilibrium conditions. The behavior for CzD6NC was calculated for the transition-state models E-300 and E-300 mol rot., characterized in part I. The vibrational frequency assignments for the deuterio system are given in Table 111. I n its dependence on the (Q+/Q) ratio, the isotope effect at high pressures is normal or inverted depending on whether there is bond tightening or bond loosening in the activated complex relative to the molecule; t,he ) ~ ethyl isocyanide is ratio here is 1.03. ( ~ I I / ~ D for found here to be 1.10, which compares very well with the theoretical value of 1.16 based on AE8 = -0.15 and6 AEo = -0.19 kcal mol-l, and is slightly greater than that measured for methyl43 isocyanide. *b The absolute value of the measured high-pressure rate constant is k ~ ,= 14.2 X sec-' = Ae-Eo"'RT = 10'3.66 e-a7.*OlRT sec-1; the theoretical A , value i s 1013.44 sec-'. Fall-off Region. The general quantum statistical formulation of Marcus and Rice (RRRM theory)'& for the unimolecular rate constant is
1.182
QY
Q"
CDz bend CD2wag CD, rock CD, twist C-C str C-N str CD3 rock CDZrock C-C-N bend C-N-C bend
2160 2120 (2) 2085 2161 ,
+
(E-300)
11.69 25.62
-
(6) Note that EOH= Es, - 0.63 from part I and EOD= Ea, 0.69 here, so that substantially AE,, = AEo. (7) (a) R. A. Marcus, J . Chem. Phus., 20, 369 (1962); (b) E. V. Waage and B. 8. Rabinovitch, Chem. Rev., submitted.
2759
KINETICISOTOPE EFFECTS IN C2H5NC-C2D5NC ~~~~
Table IV: Equivalent Classical Fall-off Shape Parameters s and n for CzDaNC and Other Molecules (230.9')
CHaNC
n
8
Expt 300 300 fig. rot.
4.6 3.3 4.4
3.3 2.6 3.2
Expt 300 300 fig. rot.
+
5.2 3.8 5.0
3.7 2.9 3.6
Expt E-300 E-300
+ mol rot.
6.1 6.0 6.4
4.5 4.5 4.8
+ mol rot.
6.8 7.0 7.5
5.1 5.2 5.6
+
CDsNC
CzHaNC
CnDaNC
Expt E-300 E-300
where ZP(E+,,.) is the sum over the degeneracy of the active vibration-rotation energy levels of the activated complex; w is the collision frequency and the strong collision assumption is used; N(E,,) is the density of the vibration-rotation energy levels of the molecule. Equation 2 ignores centrifugal effect^.'^ Equivalent classical shape parameters, Kassel's s and Slater's n, are given in Table IV. The experimental shape is well approximated by the E-300 model, as shown in Figure 1, and less so by the active molecule rotation model (Table IV). The effect of deuteration is to increase the Slater n value from 6.1 for C2HGNC to 6.8 a t 230.9", which is in close agreement with the calculated value of 7.0. Due to heterogeneity effect@ which precluded very low pressure measurements, the theoretical and experimental curves were compared as to pressure a t the arbitrarily chosen value of k/k, = 0.2, instead of at lc/k, = 0.1 which has been more customary.a~6aThe pressure correction factors which must be applied to the calculated curves to bring coincidence with experiment at k/k, = 0.2 are given in Table V for collision diameters of 5.0 A. The theoretical isotopic rate ratio k ~ / isk obtained ~ by use of eq 2 for k~ and k ~ .The theoretical and experimental ratios are compared in Figure 2. The agreement is very good until the low-pressure limit is approached, and further discussion requires consideration of the low-pressure region. The observed activation energy difference between the light and heavy systems was not measured as a function of pressure (Figure 3). It may be predicted2 that the difference AE"a = E",D - E",H is bigger a t the high-pressure than at the low-pressure limit since the parameter s(CzD5) exceeds s(c2H~)(Table IV). Recently, Lin and Laidlers have pointed out that the difference AE, first increases to a maximum as the pressure decreases below the high-pressure limit, before decreasing to the low-pressure limiting value. We
Table V : Pressure Correction Factors for Calculated CzDsNCFall-off (230.9°)a Correction factor
E-300 E-300 a
c = 5.0
+ mol rot.
1.48 2.2
A.
may readily explain this behavior in terms of the differential quantum effects described earlier.2 Since the value of the classical s parameter is larger for the deuterated than for the light system, the light system enters the fall-off region at higher pressure with concomitant decrease in E,H. Thus at a pressure just below the high-pressure limit for the light system, the observed value of A E a is E",D - ea^, which exceeds AE,, = E",D E m a ~ hence, ; AE, first increases with decrease of pressure before eventually starting to decrease. Low Pressures. At the low-pressure limit, k ~ / k ~ reduces ta
-
(3) where M is the molecular weight. For a pure secondary isotope effect, EO11 = EOD. Since the ratio of the normalizing partition functions, QD/QH, is not much larger than unity, especially at low temperatures, the enhancement of the rate constant k~ relative to k~ depends on the large difference in the quantum statistical densities of active energy states, N(Ev,), for the two molecules a t energy levels E 1 Eo, i e . , on the larger proportion of deuterated molecules that exist about Eofor the Boltzmann distribution. Owing to wall effects that are important in this system below lov2 mm in a 12-1. vessel,6 the pressure dependence of the fall-off of rate for the individual isotopes deviates from the theoretical behavior at lower pressure; thus, the isotopic rate ratio might not be expected to follow the theoretical prediction very well at such low pressures. Actually, Figure 2 shows that due to mutual compensation of the heterogeneity effects, the rate ratio continues to show reasonably good adherence to theory down to a pressure of -5 X mm; thereafter, k H / ' k ~levels off a t a value of 0.21. Because the agreement between theory and experiment is otherwise excellent, the calculated rate ratio a t P = 0 of 0.188 for AEO = 0 may be considered (8) M. C. Lin
and K. J. Laidler, Trans. Faraday Soc., 64,927 (1968). Volume 73,Number 8 August 1969
K. M. MALONEY, S. P. PAVLOU, AND B. S.RABINOVITCH
2760 Table VI : Calculated Values of Average Excess Energy of Reacting Molecules (cm-1) for Several Isocyanides (2’ = 230’)“ (E+b-o
CHaNC CDsNC CzHsNC CzDsNC
425 431 483 495
(E+)P-
m
870 1089 1575 1921
For E-300 model.
to be close to the correct value. The limiting ratio is 0.228 for AEo = -0.19 kcal. Thus, the CzH5NC-CzD5NCsystem illustrates another thermal unimolecular reaction system in which the existence of a large inverse secondary isotope effect has been found, and which is in accord with theory with regard to both magnitude and pressure dependence. The limiting ratio for the case of pentadeuteration of ethyl compares suitably with the theoretical value of 0.29 (with AEo = 0) for trideuteration of methyl isocyanide (measured = 0.28).3b E$ects of Variation of Molecular Parameters on Nonequilibrium Unimolecular Behavior. The n and s shape parameters for the methyl-methyl4 and ethyl-ethyld5 isocyanide systems were determined analytically from the shape of the fall-off by analytical computations* (Table IV). The computer program accepted values of k / k , a t various intervals of pressure and determined n and s values for the intervals by comparison with a reference set of entries calculated for a range of s and n values from the theoretical relationss of RRK (with b = Eo/RT) and Slater theory. The values of n and s reported herein supersede the earlier less accurate values38,b of n and s reported for methyl and methyl-& which were obtained by a method of visual superposition of fall-off plots. Trideuteration of methyl isocyanide has been shown experimentally to increase the values of n and s by
The Journal of Physical Chemistry
-0.6 and -0.4, respectively. Therefore, the replacement of three H atoms by three D atoms has the effect of “adding” one-half an “effective oscillator.” An increase in the chain length from methyl to ethyl isocyanide results in an increase of the observed n and s values by -1.5 and -1.2, respectively; addition of a methylene group results in approximately three times the effect of substitution by three deuterium atoms. Similarly an increase in the chain length from methyld3 to ethyl-d5 isocyanide causes n and s to increase by -1.7 and -1.4, respectively. Pentadeuteration of ethyl results in an increase in n and s of 0.8 and 0.6, respectively. Therefore, substitution of five deuterium atoms adds a little less than one “effective oscillator.” The pressure correction factors are 1.2 and 1.5, respectively, for the methyl and methyl-& 300 models, with u = 4.5 A a t 230”. The factors for ethyl and ethyl-& are 1.35 and 1.5, respectively, for the E-300 models, using u = 5.0 A. The variation of the average energy of reacting molecules with chain length has been mentioned in part I. I n Table VI is shown the calculated variation with pressure of the average energy of the reacting molecules (E+) for the members of the isocyanide series under investigation. Although the variation has been calculated for all regions of k/k,, in the interest of brevity only limiting values are given here. The quasi-constancy of (E+),,o with change in molecular structure is contrasted with that of (E+)p,,; the latter increases in a manner dependent on, but not precisely commensurate with, the change in ?a (Table IV). The RRKM theory not only predicts the change in n and s values with variation of molecular complexity but also correctly predicts the observed pressure shift of the fall-off. The effects of chain lengthening as in propyl, isopropyl, n-butyl, and t-butyl isocyanides have been discussed el~ewhere.~ (9). (a) N. B. Slater, “Theory of Unimolecular Reactions,” Cornell Universitv Press. Ithaca. N. Y.. 1959: (b) L. S. Kassel. “Kinetics of Homogeneous Gas Reactions,” Reinhold Publishing Carp., New
York, N. Y., 1982.
