J . Chem. SOC., Faraday Trans. 1, 1984,80, 267-274 The Barriers to Internal Rotation for Muonic-substituted Ethyl Radicals BY MARIA Joio RAMOS,~ DANIEL MCKENNA AND BRIAN C. WEBSTER* Chemistry Department, University of Glasgow, Glasgow G12 8QQ, Scotland AND EMIL RODUNER Physikalisch-Chemisches Institut de Universitat, CH-8057 Zurich, Switzerland Received 24th June, 1983 By fitting of the observed temperature dependence of the b-hyperfine muon-electron interaction to a theoretical model for the muonic radicals CH,MueH,, CHDMucH,, CHDMucD, and CD,MucD, values for the barrier height & restricting internal rotation are calculated. A decomposition of the barrier into pair interactions indicates the isotope effect upon the barrier to be in the sequence VpD < VpH < VFH < VpuD < TuH.On this basis the barriers to internal rotation for the radicals CH,MuCD, and CD,M&H, are predicted to be of the order 3076 and 3 186 J mol-l, respectively. Muon spin rotation (,u.s.r.) and electron paramagnetic resonance (e.p.r.) studies of several isotopically substituted ethyl radicals in the liquid phase have shown that the radicals formed exhibit temperature-dependent B-hyperfine coupling constants.' The hyperfine coupling constants for protons in analogous B-positions for alkyl radicals in liquids are known to obey the relation given by where L and M are constants (M & L) and 8 is the dihedral angle between the axis of the p,orbital at C, and the Cp-H axis. 8, specifies the equilibrium conformation of the radical by the value of 8 at the minimum of the potential barrier to internal rotation.In a classic study of internal rotation in alkyl radicals Fessenden has shown that eqn (1) is equally valid for deuterons in B-positions.2 Therefore it might be anticipated that eqn (1) will be applicable to radicals with a muon located at the p-posi tion. By fitting the experimental temperature dependence of the /I-coupling constants to a theoretical model using the quantum-mechanical averaging of the /I-coupling constants described previously, the barrier hindering internal rotation can be assayed for the radi~al.l-~ INTERNAL-ROTATION STUDIES In order to calculate the barrier V(8) to internal rotation, the experimental curve for the /I-hyperfine coupling constants at differing temperatures has been fitted to the t Present address : Departamento de Quimica, Faculdade de Cihcias, 4000 Porto, Portugal.267268 INTERNAL ROTATION IN ETHYL RADICALS curve, calculated from eqn (1) and z <AB(Y))i exp ( - Ei/k T) Cexp(-E,/RT) i = i (3) v, V(0) = - (1 - cos 20) 2 with L, M and v, as adjustable parameters. Quantitative values have been determined for the barriers to internal rotation exhibited by the different muonic radicals reported Table 1. Reduced moments of inertia for isotopic ethyl radicalsa radical I / kg m2 CH2MucH2 1.65 CHDMU~H, 1.92 CH,MucHD 2.08 CHDMU~D, 2.85 CD,MueHD 2.75 CD,MueD, 3.26 CH,D~H, 2.08 CHD2cD, 3.54 a Geometry for the radicals: dis- tances/ 10-lo m CB-Mu (1.094), CB-H angles/' MuCBC, (109.5), HC,H (120). (1.094), CBxC, (1.335), C5-H (1.083); previously, and also for the monodeuterated ethyl radical CDH,cH, together with the tetradeuterated ethyl radical CHD,t]D,.'.The muonic radicals studied here, it can be recalled, are the muonic ethyl radical, CH,MucH,, the two muonic mono- deuterated ethyl radicals, CH,MucHD and CHDMucH,, the two trideuterated ethyl radicals, CD,MucHD and CHDMucD,, and the tetradeuterated ethyl radical, CD,MucD,. Amongst these radicals two do not have a C2 symmetry rotating group, CH,MueHD and CD,MucHD. The calculation has proceeded on the assumption that the departure from symmetry for these species will not influence strongly the values obtained for the parameters. The reduced moments of inertia used in the calculation are collated in table 1 . A least-squares fit of eqn (2) to the experimental data has been performed using twenty-one wavefunctions in the calculation.These fits have been conducted in two different ways: the first allows for a simultaneous variation of the three parameters L, M and V,, the barrier to internal rotation; the second allows for the simultaneous variation of the two parameters M and V,, whilst L is equated to zero. The effect that a change in I, the reduced moment of inertia, would have upon the optimum values for V, has also been investigated. Simultaneous variation of the three parameters L, M and V , produces theoretical B-coupling constants whose values are identical to the experimental values. The resulting values for L, M and V, are listed in table 2 for all of the radicals. The radical CHD,cD, has been studied also.The resulting values for the parameters are in good agreement with those given by Fessenden.,M. J. RAMOS, D . McKENNA, B. C. WEBSTER AND E. RODUNER 269 Krusic et aL3 report that L is a contribution arising from spin polarisation while M is probably related to the hyperconjugative delocalisation of the unpaired electron onto the CX,X,X, group (Xl, X,, X, = Mu, H, D). It can be.observed from table 2 that the parameters L and'M for all the muonic radicals are larger than those for the radicals CDH,CH, and CHD,CD,. The values of the parameters suggest that both L and M are isotopically dependent. Table 2. Values for the parameters L, M and V, for muonic radicals, simultaneous variation of L, M and V, radical L/MHz M/MHz V,/J m o P - 2.8 - 0.5 - 16.3 - 17.6 - 13.1 - 20.0 - 20.0 - 23.4 156.4 151.5 19 1 .O 198.9 190.6 200.7 201.7 197.1 340 376 2845 2483 2704 2898 2927 3452 a 8, (Mu) assumed to be zero.130 120 120 140 160 180 T/K Fig. 1. Temperature dependence of the experimental three-parameter fit (0, solid line) and two- parameter fit (El, dashed line) reduced muon-electron hyperfine coupling constants for the CH,M&H, radical. The values of V , are much larger for the muonic radicals. Thus for CD,MucD, a barrier equal to 3452 J mol-l is indicated, in comparison with 376 J mol-l for the species CHD,cD,. This indicates a strong isotope effect and larger vibrational effect for muonium on the rotational barrier than is exerted by either protium or deuterium. Previously it had been suggested for these muonic radicals that the larger hyperfine interactions are associated with higher barriers to internal rotation in the radical.The V , values listed in table 2 could therefore seem anomalous for CH,MucHD and270 INTERNAL ROTATION IN ETHYL RADICALS 140 130 2! E - a P 1 120 1 - - - - a P 1 I 1 1 I I 1 1 1 1 120 140 160 180 TI K Fig. 2. Temperature dependence of the experimental three-parameter fit (0, solid line) and two-parameter fit (m, dashed line) reduced muon-electron hyperfine coupling constants for the CHDMU~H, radical. ~ ~~~~ 120 140 160 180 Tl K Fig. 3. Temperature dependence of the experimental three-parameter fit (0, solid line) and two-parameter fit (D, dashed line) reduced muon-electron hyperfine coupling constants for the CHDMucD, radical.CHDMucH,, respectively. This disparity could be attributed to the restriction of 8, (Mu) to zero in the fit. For all of the other radicals in table 2 the pattern conforms to the above stated hypothesis. Although the results are reasonable one must allow for a possible variation in the three parameters when treated simultaneously; therefore similar fits have been made equating L to zero. Fig. 1-4 show the experimental and theoretical dependences of the muon-electron /%coupling constant with this choice for the parameters. In each case the theoretical curve does not coincide exactly with the experimental curve. ThisM. J. RAMOS, D. McKENNA, B. C. WEBSTER AND E. RODUNER 27 1 12 14 16 18 TI K Fig. 4. Temperature dependence of the experimental three-parameter fit (0, solid line) and two-parameter fit (m, dashed line) reduced muon-electron hyperfine coupling constants for the CD,MucD, radical.Table 3. Values for M and 4, for muonic radicals, with L=OMHz radical M/MHz V,/J mo1-I CDH,cH, CHD,cD, CH,MueH, CH,MueHDa CHDMucH," CHDMuCD," CD,MucD, CD,MU~HD" 150.8 150.5 170.9 171.6 173.2 173.5 174.8 177.0 352 379 2710 2649 2665 2894 2906 2710 a 8, (Mu) assumed to be zero. discrepancy could be attributed to the truncation of the expansion of the hyperfine coupling constant or it reflects some distortion in the radicals at both the C , and Cp centre^.^ Final values for M and % are listed in table 3 for all of the radicals. Although there is a disparity in magnitude between the values obtained for & by the two approaches as shown in tables 2 and 3 the trend is in good accord in both cases, except for the radical CD,M&D,.For this radical & is lower than anticipated for the choice of L equal to zero. Finally, the effect of a change in I by 10% results in a change in V , of only 0.5%. This is a negligible effect. PARTITION OF THE POTENTIAL It has been assumed that the total potential V(8) is given by the truncated expression of eqn (3). This total potential term can be viewed as being representable by a sum272 INTERNAL ROTATION IN ETHYL RADICALS of terms each of which simulates the interaction of two substituents, one located at the C , nucleus and the other at the Cg nucleus: where i a n d j refer to substituents at the a-carbon and /?-carbon nuclei, respectively.2 Assuming that each of these terms Vij can be expressed in a Fourier series of even terms which can be truncated, as in v;i vij = -(1 -cos 2yu) ( 5 ) 2 Vij will represent the barrier to rotation of the pair ij noted in Therefore the total barrier to internal rotation V can be expressed by (7) v;j i j 2 v= x x -(1 -cos 2yu) Applying eqn ( 7 ) to the general case of the radical CXYZ CA, (X, Y, Z, A, z Mu, H, D) we obtain V = 2(4 VfA [ 1 - cos 2(8 + 8,X)l) + 2(8 VTA [ 1 -COS 2(8 + 8:)]) + 2(ie~ [ 1 - cos 2(8 + e?)]).(8) To calculate the maximum and the minimum of the potential V relative to the angle 8, we require dV/d8 = 0 as in = (2 VFA cos 28f + 2 eA cos 28: + 2 qA cos 2 8 3 sin 28 d V d8 - + ( 2 VfA sin 28f + 2 cA sin 28: + 2 qA sin 2 8 3 cos 28 = 0. (9) Lastly a generalised equation is obtained as v = vmax - vmin = "2 v ~ * COS 2ef + 2 GA cos 28: + 2 q~ cos 2832 + (2 VFA sin 28F + 2 cA sin 28: + 2 @A sin 28?)2]1/2 (10) where the quantity ( Vmax - Vmin) is the barrier height that has been calculated in the previous section for the muonic radicals noted by 2( VFH - VpH) = a 2( VpH - VpD) = b 2(VP"D- v y ) = c 2( VPuH - VF") = d where a, 6, c and d are the barrier heights for the radicals CH,DcH,, CD,HCD,, CD ,MuCD , and CH ,MUCH 2, respec tivel y .Table 4 presents the values taken by VpH, VFH, VpuD and VyuH as a function of VpD calculated using the barrier heights obtained by simultaneous variation of the three parameters L, M and G, and with the placement of L = 0. As can be seen from table 4 the values for the interaction between two substituents, although differing in magnitude according to the value taken by L, are placed in the order VpH < V y < VpJD < V p H .(12)M. J. RAMOS, D. McKENNA, B. C. WEBSTER AND E. RODUNER 273 Such a sequence provides support for the notion that the lighter isotope experiences a higher and the heavier isotope a lower degree of steric hindrance; the lighter isotope has in effect a higher interaction radius than the heavier isotope. The interaction VpD is anticipated to be the smallest interaction between two substituents, one atom at the a-carbon nucleus and the other at the /?-carbon nucleus, as in the sequence (13) vpD < VpH < v p < V p D < V y H . Table 4. Calculated values (J mol-l) of VpH, VFH, VFuD and VFuH as a function of VFD simultaneous simultaneous variation of variation of VFY- VpD L, M and V, MandV,;L=O 188 358 1726 1781 190 365 1355 1721 Table 5.Calculated values of the barrier heights V, for the radicals CH,MutHD, CHDMueH,, CD,MdHD, CHDMutD,, CH,MutD, and CD,MU~H, radical V,/J mol-I CH,MucHD 2963 CHDMU~H, 3030 CD,MueHD 3322 CHDMU~D, 3280 CH,MutD, 3076 CD,MU~H, 3186 Using the values for the parameters listed in table 4 for the simultaneous variation of L, M and K, the heights of the barriers hindering internal rotation have been calculated for the radicals CH,MueHD, CHDMueH,, CD,MucHD, CHDMueD,, CD,MueH, and CH,MucD,. For the first four radicals listed in table 5 the estimated barrier heights should be compared with the barrier heights noted in table 2. The trend is the same in each case except for the radical CD,MucHD, for which the value of K of 3322 J mol-l is greater than that anticipated from the placement relative to CHDMucD, in table 2. This procedure for partitioning the barrier into pair interactions permits the barriers for the radicals CH,MucD, and CD,MucH, to be predicted to be of the order of 3076 and 3 186 J mol-l, respectively. Observations upon muonic radical formation in liquid CH,=CD, are required to confirm these estimates. We thank the Instituto Nacional de Investigaciio, Cientifica, Lisbon, for a Research Studentship for M.J.R. and the S.R.C. for a Research Studentship for D. McK.274 INTERNAL ROTATION IN ETHYL RADICALS M. J. Ramos, D. McKenna, B. C. Webster and E. Roduner, J. Chem. SOC., Faraday Trans. 1, 1984, 80, 255. P. J. Krusic, P. Meakin and J. P. Jesson, J. Phys. Chem., 1971, 75, 3438. J. K. Kochi, Adu. Free-radical Chem., 1975, 5, 189. P. D. Sullivan and E. M. Menger, Adv. Magn. Reson., 1977, 9, 1. * R. W. Fessenden, J. Chim. Phyx, 1964, 61, 1570. (PAPER 3/ 1090)