Cyclohexane revisited: High pressure nuclear magnetic resonance rotating framerelaxation study of the dynamical solvent effects on the conformationalisomerization of cyclohexaneD. M. Campbell, M. Mackowiak, and J. Jonas Citation: The Journal of Chemical Physics 96, 2717 (1992); doi: 10.1063/1.462019 View online: http://dx.doi.org/10.1063/1.462019 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/96/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in High pressure nuclear magnetic resonance study of the dynamical solvent effects on the rotation ofcoordinated ethylene in an organometallic compound J. Chem. Phys. 93, 2192 (1990); 10.1063/1.459051 Highpressure nuclear magnetic resonance study of the dynamical solvent effects on internal rotation ofN,Ndimethyltrichloroacetamide J. Chem. Phys. 92, 3736 (1990); 10.1063/1.457831 Dynamical solvent effects on conformational isomerization of cyclohexane and 1,1difluorocyclohexane J. Chem. Phys. 90, 5386 (1989); 10.1063/1.456445 Dynamical effects on conformational isomerization of cyclohexane J. Chem. Phys. 75, 1571 (1981); 10.1063/1.442193 Molecular Motions in Several Solids Studied by Nuclear Magnetic Relaxation in the Rotating Frame J. Chem. Phys. 52, 5525 (1970); 10.1063/1.1672820

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Cyclohexane revisited: High pressure nuclear magnetic resonance rotating frame relaxation study of the dynamical solvent effects on the conformational isomerization of cyclohexane

D. M. Campbell, M. Mackowiak, and J. Jonas School o/Chemical Sciences, and Material Research Laboratory, University 0/ Illinois, Urbana, Illinois 61801

(Received 3 July 1991; accepted 30 October 1991)

The main goal of this study is to extend the dynamic range of isomerization rates for cyclohexane in order to determine with high accuracy whether the barrier height to isomerization is pressure dependent. Therefore, the effect of pressure and temperature on the conformational isomerization of cyclohexane in carbon disulfide solvent has been investigated using the NMR (nuclear magnetic resonance) rotating frame relaxation technique. This technique, used for the first time in pressure studies of chemical exchange, allows the measurement of isomerization dynamics over a wide range of pressures and temperatures. By combining the rotating frame and NMR line shape techniques and generating the isoviscosity plots, it is shown that the barrier height to isomerization is independent of pressure. Since the experimental isomerization rate is accelerated by pressure, the viscosity dependence of the reduced transmission coefficient shows that the isomerization falls into the energy controlled (inertial) regime of the Kramers model in agreement with our earlier experimental findings. These experimental results, as interpreted in terms of stochastic' models of isomerization reactions, indicate a strong coIlisional coupling and the presence of dynamical solvent effects.

I. INTRODUCTION

The dynamical solvent effects on the unimolecular iso­merization dynamics have been investigated in many theo­retical l-

9 and experimental studies. 10-18 According to theo­retical models l

-3

•9 describing the dynamical solvent effects

on reaction rates in liquid solutions, the reaction coordinate is coupled to the solvent, enabling the system to gain suffi­cient energy to cross the barrier, lose energy, and become trapped into the product well. In absence of electrostatic interactions, this coupling is produced by collisions between the solvent and solute molecules. In contrast to classical transition state theories for isomerization reactions, the sto­chastic models propose a dependence of the transmission coefficient K upon the so-called "coIlision frequency" a, which reflects the actual coupling of the reaction coordinate to the surrounding medium. According to theoretical mod­els, the transmission coefficient K is found to be a strong non monotonic function of a with two different limits. Acti­vation due to collision rate is limiting and K is proportional to a for the energy-controlled regime at low collision frequen­cies. At high collision frequencies in the diffusive regime, particles which have crossed but not yet cleared the barrier may suffer collisions and recross the barrier. The reaction in this limit is said to be diffusion controlled and the rate is inversely proportional to a. Between these two regimes there is a non monotonic transition, Kramers turnover (cross­over).19

Most of the systems studied in condensed media fall into the intermediate to high-friction regimes. 12.16 This behavior is consistent with the prevalent view that intramolecular vi­brational relaxation is much faster than intermolecular ener­Y9 transfer. The belief was that the high dimensionality and rapid vibrational energy transfer between the modes of a

polyatomic molecule would eliminate the possibility of the inertial regime in a unimolecular isomerization in dense liq­uid solvents. However, in our laboratory we have shown that the cyclohexane isomerization exhibits the entire range of behavior from the low-friction to the high-friction regime. \0

Chandler and co-workers,20 using stochastic dynamics com­puter simulations, have shown that the cyclohexane data can be explained if the intramolecular energy flow is at least par­tially inhibited, so that the reaction coordinate's coupling to the solvent environment dominates. The idea that the obser­vation of the inertial behavior depends strongly on the rela­tive strength between the inter- and intramolecular cou­plings was further confirmed by our experiments 18 on an organometallic compound. The observation of the inertial regime in this system may be a consequence of the so-called heavy metal atom bottleneck effect which reduces the intra­molecular energy transfer within the molecule.

All theoreticafo and experimental evidence21 points out that the inefficient internal energy flow is the critical factor in order to observe the inertial regime or the Kramers turnover for a system in dense solvent fluid. Our earlier stud­ies of cyclohexane isomerization used the NMR (nuclear magnetic resonance) line shape technique which has some inherent limitations as far as the accessible dynamic range of rates is concerned. As we pointed out,22 for this NMR line shape technique to be applicable, molecular motions must fall within a narrow timescale and this restricted range of measurable rates leads to a relatively large error in the acti­vation parameters determined. Since the rate depends ex­ponentially upon the barrier height, one must 'rule out the possibility that the barrier to isomerization is pressure de­pendent. For example, for cyclohexane in carbon disulfide a pressure induced decrease of only 0.25 kcal mol - 1 in the free energy of activation over a 5 kbar pressure range would ac-

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2718 Campbell, Mackowiak, and Jonas: Conformational isomerization of cyclohexane

count for the observed increase of the reaction rate. \0 There­fore, it is necessary to measure isomerization rates over a wide range of temperatures, pressures, and viscosities in or­der to determine activation parameters with high accuracy. Fortunately, by using the NMR rotating frame technique to measure isomerization rates,23 it is possible to overcome the inherent weakness of the NMR line shape analysis and ex­tend greatly the dynamic range of experimental rates, as the highest measurable rates by the NMR rotating frame meth­od are greater than 2 orders of magnitUde above those acces­sible by other NMR methods. Thus it would be possible to cover experimentally the high temperature and/or low vis­cosity regions, where the inertial regime is most likely to occur. This fact is of particular importance in experiments searching for new molecular systems showing an inertial be­havior.

In this paper we present the results of pressure effects on the conformational chair to chair isomerization rate of cy­clohexane. Rotating frame spin-lattice relaxation time Tip

measurements were made to determine these rates and com­bined with previous results 10 using the line shape fitting analysis technique. The data indicates that this system is in the inertial to turnover regions in the Kramers' model. By increasing the temperature we were able to shift the system further into the inertial regime by lowering the viscosity which is proportional to the collisional frequency. At this temperature, the interconversion of cyclohexane is too fast to be measured by the line shape technique, therefore, we used the Tip method to measure isomerization rates and their pressure dependence. Using the same solvent and changing its viscosity by pressure and temperature is more advantageous than the use of different solvents. Different solvents of the same shear viscosity may not have the same solvent interactions, and this could have a large effect on the reaction rate measured.

There are several motivations for this study. First, we want to apply the NMR rotating frame technique to measure isomerization rates over a wide range of temperatures and pressures in order to determine activation parameters with higher accuracy. The advantages and limitations of this method will be discussed. Second, we want to concentrate on one solvent (CS2 ) studied over a wide range of temperatures and pressures to generate isoviscosity plots. These plots would determine if the barrier to rotation is pressure depen­dent. Third, using the Tip technique we should be able to shift the system farther into the inertial regime by increasing the temperature, and thus lowering the viscosity. Finally, we would like to compare our new experimental results ob­tained at low viscosity with the stochastic models for the isomerization in solution in the inertial regime.

II. EXPERIMENT

Both cyclohexane and carbon disulfide were obtained from Aldrich Chemical Company and used without further purification. Solutions were made 5 mole % after each com­ponent had been degassed by at least six cycles of the freeze­pump-thaw method. The components were then mixed un­der an inert atmosphere and loaded into the high pressure sample cell.

All the IH FT NMR experiments were performed on a home built spectrometer system operating at 180 MHz equipped with an Oxford Instruments Company supercon­ducting magnet. The temperature of the sample was mea­sured by a copper-constantan thermocouple inside of the high-pressure vessel near the sample cell. Thermal stability was maintained by the use of a MG W Lauda UItra-Kryomat K-120W temperature bath. Temperatures were estimated to be accurate to ± 0.2 K. The pressure range studied was from 1 bar up to 5 kbar. The high-pressure probe, described previously,t4 was retuned, and the magnetic field homo­geneity was optimized after every change in pressure or tem­perature.

The rotating frame spin-lattice relaxation times Tip

were measured by applying a ( 1T /2) x pulse followed by a 90· phase-shifted spin-locking pulse of variable duration. Ex­change contributions to proton Tip relaxation were mea­sured as a function of temperature, pressure, and HI field strength. In contrast to present commercial NMR instru­ment design, a much wider range of HI values must be used in order to cover a broad range of exchange rates. In our experiment the amplitude of the spin-locking pulse was changed by 2 orders of magnitude in the interval 0.015-1.5 G. Therefore, special modification to our spectrometer was required. The modification is achieved by the use of a vari­able rf attenuator. This device consisted of a current con­trolled mixer with a variable current source. The smaller the current to the mixer the greater the attenuation. The system does not require software modification of a spectrometer and the phase was verified to be independent of the level adjust­ment of the spin-locking field amplitUde.

The strength of the lock field, H sI ' was calibrated by two different methods. Measurements of the rf voltage of the 1T /2 pulse at the probe allowed the HI field strength (in Gauss) to be determined from the following relation:24

1TVsI Hsi = ,

2yV"'/21'p (1)

where VsI is the rf peak to peak voltage, y is the proton mag­netogyric ratio (in rad/G), V"./2 is the rfpeak to peak vol­tage of the 1T/2 pulse, and 1'p is the 1T/2 pulse length. The rf field strength can also be calibrated by disconnecting the excitation rf pulse and using the spin-locking rf pulse as an excitation pulse.

In weak rffields the phasing of the Fourier transformed spectra is not straightforward. The individual nuclear mag­netizations for spins possessing different chemical shifts pre­cess about their respective effective fields, and during the spin-locking period phase differences can occur. The soft­ware zero and first order phase corrections are no longer adequate. To avoid this problem the power spectrum has been measured in our experiments. Values of Tip were ob­tained by fitting the power spectrum intensities to the expression P = Po exp( - tsl/Tlp ) + Pr' where Pr repre­sents a residual signal present at long spin-locking times, tsl

(usually, P r < 0.03po ). Each decay curve consisted of at least 20 spin-locking times. The frequency offset never exceeded 30 Hz. All spectra were accumulated several times and Tip

measurements were repeated at least three times at each of

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Campbell, Mackowiak, and Jonas: Conformational isomerization of cyclohexane 2719

the rotating field strengths. The standard deviations were calculated to give estimated errors of ± 7%-8% in Tip val­ues. Proton TI values were measured by the inversion-recov­ery method.

The shear viscosity of the sample was measured using a high pressure rolling ball viscometer. At a given temperature and pressure the viscosity 7J of the liquid is proportional to the roll time of the ball troll and is given by

(2)

where C is the calibration constant of the tube and ball sys­tem. The calibration constant was determined using viscos­ity data for acetone at 253 K. The roll times are detected by an inductively coupled radio frequency coil arrangement and are consistent to ± 1 %. The viscosity is estimated to be accurate to within ± 2%.

III. RESULTS AND DISCUSSION

A. Rotating frame nuclear spin-lattice relaxation studies of conformational exchange

The chair to chair isomerization of cyc10hexane can be characterized by two degrees of freedom. The chemical ex­change contribution to the Tip relaxation of the system of spins exchanging between two sites of different chemical shift is determined by the exchange rates and the chemical shift differences between those sites.23 For a given exchange rate, larger chemical shift fluctuations have larger effects.

For a system of I spins having a chemical shift OtJJ from the average Larmor frequency wo, the fluctuating part of the spin Hamiltonian, ow(t)Iz , can provide an efficient relaxa­tion mechanism with correlation time 'Te' This mechanism can relax spins aligned along the rotating rf field HI .

Because chemical shift fluctuations have a maximal ef­fect at the frequency of precession about the spin-locking field, the exchange contribution can be dependent on the strength of that field. According to the two-site exchange model23

(_1_) = _O_{J)_2 _-z_k_-z-'

Tip ex 2 4k + WI (3)

where (1/Tlp )ex is the exchange contribution to the Tip relaxation time, (J)I is the precession frequency about the spin-locking field, o{J) is the chemical shift difference be­tween the sites, and k = 1/2'T, is the probability per unit time of transition from one site to the other.

The observed relaxation time (Tip )obs will also have contributions from other mechanisms due to dipolar cou­pling, spin-rotation interactions, etc., but their contributions to Tip and TI are equal and independent of {J)o and {J)I' as­suming that the correlation time are short ({J)o'T, ~ 1). Therefore, the contributions to Tip due to chemical ex­change may be obtained by subtracting the contributions due to these mechanisms from the observed value

(4)

For cyclohexane where protons undergo exchange be­tween equatorial and axial sites which are equally probable, Eq. (4) predicts a linear dependence of ( Tip) ex on (ut with a

gradient of 2/[k(O{J)2] and intercept at {J)I = 0 of (Tip )ex = 8k I(O{J) )2. Measurements of the gradient and in­tercept allow independent determination of values for both k and o{J), which may be extracted from a plot of ( Tip) ex VS {J)i •

Calculated exchange contributions to the rotating frame relaxation rate according to the two-site, equal probability model are presented in Fig. 1. As follows from Eq. (4), the highest exchange contribution is observed for WI = 2k. With increasing spin-locking field strength, the exchange contri­bution decreases rapidly. The range of exchange rates that give rise to significant contributions to Tip is determined also by the chemical shift difference between the sites. For a given chemical shift the {J)I range has to be adjusted for each particular value of exchange rate to be measured. Thus there is a window of observability for exchange motions affecting Tip' The upper limit to the exchange rate which can be deter­mined by this method may be called the "no-slope limit." As predicted by Eq. (4), the slope of Tip vs {J)i plot (which is inversely proportional to k) for high exchange rates becomes too small to be measured accurately. For rotating field strength higher than 4 X 104 radls effective exchange contri­butions are too small to give a detectable dependence of ( Tip) ex on (J)i in cyc1ohexane. There are also some serious instrumental limitations. Higher exchange rates require very high spin-locking fields. In the case of cyclohexane we ap-

60

4-0

20

........ 0 ... I

!1l 6 --.ct

()

H " G) ........ CI. ... 2. Eo-<

.......... ... -- 1.6

1.2

O.B

0.4-

0.0 10 1 10· lOS 10' 1011 1011

k (8-1

)

FIG. 1. Theoretical contributions of conformational exchange to rotating frame relaxation rate for parameters in the range of the present study: chem­ical shift difference between sites 83 Hz and rotating field strengths between 5 X 102 and 7 X 104 rad/s. The vertical bars represent the range of param­eters covered by the experimental for given values of exchange rate k. The values of the rotating field strength expressed in rad/s are the following: a-5X102

, b-I X 103, c-2X103

, d-4XlO3, e-7x103, f-lXlO\ g-2XIO\ h-

3 X 104, i-4 X 104

, j-7 X 104•

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2720 Campbell, Mackowiak, and Jonas: Conformational isomerization of cyclohexane

plied an rf field of about 1.5 G in order to measure rates of 5 X 104 S - I. At such a high rf field special attention must be given to the probe construction and diodes. Also, tempera­ture of the sample must be carefully monitored to avoid heat­ing by the long spin-locking fields.

The lower limit to the exchange rate which can be mea­sured by rotating frame relaxation will depend upon the ho­mogeneity of the magnetic fields Ho and HI . The lower limit is determined by the line broadening associated with short rotating frame relaxation times in weak rf fields, because calibration of very weak rf fields is difficult and may lead to substantial experimental errors. In order to perform these experiments accurately, special attention must be paid to minimizing fluctuation in phase, amplitude, and frequency of the applied rf field. Generally, it is also clear that the experiments themselves are much more time consuming since they involve a systematic variation and calibration of HI , and strict control of many experimental parameters. It may well be that these inherent experimental difficulties led to the very limited application of this method. So far the conformational exchange by rotating frame relaxation has been studied in cyclohexane only for k < 103 S - I and at am­bient pressure.2S Bleich et al.26 and Kopple et al.27 have previously discussed the use of Tip for studies of peptide conformational exchange, where a large value of chemical shift makes these systems more appropriate for Tip examina­tion, as compared to cyclohexane.

B. Variable-temperature measurements at atmospheric pressure

According to transition state theory (TST), the unimo­lecular isomerization rate constant is defined as

koT (-aG""') nfR kTST =K--exp --,

h RT f"'" (5)

wherefR andf"'" are the activity coefficients of the reactants and the transition state, respectively, K is the transmission coefficient, aG "'" is the Gibbs free energy of activation, and the other symbols have their usual meaning. The activity coefficients are defined such that their standard states at infi­nite dilution in solution are equal to unity. For adequately diluted solution, one can assume that nfRlf"'" is unity and independent of both temperature and pressure. The trans­mission coefficient is also taken to be unity. With these as­sumptions, the traditional TST equation is derived

kaT (-aG""') kTST = -- exp .

h RT (6)

The stochastic models l-

3 reintroduce the transmission coefficient as

(7)

where kobs is the observed isomerization rate. The transmis­sion coefficient is found to be a strong nonmonotonic func­tion of the coupling strength between the reaction coordi­nate and the surrounding medium.

The activation parameters for isobaric conditions may be obtained from an Arrhenius-type plot. The temperature dependence of the chair to chair isomerization rate of cyclo-

4

2

• T 1 P measurements

o line shape analysis

o~~~~~~~~~~~~~~~

3.0 3.5 4.0 4.5 5.0 1000rr (K-1)

FIG. 2. The temperature dependence of the chair--<:hair isomerization rate of cyclohexane at ambient pressure, obtained from measurements of rotat­ing frame relaxation times (,,). The results of earlier line shape analysis (0) from Ref. 9 are also shown.

hexane in CS2 solvent at ambient pressure is shown in Fig. 2. The results of earlier NMR line shape analysis from Ref. 10 are also reported for comparison. As shown in Fig. 2, the highest measurable rate by the NMR rotating frame method is about 5 X I (f s - I.

Analysis ofEq. (4) and Fig. I shows that in this case we are approaching the no-slope limit for a given value of the chemical shift (83 Hz) in cyclohexane. For higher exchange rates the slope of Tip vs wi plot becomes too small to be measured accurately. Therefore, we chose to perform vari­able pressure experiments at 263 K for two reasons. First, we wanted to achieve sufficient overlap in the viscosities of the solution with our data and the data taken previouslylO to generate isoviscosity plots. Second, this temperature would limit experimental problems of sample heating and burned out diodes due to long duration of high power spin-locking pulses, while still providing for a significant decrease in the viscosity to explore the inertial regime in cyclohexane at high exchange rates. At 263 K and atmospheric pressure the exchange rate is about 6 X 103

S - I. As follows from Fig. 1, at this rate the exchange contributions to Tip values are sub­stantial and pressure experiments can be performed with high accuracy.

Thus the inherent weakness and limitations ofthe NMR line shape analysis approach may be overcome by using the NMR rotating frame technique to measure isomerization rates in a wide range of temperatures and to determine acti­vation parameters with higher accuracy. In fact, by combin­ing the NMR line shape and Tip techniques, we were able to double the temperature range investigated by our earlier study.IO The size of the experimental point for the highest measurable rate reflects the larger error for high rates as determined by the Tip technique. However, the activation energy derived from the Arrhenius plot in Fig. 2 is 11.12 kcallmol, which is in good agreement with reported litera­ture values. 10

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Campbell, Mackowiak, and Jonas: Conformational isomerization of cyclohexane 2721

0.5 ~---.------,----,-----,

0.'

,...., 1/1

'-" )(

• 0.3 ,...., <I. ....

E-I '-'

0.2

0.1 L-__ ~ ___ ~ ___ ~ __ ~

o 5 10 20

2 2-2 ("'1) (rad s

FIG. 3. The exchange contributions to the NMR rotating frame spin-lattice relaxation time (Tip )mh of cyclohexane in CS2 at 263 K as a function of spin-locking field strength at: (0)-1 bar, (\1)-1.5 kbar, (0)-2.3 kbar, (.1)-3.5 kbar, (<»-4.7 kbar.

C. High pressure measurements

The exchange contributions to the NMR rotating frame spin-lattice relaxation time ( Tip) exch of cyclohexane in CS2

were measured at 263 K as a function of spin-locking field strength at several pressures varying from 1 bar to 4.7 kbar. As shown in Fig. 3, with increasing pressure the slope of Tip

vs {()~ plots decreases whereas their intercepts at {()I = 0 in­crease. Measurements of the gradient and intercept allow independent determination of values for both k and {j{() as a function of pressure. Results of calculations, using Eq. (4), are presented in Table I. In the limit of experimental error the chemical shift {j{() is independent of pressure in the range investigated. The isomerization rate exhibits a strong accel­eration with pressure. The experimental data was fit to a linear equation

In k(P) = Ao + Al P, (8)

where P is the pressure in bar and Ao and A I are adjustable parameters. By differentiating the above formula one can extract the observed activation volume, I:::.. V ~s, from the fol­lowing relationship:

TABLE I. Pressure dependence of the isomerization rate and chemical shift of cyc10hexane in CS2 solvent at 263 K measured by the NMR rotating frame relaxation technique.

P (kbar)

0.001 1.500 2.300 3.500 4.700

k (s -I)

6190 7450 7930 8770 9950

ov (Hz)

83 83 83 82 82

-. 3 p.. C) ....... ~ 2

1

oL_---' __ --'-__ -L... __ -'-__ .L..-....J

a 1000 2000 3000 ,(,000 5000

Pressure (bar)

FIG. 4. The shear viscosity coefficient 'TJ as a function of pressure for 5 mole % cyclohexane in CS2 and pure CS2 at various temperatures: (0)

T= 286 K, pure CS2 ; (e) T= 286 K, 5 mole %; (\1) T= 263 K, pure CS2 ; ( .... ) T= 263 K, 5 mole %; (D) T= 236 K, pure CS2 ; (_) T= 236 K, 5 mole %; (.i) T= 225 K, pure CS2 ; (<» T= 218 K, pure CS2 ; (+)

T= 213 K, pure CS2 •

I:::..V" = _RT({jlnk(p») . obs {jP T

(9)

Using the data given in Table I we obtained I:::.. V ~s = - 2.0 cm3/mol.

Viscosity measurements of both 5 mole % cyclohexane in carbon disulfide and pure carbon disulfide were carried out from I bar to 5 kbar at 286 and 263 K, and from 1 bar to 3 kbar at 236 K. The results are shown in Fig. 4. As may be expected, the viscosity of 5 mole % cyclohexane in CS2 is slightly higher than that for pure CS2 , but the difference between the two solutions was always less than 5%. Such a small difference cannot lead to any misinterpretation of the isomerization dynamics. We therefore can combine our re­sults with the viscosity values obtained at lower tempera­tures by Hasha et a/. lo when reporting isoviscosity plots.

In Fig. 5 the viscosity dependence of the normalized isomerization rate is plotted at various temperatures. It is clear that the isomerization rate of cyclohexane in CS2 ex­hibits a stronger viscosity dependence in the less viscous sol­vent (at high temperature and/or high exchange rate) than that in the more viscous solvent (at lower temperature). This is an indication of the fact that at low viscosity cyclo­hexane is further in the inertial regime. The NMR rotating frame relaxation technique allows one to extract informa­tion from systems more deeply in the inertial regime.

For large amplitude isomerization motion, and for Eo > R T the rate constant has been related to the viscosity by the following expression:28

k = F( 'TJ} exp ( - Eo / R T), ( 10)

where F( 1]) is a function of the solvent shear viscosity 1] and

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2722 Campbell. Mackowiak. and Jonas: Conformational isomerization of cyclohexane

-.-.. ~ as

.0 -t ....... ..!l4 ........ .-.. P-c ....... ..!l4 ........ J:l -

0.5 ,...---..,-----,r-----,------r---,

0 • .(.

0.3

0.2

0.1

• o ". -... ;. ...

/o,,~ ;I" ,' ••••

/0;;,'''-; "

/ ; "[J /0 ; ,v

a' v~c'

/ b'~'" 0/ '1'", / ',' , ,

9 /v,'r/ , " , ,

I '..-, ,

QV,'9:' I ': , '

I ':' , ' ~ ,'Y,pi

I ' : , '

I ' ..-, :' 0.0 '---_-e-9Ei--'-----1------'L-----'

o 1 2 3 5

viscosity (cP)

FIG. 5. The normalized isomerization rate as a function of viscosity for cyclohexane in CS2 : (.)-T= 263 K, (O)-T= 225 K, (\7)-T= 218 K, (0)-T = 213 K. Closed symbols from NMR rotating frame reiaxation measurements; open symbols from NMR line shape (Ref. 9).

Eo is the internal barrier height. According to this formula, the energy barrier effects can be separated from the frictional effects of the solvent. The accurate activation energy should be determined by the temperature variation of the rate at fixed viscosity. If different isoviscosity plots yield the same value of Eo then the activation barrier would have to be pressure independent for a given solvent.

..!l4 bD o -

The isoviscosity plots ofln k vs liT for cyclohexane in

5 .------,------,-----,

3

2

1 3.5 4.0

'l1 = 1.34 cP 'l1 = 1.50 cP 'l1 = 1.75 cP 11 = 2.00 cP

4.5 5.0

FIG. 6. Theisoviscosity plotsofIn kvs I!T forcyclohexane in CS2 : (0)­

'1/ = 1.34 cP, (\7)-'1/ = 1.50 cP, (0)-'1/ = 1.75 cPo (1l.)-'1/ = 2.0 cPo

CS2 solvent are shown in Fig. 6. The points on the lines were extracted from the pressure fits of viscosity and rate con­stants. The parallel relationship among the four lines for viscosities ranging from 1.34 to 2.0 cP clearly indicates that the internal barrier height is not sensitive to the solvent vis­cosity. From the isoviscosity plots we get the activation ener­gy which is 11.28 ± 0.01 kcallmol for all four viscosities. High accuracy of the isoviscosity data, reflected by the value of correlation coefficients close to one, is remarkable. Since these plots have identical slopes we have ruled out the possi­bility that the barrier height is a function of pressure in the range studied. The pressure induced decrease in the energy of activation should be about 0.25 kcaI!mol in order to ac­count for the observed increase of the reaction rate over a 5 kbar pressure range. Such a significant decrease of the acti­vation energy would be detectable from our isoviscosity plots. However, they remain constant within ± 0.01 kcaI! mol. Therefore, the pressure induced acceleration of the iso­merization rate of cyclohexane is due to collisional frequen­cy changes characteristic for molecular systems in the iner­tial regime.

A practical way to discuss the isomerization dynamics is provided by the relationship between the transmission coef­ficient K and the solvent viscosity. To obtain information about the transmission coefficient we need to evaluate the reduced rate constant, since K and kTST are not independent­ly observable. The normalized transmission coefficient can be obtained from

K(Tf) = k(rJ} exp(P-PO)f!..vfST) , (11) K(Tfo) k(Tfo) RT

where K( Tfo ) and k( Tfo ) are the transmission coefficient and rate constant at a reference viscosity Tfo and corresponding pressure Po. a v fsT is the transition state theory volume

1.1

2 3 4 5 6 viscosity (cP)

FIG. 7. The normalized transmission coefficient as a function of viscosity for cycIohexane isomerization in CS2 solvent. (.)-NMR rotating frame relaxation data at 263 K, (O)-NMR line shape data at 225 K (Ref. 9). The lines are the best fits to the stochastic model (solid line-T = 263 K. broken line-T= 225 K).

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Campbell, Mackowiak, and Jonas: Conformational isomerization of cyclohexane 2723

difference between the transition and initial states which should be pressure independent. Figure 7 shows the depend­ence of the normalized transmission coefficient upon viscos­ity as generated from the experimental data for cyclohexane isomerization in CS2 solvent. As in our previous work lo the value of aVtsT = - 1.5 cm3 mol-I has been assumed. We emphasize that in Fig. 7 the two lines are experimental K( "l )/K( 1.5 cP) vs "l plots for two different temperatures as obtained from Eq. (11). The plots indicate the strong corre­lation between the reduced transmission coefficient and sol­vent viscosity. Of particular importance is the fact that the transmission coefficient increases with "l by about 20% and that the turnover occurs between 2.5 and 3.0 cPo The results obtained by the NMR line shape analysis 10 at 225 K are also shown for comparison. The data taken at 263 K using the NMR rotating frame relaxation technique clearly indicate that cyclohexane is in the inertial regime ofKramers' model.

As pointed out by Hanggi et al.,4 the regime of validity of Kramers rate expressions for weak friction and for moder­ate-to-Iarge friction depends in a characteristic way on the two relevant dimensionless parameters: the inverse Arrhen­ius factor kB T I Eo and the friction strength rl{llb (where r is the damping relaxation rate and (llb is the angular frequen­cy of the unstable state at the barrier). The two dimension­less parameters, k BTl Eo and r I OJ b' therefore characterize the regimes of validity of the different results for the rate in the spatial-diffusion limited regime and in the energy-diffu­sion (inertial) limited regime. This behavior can be charac­terized by the classical rate-phase diagram. For rlOJb> 1 (high viscosity) the thermal equilibrium prevails through­out the escape process, and the rate k becomes spatial-diffu­sion limited. The escape dynamics becomes controlled by energy diffusion (inertial regime) whenever the condition rlOJb > kB T lEo starts to fail. For rlOJb = kB T lEo a turn­over region occurs. Therefore, the inertial regime is more pronounced for molecular systems where the k BTl Eo > r I OJ b condition is fulfilled. NMR rotating frame relaxation technique allows characterization of high-tem­perature and low-viscosity regions where the inertial behav­ior is most likely to be observed.

IV. CONCLUSIONS

In this study we report the results of the NMR rotating frame relaxation study of the isomerization rate of cyclohex­ane in carbon disulfide at various temperatures and pres­sures. It is the first time that the TIp technique under high pressure has been applied for the chemical exchange mea­surements. By combining the TIp and NMR line shape tech­niques and generating the isoviscosity plots we were able to show that the barrier height to isomerization is independent

of pressure. The isomerization rate constant in the pressure range studied was a monotonic linear increasing function of pressure indicating that cyclohexane in this viscosity, tem­perature, and pressure range is in the inertial regime of the Kramers' model. The viscosity dependence of the reduced transmission coefficient suggests that the reaction coordi­nate is weakly coupled to the surrounding medium and the isomerization dynamics shows an energy controlled behav­ior.

ACKNOWLEDGMENTS

This work was supported in part by the National Science Foundation under Grants Nos. NSF CHE 90-17649 and NSF DMR 89-20538.

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