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(Eingegangen am 19. Marz 1987, E 6467 endgiiltige Fassung am 18. M a i 1987)
High Pressure NMR Study of the Molecular Dynamics of Liquid FIuoroform and Deutero-Fluoroform
E . W. Lang, F . X. Prielmeier, H . Radkowitsch, and H . - D . Liidemann
Institut fur Biophysik und Physikalische Biochemie, Universitat Regensburg, Postfach 397, D-8400 Regensburg Diffusion / High Pressure / Liquids / Spectroscopy, Nuclear Magnetic Resonance / Transport Properties
The 2D - , 1H - and , 9F-spin-lattice relaxation rates Ri have been investigated in fluid fluoroform in the temperature range 150 K < T <
450 K and at pressures up to p = 200 M P a . Previous measurements of the self-diffusion coefficient D have been supplemented to cover the same p, r-interval. Within the rough hard sphere (RHS) approximation a temperature independent RHS diameter d and a strongly temperature dependent rotation-translation coupling ART are obtained. Both parameters are also compared with those obtained in a series of related halomethanes. The total 2D - , 1H - and 1 9F-relaxation rates in C D F3 and C H F3 are decomposed into their respective quadrupole, dipolar and spin-rotation contributions and orientational and spin-rotational correlation times are extracted from these rates. It is shown that they are in good agreement with the predictions of the Fokker-Planck-Langevin model. Inertial effects influence the molecular dynamics at high temperatures and low densities. The agreement of self-diffusion coefficients and orientational correlation
times with MD-simulation data is very satisfactory.
Introduction
T h e molecular dynamics of l i q u i d fluoroform have been studied in recent years by a variety of experimental tech- niques such as N M R , dielectric relaxation and absorption, R a m a n scattering, and IR-spectroscopy [1 — 10]. F r o m the high frequency methods orientational correlation functions
may be obtained over a limited time range and details of the short time dynamics of orientational fluctuations un- ravelled. N M R studies cannot yield such detailed informa- tion since i n the short correlation time limit only the area of the relevant correlation function is determined and not its shape. However, if spin-rotation interactions dominate, Ber. Bunsenges. Phys. Chem. 91, 1025-1033 (1987) - © V C H Verlagsgesellschaft mbH, D-6940 Weinheim, 1987.
0005-9021/87/1010-1025 $ 02.50/0
the N M R method offers the unique possibility to obtain correlation times of angular velocity fluctuations. Hence dy- namic parameters deduced from N M R - e x p e r i m e n t s include the self-diffusion coefficient Z), the molecular orientation cor- relation time T02 and the correlation time xa) of the molecular angular velocity. Together these parameters provide v a l u - able insight into the state of molecular m o t i o n .
In the past l i q u i d fluoroform [1—3] and deuterofluoro- form [4] have been investigated with relaxation rate and Overhauser effect measurements. The dominant interactions were found to be dipole-dipole and spin-rotation interac- tions in the former and quadrupole interactions i n the latter.
Since C H F3 is a symmetric-top molecule, although of almost spherical shape, in principle two orientational correlation times may be expected. However, a comparison of the ex- perimental results obtained in C H F3 and C D F3 revealed that the orientational fluctuations are nearly isotropic [ 4 ] . A l m o s t a l l investigations of fluoroform hitherto have been conducted under saturation vapor pressure. The present i n - vestigation is a study, by N M R methods, of C H F3 and C D F3 in the neat l i q u i d phase over the temperature range 150 K to 450 K and a pressure range 0.1 M P a to 200 M P a .
Experimental
Spin-lattice relaxation times were measured by the inversion- recovery method with alternating phase [11] on a Bruker M S L - 300 multipurpose solid-liquid N M R spectrometer operating at 300.13 (MHz) for 1H , 282.35 (MHz) for 1 9F and 46.07 (MHz) for 2H and on a Varian X L 100 high resolution spectrometer operating at 100.1 (MHz) for 1H , 94 (MHz) for 1 9F and 15.35 (MHz) for 2H . Self- diffusion coefficients D were determined on the MSL-300 spectro- meter equipped with a Bruker gradient unit and home-built Helm- holtz coils were used to apply pulsed field gradients. The experi- mental procedure and high-pressure equipment have been described elsewhere [12]. The relaxation times and the self-diffusion coeffi- cients are judged reliable to ± 5 % .
The pressure was measured with a Heise bourdon gauge (Heise, Connecticut, USA) with a precision of +0.6 (MPa). The tempera- tures were measured with a 0.5 (mm) Chromel-Alumel thermocou- ple (Philips, KasseL F R G ) and are considered reliable to +1 (K).
Substances
Fluoroform (98%) was purchased from Baker Chemikalien (GroB-Gerau, F R G ) and deuterated fluoroform (98% 2H) was bought from IC Chemikalien (Munchen, F R G ) . The gases were dried over molecular sieve (3 A) and stored in a glass flask. Prior to use residual oxygen was removed by at least five freeze-pump- thaw cycles to a final pressure of 5 (mPa).
Results and Discussion The Self-Diffusion Coefficient of Fluoroform
In the absence of strong, anisotropic attractive interac- tions the structure and dynamics of liquids is dominated by short-ranged repulsive interactions. T h u s the concept of the hard-spheres fluid as a zeroth order a p p r o x i m a t i o n to real liquids has been developed. F o r smooth hard spheres the self-diffusion coefficient may be expressed as [13]
Table 1
Molecular constants in fluoroform
DQUQ —
3 / kT
m-n -d-P(gd3) (I)
1) distances and angles rC F = 1.39-10-I O(m) rC H= 1.09-10-, 0(m) rF F =2.166-10-, 0(m) rF H = 1.997-10-, 0(m)
2) dipole moment // = 1.649 (Dy)
3) principal moments of inertia C H F3
/,. = 81.60- 1 0 -4 7( k g - m2) / = 148.40-10"4 7(kg-m2)
IhiW
= 96 • IO™4 7 (kg-m2)
(FCF) = (FCH) =
109 109
<-h¥'
C D F . ;
85.61 • 10 (kg • m2) 148.40 • IO"4" (kg • m2)
99.67 • I O "4 7 (kg • m:)
4) principal spin-rotation tensor components
1 9F : CX = 8.83 (kHz) C =4.83 (kHz)
1H : C1 =0.667 (kHz) C =0.167(kHz)
5) deuterium quadrupole coupling constants
— ^ - = 1 7 0 . 8 ± 2 ( k H z ) gas
h
159 ± 5(kHz) liquid
packing fraction of the molecules have been obtained by forcing E q . (1) to fit molecular dynamics results [14] of the hard sphere fluid. R o u g h hard spheres may represent non- spherical molecules or spherical molecules with anisotropic intermolecular interactions. A c o u p l i n g of rotational and translational fluctuations must then be taken into account leading to a decrease of DSHS. Arguments have been put forward to account for the R - T - c o u p l i n g simply by intro- ducing a coupling parameter A such that [15]
flex, A R H S — ZX5 (2)
with d the diameter of the spheres and g the number density.
T h e coefficients of an empirical p o l y n o m i a l P(Qdi) in the
Table 2 compiles the self-diffusion coefficients measured to- gether with the densities taken from the Refs. [16,17]. F i g . 1 shows the isobaric temperature dependence of the self-dif- fusion coefficient together with the data obtained previously [12]. F i g . 2 shows the isothermal density dependence of the self-diffusion coefficient D. The hard sphere diameter d may be obtained from a non-linear least squares fit ( N A G E04 F C F ) to Eqs. (1) and (2). A temperature independent d i - ameter d = 3.95 A results for C H F3. T h i s compares favor- ably with the slightly smaller diameter d = 3.57 A obtained in C H3F [18]. The temperature dependence of the slope (QDfdg)r is predicted by E q . (1) to be p r o p o r t i o n a l to T12. In most liquids, however, a stronger increase with temper- ature of the slope in the D versus g plot is found. Hence the coupling of rotational and translational motions must de- pend on temperature. In fluoroform the R - T c o u p l i n g pa- rameter ART assumes values ~ 0 . 3 at l o w temperatures and increases towards 1.0 at the highest temperatures measured.
T h i s indicates an increased decoupling of rotation and trans- l a t i o n w i t h increasing temperature. Table 3 compares the hard sphere diameter and the R - T c o u p l i n g parameter A of
Table 2
Self-diffusion coefficient D and mass density g of fluoroform ( 7 [ K ] , p ( M P a ] , D [ 1 ( T9 m2 s~'], q [kg • m"3])
142
154
168
177
188
208
222
250
262
291
322
353
383
433
Table 3
Rough hard sphere diameter d[nm] and rotation-translation coupling pa- rameter Ar t of fluoroform and related halomethanes
S V P 20 50 100 150 200
0.84 0.78 0.69 0.52 0.40
1610 1650 1680 1730 1760
1.25 1.15 0.96 0.74 0.57
1570 1610 1650 1700 1740
1.90 1.70 1.45 1.10 0.86 0.70
1530 1560 1610 1670 1710 1740
2.35 2.00 1.65 1.30 1.05 0.89
1500 1540 1590 1650 1700 1720
2.85 2.50 2.20 1.70 1.37 1.15
1460 1500 1560 1630 1680 1710
4.40 3.70 3.10 2.50 2.10 1.70
1380 1440 1510 1580 1640 1670
5.50 4.70 3.65 2.95 2.40 2.10
1330 1390 1470 1550 1610 1650
8.00 6.80 5.60 4.40 3.60 3.20
1180 1280 1390 1490 1560 1610
8.00 6.20 4.70 3.80 3.40
1250 1350 1460 1540 1590
11.20 8.50 6.40 5.20 4.10
1130 1270 1400 1490 1540
17.00 11.50 8.20 6.70 5.60
980 1180 1340 1430 1490
24.00 15.00 10.50 8.50 7.00
800 1090 1270 1370 1440
32.50 19.00 13.00 10.00 8.30
650 1010 1210 1320 1390
53.00 26.50 18.50 13.00 10.50
860 1100 1230 1320
C H3F C H F3 C H3C l C H2C l2 C H C l3
T(K) 150-440 1 4 2 - 4 2 0 185-440 186-406 230-400
d (nm) 0.357 0.395 0.40 0.44 0.48
0.4-1.0 0 . 3 - 1 . 0 0.4-0.75 0.3-0.8 0.4-0.8
Table 4
Comparison of self-diffusion coefficients D and integral orientational corre- lation times T2., as obtained by M D simulations with experimental results of
fluoroform
M D Expt.
TLK.-] 250 136 250 136
q [kg • m- 3] 1390 1640 1390 1640
D [ I O9I T i2S -1] 4.1 0.5 5.5 0.58a
*2./ [PS] 0.55 3.0 0.58 2.6a)
fluoroform w i t h methylfluoride [18] and related halome- thanes [19]. F i g . 3 compares the temperature dependence of the parameter ART in fluoroform, methylfluoride [18] and C a r b o n tetrafluoride [20]. Recently B o h m et al. [21] de- vised an intermolecular pair potential and applied it to a number of molecules i n c l u d i n g those listed i n Table 3. T h e i r calculated self-diffusion coefficients are i n fair agreement w i t h experiment. N o t a b l y their self-diffusion coefficient i n fluoroform is i n good agreement with those obtained i n this study. However, b o t h are in substantial disagreement w i t h results obtained earlier by Chaffin et a l [3]. Table 4 gives
(O(T))0
IOj H 1
Fig. 1
Isobaric temperature dependence of the self-diffusion coefficient D in C H F3. ( • XL-100 (1H), O MSL-300 (, 9F))
(10~9m2/s)
Obtained by slight extrapolation.
k g / m3) Fig. 2
Isothermal density dependence of the self-diffusion coefficient D in C H F3. - Best fit curve according to Eqs. (1) and (2)
a c o m p a r i s o n of dynamic parameters obtained w i t h molec- ular dynamics calculations and experimental results ob- tained i n our group [12,18,19].
uzJ i 1 1 1 1 1 1 1 1 1 1 r
OM 0 6 0.8 1.0 1.2 U 1.6
— • T Z Tc Fig. 3
Temperature dependence of the rotation-translation parameter AR J
in C H3F (O), C H F3 ( x ) and C F4 ( A ) [20]
Deuterium Spin-Lattice Relaxation Rates in C D F3 T h e spin-lattice relaxation of 2D nuclei is almost exclu- sively due to fluctuating electric Field gradients interacting with the electric quadrupole moment of the nucleus. Pre- suming rigid molecules of spherical shape the relaxation rate is given i n the short correlation time limit i n a laboratory- Fixed frame as [22]
0 50 100 150 200
—*~ p (MPQ)
Fig. 4
Isothermal pressure dependence of the deuterium spin-lattice re- laxation time (T1(2H) in C D F3
hertz with eQ the electric quadrupole moment and eq™ the largest component of the electric field gradient (efg) tensor along the C - D b o n d in a molecule fixed axis system. The efg asymmetry parameter \\ = (q™ — ^JvJ)/g~ is generally small a n d w i l l be neglected. F i g . 4 shows the isothermal pressure dependence of the 2D - T1 i n C D F3.
The orientational correlation time x(K2 is defined as the integral over the n o r m a l i z e d orientational correlation func- tion of elements of the W i g n e r rotation matrix specifying the orientation of the m a i n axis system of the efg tensor relativ to the l a b o r a t o r y frame. In principle, because C D F3
is a symmetric top a n d because the deuterium nucleus is o n the symmetry axis of the molecule, x(K2 monitors only o r i - entational fluctuations of the C3- a x i s .
T h e deuterium quadrupole c o u p l i n g constant has been obtained i n the gas phase to 170.8 ± 2 k H z [23]. U t i l i z i n g correlation times extracted from R a m a n line shape studies [10] a n d deuterium relaxation rates [4] a substantially lower value of 159 ± 5 k H z has been reported for l i q u i d C D F3. T h i s is i n g o o d accord with the general finding that deuterium q u a d r u p o l e c o u p l i n g constants are reduced by
~ 1 0 % roughly in going from the gas phase to the solid state. T h e latter value w i l l be used in this study to calculate the correlation times T^2.
2.0-
— • • I O3/ T (K"1)
Fig. 5
Isochoric temperature dependence of the orientational correlation time X0.2 in C D F3
F i g . 5 shows their isochoric temperature dependence. A t constant density the temperature dependence reflects the sole influence of the kinetic energy of the molecules upon orientational fluctuations. Because intermolecular torques fluctuate more r a p i d l y at higher temperatures, orientational correlations decay o n a shorter time scale leading to de- creasing correlation times. The latter are seen to be longer than those obtained i n methylfluoride indicating stronger intermolecular torques retarding rotational motions. Indeed fairly high root mean square intermolecular torques have been found to prevail i n fluoroform along the orthobar [ 7 ] . T h e isochoric temperature dependence of x(K2 follows an A r - rhenius law w i t h a density independent activation energy E * = 2.75 k J / m o l . T h i s is about half the value [E£ = 4.6
\
3.0 ( ^ e , 2 ( 9 ) )T
(ps) 2.0-
1.0- 0.8
0 . 6 -
OM
0.2
1 0 0 0 1 2 0 0 U 0 0 1 6 0 0 1800~
. Q ( k g / m ° ) 3^
Fig. 6
Isothermal density dependence of the orientational correlation time T0,2 in C D F3
k J / m o l ) obtained from dielectric data a l o n g the orthobar [ 7 ] . T h e ratio E£/E* - 1.6 i n methylfluoride [ 1 8 ] a n d fluoroform. C o m p a r e d to methylfluoride the isochoric tem- perature dependence ( £ * = 2 . 4 0 (kJ/mol)) as well as the isothermal density dependence of orientational fluctuations (Fig. 6 ) is found to be slightly stronger i n fluoroform. A t constant temperature the correlation times increase i n a nonlinear fashion w i t h density, reflecting the retarding i n - fluence of m o l e c u l a r torques u p o n the reorientation process at higher p a c k i n g fractions. T h e increase is the more p r o - nounced the lower the temperature (see F i g . 6). T h e fairly short correlation times T ^2 a n d the l o w barrier to rotation suggest the importance of inertial effects d u r i n g the reorien- tation process. A n i n d i c a t i o n may be obtained by c o m p a r i n g 'the orientation correlation times T02 w i t h the free rotor cor-
r e l a t i o n time TF =
k^T' ^ mt^ 0 T ( l 2^T { ^ r e~
1H and 1 9F Spin-Lattice Relaxation Rates in C H F3
T h e spin-lattice relaxation of the protons a n d fluorine nuclei proceeds m a i n l y v i a magnetic dipole-dipole interac- tions a n d spin-rotation interactions. T h e d i p o l a r interac- tions have to be decomposed into intra- a n d intermolecular interactions. I n the short correlation time limit the various d i p o l a r relaxation rates are given, again presuming r i g i d molecules of spherical shape, as [ 2 2 ]
a) dipole-dipole, i n t r a
Ho n V > j
(4)
w i t h I(S) representing the spin q u a n t u m number of the fluorine (proton) nuclei and n the number of spins per molecule.
b) dipole-dipole, inter
8TX_
9 77
4n h2
ni V - i U <
*
i , 1 6 / Ho\2 ns v( 5 )
T h e expressions for the intermolecular relaxation rate m a y be obtained w i t h a diffusion equation with reflecting b o u n d - ary conditions [ 2 4 , 2 5 ] . A n y off-center effects have been ne- glected as they contribute less than 1 0 % to the leading term [ 2 6 ] . T h e relative diffusion coefficient has been replaced by twice the measured self-diffusion coefficient. The distance of closest approach a has been taken from radial pair-distri- b u t i o n functions as obtained by the molecular dynamics simulations o f B o h m et al. [ 2 1 ] .
T h e i n t r a m o l e c u l a r d i p o l a r relaxation rates m a y be o b - tained from E q . ( 4 ) if x0d2 is replaced by T0 2 2. T h i s should be a reasonable a p p r o x i m a t i o n because C D F3 a n d C H F3 pos- sess very similar moments of inertia. A l s o intermolecular interactions cannot differ much, hence one may not expect a significant isotope effect o n the correlation times. F u r - thermore orientational fluctuations i n fluoroform have been s h o w n to be nearly isotropic a n d the numerical identity tfii = T$2n a s ^een demonstrated [ 4 , 1 0 ] .
T h e spin-rotation relaxation rate is given for a spherical top by [ 2 7 ]
orientational processes are diffusive a n d inertial effects may be neglected. If T0>2IT{ ~ 1 inertial effects severely influence
\ orientational correlation functions. I n C D F3 at low temper- Jatures the ratio T f t2/ if yields values ~ 6 whereas at high
temperatures values ~ 1 . 5 are found. Hence i n analogy to methylfluoride the molecular dynamics of fluoroform do not correspond to a small step diffusive process over most of the p, 7-range investigated. F i n a l l y a c o m p a r i s o n of TFU w i t h molecular dynamics results [ 2 1 ] indicates g o o d agreement as has also been found i n the case o f the self-diffusion data (see Table 4 ) .
2knT
< / > I VClrTco + 2 ( A C ) ? - TS R] (6)
w i t h </> = Q Z I / / , - )- 1 the mean moment of inertia of the m o l e c u l e . C0 = i ( C) l + 2 CJ L) a n d A C = i ( C11- CJL) give the isotropic a n d the anisotropic part of the spin-rotation tensor i n terms of the parallel and perpendicular component of the tensor i n its m a i n axis system. T(U is the correlation time for fluctuations o f the molecular angular velocity a n d Ts r is the correlation time for the angular velocity-orienta- t i o n a l product correlation function which characterizes a n -
isotropic s p i n - r o t a t i o n a l interactions. E q . (6) is equivalent to an expression given by H u b b a r d [28,29]
l??R = ^ | I < y > l i ( C i f , + 2 C i, ) - T sR ( 7 )
if due account is taken of the different definitions of iS R. E q . (7) has been used to calculate the c o r r e l a t i o n times TSR .
Separation of the 1 9F Relaxation Rates in C H F3 and C D F3
T h e 19F spin-lattice relaxation rate is a sum of different contributions a c c o r d i n g to
R1 (1 9F ) = R Js r(1 9F ) + RiS (1 9F -1 9F ) + R S (19F -1H ) (g) + R S (19F -19F ) + R in d eJ (1 9F -1H ) .
R e l a x a t i o n due to anisotropic chemical shielding of the flu- orine nucleus has been s h o w n to be negligible [ 3 , 4 ] . T h i s is c o r r o b o r a t e d by the agreement of b o t h sets of data o b - tained at ambient pressure, where no field dependence c o u l d be detected. In C D F3 the d i p o l a r interactions between the
1 9F - and the2H nuclei may be neglected because of the m u c h smaller gyromagnetic ratio of the deuterium nucleus. T h e pressure and temperature dependence of the experimental
Fig. 7
Isobaric temperature dependence of the fluor-19 spin-lattice relax- ation time in C H F3 and C D F3
Fig. 8
Temperature dependence of the various relaxation rates contrib- uting to the total fluor-19 spin-lattice relaxation rate in C D F3 at
two pressures ( • K1(1 9F ) , A Rf\ O KinSd, * KinJdi)
1 9F spin-lattice relaxation times are shown in F i g . 7 for C H F3
and C D F3. The spin-rotation interaction dominates over most of the temperature range investigated. A t the highest temperatures the spin-rotation relaxation rates are almost identical with the total rate measured. A l s o at these tem- peratures the relaxation rates observed in C H F3 and C D F3
are identical. T h i s is to be expected because of the small isotope effect u p o n the components of the moment of inertia tensor I. A t l o w temperatures the relaxation rates are larger in C H F3 than in C D F3 because of the a d d i t i o n a l 1 9F -1H d i p o l a r contributions to the total rate measured. T h e intra- and intermolecular d i p o l a r rates have been calculated as described above using a distance of closest a p p r o a c h a (1 9F -
1 9F ) = 2.5 A and a (1 9F -1H ) = 2.2 A and are s h o w n for two pressures i n F i g . 8. Subtracting these contributions from the total rate measured, the spin-rotation relaxation rate may be obtained. T o calculate correlation times TSR with E q . (7) independent information about the components of the spin- r o t a t i o n tensor is necessary. Reynders et al. [30] measured the hyperfine structure of the J = 1 —> 0 r o t a t i o n a l tran- sition w i t h a high-resolution beam-maser spectrometer and
3
report the c o u p l i n g constant C1 0(1 9F ) = Z Hdiiii + C1bh) i = 1
= 10.6 + 1 (kHz) in the p r i n c i p a l axis system of the moment of inertia tensor, while K u k o l i c h et al. [23] obtained a hy- perfine constant for the same transition O f C i0(1 9F ) = 5.4 + 1.5 (kHz). Besides the large discrepancy between both re- sults, these c o u p l i n g constants are insufficient to determine all components of the spin-rotation tensor C M o l e c u l a r beam determination of the spin-rotation constants as ob- tained by Follett [31,32] yield i ( Cv v + Cv v + Czz) = 7.5
± 1.2 k H z and H CVV + Cv v - Czz) ^ 4 k H z . These may be rearranged to give Czz = Cll = 4.83 (kHz) and \ (Cxx + Cyy) = C1 = 8.83 (kHz) to be used in E q . (7) to calculate spin-rotation correlation times xS R. These c o u p l i n g c o n - stants are in better accord with the results of Reynders et al. [30] than with those obtained by K u k o l i c h et al. [23].
D o n g and B l o o m [33] and A r m s t r o n g and C o u r t n e y [32]
obtained an effective spin-rotation c o u p l i n g constant C2 n- (1 9F ) = 48.1 + 2.1 ( k H z )2 and 76 < C2 T(1 9F ) < 94 ( k H z )2
respectively from 1 9F - Ti measurements of gaseous C H F3, whereas C2 f y = 63.45 ( k H z )2 is calculated by Sanctuary [34]
w i t h the components Cll, C± as evaluated from the results of F o l l e t t [31].
F i g . 9 shows the isothermal density dependence and the isochoric temperature dependence of the reduced correlation times r fR = (IcTf(I))12 • Ts r. W i t h increasing density T ?R
decreases due to the stronger perturbing influence of m o - lecular torques at higher p a c k i n g fractions. T h e decrease is the more pronounced the lower the temperature and/or the higher the density. The isothermal density dependence of Ts r is stronger in fluoroform than in methylfluoride at c o m - parable temperatures. A t the lowest temperatures the c o n - d i t i o n TS R ^ T(U holds corresponding to a rotational diffu- sion process. The isochoric temperature dependence of TS R
(see F i g . 10) shows an Arrhenius-dependence with an ap- parent, density independent, activation energy of £ * = 2.05 (kJ/mol). T h i s is ~ 3 / 4 of the activation energy £ * = 2.75 (kJ/mol) found for the isochoric temperature dependence of Tft2- It is interesting to note that i n methylfluoride an ap-
parent a c t i v a t i o n energy E* ~ 2.11 (kJ/mol) has been found at a density g = 1.0 (g/cm3) w h i c h is about the highest density obtained in C H3F whereas it is roughly the lowest density obtained in C H F3. It should be noticed, however, that the n u m b e r densities corresponding to a mass density range Q: 3 0 0 - 1 1 0 0 ( k g / m3) in C H3F and Q: 9 0 0 - 1 7 0 0 ( k g / m3) covered by the experiments are practically identical in both substances. A t lower densities deviations from an Arrhenius-dependence have been observed in methylfluoride w i t h a weaker temperature dependence at lower densities contrary to the density independent activation energy found in fluoroform.
CU-
OOl4-I , , , , , , , ,—
900 1000 1100 1200 1300 U00 1500 1600 1700 g (kg/m3)
Fig. 9
Isothermal density dependence of the reduced spin-rotation cor- relation time TfR
( U -
H 1 1 1 1 1 1 1 1 r 2 3 4 5 6
— ^ IO3/ T (K"1)
I Fig. 10
^lsochoric temperature dependence of the reduced spin-rotation cor-
\ relation time t fR
Separation of the 1H Relaxation Rates in C H F3
The p r o t o n spin-lattice relaxation rate is a sum of different contributions a c c o r d i n g to
K1(1H ) = K1 iTda(1H-1 9F) + Ki 1 nS (1H -1 9F ) ( 9 ) + Ki 1 nS (1H -1H ) + KS R.
The pressure and temperature dependence of the p r o t o n spin-lattice relaxation times T1 is shown in F i g . 11. The
Fig. 11
Isobaric temperature dependence of the proton spin-lattice relax- ation time Tx (1H) in C H F3
intermolecular d i p o l a r r e l a x a t i o n rates have been calculated with E q . (5) using the measured self-diffusion coefficient D a n d distances of closest a p p r o a c h A (1H -1H ) = 3.6 A and Gf(1H-1 9F) = 2.2 A as estimated from the r a d i a l pair distri- b u t i o n functions [ 2 1 ] . T h e densities have been taken from Refs. [16,17]. T h e i n t r a m o l e c u l a r d i p o l a r relaxation rates have been calculated w i t h E q . (4) and w i t h Tod2 replaced by T#2 because of the near isotropy of orientational fluctuations in fluoroform a n d because of the negligible isotope effect.
The spin-rotation relaxation rates K s r have been calculated with E q . (7) w i t h T £r as deduced from the fluorine spin- rotation relaxation rates. Several determinations of the hy- perfine constants of C H F3 have been reported in the liter- ature. Reynders et al. [30] obtained C j ^ = C l = 0.0 ± 1 (kHz) whereas K u k o l i c h et al. [23] obtained for the same transition Cjo* = 24.5 ± 3 ( k H z ) i n sharp disagreement with the results of Reynders et al. [ 3 0 ] . F o l l e t t [31] reports Ca = $(CXX + CYY + CZZ) = 0.5 ± 0.1 ( k H z ) and Cd = i ( Cx x
+ CYY) - CZZ = 0.5 ± 0.1 ( k H z ) yielding Ci i = 0.167 ( k H z ) a n d CL = 0.667 ( k H z ) i n g o o d agreement w i t h the results obtained by Reynders et al. [ 3 0 ] . D o n g a n d B l o o m obtained an effective spin-rotation constant Ce2 f f(1H) = 0.40 ± 0.05 (kHz), whereas A r m s t r o n g a n d C o u r t n e y [32] give 0.2 <
Ce2fl-(1H) < 0.3 ( k H z )2. U s i n g the results of F o l l e t t [31]
Ce2 f f(1H) = 0.3 ( k H z )2 is calculated i n reasonable agreement
w i t h the estimates given above. T h e calculated relaxation rates K f1 1(1H ) are very small, thus m a y be neglected at a l l but the highest temperatures. T h e experimental K f1 1(1H ) ob- tained by subtracting a l l d i p o l a r relaxation rates from K fx p(1H ) are 2 — 3 times larger than the rates estimated with E q . (7). T h i s m a y be partly due to the large errors i n v o l v e d in estimating experimental K Js r(1H ) as a difference of large numbers and partly due to the a p p r o x i m a t i o n s inherent i n the calculation of KJIddr(1H). F i g . 12 gives the temperature dependence of the various c o n t r i b u t i o n s of E q . (9) to the total rate at three different pressures.
Orientational (T2) and Spin-Rotational (Ts r) Correlation Times in Fluoroform
O n l y integral orientational c o r r e l a t i o n times c o u l d be de- termined in this study. T h e orientational correlation func-
Fig 12
Temperature dependence of the various relaxation rates contributing to the total proton spin-lattice relaxation rate in C H F1 at three pressures ( • R1(1H), O Kjn d eJ, x Rjn d rJ, A RfR)
5 . 0 -
0.1 H 1 1— i — i — i i i i i 1 i — i— i —I i i i i
0.01 0.1 1.0
* Fig. 13
Reduced orientational correlation times T* versus reduced spin- rotational correlation times T*R. Full curve represents the prediction according to the Fokker-Planck-Langevin model. Straight line rep- resents the Hubbard relation T ? - T JR = 6"1 according to the Debye
model
tions, however, show features characteristic of high torque liquids [7,35] with torsional oscillations at short times typ- ical for motions in a cage and an exponential long-time-tail signifying M a r k o v i a n behaviour as a simple consequence of the l o n g range isotropy of the l i q u i d . Integral correlation times are mostly dominated by the M a r k o v i a n nature of orientational fluctuations. Spin-rotation relaxation relates to fluctuations of the angular velocity and of the orientation of the molecules. A g a i n only integral spin-rotation corre- l a t i o n times T8R of the angular velocity-orientational product correlation function can be determined from an N M R ex- periment. A l s o it is only within certain m o t i o n a l models that these bivariate correlation functions have been obtained.
Because orientational correlation functions may also be ob- tained w i t h i n the realm of these models, a comparison of both T2 and Ts r may provide insight into the state of mo- lecular motions. The F o k k e r - P l a n c k - L a n g e v i n ( F P L ) model [27,29,36] w i l l be shown to be i n reasonable agreement with the experimental results. In this model a single parameter,
Tw, is changed to produce the limits of free rotation and rotational diffusion. T h e model assumes successive uncor- r e c t e d instantaneous collisions w i t h the angular impulse of each collision being so small as to change the angular m o - mentum only infinitesimally. However, r a n d o m uncorre- c t e d collisions cannot cause a reversal of the angular m o - mentum as is observed i n high torque molecular liquids and indeed has been observed in fluoroform also [ 6 , 7 , 9 ] . Hence xui must be considered a lower limit to the t tIifetimev of the angular velocity correlation function. F i g . 13 shows a graph of T * = T2 • (kTf(I))1'2 versus TsV T h e full curve gives the dependence of x*(Tsr) according to the F P L - m o d e l . A t the lowest temperatures the reorientational motions are in the rotational diffusion limit ( TS R = T * <^ Tt), hence the H u b - bard relation T * - T * = 6'1 is obeyed. In general reorienta- tional motions i n fluoroform deviate less from the rotational diffusion limit than do these motions in methylfluoride [18]
investigated in the same range of number densities Q and reduced temperatures TfTi.. F i n a l l y it is instructive to note that the rough hard sphere model of rotational motions [37,38] gave values of T * calculated via the relations
J _ _ _ J 1_
~x% K+ 1 t i
1 8 ( n - Q ) \2 ,
- Z = T ~l ' QA"
T * 3 V ma- J
</>
with a = 3.95 A the hard sphere diameter as obtained from a fit of the rough hard sphere model to the measured self- diffusion coefficients D and with ghs(a) calculated via the C a r n a h a n - S t a r l i n g a p p r o x i m a t i o n [39], w h i c h are a factor of 2 —5 larger than the Tsr deduced from the experimental results in both fluoroform and methylfluoride.
The technical assistance of Mr. R. Knott, E. Treml, G . Wuhrl and S. Heyn made this work feasible. Financial support by the Deutsche Forschungsgemeinschaft, the Fonds der Chemischen In- dustrie and the Friedrich Ebert Stiftung is gratefully acknowledged.
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(Eingegangen am 18. M a i 1987) E 6496
Plancks Strahlungsgesetz und die Quantisierung der Entropie
Herbert W. Zimmermann
Institut fur Physikalische Chemie der Universitat Freiburg i . Br., AlbertstraBe 21, D-7800 Freiburg, Federal Republic of Germany Quantum Mechanics / Statistical Mechanics / Thermodynamics
In order to derive Planck's law of black-body radiation we consider a gas of photons which is in equilibrium with its surroundings at temperature T. Instead of the usual energy quanta of the photons e = hv we introduce entropy quanta a = e/T. Now, e and T are used from the beginning of the calculations and it is easy to derive Planck's distribution law of the photon gas, avoiding Lagrange's method of undetermined multipliers. Using a instead of s we get the entropy-distribution of the black-body radiation which may be transformed by /: = hv into the usual energy-distribution formula. Conventionally, entropy is a property of an ensemble of particles. But the entropy
quantum a is a property of the particle itself which is in thermodynamic equilibrium with its surroundings.
Einleitung
In den GIeichungen der W a r m e s t r a h l u n g begegnen sich Quantentheorie u n d T h e r m o d y n a m i c P l a n c k [ 1 ] ist es be- kanntlich als erstem gelungen, unter der A n n a h m e v o n Energiequanten die Energieverteilung der H o h l r a u m s t r a h - Iung zu berechnen. I m folgenden sollen die Gesetze der W a r - mestrahlung nochmals behandelt werden, jedoch unter ei- nem anderen Gesichtspunkt, der vor allem die thermody- namische Seite des Quantenproblems zur G e l t u n g bringen soil.
D a z u untersuchen wir ein Photonengas i m H o h l r a u m ei- nes schwarzen Strahlers, das bei der Temperatur T i m Gleichgewicht mit seiner U m g e b u n g steht. D i e P h o t o n e n
werden v o n den W a n d e n des H o h l r a u m s reversibel emittiert und absorbiert. B e i E m i s s i o n eines Photons der Frequenz v w i r d dem Photonengas die Energie s = hv zugefuhrt. D a das P h o t o n aber auch i m Strahlungsgleichgewicht mit seiner U m g e b u n g steht und reversibel isotherm erzeugt wurde, er- fahrt das Photonengas gleichzeitig den formalen E n t r o p i e - zuwachs cr = e/T. D i e P h o t o n e n sind diskrete Teilchen.
Ihnen ist dann nicht nur eine diskrete Energie sondern auch Entropie zuzuordnen. In A n a l o g i e zu den Energiequanten e = hv stehen jetzt die Entropiequanten a = e/T
D i e V o r s t e l l u n g v o n Entropiequanten a ist ungewohnlich.
Sie beinhaltet die Hypothese, daB die E n t r o p i e auch eine Eigenschaft einzelner Teilchen ist und nicht nur eines E n - Ber. Bunsenges. Phvs. Chem. 91, 1033-1036 (1987) - (© V C H Verlagsgesellschaft mbH, D-6940 Weinheim, 1987.
0005-9021/87/1010-1033 $ 02.50/0
