J. Phys. Chem. 1981, 85, 2169-2180
much closer in magnitude to our fixed-nuclei cross sections. We believe that this dramatic effect due to vibrational averaging in the region of the resonance is unphysical and is an artifact of the CMSM model. One can hence expect that the CMSM model would overestimate non-FranckCondon effects in the calculation of vibrational branching ratios in the region of resonances. This feature of the CMSM model had already been noted for vibrational branching ratios in the photoionization of the 3ag level in N2.5 Finally, in Figure 3 we compare our calculated photoionization cross section for the C2 2 , + ( 4 a ~ ~ion ) of COz with the (e,2e) measurements of Brion and Tan.15 These experimental results do not indicate any significant enhancement in the C2Bg+cross section around 40 eV. This (15) Brion, C. E.; Tan,K. H. Chem. Phys. 1978, 34, 141.
2169
discrepancy may be due to interchannel coupling or other effects which could Teduce its apparent magnitude to a level which would be difficult to identify easily in current experiments. A more detailed description of our calculations for the photoionization cross sections of C 0 2 including the energy dependence of the asymmetry parameters is in preparation. Acknowledgment. This research is based upon work supported by the National Science Foundation under Grant No. CHE79-15807. R. R. L. acknowledges support from an Exxon Educational Foundation Fellowship. The research reported in this paper made use of the DreyfusNSF Theoretical Chemistry Computer which was funded through grants from the Camille and Henry Dreyfus Foundation, the National Science Foundation (Grant No. CHE78-20235), and the Sloan Fund of the California Institute of Technology.
FEATURE ARTICLE A Molecular Quasi-Hydrodynamic Free-Space Model for Molecular Rotational Relaxation In Liquids Janis L. Dote, Danlel Kivelson," and Robert N. Schwartrt Department of Chemistry, University of California, Los Angeles, California 90024 (Received: March 2, 198 1)
The molecular reorientation time T~ can be described by the quasi-hydrodynamic relation T~ = (Vp~/kgT)fBtic& where V , is the volume of the rotating molecule, q is the viscosity, fatick is the Perrin stick factor dependent upon molecular shape, and C is a measure of the coupling between the rotating molecule and its surroundings. We have developed a theory for C based on the existence of free spaces in the hydrodynamic continuum in which the molecule can rotate, and we describe these spaces in terms of bulk properties of the liquid. Using available experimental data we have compared our free-space model to a number of existing theories and find that the free-space model correlated the data appreciably better than did the others. Although the results of our free-space model are not rigorous, the overall description in terms of bulk properties of the molecular inhomogeneities in the fluid which affect rotations probably constitutes a reasonable first-order correction to a hydrodynamic picture.
Introduction Rotational relaxation of molecules in liquids has been studied by a variety of experimental techniques. In most experiments, only a single characteristic time is determined and this is inadequate to answer many important questions concerning the mechanisms of rotational relaxation. For example, in a typical NMR experiment a single particle correlation time r2 is determined, r2 being given as
-
(P,(cos 0(t))Pz(cos
dt
(1)
where P2is a second-rank Legendre polynomial, 0 is the time-dependent angle of orientation of a given molecule, and the angular brackets indicate an equilibrium ensemble +Universityof Illinois at Chicago Circle, P.O. Box 4348, Chicago,
IL 60680.
average. In a Raman scattering experiment, one studies the frequency-dependent spectral intensity 12(w),where
only the line width of the central quasi-Lorentzian can be unambiguously interpreted because the non-Lorentzian wings may be attributed to mechanisms other than molecular re0rientation.l If the rotational motion is diffusional, Le., if (P2(t)P2(0))is exponential even at rather short times, then T2-l = r2 (3) Infrared experiments yield spectral information similar to that described in eq 2 with p2(c0s 0) replaced by a first(1) D. Kivelson and P. Madden, Annu. Reu. Phys. Chem., 31, 523 (1980).
0022-3654/81/2085-2169$01.25/00 1981 American Chemical Society
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The Journal of Physical Chemistry, Vol. 85, No. 15, 1981
rank Legendre polynomial P,(cos 0); again only the line width rlcan be adequately determined, and if the rotational motion is diffusional, then2 3 r 1 = rz (4) ESR experiments3 show promise of yielding subtle information concerning rotational mechanisms because I J w ) at both frequencies w N 0 and w N r2-lcan be determined, where the former yields 7 2 and the latter rz;the results to date are somewhat ambiguous. For the purpose of this article we will consider rotational diffusion only. It is often convenient to express T~ (5)
where V, is the volume of the rotating molecule (called the “probe” or the “tagged” particle), 7 is the coefficient of shear viscoisty of the solution, kB is the Boltzmann constant, Tis the absolute temperature, fstiais a well-specified dimensionless hydrodynamic frictional coefficient for stick boundary conditions and dependent only on the shape of the rotating molecule, C is an experimentally determined dimensionless parameter, and 720 is an experimentally determined parameter which has sometimes been associated with the free-rotor correlation time, called the inertial correlation time. For a given system, the dominant temperature dependence in eq 5 is described by ( q / T ) since the other parameters are relatively temperature independent. In some systems T~~ appears to be negligible. In these cases, eq 5 can be rewritten as
Although the dimensionless parameter C appears to be reasonably independent of temperature and pressure, it has a marked dependence upon the solute, the solvent, and the concentration.’ For spherical top molecules, only one correlation time exists, while for symmetric top molecules the value of 72, fstick, C, and T~~ can be different for rotations parallel (11) and perpendicular (I) to the symmetry axis; in most experiments only 72’ is determined. For asymmetric top molecules there are three rotational correlation times and though in a few cases all three have been determined, usually a single weighted average time is presented. In this article, we discuss a number of hydrodynamically based theoretical interpretations for C, including a freespace model which we have developed in an attempt to unify the description of a great collection of available data. In addition, we compare these theories with available experimental data on simple systems wherein the rotating molecules under study are linear or spherical or symmetric tops. We consider both the neat liquids and dilute solutions of the probe. We find that our free-space model gives a reasonably semiquantitative description for a wider range of data than do any of the other theories. We do not attempt to interpret T ~ O ;but as indicated by some of the data analysis, the explanation of 720 may be incorporated in the explanation of C. Hydrodynamic Description Debye5 showed that the reorientation of a dipole in a viscous medium can be described as a random walk traW. A. Steele, Adu. Chem. Phys., 34, 1 (1976). J. S. Hwang, R. P. Mason, L. P. Hwang, and J. H. Freed, J.Phys. ,., 79, 489 (1975); W. J. Lin and J. H. Freed, ibid., 83, 379 (1979). G. R. Alms, D. R. Bauer, J. I. Brauman, and R. Pecora, J. Chem.
.”(
UULl
,La,*,.
‘ye,“Polar Molecules”, Dover, New York, 1929, pp 72-85.
jectory on the surface of a unit sphere. This result can be generalized to give the reorientation time q for a rotation matrix of order 1: 71 = [ / & 1 + l)kBTwhere [ is the rotational friction coefficient. For a sphere with stick boundary conditions Debye related this result to q by means of the hydrodynamic Stokes law and derived eq 6 with
This result is valid for a spherical Brownian particle, a particle that is very large compared to the surrounding solvent molecules. Perrid extended the theory to Brownian ellipsoids; if we once again set c = 1,we now find fstjek 2 1, increasing with the nonsphericity of the rotating particle. Youngren and Acrivos7 developed numerical techniques for calculating fstick for particles of arbitrary shape. Hu and Zwanzig8 considered a rotating spheroid with “slip” boundary conditions where the rotating particle displaces the surrounding fluid which, in contrast to the stick case, exerts no drag force on the particle. For slip boundary conditions Cali, is zero for a sphere and approaches unity for increasingly nonspherical spheroids. Youngren and Acrivos7 extended this theory to particles of arbitrary shape. Ahn! using a theory introduced by Basset,lO considered intermediate boundary conditions between stick and slip for a sphere. In this theory
where V, is the volume of the sphere, 7 is the shear viscosity of the solution, and p is a parameter which is independent of probe size and indicates the boundary conditions; if P = 0, slip boundary conditions pertain and C = 0, and if p m, stick boundary conditions pertain, and c = 1. The theory proposed by Ahng has not been generalized to nonspherical particles. Therefore, we formulate an interpolation formula for spheroids which reduces to Ahn’s expression eq 8 for spheres, to Csli for slip boundary conditions and arbitrary shape, a n 8 to C = 1 for stick boundary conditions and arbitrary shapes. In this formulation
-
fstick
(9a)
as given by Perrin6 or Youngren and A c r i v o ~and ,~
CShpis the value of C for the given shaped particle under slip boundary conditions. This formula is similar, though slightly different in detail, to that given by Tanabe.ll (6) F. Perrin, J. Phys. Radium,5, 497 (1934). (7) G. K. Youngren and A. Acrivos, J . Chem. Phys., 63,3846 (1975). (8) C. M. Hu and R. Zwanzig, J. Chem. Phys., 60,4354 (1974). (9) M. Ahn, Chem. Phys. Lett., 52, 135 (1977). (10) A. B. Basset, “A Treatise on Hydrodynamics”,Vol. 2, Dover, New York, 1961.
The Journal of Physical Chemistiy, Vol. 85, No. 15, 1981 2171
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Real Brownian particles are not truly spherical and smooth but are corrugated on a molecular scale, Le., on a scale comparable to the size of the solvent molecules. The effect of this “roughness” is to couple the Brownian particle with the neighboring layers of the solvent; this results in 0> 0. Thus for sufficiently large particles, one always obtains stick boundary conditions. In eq 5 and 6, the dimensionless factor fstick, which can be well-specified for molecules of known shape, accounts in part for the asymmetry of the probe; C can then be ascribed in large part, but not entirely, to other effects such as nonstick boundary conditions, poor contact between probe and solvent due to fluid inhomogeneity, and nonhydrodynamic dissipation due to the noncontinuous nature of the fluid surrounding the probe.
Subslip and Molecularity Conventional hydrodynamics yields values for C between Cslipand unity; however, experimentally C is sometimes found to be less than Cslip. When this is true the probe particle appears to be rotating with “subslip” boundary conditions, a phenomenon which cannot occur within a hydrodynamic framework but can be interpreted as evidence of the molecularity of the system. These subslip results arise when the probe molecule is smaller than the solvent molecules, and consequently the surrounding solvent no longer acts as a continuous homogeneous fluid. Cavities, spaces, or holes, dependent upon the shape and size of the solvent molecules, are formed in the fluid and can affect the rotational relaxation of the probe molecule. If the volume of a probe molecule is considerably less than the volume of the cavities present in the fluid, the small probe might rotate rather freely, with the result that C = 0 regardiess of the shape of the probe; in this case one might find C < Cslip. Gierer-Wirtz Microviscosity Theory Gierer and Wirtz12 took account of the discontinuous nature of the surrounding fluid for a dilute solution of spherical probe molecules immersed in spherical solvent molecules. They visualized the probe at the center of concentric shells of fluid, each of width equal to the diameter of the solvent molecule. The liquid in each shell has constant angular velocity because each shell consists of a single layer of rigid molecules. The angular velocity of successive shells decreases as though the flow between the shells were laminar, and at a distance far from the probe the angular velocity vanishes. We formulate the Gierer-Wirtz theory to incorporate a more general description of the boundary conditions. We relate the angular velocity w1 of the first neighboring shell to the angular velocity wo of the probe by means of a “sticking factor” u: w1
= (Two
(10)
When u = 1 we have “true” stick and when u = 0 we have slip boundary conditions. As shown in the Appendix, the procedure of Gierer and Wirtz leads to
where m is the shell number and V, is the volume of the solvent molecule. Gierer and Wirtz do not allow u to be arbitrary; they obtain (11) K. Tanabe, Chem. Phys., 31, 319 (1978). (12) A. Gierer and K. Wirtz, 2.Nuturforsch. A , 8, 532 (1953).
5 [ l + 2m(V,/Vp)’~3]-4
uGW
=
m=l
C [I + 2m(V,/Vp)1/3]-4
(12)
m=O
which implies a variation in boundary conditions with varying (V,/ V,). As shown in the Appendix, in the limit (V,/V ) > 1, uGW and the corresponding value of C approach zero as anticipated. Simple approximate expressions for uGW and C can be obtained by separating out the m = 0 and m = 1 terms in the sums; providing V, < V,, the remaining sums can be approximated as integrals. The results are c = aCo (13) 1
aGW
=
1
+ 6(V,/Vp)”3Co
where
For neat liquids V, = V,, and these expressions yield Co 12.3, uGW 0.013, and the Gierer-Wirtz value of CGW N 0.16. The very small value of uGWsuggests nearly slip boundary conditions and the large value of Cosuggesta that solvation takes place. This can be understood more clearly by examining the case where V, = V,, and the solvent sticks very well to the probe; in this case u = 1 and eq 15 and 13 yield C N 12. The fact that C exceeds unity is evidence of solvation, but the fact that it is less than the fully solvated value of 27 is a consequence of the reduction due to the stepwise falling off, correlated with appreciable solvent size, of the angular velocity. The stepwise falling off of angular velocity and the consequent reduction in C below the hydrodynamic value characterizes the Gierer-Wirtz theory. This theory, generalized in the manner described above, accounts for boundary conditions which can lead to a description of solvation. The theory does not take into account the “cavities” or “free spaces” created by solvent about the probe molecule. However, the sticking parameter Q can be related to these free spaces since they do influence probe-solvent contact interaction. Although the theory has been developed for spheres, by allowing fstick in eq 6 to assume its appropriate value for nonspherical molecules, one might expect the applicability of the theory to be extended somewhat. Intermolecular Spaces The Gierer-Wirtz theory accounts for the finite size of the solvent molecules on the stepwise decrease of flow angular velocity about the probe molecule. However, it does not account for the relatively poor physical contact that can exist between the probe and solvent due to the free spaces between the solvent molecules. The free space is associated with the isothermal compressibility KT of the fluid. Vold, Vold, and Szeverenyi13suggest that C-l should be monotonic in KT, and others14proposed that C-’ should (13) R. R. Vold, R. L. Vold, and N. M. Szeverenyi, J. Chem. Phys., 70, 5213 (1979). (14) A. M. Goulay-Bize, E. Dervil, and J. Vincent-Geisse, Chem. Phys. Lett., 69, 319 (1980).
2172
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The Journal of Physical Chemistry, Vol. 85, No. 15, 1981
be linear in the inverse of the free volume Vp An approximate theory of the effect of intermolecular spaces on the rotational parameter (C) can be developed from Frenkel’s15model for liquid behavior. The space in which the probe molecule rotates is equal to the volume of the probe molecules plus the volume of the free space surrounding the probe. Because of the nonsphericity of the probe molecule and its interactiveness with the surrounding solvent, the probe will have an effective rotational volume 4 V,; if the attractive solute-solvent interactions are small and the solventsolute interactions are primarily geometric, to a first approximation 4 might be taken as the hydrodynamic frictional coefficient for slip boundary conditions,fslip. If AV is the smallest volume15 of free space per solvent molecule, the total free space surrounding the probe molecule is AV for very small probe molecules, and AV multipled by the number (-4(V,/V,)2/3) of sovlent nearest neighbors of the probe for large probe molecules. To satisfy the two limiting cases, the space in which the probe rotates is taken as (4V AV[4(V,/V,)2/3 + 11). C is the probability of contact and is associated with the ratio of the effective volume (4V) of the probe to that of the rotational volume. C is thus expressed as
+
where
This theory has the same dependence upon V, in the hydrodynamic limit (V, >> V,) as does the Ahn theory. Note that for symmetric tops, y is the same for motions about or perpendicular to the symmetry axis, but if C#J is approximated by falip, then C is markedly dependent upon the axis of rotation. We use Frenkel’s15 expression for KT in terms of A V
vf
AV
vO
kBT
KT=--
where Vf is the free volume and Vo is a close-packed volume of the solvent molecules, and we incorporate the Frenkel-Batschinski relationship for vis~osityl~-~’ q = Vo(BVf)-l
(19)
where B is a parameter independent of temperature and slightly dependent on the system. We can then rewrite the expression for y in eq 17: 7 = B ~ F J T K T ~ [ ~ ( V , / V+, )l]/vp ~/~
(20)
Equation 16 with eq 20 describes a quasi-hydrodynamic model; the dynamic part of the rotational relaxation occurs by the hydrodynamic transfer of angular momentum from the probe to the surrounding fluid as described in eq 6, but the coupling of the probe to the surrounding fluid involves the contact and interactiveness between the probe and the nearest-neighbor solvent molecules. This model has some similarities to that of the Hynes-Kapral-Weinberg18J9model described in a later section. We also note (15)J. Frenkel, “Kinetic Theory of Liquids”, Dover, New York, 1955. (16)A. J. Batschinski, Z. Phys. Chem., 84,643 (1913). (17)J. H. Hildebrand, Science, 174,490 (1971). (18)J. T.Hynes, R. Kapral, and M. Weinberg, J. Chem. Phys., 61, 3256 (1977). (19)J. T. Hynes, R. Kapral, and M. Weinberg, J. Chem. Phys., 69, 2725 (1978).
that our theory relates to the theories of Ahn? Vold et and Goulay et al.14 which set C-l linear in q, KT-~, and Vel, respectively, but that in our more “complete” analysis the linearity is expressed in terms of K T ~ , K T V ~or~ ,Vfq. Though both our theory and that of Gierer-Wirtz depend upon relative probe-solvent sizes, they describe quite different phenomena: our theory accounts for the degree of contact between the probe in a cavity with the surrounding solvent, whereas the Gierer-Wirtz theory accounts for a stepwise decrease of angular velocity due to the finite size of the solvent molecules.
Compressible Hydrodynamic Fluid. Probe-Solvent Intermolecular Potential Steele related C to the intermolecular torques and hence to the anisotropic part of the intermolecular potential, thereby taking account of probe-solvent interactions.2o Recently Peralta-Fabi and Zwanzig21 investigated the situation where the boundary conditions describing the interaction of a probe molecule with the surrounding continuous fluid are replaced by a potential function. The behavior of the fluid in the force field of the probe is described by the hydrodynamic equations for a compressible fluid. The work of Peralta-Fabi and Zwanzig indicates that a force field approach is feasible. Keyes et al.22 have recently pursued this approach. Masters and Madden23have obtained solutions for “hard potentials” spanning the range between slip and stick boundary conditions. Spin Angular Momentum. Hill Theory The Gierer-Wirtz theory and the intermolecular free space theory both assume that the angular momentum of the rotating probe molecule is transferred to the flow angular momentum of the surrounding fluid. The probe’s angular momentum can also be transferred to the spin or intrinsic angular momentum of the surrounding solvent molecules. Hill’s24mutual viscosity theory accounts for this transfer of spin angular momentum; however, it appears to neglect the transfer to flow angular momentum and the associated long-range or hydrodynamic effects. Hill’s theory is based on the Andrade25theory of viscosity. The molecules in a liquid vibrate with a high frequency v about equilibrium positions which vary slowly in time. The viscosity of the liquid or transfer of linear momentum arises from collisions between neighboring molecules which are momentarily closely associated so that no linear motions are possible. Hill extended Andrade’s viscosity model to include both mixtures and the effect of the transfer of angular momentum. The Andrade and Hill theories are derived for spherical molecules but have also been applied to rigid, axially symmetric molecules; however, they would not be expected to hold for flexible long-chained molecules nor for mixtures of molecules differing considerably in size. For a second-rank tensor, the Hill model defines C as
where X , is the mole fraction of probe molecules, q is the solution viscosity, qsp is the mutual viscosity, qP is the probe (20)W. A. Steele, J. Chem. Phys., 38,2404 (1963). (21)R.Peralta-Fabi and R. Zwanzig, J. Chem. Phys., 70,504(1979). (22)T. Keyes, M.Terumitsu, and J. Mercer, preprint. (23)A. J. Masters and P. A. Madden, J. Chem. Phys., 74,2450,2460 I1 981). \_.__,.
(24)N.Hill, Proc. R. SOC. London, Ser. B, 67,149(1954);h o c . R. Soc. London, Ser. A.,240,101(1957). (25)E.N. da C. Andrade, Phil. Mag., 17,497,698 (1934).
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The Journal of Physical Chemistty, Vol. 85,No. 15, 1981 2173
viscosity, and ,us, b,,and us, are dimensionless parameters defined as
”)
--(
IPP
wpp
R=
113
= MPVp2l3 47
n is the solvent number density, g(R,)is the pair distribution function a t the solvent-solute contact distance R, = [3/(47r)]1/3(Vs1/3 VP’/3)
+
where M, and Mpare the molecular masses of the solvent and probe molecules, respectively, Ippis the moment of inertia of the probe, I,, is the moment of inertia of a solvent molecule taken about the center of a probe molecule when the two are in collisional contact, and V, and V, are the volumes of the solvent and probe molecules, respectively. The mutual viscosity vsp is defined as24 ?sp
- (v[XpVp1/3+ (1- X,)V,1/3]
-
77pxpzvp1/3
v,(l - X,)2V,1/3)/[Xp(l- X,)(Vp’/3+ V,1/3)] (23)
In this paper, two limiting cases have been discussed, the very dilute solution (X, > V,, Mp >> M,, X, ~ 1. ] However, for such fluids, they find that the rotational motion is not diffusional so that eq 17 does not hold. In the Hynes, Kapral, and Winberg theory, p takes account of spin as indicated by the 6 factor, whereas the free space model does not. In eq 30, the terms 6(6 l)-land [M,/(M, + M,)]1/2 arise from the dynamic transfer of angular and linear momentum, respectively, between the probe molecule and the surrounding solvent; these terms do not appear in the free-space expression (eq 20) for 0. Though both theories take into account the effective contact between the probe and solvent, the free space model makes use of bulk characteristics (Le., compressibility, viscosity) of the solvent, while the Hynes, Kapral, and Weinberg theory makes use of difficult-to-obtain statistical quantities.
+
Correlation Functions For diffusive motion, the rotational relaxation time is given by the expressionz6 72
= J m d t ([exp i(1
-
P)&t]
72
X
e),
(1 - P)P~(cos (1 - P)P~(cos B))/(IP~(cos @I2) (32)
where L is the Liouville operator and P is the MoriZwanzig.operator which projects onto P2(c0s19). The term (1- P)IP2(cose), where I is the moment of inertia, consists of torque-like terms plus kinetic terms, the latter being bilinear in angular velocity. We can express (1- P)P2as (1 - P)Pz = Hspin+ H,,, (33) where Hflow =
Hspi,
xhhl h l ( r i j ) a iO , ( R 1 ) a h lo(ilij)
1 . lij =l j2f O hikldj m i j
(34)
si, m i n i ( n l ) a in. . ( ~ j ) a ~ , , ~ (+ ~ i j ) J
.I
11
kinetic terms ( 3 5 )
where
(26) D. Kivelson, M. G. Kivelson, and I. Oppenheim, J. Chem. Phys.,
52, 1810 (1970); D. Kivelson, Chem. SOC.Faraday Symp., 11, 7 (1977).
The Journal of Physical Chemistry, Vol. 85, No. 15, 1981
2174
c
/
/
Dote et al.
%
8.01
L / / / L*
/
I
t
.x '
/
d/
i
n-C..H..
1 1
05
I
10
1.5 T~
Flgure 1. T~ vs. v / T for neat CS227329at different temperatures (X) and for dilute solutions30of CSz (0)in n-alkanes at 303 K. The solid and broken lines are linear least-squares fits for the dilute solutions and the neat liquid, respectively. 720 = 0 for neat CSz. Values of 9 obtained from ref 31 and 32.
ill and i12 are the Euler angles specifying the orientation of the probe molecule and the jth solvent molecule, respectively, while illjspecifies the angular position of the jth particle relative to that of the probe molecule; a>&, is a Wigner rotation function, rlI is the distance between the probe and the j t h solvent molecule. H",, arises from the interaction of the probe molecule with the surrounding hydrodynamic fluid; the internal structure of the solvent molecules and the transfer of probe angular momentum to the spin of the solvent molecules are largely neglected in this term. Hfl,, should give rise to the hydrodynamic contributions to r2. Hspinis strongly dependent upon the transfer of the probe's angular momentum to the spin of the solvent molecules. The problem of relating the correlation function expression to the hydrodynamic results is discussed elseThe models presented in this article do not attempt to treat all aspects of the problems systematically. Analysis a n d Discussion of Experimental D a t a In this section, we correlate available experimental data with some of the hydrodynamic and quasi-hydrodynamic theories presented above. The input necessary t o study the applicability of the Hill theory is not available, nor for most systems considered is that for the Hynes, Kapral, and Weinberg theory. We restrict our discussion to linear, symmetric and spherical top molecules. The comparisons between theory and experiment are carried out graphically. The data on which these comparisons are made have been taken from the literature; tabulations of these data, processed for the present analysis, are available from the authors. Carbon Disulfide We first consider the rotation of carbon disulfide (CSJ for which a considerable amount of data has been assembled; Raman,27l3Cz8and 33S29 NMR for the neat liquid as a function of temperature, and 33SNMR30 for dilute so(27) T. L. Cox,M. R. Battaglia, and P. A. Madden, Mol. Phys., 38, 1539 (1979). (28) H. W. Spiess, D. Schweitzer, U. Halberlen, and K. H. Hawser, J. Magn. Reson. 5, 101 (1971). (29)R.R.Vold, S. W. Sparks, and R. L. Vold, J. Magn. Reson., 30, 497 (1978).
I
I
20
25
30
(THEORY)
Flgure 2. Tz(expt) vs. T,(theory) for neat CS,27829 at different temperatures for the Hu-Zwanzig model (0),the Gierer-Wirtz model (A), and the free-space model (*) with 4 = f,,,,. The T~ values for the Debye-Perrin model are too large to fit on the graph. f* = 0.32and fstlck= 1.44for an equivalent prolate ellipsoid of axial ratio 0.52;V , = 52.5 A3; 33 V , obtained from molar volumes;3i k, from ref 34;and B = 14."
i
3.0t
i -
W
I
'" 1 0
io
20 T~
30
40
50
(THEORY)
Flgure 3. ~,(expt)vs. T,(theory) for dllute solutions of Csz3Oin ~-c~H,, (I), CSz (21, C(CzHcJ4 (3),n-Ci2Hx (4h n-C14H30 (51,and n-Ci~H3,(e), all at 303 K for the Hu-Zwanzig model (0),the Gierer-Wirtz model
(A), and the free-space model (") with 4 = f,,,,. The r2 values for the Debye-Perrin model are too large to fit on the graph. f,, = 0.32 and f,, = 1.44for an equivalent prolate ellipsoid of axial ratio 0.52; V , = 52.5 A3; 33 V , obtained from molar v o i u m ~ s K~, ~from ~ ~ ref ~ ~34; and Bfrom ref 15,16, and 17.
lutions of CS2 in a series of alkane solvents at a constant temperature. The reorientation time T~ is plotted against v / T in Figure 1; it varies linearly with T for the neat liquid and passes through the origin (7) = 0), but at a given temperature in a series of dilute solutions in alkanes, T~ is nearly independent of q/T and thus has an apparent nonzero intercept ( T ~ O= 1 ps). In Figure 2, we compare the experimental values for T~ in neat CS2 with those predicted by the various theoretical models, while in Figure 3 we do the same for dilute CS2 in a series of n-alkanes. If CS2 is treated as a prolate spheroid with an axial ratio of 0.52, the Debye-Perrin stick hydrodynamic model with C = 1 yields theoretical 7*'s far greater than the experimental values for both neat CS2 and dilute solutions in n-alkanes; this is not surprising since this theory should hold well only for Brownian particles, particles large compared to the solvent molecules. The Hu-Zwanzig hydrodynamic model with slip boundary conditions predicts (30) B. Ancian, B. Tiffon, and J. E. Dubois, Chem. Phys. Lett., 65,281 (1979).
The Journal of Physical Chemistty, Vol. 85, No. 15, 1981 2175
Feature Article
TABLE I: Comparison between Experimental and Theoretical Reorientation Time 1
X = -
system CS,l(neat) CS,L(yln) C,H, ( n e a t ) C, H,l( soln ) without CD,OD and C,Cl, C,H, l'(neat) ~C,H,II (soln) without CD,OD and C,Cl,
{
CCl,( soln ) I,(soln ) H-C=C-C-N (soln) N=C-C=C-C=N (soln)
m -
DebyePerrin
22 150 37 71 69 46 5 442 495 28 29 33 371 136 199 282
m
1
2: [(7,(theory)/7,(expt))n n=l
HuZwanzig
YoungrenAcrivos
0.06 0.11 0.26 1.3 1.2
1 I'
GiererWirtz
freea space
free space
HKW (Tanabe)
0.10
0.019 0.036 0.04 0.16 0.087 0.19 0.069 0.034
0.074b 0.16b 0.10b
0.32 0.66 0.68 0.074 0.017 0.014
0.17 0.026 0.19 0.20 2.4 2.9 2.2 0.038 0.17 0.12 1.3 0.20 0.22 0.40
0.07 0.66 0.65
1.1 1.1
1.2 0.32 0.39 0.35 0.58 7.3 2.0
@ = fslip, determined a @ = fSlie, determined by the Hu and Zwanzig model.* @ determined by plotting C-' vs. y and fitting with a linear least-squares fit with
0.06c 0.16c 0.14c 0.13c
0.21 0.17
0.12 0.067 0.026 by t h e Youngren and Acrivos model.' C-' = 1 a t y = 0.
TABLE 11: Data for Neat C,H, 1o''KT,'
T,K
pressure, bar
q:
CP
721,a
PS
T2'1,a
ps
Vs,bit3
cm2/dyn
B,d cP-'
0.633 2.08 0.891 138 9.71 14.8 50 14.8 0.438 1.38 0.694 143 12.2 50 0.845 2.16 133 14.0 1000 0.825 5.9 1.473 3.33 1.12 3.80 13.4 2000 127 14.8 0.322 1.02 0.534 148 15.38 363.13 50 14.0 0.601 1.59 0.656 136 6.6 1000 0.997 2.42 129 4.17 13.4 0.817 2000 0.441 14.8 0.259 0.80 154 17.86 393.13 50 0.513 140 7.35 14.0 1000 0.472 1.22 0.756 1.74 0.641 132 4.37 13.4 2000 a Obtained from ref 1 M c - r volume calculated from L-.e so e n t density c-tainel. .-om ref 11. IsoLermal compressibility extrapolate( 'rom data obtained from ref 31 and 34. Calculated by using eq 19."-17 303.13 333.13
values of r2 just slightly higher than those observed for neat CS2,but for dilute solutions in n-alkanes, the predicted s7 ; are appreciably larger than the experimental ones, Le., the experimental ones are subslip. The T ~ ' Spredicted by the Gierer-Wirtz theory are in fair agreement with the experimental values for neat CS2 and for dilute solutions of CS2 in the lower n-alkanes, but deviate for solutions in the higher n-alkanes; this breakdown in the Gierer-Wirtz model may be due to the nonsphericity of the CS2 molecule and to the nonrigidity of the n-alkanes molecules and not necessarily to a basic defect in the principle of concentric shells of given angular velocity. Our free-space model yields values of 72 which are in reasonable agreement with the experimental ones for both neat CS2 and the dilute solutions of CS2 in n-alkanes. Our theoretical values are based on 9 = fslip, as determined by Hu and Zwanzig8for a prolate ellipsoid; slightly different values of 9 and V might improve the agreement between experiment and theory. A quantitative measure of discrepancy between theory and experiment can be given by x: m
x = n=l C([72(theory)/.r,(expt)l,
- 1)2/(m - 1)
(36)
where the subscript n indicates the nth datum point. Values of x for each theory are given in Table I. The free-space model correlates the data better for both the neat liquid and the dilute solutions than do the other models. Benzene We next consider benzene (C6H6) for which Raman data for both the tumbling and spinning motions (rotation
0
I
I
I
I
io
20
3.0
40
?T I
x
5.0
io3 (CP K - ' )
Figure 4. T~~ vs. t)lTfor neat C6Heii at different temperatures and pressures (X) and for dilute solutions35 of C6H6(0)at 293 K. The broken line is a hear least-squares fit for neat C6He. r2l0 = 0.5 ps for neat C6H6. Values of 9 obtained from ref 11, 31, and 32. See Table 11.
perpendicular and parallel to the symmetry axis of the molecule, respectively) have been obtained for the neat liquid as a function of temperature and pressure" and for dilute solutions of C6H6in a variety of organic solvents at a constant t e m p e r a t ~ r e .For ~ ~ the tumbling motion, a plot (31) E. W. Washburn, Ed., "International Critical Tables", McGrawHill, New York, 1928.
The Journal of Physical Chemistry, Vol. 85, No. 75, 7981
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Dote et al.
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io
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i 50
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Flgure 5. ~ ~ ‘ ( e x p tvs. ) T,’(theOry) for neat C6H611at dlfferent temperatures and pressures for the Debye-Perrin model (O), the HuZwanzig model (0),the Youngren-Acrivos slip model (6)with f ~ , , , , = 0.197, the Glerer-Wirtz model (A), the free-space model (*) with 4 = 0.238, the free-space model (@) with 4 = 0.197, and the = 1.110 for an equivalent HKW model (V). fLSb = 0.238 and f’,, oblate ellipsoid of axial ratio 1.92 and V, = 80.33 A? 33 See Table 11.
’
30
20 q/T
X
40
50
IO3 (cp K - ’ )
Flgure 7. T2i1 vs. q l r f o r neat C~H,’~at different temperature and pressures (X) and for dilute solutions35of C6H6( 0 )at 293 K. The broken line is a linear least-squares fit for neat C6H6. T$O = 0.35 ps for neat C6H6. Values of 7 obtained from ref 11, 31, and 32. See Table 11.
‘1
t 1.5
!
- t i! X
W
v
=bN
05
1
A
c t
1 10
1
1
15
20
25
T:
(THEORY)
I
I
30
35
Flgure 6. ~ ~ ‘ ( e x p tvs. ) ~~‘(theory)for dilute solutions of C6Hs35in CO(CDd2 (11, C6D12 (2), CDpClp (3), C~CII(4), CCI, (5)s CS2 (61, C& (7), CD,OD (8),and CDCI, (9) at 293 K for the Hu-Zwanzig model (0), the Gierer-Wirtz model (A), the Youngren-Acrivos slip model (6) with fLSy, = 0.197, the free-space model (*)with 4 = 0.238, the freespace model (@) with = 0.197, and the HKW model (V).The T2’ values for the Debye-Perrin model are too large to fit on the graph. fLdlp= 0.238 and fLstick= 1.110 for an equivalent oblate ellipsoid of axial ratio 1.92; V, = 80.33 A3; 33 V, obtained from molar volu m e ~K~; from ~ ~ ref ~ 34; ~ ~and B from ref 15, 16, and 17.
+’
of 72’ vs. q/T is given in Figure 4. (See Table 11). For the neat liquid, T~~ is linearly dependent on q/T, while the data for different solutions at a given temperature are scattered for the different solvents. These results are similhr to those for CS2,though the scatter in the benzene solution data is greater; this scatter is due to the wide variety of solvents used in the benzene experiments compared with the closely related n-alkane solvents used for CS2. In Figures 5 and 6 the experimental T ~ ’ is compared with 72”s predicted by the Debye-Perrin, the Hu-Zwan(32) “Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds”, American Petroleum Institute Project No. 44, Carnegie Press, 1953. (33) A. Bondi, J. Phys. Chem., 68,441 (1964). (34) R. C. Weast, Ed., “Handbook of Chemistry and Physics”, 52nd ed, The Chemical Rubber Co., Cleveland, 1971. (35) K. Tanabe and J. Hiraishi, Mol. Phys., 39,493 (1980).
0
05
10 T;
15
20
25
(THEORY)
Flgure 8. T$(expt) vs. T$(theOty) for neat C&i6” at different temperatures and pressures for the Gierer-Wirtz model (A), the free-space model (”) with 411 = 0.086, and the HKW model (V).The T~ values for the Debye-Perrin model are too large to fit on the graph. filstick = 1.378 for an equivalent oblate ellipsoid of axial ratio 1.92 and V, = 80.33 A3.33 4 11 obtained by plotting ( Cll)-‘ vs. y and fittin the curve with a linear least-squares fit with (Cl,)-’ = 1 at y = 0. (4 = 0.086.) See Table 11.
f
zig, the Youngren-Acrivos, the Gierer-Wirtz, and our free-space model, as well as the Hynes-Kapral-Weinberg model as re-formulated and extended by Tanabe.11*36For both the neat liquid and the dilute solutions, the DebyePerrin and the Hu-Zwanzig models, both with C6H6as an oblate spheriod with an axial ratio of 1.92, predict values higher than the experimental values; thus subslip boundary conditions apply. For the neat liquid, the YoungrenAcrivos theory (which assumes a more realistic shape for benzene), the Gierer-Wirtz theory, and the free space theory (with c$I= fLslip for an oblate spheroid and with cpL = fl,, determined by Youngren and Acrivos’) all show similar agreement with the experimental data, whereas the Hynes-Kapral-Weinberg theory deviates seriously at higher solution viscosity. For dilute solutions, the freespace theory correlates better with the experimental data than does the Gierer-Wirtz, the Hynes-Kapral-Weinberg, and the Youngren-Acrivos models. All the theories yield poorer results for C6H6than for CS2,probably because the CS2 measurements were made in a series of n-alkanes whereas the benzene data were obtained in a wide assortment of unrelated solvents. In particular, the data in
Feature Article
The Journal of Physical Chemistry, Vol. 85, No. 15, 1981 2177
I
-'""I/ 04
-1
06
08 T!
12
10 (THEORY)
14
Flgure 9. T#(expt) vs. T$(theory) for dilute of C6H6 in CD&D (1)-CBD, (2),CSp (31,CDCIs (4),CsD1, (5),CO(CD,), (61,CCI, (7),CDzCIz(e),and C,CI, (9)at 293 K for the Gierer-Wirtz model (A), the fre -space model ( * ) with 4 11 = 0.086,and the HKW model (V). The T~~ values for the Debye-Perrin model are too large to fit on the graph. f$, = 1.378 for an equivalent oblate ellipsoid of axial ratio 1.92; V, = 80.33 A,; 33 V , obtained from molar v o l ~ m e s ; ~K~' -from ~~ ref 34;and Bfrom ref 15, 16, and 17. $11 obtained as in Figure 8.
t
6.0
c
I
I
I
I
I
I
l
0
I
I
1.o
l
,
2 .o
1
1
l
3.0 T*
/
4.0
l
l
5 .O
6.0
(THEORY)
Flgure 11. r,(expt) vs. T,(theory) for neat CDC1,38 at different temperatures for the Hu-Zwanzig model (0),the Gierer-Wirtz model (A), and the free-space model (*) with 4 = 0.465 where 4 is determined by a linear least-squares fit of C1vs. y with C-' = 1 at y = 0. The values for the Debye-Perrin model are too large to fit on the graph. fsYp= 0.08 and f,, = 1 .O1 for an equivalent oblate ellipsoid of axial ratio 1.43;Vp = 71 A,; 33 V, obtained from molar volumes;34K~ from ref 34;and B = 14.'7
I " " " " ' ~ r I
5't01
O9 0'8
A /
t9
4.0
r
0
1
1
1
1
I
I
I
10
20
30
40
50
60
70
80
r i / ~x io3 (CP K - ~ I
i
Figure 10. T~ vs. q/ Tfor neat CDC1338at different temperatures ( X ) and for dllute solutions37of CDCI, ( 0 )at 293 K. The broken line is a linear least-squares fit for neat CDCI,. 720 = 0.42 ps for neat CDCI,. Values of q obtained from ref 31 and 32.
CD30D and CZCl4were particularly anomalous; if these solvents are overlooked, the experimental and theoretical results correlate much better. (See Table I.) It should also be pointed out that for neat benzene, a T~~ vs. q / T plot yields an appreciable positive intercept, T ~ * O , at TIT 0. For the spinning motion, a plot of ~ ~ vs. 1 T/T 1 is given in Figure 7. The results are similar to those for the tumbling motion. Plots of the experimental T.,#'s vs. the theoretical T#'S are given in Figures 8 and 9, and the measure of discrepancy x between the experiment and theory is given in Table I. We determined 411 in the free-space model, as indicated by eq 16, by plotting the "measured" - ~ the calculated value of y and by using value of ( C I ~ ) vs. a linear least-squares fit with (C -l = 1 at y = 0; we obtained $11 = 0.086, which should e accepted with the realization that j l l ~ p= 0 for a spheroid but should be nonzero for a knobby structure such as benzene. Once again it is seen that the free-space model does quite well. For both the neat liquid and the dilute solutions, the Debye-Perrin model (with benzene as an oblate spheroid with an axial ratio of 1.92) and the Gierer-Wirtz model predict values for ~ 2 ( 1higher than the experimental values. The HynesKapral-Weinberg model correlates the data in this case even better than does the free-space model. The intercept T$O is comparable to T ~ ~ ~ .
-
t
4
0
io
20
30
40
50
60
r2 (THEORY) Figure 12. T2(expt) vs. T,(theory) for dilute solutions37of CDC13 in n-CBH1, (I),CSp (2),CBHi, (3),CDCI, (4),CH30H (5),CHCl3 (6 co(CH,), (7),CCI, (e),and C6H6(9) for the Hu-Zwanzig model (0),the Gierer-Wirtz model (A), the free-space model (*) with 4 = 0.465,and the HKW model (V).The T , vaules for the Debye-Perrin model are too large to fit on the graph. fsllp= 0.08 and f , = 1.01 for an equivalent oblate ellipsold of axhl ratio 1.92;V, = 71%; V, obtained from molar v ~ l u r n e s ; ~K~~ obtained , ~ ~ ~ ~from , ~ ~ref 34;and 8 from ref 15, 16, and 17. 4 obtained as in Figure 11. 19
Chloroform The rotation of deuterated chloroform (CDC13)has been studied by Raman scattering37from dilute solutions of CDC13 in a variety of organic solvents at a constant temperature, and by 2H NMR3s for the neat liquid as a (36) J. Timmermans, "Physico-Chemical Constants of Pure Organic Compounds", Vol. 2, Elsevier, Amsterdam, 1965. (37) K. Tanabe and J. Hiraishi, Adu. Mol. Relaxation Interact. Processes, 16, 303 (1980).
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The Journal of Physical Chemistry, Vol. 85,No. 15, 1981
Dote et al.
1 1 1 1 1 1 1 1 1 ' 1 1 1 1
1.01 1 1
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40
1
1
l
80
6.0
I
1
1
100
1
1
12.0
TIT x io3 (CP K-I)
x lo3 (CP K - I )
Flgure 13. r2 vs. q1Tfor dilute solutions of H-C%-C=N3' at 297.5 K. The solid line is a linear least-squares fit to the data. Values of q obtained from ref 39.
160
0
Figure 15. r2 vs. q1 Tfor dilute solutions of CC1,30 at 294 K. The solid line is a linear least-squares fit to the data. Values of q obtained from ref 30.
1
Rot
801
i4
TOLUENE
,@J
c 0
11
n-CgHlq
1
1
20
1
1
1
40
1
6.0 ?IT x
1
1
80
1
1
100
1
1
120
io3 (CP K - ' )
Flgure 14. r2 vs. qITfor dilute solutions of Ns-C=C-C=N13 at 297.5 K. The solid line is a linear least-squares fR to the data. Values of 11 obtained from ref 13.
function of temperature. A plot of r2vs. q/T is given in Figure 10. The results are similar to those for benzene. Plots of the experimental r2's vs. the theoretical 72's are given in Figures 11 and 12, and the x values are given in Table I. For both the neat liquid and the dilute solutions, the Debye-Perrin and the Hu-Zwanzig models (with CDC13 as an oblate spheroid with an axial ratio of 1.43) predict values higher than the experimental values; thus subslip conditions apply. As for benzene, the Gierer-Wirtz and the free-space models are in reasonable agreement with the experimental results for both the neat liquid and the dilute solutions. For the free-space model, the parameter C#J is determined by plotting C-l vs. y, as explained above in the discussion for benzene. The HKW theory does not agree as well with the data as does the free-space model. If the result for the benzene solution is omitted, the value of x is improved appreciably for most of the theories. (See Table I.) Note that the intercept r20 is appreciable for CDC13.
0
40
20
60 80 T I T x lo3 ( C P K - I )
100
120
Flgure 16. T~ vs. qlTfor dilute solutions of ';1 at 298 K. The solid line is a linear least-squares fit to the data. Values of q obtained from ref 32. I
*4
A4
-
3 0A3
t3
/03
a X
I
0
10
20 T~
30
40
50
(THEORY)
Carbon Tetrachloride, Cyanoacetylene, Dicyanoacetylene, and Iodine We also examined the rotation of carbon tetrachloride (CC14),30 cyanoace tylene (HC=C-C=N) ,39 dicyano-
Figure 17. r2(expf) vs. ~,(theory) for dllute solutions3' of H - C S C G N in n-C8H14(l), C8H12(2), toluene (3),and n-CleH34 (4) for the Hu-Zwanzig model (0),the Gierer-Wirtz model (A),and the free-space model ( * ) with 6 = fsllp. The r2 values for the Debye-Perrln model are too large to fit on the graph. f,, = 0.38 and f,, = 1.565 for an equivalent prolate ellipsoid of axial ratio 0.48; V , = 50.3 A3; 33 V, obtained from molar volume^;^' K~ from ref 13; and Bfrom ref 15, 16, and 17.
(38)D.L.VanderHart, J. Chem. Phys., 60,1858 (1974). (39)N.M.Szeverenyi, R. R. Vold, and R. L. Vold, Chem. Phys., 18, 31 (1976).
acetylene (NdX=C-C=N),l3 and iodine (LJ40 in dilute solutions of alkane solvents. For all four cases, T~ appears
The Journal of Physical Chemistry, Vol. 85, No. 15, 1981 2179
Feature Article
o2
l
0
l
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’
1
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1
1
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8.0
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10.0
1
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/
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/
A4
r2 (THEORY) Flgure 19. ~ ~ ( e x pvs. t ) T(theory) for dilute solutions30of CCI, in n-C6H14
(l), CCI, (2), n-Cj2H26 (3), n-C14H30 (4), and t?-C&i34(5) for the Gierer-Wlrtz model (A)and the free-space model ( * ) with q5 = 0.158
4 is determined by a linear least-squares fit of
0
10
20 T~
Flgure 18. ~ ~ ( e x pvs. t ) T2(theory) for dilute s o l ~ t l o n sof~ N=C-C= ~ C-CEN in n-C,H,, (l), C&+ (2), C6HB(a), and toluene (4) for the the Gierer-Wirtz model (A), and the free-space Hu-Zwanzig model (0), model ( * ) with 4 = fslIp.The 7, values for the Debye-Perrin model are too large to fit on the graph. fSSp= 0.72 and f,,, = 1.94 for an equivalent prolate ellipsoid of axial ratio 0.39; V , = 74.82 A3; 33 V , obtained from molar volume^;^' K~ from ref 13; and Bfrom ref 15, 16, and 17.
where
/
1
12.0
T~ (THEORY)
t
L
C-l vs. y with
C-’= 1 at y = 0. The T~ values for the Debye-Perrin model are too = 1.0 for an equivalent large to fit on the ra h. f,, = 0.0 and ,f !? V’, obtained from molar K~ from sphere; V , = 85
i3;
ref 33; and Bfrom ref 15, 16, 17.
to be more or less independent of v/T (see Figures 13-16). In Figures 17-20, we plot the experimental T ~ ’ Svs. T ~ ’ Sas given by the Debye-Perrin, the Hu-Zwanzig, the GiererWirtz, and the free-space models. With CC14, H-C=CC r N , N=C-C=C-C=N, and I2 taken as prolate spheroids with axial ratios of 1.00, 0.48, 0.39, and 0.62, respectively, the Debye-Perrin and the Hu-Zwanzig models predict values much higher than the experimental values (subslip). We find that the Gierer-Wirtz theory does not account as well for the q ’ s determined in solvents of the higher numbered n-alkanes as does the free-space model. (See Table I.) For cyanoacetylene, dicyanoacetylene, and iodine the theoretical values for the free-space model are based on = fBlip as determined by Hu and Zwanzig8for prolate ellipsoids. For carbon tetrachloride q5 is determined by plotting C-’ vs. the parameter y , as discussed above; the result C#J = 0.158 is to be compared with the value fBlip (40)M. R.Battaglia and P.A. Madden, Mol. Phys., 36,1601 (1978).
30
40
50
(THEORY)
Flgure 20. T,(expt) vs. T,(theory) for dilute solutions40of I2 in n-C2H1, (11, n-C,H16 (21, n-C8H18(31, n-C10H22 (4), C6H12 (5),n-C12H2,(61, and n-C16H3, (7) for the Hu-Zwanzig model (0), the Gierer-Wirtz model (A),and the free-space model ( * ) with q5 = fsMp.The T~ values for the Debye-Perrin model are too large to fit on the graph. f,,, = 0.16 and f, = 1.26 for an equivalent prolate ellipsold of axial ratio 0.62; V, = 76.44 A3; 33 V , obtained from molar volumes;32K~ from ref 34; and Bfrom ref 15, 16, 17.
= 0 for a sphere, but it should be remembered the CCll is a knobby sphere.
Conclusion The molecular reorientation time 7 2 can be described by eq 6; the effect of molecular asymmetry and geometry is partially described by the parameter fBtick, and the coupling between the molecule and the surrounding fluid is given by C. We developed an approximate theory for C based on the existence of free spaces in the hydrodynamic continuum, and we described these spaces in terms of the bulk properties of the fluid. The value of C depends upon a parameter 4 which we have set equal to fBlip; this parameter incorporates additional shape and asymmetry effects. Using available experimental data, we compared our free-space model to the hydrodynamic stick and slip, the Gierer-Wirtz and the Hynes-Kapral-Weinberg theories. Overall, the free-space model consistently correlated the data for both the neat liquid and dilute solutions better than did the other models. (See Table I.) Although our free-space model is an approximate theory, the overall description in terms of bulk properties of the fluid and of the space in which the probe molecule rotates probably constitutes a reasonable first-order correction to a hydrodynamic picture. We have ignored T ~ Oin eq 5 and have assumed it to be an experimental artifact arising from the fact that C is a function of T and 7; though we obtain reasonable agreement between experiment and theory with this assumption, better fits might be obtained if the T~~ values where available for each solution. Though we did not feel that the increased complexity would be worth the improvement, some obvious refinements could be included in the free-space model. 4 need not be set equal to fslip, but could incorporate more information about the attractive forces between solvent and probe; perhaps the neglect of these interactions accounts for the poor results for the methanol solutions. Setting the volume of space in which the probe molecules rotates equal to (4V + AV[4(V /VJ2/3 + 131 is quite clearly inexact and could be improvecfby considering the structure of the liquid and the geometry of the probe and solvent molecules. However, as it stands the theory seems to account for some general trends and it is unlikely that these modifications would yield substantial improvements.
J. Phys. Chem. 1981, 85, 2180-2182
2180
Acknowledgment. We thank the National Science Foundation (Grant CHE 77-15387) for its support.
Appendix We formulate the Gierer-Wirtz theory to incorporate a more general description of the boundary conditions. We relate the angular velocity w1 of the first neighboring layer to the angular velocity wo of the probe by means of a “sticking” factor u, as indicated in eq 10. In the GiererWirtz theory,12 the angular velocity u1 is expressed as
where T is the angular torque acting on the probe and m is the shell number. The reorientation time r2 is written as
We use eq Al, A2, and 6 to obtain an expression for C.
The factor u is not arbitrary in the Gierer-Wirtz theory. To obtain the Gierer-Wirtz uGWvalue, we note that the authors have
Combining eq 10, Al, and A4 we find that
L-r
ARTICLES Laser-Induced Fluorescence Analysis of Vapor-Phase Pyrene T. J. Whltaker” and B. A. Bushaw Pacific North west Laboratory, Barteiie Memorial Institute, Richiand, Washington 99352 (Received: March 20, 1980; In Final Form: June 20, 1980)
Collisionalquenching of vapor-phase pyrene fluorescence by atmospheric constituents is studied by both quantum yield and lifetime measurements. Oxygen is found to be the only gas studied which shows appreciable quenching characteristics. The rate constant for the bimolecular quenching of pyrene by 0 2 is found to be (1.90 f 0.08) x L mol-’ s-l, In an oxygen-free environment, detection of 1ppb pyrene is demonstrated by using simple fluorescence techniques. A detection limit of less than 100 parts per trillion is shown to be feasible.
Introduction Large polynuclear aromatic hydrocarbons (PAHs) strongly absorb light a t wavelengths corresponding to a transition from the ground state to any of the excited singlet states. Fluorescence emission, however, is usually dominated by wavelengths corresponding to the lowest excited state, S1. This allows laser-induced nonresonant fluorescence measurements to be made with a great deal of sensitivity since the scatter from the shorter wavelengths of the exciting laser may be easily filtered from the detector. However, in situ fluorescence analysis of atmospheric PAHs is subject to problems of collisional quenching from atmospheric constituents. This paper reports the effect of collisions with 02,N2, and He on laser-induced fluorescence analysis of vapor-phase pyrene. In the case of oxygen, both quantum yield and excited state lifetime are plotted in a Stern-Volmer type plot to determine the collisional quenching rate constant. Pyrene was chosen as a representative PAH because it exhibits characteristics common to many molecules of (1)W. R. Ware, J . Phys. Chem., 6 5 , 455 (1962).
environmental interest. We know from condensed-phase studies1t2and work by Chihara and Baba3y4on the vapor phase, that oxygen efficiently quenches fluorescence from excited states of pyrene. Since we wish to measure the exact quenching constant under conditions similar to those of an in situ analysis scheme for PAHs, we have chosen 266 nm as the exciting radiation wavelength. This is a region where most PAHs absorb strongly, and it also corresponds to a frequency quadrupled Nd:YAG laser. The power and reliability of the Nd:YAG laser make it ideal for analytical studies. In the case of pyrene vapor, the 266-nm wavelength excites the 0-0 band of the third excited singlet S3. This energy is rapidly redistributed via intramolecular conversion to low-lying vibrational levels of SlS5-l4Fluorescence from this level (Arnm ~ 4 0 0nm) (2)L.K.Patterson, G. Porter, and M. R. Topp, Chem. Phys. Lett., 7 , 612 (1970). (3) K. Chihara and H. Baba, Chem. Phys., 25, 299 (1977). (4)K.Chihara and H. Baba, Bull. Chem. SOC.Jpn., 48,3903 (1975). (5) H.Baba and M. Aoi, J. Mol. Spectrosc., 46,214 (1973). (6)H.Baba, A. Nakajima, M. Aoi, and K. Chihara, J. Chem. Phys., 55, 2433 (1971).
0022-3654/81/2085-2180$01.25/00 1981 American Chemical Society
