A general expression for weak collision unimolecular rate coefficients

Apr 6, 1989 - The effect of weak collisions is discussed and parametric equations are ... exact strong collision RRKM theory and a V equation” was m...
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J. Phys. Chem. 1990, 94, 567-576

567

A General Expression for Weak Collision Unimolecular Rate Coefficients in the Falloff Region Z. Pawlowska and I. Oref* Department of Chemistry, Technion- Israel Institute of Technology, Haifa 32000, Israel (Received: April 6, 1989; In Final Form: June 9, 1989)

A general analytical expression (interpolation formula) for unimolecular rate coefficients is given for strong and weak collisions. Quadricyclane, cyclobutene,cyclobutane, cyclopropene, CH2DCH2CI,and C2HSBrwere investigated. RRKM master equation calculations were performed and compared with results of an analytical “ J equation” in the temperature range 298-2500 K and the pressure range 104-108 Torr. The effect of weak collisions is discussed and parametric equations are given to enable calculations of the unimolecular rate coefficient for any temperature, pressure, and weak collider with a given (A,!?).+ The expressions given can be used in large-scale calculations such as dynamic modeling of high-temperature combustion and atmospheric chemistry.

Introduction

In a previous paper (I)I an equation was derived for the evaluation of unimolecular rate coefficients as a function of three parameters: the low-pressure rate coefficient ko, the high-pressure rate coefficient k,, and J3/2, a constant evaluated at the pressure where the reaction order is 312. The equation derived can replace RRKM calculations in situations where large-scale modeling of atmospheric or combustion reactions is involved and makes the former impractical to use on a routine basis. At present, motivations for studies of unimolecular reaction rate coefficient calculations move in two diverse routes. On the one hand, classical RRKM calculations are becoming more and more detailed and involved, attempting to provide information on state to state rate coefficients and vibrational-rotation distributions. Energy and angular momentum resolved rate coefficients are obtained for specific assumed tran~ition-state~-~ configurations and compared with e ~ p e r i m e n t . ~In other cases, phase space theory (PST)6 has been used to estimate product state distributions’ as was an adiabatic channel model (ACM).8 On the other hand, large-scale dynamic modeling of atmospheric or combustion processes involves hundreds of reactions, many of them unimolecular, such that detailed RRKM, PST,or ACM calculations are impractical. A pragmatic approach to estimate rate coefficients in the falloff region for such modeling effort is to use a parametric equation. The simplest one is the one-state strong collision Lindemann expression. Its deficiencies are well-known: the unimolecular process is a multilevel one9 and many bath gas molecules are weak colliders, which changes the shape of the plot of rate coefficient vs pressure.I0 Johnston” has used Kassel’sI2 s as a fitting parameter to take partial account of these effects. Troei3has suggested empirical corrections to the Lindemann falloff ( I ) Oref, 1. J . Phys. Chem. 1989, 93, 3465. (2) Marcus, R. A. Chem. Phys. Lett. 1988, 144, 208. (3) Wardlaw, D. M.; Marcus, R. A. Adu. Chem. Phys. 1988, 70, 231 and references therein. (4) Klippenstein, S. J.; Marcus, R. A. J. Phys. Chem. 1988.92, 3105 and references therein. ( 5 ) Klippenstein, S. J.; Kundkar, L. R.; Zewail, A. H.; Marcus, R. A. J . Chem. Phys. 1988, 89, 4761. (6) Pechukas, P.; Light, J . C. J . Chem. Phys. 1965.42, 3281. Pechukas, P.; Rankin, R.; Light, J. C. Ibid. 1966, 44, 794. (7) Qian, C. X.W.; Noble, M.; Nadler, 1.; Reisler, H.; Wittig, C. J. Chem. Phys. 1985, 83, 5573. (8) Quack, M.; Troe, J . Ber. Bunsen-Ges. Phys. Chem. 1974, 78, 240. Troe, J. J. Phys. Chem. 1984,88, 4375. (9) Robinson, P. J.; Holbrook, K. A. Unimolecular Reactions; Wiley: New York, 1972. Forst, W. Theory of Unimolecular Reactions; Academic Press: New York, 1973. (IO) Tardy, D. C.; Rabinovitch, B. S. Chem. Reu. 1977, 77, 369. ( 1 1 ) Johnston, H. S. J . Chem. Phys. 1952, 20, 1103. (12) Kassel, L. S. J. Chem. Phys. 1953, 21, 1093.

0022-3654/90/2094-0567$02.50/0

formula which compensate for its deficiencies by introducing a broadening function F with separate strong and weak collision corrections. Gardineri4 has used a Minkowski metric expression for the rate coefficient. The shortcomings of the above procedures are rooted in their empiricism. In I, an alternative expression based on basic understanding was developed. A comparison between exact strong collision RRKM theory and a “J equation” was made for a set of sample reactions. It was found in all cases that the agreement between RRKM and J equation calculations was good. However, for testing the model’s usefulness, master equation calculations for weak colliders should also be compared with the J equation. Such comparisons are reported in the present work. Theory

It was shown in I that the unimolecular rate coefficient for a strong collider can be represented by the expression -(k, k =

+ ko) + [ ( k , + ko)]’ + 4(J - 1)k,koli/’ 2(J- 1 )

(1)

where ko and k , are the pressure-dependent low- and high-pressure rate coefficients,respectively, and J is a parameter. At the pressure P3/2 where the reaction order is 312, k , = koP312, and the value of J is

where k3/2 is the value of the rate coefficient at P = P3/2 and k’ is the pressure-independent second-order low-pressure rate coefficient. In principle, eq 1 should be useful empirically to calculate rate coefficients in the weak collision regime as well, although the values of ko and J will differ from the strong collision case. Equation 1 is then modified for the weak collision case such that at P312 kWC =

- ( k , - kOWc)+ [ ( k ,

+ kOWC)2 + 4(J3/zWc- l)k,koWc]l/’ 2(J3/2WC - 1) (3)

J3/zWcis again evaluated by the use of eq 2. k3/IWC and kOWC are calculated by solution of the weak collision master equation with an assumed transition probability model and energy and angular momentum dependent RRKM rate coefficients. J as well as k , and ko are functions of temperature, internal energy of the (13) Troe, J. Ber. Bunsen-Ges Phys. Chem. 1983, 87, 101. (14) Gardiner, Jr., W. C. Proceedings of the 12th IMACS World Congress on Scientific Computations, Paris, 1988.

0 1990 American Chemical Society

568 The Journal of Physical Chemistry, Vol. 94, No. 2, 1990

Pawlowska and Oref

TABLE I: Reactions Studied

quadriiyclane 2.

norbornadiene

I--@ 4 Ar

cyclobutene 3.

cyclobutane 4.

A cyclopropene

butadiene

-

___)

5.

CH2DCHzCI

2= ethylene

4

H3C-CEC methyl acetylene H2C= C= CH2

allene

CHDCH2

+

HCI

CH2CH2 + DCI

6.

C2H5Br

C6H6

C2H4

+

HBr

molecule, threshold energy for decomposition, and average energy transferred in a weak collision. The temperature alone does not represent the full energetic situation of a molecule since the number of internal vibrational-rotational modes will determine the overall average internal energy, ( E ) . Thus, the parameter ( E ) / E ocan be used to unify calculation procedures for large and small molecules with varying values of Eo. We will explore the dependence of J on this ratio. Since, however, temperature is a pragmatic experimental variable, the J dependence on it will be explored as well. Because there is a large number of bath molecules with varying values of average energy transferred per down collision ( AE)d, it is reasonable to separate the reactant characteristics from those of the weak collider. The former are embodied in J3l2=and the latter in J312~’.J3/ZWcwill thus be given as a function of J 3 / 7 for various values of ( AE)d. In the next section rate coefficients as a function of pressure, temperature, and ( m ) d , for six reactants will be calculated by using eq 3 and compared with the solution of a weak collision master equation with exponential transition probability.

-

L

4

I

I

-

8

2

8

I

(

1

2

0

,

I

,

I

6

4

LOG P (Torr)

Figure 1. log k / k , vs log P in Torr for cyclobutene isomerization with He, Ar, and benzene as colliders. Full line, k calculated with the J equation; dashed line, exact RRKM calculations. In places where there is only one line, the results of the two calculations overlap.

I

He

C6H6

Results and Discussion

The six representative unimolecular reactions listed in Table

b-500K

I were investigated. Reactions 1, 2, and 4 are isomerizations and 3, 5, and 6 are bond fission reactions. RRKM calculations were

-6

performed under the strong collision assumption while in the presence of weak colliders the master equation9J0J5J6was solved by using the exponential transition probability model P ( A E ) = exp(-AE/a) with a = 250, 500, and 1200 cm-’ appropriate for He, Ar, and benzene bath gas, respectively.I0 AE is the energy transferred between reactant and bath molecules and a is the average energy transferred in a down collision ( AE)d. Figures 1 and 2 show the results of sample calculations of the dependence of the unimolecular rate coefficients of the cyclobutene and cyclobutane reactions on pressure for these three colliders at various temperatures. While the two reactants have almost the same number of internal modes and collision cross section, cyclobutene has only half the threshold energy for reaction of cyclobutane and represents an isomerization whereas cyclobutane represents a fission reaction. Table I1 gives the molecular constants used in the RRKM calculations of k and Table III lists the parameters used in the calculation of k from the J equation. The computational time of the master equation depends on the grain size,

-10

(15) Tzidoni, E.; Oref, I. Chem. Phys. 1984,84, 403. (16) Oref, 1.; Tardy, D. C . J . Chem. Phys. 1989, 91, 205. (17) Lishan, D. G.; Reddy, K. V.; Hammond, G . S.;Leonard, J. E. J Phys. Chem. 1988, 92, 656

- 6 - 4 - 2

c -800K d-IOOOK e-1200K f -1500 K g -2003K

0

2

4

6

8

LOG P (Torr)

Figure 2. log k / k , vs log Pin Torr for cyclobutane fission with He, Ar, and benzene as colliders. Full line, k calculated with the J equation; dashed line, exact RRKM calculations. In places where there is only one line the results of the two calculations overlap.

temperature, and energy-transfer model used and it is much longer than the J equation calculations which is practically instantaneous once the basic parameters are tabulated. It can be seen that the agreement between k(RRKM) and k, in Figures 1 and 2 is good. Figures 3-8 give the percent deviation of the rate coefficient k, calculated by using eq 3 (the J equation) from the value determined by master equation calculations. As can be seen, above P3/2 the deviation is only a few percent over the pressure and temperature range. For all cases studied it is less than &IO%. The stronger the collider, the smaller the deviation. Below P312 the deviation is small at low temperatures and larger at higher temperatures. The reasons for the deviations at higher temperatures will be discussed below. At the shock tube “reaction

The Journal of Physical Chemistry, Vol. 94, No. 2, 1990 569

Unimolecular Rate Coefficients 40t

He

r

40

20 0 -20

/ =

-401

t

-3

-.

Ar . ..

40

Q

20

a \

-

/”

-4%

-

0

g0

-20

‘6 H6

20

0 -20

-40r l

l

1

1

1

- 4 - 2

1

-40

c -800 K d -1000K e-I100 K

1

1

2

0

1

I

l

l

l

4 6 LOG P(Torr)

l

8

l

l

t

He

g--XX)OK I

I

I

- 4 - 2

IO

Figure 3. Percent deviation of k calculated by the J equation from k(RRKM) vs log P in Torr for quadricyclane isomerization in He, Ar, and benzene as inert colliders. The dots indicate the P3,2 point at each temperature.

40

1 4

l

e -1200K f -1500 K I

I

I

I

2

0

I

I

4

I

I

I

I

I

,

IO

8

6

LOG P(Torr) Figure 5. Percent deviation of k calculated by the J equation from k(RRKM) vs log P in Torr for cyclobutane fission in He, Ar, and benzene as inert colliders. The dots indicate the P,,, point at each temperature.

40t

e

He

n

20

0 -20

/

/

-401

-

e1

i a b

c

Ar

/

Ar

A

d e

x

0

c6 H6

ab

-40-

I

c d e

b--500 C--800

K K

d -I 000 K e--l500K f . -2000K

LOG P ( T o r r )

Figure 4. Percent deviation of k calculated by the J equation from k(RRKM) vs log P in Torr for cyclobutene isomerization in He, Ar, and benzene as inert colliders. The dots indicate the P,,2 point at each temperature.

window” which is in most cases (but not all) above P3,* the J equation is more than satisfactory. For example, the values of

Q -L

I

L

- 4 - 2

l

~

l

,

l

l

L

l

,

2500 K ,

,

l

,

4 6 8 1 0 LOG P(Torr1 Figure 6. Percent deviation of k calculated by the J equation from k(RRKM) vs log P in Torr for cyclopropene isomerization in He, Ar, and benzene as inert colliders. The dots indicate the P,,2 point at each temperature.

0

2

,

570 The Journal of Physical Chemistry, Vol. 94, No. 2, 1990

Pawlowska and Oref 40-

He

I

C?

H

-401-

a-298K b-mK

d -lOOOK e-1500K

, -60f-

L

L

/

I

l

- 4 - 2

l

1

0

1

1

2

1

1

I

,

f--2000K

I

4

4 LOG P(Torr)

Figure 7. Percent deviation of k calculated by the J equation from k(RRKM) vs log P in Torr for CHlDCH2Cl fission in He, Ar, and benzene as inert colliders. The dots indicate the P3/2 point at each

temperature.

8

I

8

- 4 - 2

8

I

2

0

I

I

I

'

1

1

4 6 LOG P (Torr)

'

1

~

i

10

8

a ) U:250 cm-' b) a.500 cm-I

3 0

9.

A

-

7

L

I

,

,

500

300

N ( t ) = C&,e-Ali'

,

,

700

,

I

900

,

,

IlOO

I

I

1300

,

1500

TEMPERATURE, K

r l

where a,, and X j are the eigenvectors and eigenvalues, respectively. Therefore, caution is in order when a comparison of the J equation and "exact" master equation solution is made, since the latter can deviate by as much as 50%. This is also true for strong collisions where the conventional RRKM expression fails at high temperatures.16 In the present work we compare J equation results with pseudo-steady-state master equation calculations; therefore an upper value for the temperature is required in order to make comparison meaningful. The value chosen is the temperature at which the average internal energy of the molecule divided by Eo is equal to 0.5 ( ( E ) / E o= 0.5). Table IV gives the temperature T* at which the above ratio is obtained for the six test reactions. The table also gives the number of internal modes, s, of each molecule and the effective number of modes serfdefined by the expression seff= ( E ) / R T * . As can be seen seffvaries between s / 2 and s/3.5. The larger the molecule and the smaller the value of Eo the smaller the value of seff Thus, quadricyclane with 39 normal modes and a low value of Eo of 33.6 kcal/mol has serfof only 1 1.5 while cyclopropene, a smaller molecule, with s = 15 and Eo = 42.5 kcal/mol has serfof 7 , about half the number of its internal modes. The dependence of P3/*on temperature can be obtained by / ~ temperature ) for weak and for strong plotting log ( ~ 3 / 2 T ' vs colliders. 0 3 1 2 is the collision frequency at P 3 / 2 . The plots are

8

Figure 8. Percent deviation of k calculated by the J equation from k(RRKM) vs log P in Torr for C2H5Brfission in He, Ar, and benzene as inert colliders. The dots indicate the P,,?point at each temperature.

k / k , at 1000 K at P3j2are 0.1 for cyclobutene and 0.01 for cyclobutane. As the temperature increases, the population shifts to higher values of internal energy. Eventually, a large part of the population is above Eo. In such a case the steady-state assumption fails. The master equation solution inludes a set of eigenvalues all of the same order of magnitude. The time-dependent population, N ( t ) , has the formI5

I

g -2500K

i

-2000K

9/ - 2 I2 0 0, K I I 6 8 IO

1

Figure 9. log ( w ~ / ~ T vs'temperature /~) for cyclobutene isomerization in the presence of strong and weak colliders with ( A E ) d = 250, 500, and 1200 cm-I.

a) ~ 1 . 2 5 0cm-l b) a=500 cm-' c ) a=1200cm-l d) Strong Collider

00006

00016

00326

00036

I / T IK-')

Figure 10. Logarithm of the second-order rate coefficient vs 1/T for cyclobutene isomerization in the presence of strong and weak colliders.

practically linear up to temperatures where (,!?)/Eo = 1.O. Figure 9 shows such a plot for cyclobutene isomerization. Table V gives the best fit parameters to a straight line equation of the form log

( W 3 / 2 ~ 1 / 2= )

QT+ b

(4)

Equation 4 and equation 5 below enable calculations of k b at any temperature. This is done in the following manner: 0 3 / 2 and

The Journal of Physical Chemistry, Vol. 94, No. 2, 1990 571

Unimolecular Rate Coefficients

TABLE It: Frequencies and Degeneracies of Molecules and Complexes Used in the RRKM Calculations Cyclobutane ( E , = 62.7 kcal/mol, log A(298 K) = 14.69, log A (1000 K) = 15.6) 1220 (4)

1000 (2)

926 ( I )

1000 ( I )

898 ( I )

850 ( I )

Quadricyclane” (Eo= 33.6 kcal/mol, log A(298 K) = 13.27) 2876 (2) 1250 (IO) 925 (6) 775 (4) 600 (4)

500 (2)

2980 (2) 900 (2)

2950 (2) 750 (2)

2890 (4) 741 (1)

1445 (4) 627 (1)

2980 (8) 600 (2)

1445 (4) 380 (1)

1260 (2) 350 (2)

1250 (2) 150 ( I )

3087 (4) 450 ( I )

2949 (2) 825 (4)

3087 (4) 451 ( I )

2949 (2) 1350 (2)

2876 (2) 800 (2)

3100 (6)

1650

Cyclobutene ( E , = 32.2 kcal/mol, log A(298 K) = 13.52) 1450 (3) 1150 (3) 1000 (6) 800 (2)

3100 (6)

1650

1450 (3)

1250 (6) 825 (4)

1150 (3)

1260 (4) 197 ( I )

Complex 1100 (2)

Complex 925 (5)

Complex 1000 (5)

775 (4)

600 (4)

740 (2)

500 (2)

700 (3)

800 (2)

700 (3)

Cyclopropene (Eol = 42.5 kcal/mol, log A(298 K) = 13.76, Eo2 = 48.5 kcal/mol, log A(298 K) = 13.53) 3152 ( I ) 2909 ( I ) 1653 (1) 1483 ( I ) 1105 ( I ) 905 (1) 996 (1) 815 ( I ) 3116 ( I ) 1043 (1) 1011 (1) 769 ( I ) 2995 ( I ) 1088 ( I ) 569 ( I )

(I) (I) (I) (I)

3004 ( I ) 974 ( I ) 3046 (1) 1021 (2)

2925 927 2932 957

3020 ( I ) 1446 ( I ) 243 ( I )

2985 ( I ) I380 ( I ) 294 ( I )

3055 ( I ) I380 ( I )

3045 ( I ) I300 ( I )

2814 794 2854 937

(I) (I) (I) (I)

2105 784 2355 862

(I) (I) (I) (I)

Complex 1744 ( I ) 473 ( I ) 1556 ( I ) 644 ( I )

1166 ( I )

I384 (1) 414 (1) 420 ( I ) 592 ( I )

1149 (1)

C2H5Br ( E , = 52.2 kcal/mol, log A(298 K = 13.40) 2975 ( I ) 2963 ( I ) 2927 ( I ) 562 ( I ) 1065 ( I ) 244 ( I ) 1023 ( I ) 1446 ( I )

3060 ( I ) 1005 ( I )

3040 ( I ) 995 ( I )

Complex 2200 ( I ) I010 (1)

395 ( I ) 990 (1)

1028 ( I )

960 ( I ) 780 ( I )

I456 ( I ) 1185 ( I )

1385 ( I ) 807 (1)

1400 ( I ) 479 ( I )

200 ( I )

CH2DCH2Cl (E,, = 55.6 kcal/mol, log A(298 K) = 12.65, EO2= 56.9 kcal/mol, log A(298 K) = 12.97) 2940 (4) 2160 ( I ) 1340 (4) 1270 (2) 960 (3) 720 (2) 330 ( I ) 200 ( I ) 3000 (3) 3000 (4) a

2200 (2) 2088 ( I )

1380 (2) I380 ( I )

1115 ( I ) 1100 ( I )

Complex 960 (5) 960 (4)

850 ( I ) 250 (3)

820 ( I ) 750 ( I )

645 ( I ) 570 ( I )

403 ( I ) 501 ( I )

From ref 17.

k, are evaluated at the desired temperature; these values are inserted in eq 5 and k b is obtained. As mentioned before, qI2 is a function of the substrate molecule only and independent of p and Q of the bath molecule. P312,however, is a function of p and u through its relation to w312; e.g., w312 = m 2 ( 8 k T / Figure 10 shows a plot of log k’b vs T I for cyclobutene and various CY over a wide temperature range. k’b is defined by k‘b = k , / ~ ~ / Since ~ . both k, and w312are independent of p and u, so is k”,,. It is dependent on CY as expected and as can be seen from Figure 10. The collisional efficiency, 0, decreases with the temperature as can be seen from the nonlinearity of the plots in Figure 10 as the temperature increases. The linear behavior at low temperatures is very helpful in the usage of the J equation as well as in the general application of unimolecular theory. In the latter case it is possible to estimate P312 in the falloff region at any temperature. This is an important aid in shock tube and combustion work at high temperature where the degree of falloff is not always known. At P312

and the common definition of the low- and high-pressure unimolecular rate coefficients are

where k(E) is the energy and angular momentum dependent rate coefficient and B(E) is the Boltzmann distribution of internal energies.

The last term denotes Boltzmann-averaged k(E). The expression in the numerator is simply the Arrhenius expression. Both numerator and denominator are independent of p and u of the bath molecule hence w , / ~ is independent of the two. The temperature dependence of J312 was given in I by the expression Ce-E/RT

J3/2=

2

(7l)

(7)

where C, E, and m are constants. This equation gives very good fit up to values of (E)/Eo= 0.5. This is shown in Figure 11 where J312 for weak and strong collisions with cyclobutene is plotted vs temperature. The values of the constants for the six molecules investigated are given in Table VI for a strong collider and for He, Ar, and benzene as weak colliders. Equation 7 appeals to the “connoisseur” since it is rooted in basic principles.’ However, it fails badly for values of ( E ) / E o >> 0.5. As discussed before, there is a good physical reason for it, namely the pseudosteady state is not obtained at high tem-

512

Pawlowska and Oref

The Journal ofPhysica1 Chemistry, Vol. 94, No. 2, 1990

TABLE 111: Summary of Calculated Parameters Used in the J Equation C yclobutane

k, k , IC

J3IP

1000 K 2.5+1 7.4-5 128.9

1200 K 7.7+3 3.2-3 169.6

1500 K 2.5+6 6.8-2 172.4

2000 K 8.5+9 5.1-1 87.0

1.3+7 1.9-6 5.7-2 278.7

1.6+8 1.4+ 1 4.8-5 4.9-2 370.7

5.2+9 5.0+2 4.8-4 5.2-2 332.8

1.1+12 1.3+5 7.4-4 9.2-2 97.9

4.5+6 7.0-1 5.6-6 6.1-2 240.7

5.8-1-7 9.9 1.3-4 5.6-2 286.6

2.1+9 4.0+2 I .2-3 6.7-2 195.6

4.3+11 9.4+4 2.0-3 1.1-1 57.1

1.5+6 1.4-1 1.7-5 6.7-2 192.4

I .6+7 1.6 4.6-4 6.0-2 247.4

5.0+8 5.6+1 4.9-3 6.4-2 21 5.9

8.4+10 1.1+4 1 .o-2 1.2-1 57.6

1000 K

1100 K

3.2+6 3.7-2 14.1

1.5+7 9.1-2 15.0

1200 K 5.2+7 1.7-1 14.9

1300 K 1.5+8 2.8-1 13.8

5.4+9 4.3+2 5.9-4 1.6-1 29.2

1.4+10 1.2+3 1.o-3 1.6-1 28.9

3.5+10 3.0+3 1.5-3 1.6-1 26.5

8.3+10 7.4+3 I .8-3 1.7-1 22.5

1.8+9 2.7+2 1.8-3 1.6-1 25.8

4.4+9 7.2+2 3.3-3 1.6-1 26.2

1.1+10 1.8+3 4.8-3 1.7-1 24.7

2.5+10 4.4+3 6.1-3 1.8-1 21.6

5.1+8 4.6+1 6.3-3 1.8-1 21.7

1.2+9 1.1+2 1.2-2 1.7-1 23.3

2.6+9 2.6+2 2.0-2 1.7-1 23.3

5.7+9 5.8+2 2.7-2 1.8-1 21.6

3.6+4 1.6-4 13.5

1500 K 7.4+7 3.6-2 19.8

2000 K 3.3+9 2.6-1 17.6

2500 K 3.1+10 5.7-1 10.2

6.9+9 9.7+2 5.3-6 1.5-1 29.7

2.0+ 1 1 3.5+4 3.7-4 1.3-1 41.9

3.0+12 6.0+5 1.1-3 I .5-1 30.0

2.5+13 5.5+6 1.3-3 2.0-1 16.1

2.4+9 6.0+2 1.5-5 1.7-1 24.5

6.5+10 2.0+4 1.1-3 I .4-1 37.0

9.4+11 3.3+5 3.5-3 1.6-1 27.8

7.5+12 2.9+6 4.2-3 2.1-1 14.1

8.6+8 1.0+2 4.2-5 1.8-1 19.4

1.7+10 2.5+3 4.3-3 I .5-1 32.2

2.0+ 1 1 3.4+4 1.6-2 I .6-1 28.1

1.5+12 2.7+5 2.1-2 2.0-1 15.2

298 K 4.9-330 1.7-35 8.2

500 K 6.9-14 3.6-17 21.4

800 K 5.4-3 1.2-7 73.4

1.1+3 4.8-5 4.4-36 2.7-1 7.2

1.5+4 8.4-4 4.6-18 1.5-1 33.3

9.1 +5 6.4-2 5.9-9 7.3-2 161.8

6.2+2 5.3-5 7.8-36 2.9-1 5.9

7.1+3 7.8-4 9.8-18 1.6-1 26.4

3.4+5 4.8-2 1.5-8 8.0-2 133.2

4.1+2 2.0-5 1.2-35 3.1-1 5.0

3.7+3 2.4-4 1.9-1 7 1.8-1 21.3

1.3+5 1.1-2 4.0-8 8.8-2 106.3

298 K 2.7-9 1.5-17 3.2

500 K 1.6-1 4.0-8 4.6

800 K

5.1+4 2.1-3 10.0

7.2+6 3.1-1 3.7-18 4.1-1 2.0

3.7+7 2.1 4.4-9 2.8-1 6.8

7.2t8 5.0+1 7.1-5 1.7-1 22.7

4.0+6 3.4-1 6.7-18 4.3-1 1.7

1.6+7 1.8 9.9-9 3.0-1 5.4

2.5+8 3.5+1 2.0-4 1.9-1 18.9

2.6+6 1.3-1 1.0-17 4.5-1 1.5

8.3+6 5.3-1 2.0-8 3.3-1 4.3

9.1+7 7.3 5.6-4 2.0-1 15.0

298 K 9.3719 9.6l26 3.4

500 K 7.9-6 3.6-13 4.2

800 K

1000 K

1.3+2 1.5-6 9.2

3.7+7 2.9 2.5-26

1.6+8 1.6+1 4.8-14 2.8-1 6.5

1.6+9 2.0+2 8.6-8 1.9-1 19.1

2.1+7 2.8 4.4-26 4.1-1 2.1

7.6+7 1.3+ 1 1 .O-13 3.0-1 5. I

6.1+8 1.3+2 2.2-7 2.0-1 15.4

1.4+7 8.9-1 6.7-25 4.3-1 1.8

4.1+7 3.4 1.9-1 3 3.3-1 4.1

2.5+8 2.6+1 5.4-7 2.2-1 11.9

In He O3/2 p3/2

k bwc k3/i/kJ3/2wc

1.o

In Ar w3/l

p3/2

K b"C k3/2/kJ3/2wc

In C6H6 O3/2

p3/2

k bwc k3/2/kJ3/2wc

Cyclobutene

k k? J3/2Pc

In He O3/2

p3/2

k bWc k3/2f k -

J3/2wc

In Ar @3/2 p3/2

k b"' k3/2/kJ3/2wc

In C&6 O3/2 p3/2

k b"' k3/2k-

J3/2wc

Cyclopropane

k, k b" J3/2K

In He a312 p3/2

k b"'

In Ar w3/2 p3/2

k bWc k3/2/kJ3/2wc

In C6H6 O3/2 p3/2

k bWc k-

k3/1/

J312wc

The Journal of Physical Chemistry, Vol. 94, No. 2, I990 573

Unimolecular Rate Coefficients

TABLE I11 (Continued) Quadricyclane 600

1.3-12 3.6-15 5.1

500 K 2.2-2 6.6-6 21.2

1.7+3 7.3-5 7.8-16 3.1-1 5.1

4.3+4 2.4-3 5.3-7 1.4-1 35.6

2.8+5 1.7-2 2.7-5 1.1-1 64.2

w3/2 p3/2

9.1+2

k'owc k3/2/k.

1.5-15 3.3-1 4.2

1.7+4 2.1-3 1.3-6 2.3-1 29.0

1.0+5 1.3-2 7.4-5 1.2-1 54.1

5.6+2 3.2-5 2.3-15 3.5-1 3.5

8.0+3 5.9-4 2.8-6 1.7-1 22.8

4.0+4 3.2-3 1.9-4 1.3-1 42.9

298

k. k b" J3/2"

K

K

1000 K 1.1+6 3.9-1 42.4

1100 K

4.9+2 9.6-3 54.2

800 K 1.3+4 6.5-2 53.2

1.9+6 1.3-1 2.5-4 9.7-2 85.3

1.2+7 8.4-1 1.1-3 9.0-2 100.8

5.3+8 4.2+1 2.0-3 1.3-1 43.1

3.0+9 2.5+2 2.7-3 1.6-1 27.0

6.5+5 9.4-2 7.5-4 1 .0-1 75.4

4.3+6 6.6-1 3.0-3 1.0-1 77.3

1.6+8 2.7+1 6.8-3 1.4-1 39.2

8.6+8 1.5+2 6.1-3 1.7-1 23.2

2.2+5 1.9-2 2.2-3 1.1-1 62.1

1.2+6 1.1-1 1.1-2 1.1-1 71.3

3.2+7 3.4 3.2-2 1.3-1 45.5

1.7+8 1.9+1 3.2-2 1.6-1 25.1

700 K

7.6 5.8-4 37.0

5.3+6 5.9-1 31.7

In He w3/2 p3/2

k bwc k3/2lk. J3/2wc

In Ar 8.5-5

J312wc

In C6H6 W3/2 p3/2

k bWc k3lilkJ3/2wc

CHZDCHlCl k.

k t WC J3/2r

298 K 4.9-29 5.7-33 3.5

500 K 2.6-12 1.0-16 5.8

800 K 6.7-3 4.0-8 17.1

1000 K 9.2 1.6-5 31.5

3.1+4 1.3-3 1.5-33 3.7-1 3.0

1.8+5 1 .0-2 1.4-17 2.3-1 10.8

2.7+6 2.0-1 2.3-9 1.4-1 39.6

1.7+7 1.3 5.5-7 1.1-1 65.9

1.7+4 1.5-3 2.6-33 3.9-1 2.5

8.6+4 9.8-3 3.0-17 2.5-1 8.6

1.1+6 1.6-1 6.0-9 1.5-1 32.2

6.0+6 9.7-1 1S-6 1.2-1 54.8

1.1+4 6.0-4 4.0-33 4.0-1 2.2

4.6+4 3.1-3 5.6-17 2.7-1 6.9

4.6+5 3.9-2 1.5-8 1.6-1 25.4

2.1+6 2.0-1 4.3-6 1.3-1 43.7

1100 K 1.3+2 1.2-4 39.1

1300 K 7.6+3 2.3-3 50.7

1500 K 1.5+5 1.5-2 64.2

2000 K 1.9+7 1.9-1 56.0

2500 K 6.2+7 3.0-1 47.1

2.1+8 1.9+1 3.7-5 9.4-2 93.5

9.7+8 9.3+1 1.6-4 9.3-2 95.7

2.3+10 2.6+3 8.2-4 1.1-1 64.3

6.6+10 7.8+3 9.3-4 1.3-1 46.2

6.9+7 1.3+1 1.1-4 1.0-1 80.4

3.2+8 6.3+1 4.8-4 9.9-2 83.0

8.4+9 1.9+3 2.2-3 1.3-1 46.2

2.5+10 6.1+3 2.4-3 1.5-1 33.1

2.0+7 2.2 3.7-4 1.1-1 66.9

8.5+7 9.9 1.8-3 1.0-1 72.4

1.9+9 2.5+2 1.0-2 1.2-1 51.8

5.3+9 7.4+2 1.2-2 1.4-1 35.9

In He O3/2

p3/2

k bwc k,p/k, J3/2wc

4.0+7 3.3 3.2-6 1.0-1 77.9

In Ar w3/l p3/2 k '0"'

k3/2/k, J3/ZWc

1.4+7 2.3 9.4-6 1.1-1 65.6

In C6H6 w3/2

p3/2 k '0"'

k3/2/k. J3/2wC

4.6+6 4.6-1 2.8-5 1.2-1 52.8

C,HIBr k, k bW J3/2r

298 K 6.8-26 7.5-31 2.8

500 K 2.8-10 1.3-1 5 4.0

800 K 1.4-1 1.4-7 9.4

3.5+5 1.4-2 2.0-3 1 3.7-1 2.8

1.6+6 9.3-2 1.8-16 2.7-1 6.9

1.7+7 1.4 8.1-9 1.7-1 22.5

2.0+5 1.7-2 3.5-31 4.0-1 2.3

7.0+5 9.5-2 3.8-16 3.0-1 5.4

6.6+6 1.2 2.1-8 1.9-1 18.1

1.3+5 6.9-3 7.5-31 4.2-1 1.9

4.0+5 3.1-2 1.3-1 5 3.2-1 4.3

2.7+6 3.0-1 1.4-7 2.1-1 14.1

1000 K 1.1+2 3.8-5 15.4

1500 K 7.5+5 2.3-2 27.6

2000 K 6.1+7 2.3-1 24.1

2500 K 7.0+8 5.6-1 11.9

8.3+7 7.9 1.3-6 1.4-1 37.3

3.4+9 4.3+2 2.2-4 1.2-1 57.2

6.7+10 1.0+4 9.2-4 1.4-1 39.6

7.2+11 1.3+5 9.7-4 2.0-1 15.4

3.0+7 6.2 3.6-6 1.5-1 30.9

1.1+9 3.1+2 6.8-4 1.2-1 49.8

2.4+10 8.0+3 2.6-3 1.5-1 29.7

2.5+11 1.0+4 2.8-3 2.1-1 12.6

2.9+8 5.0+1 2.3-2 1.3-1 43.6

5.1+9 1.1+3 2.3-1 1.5-1 33.5

4.9+10 1.2+4 5.6-1 2.3-1 11.8

In He w3/2

p3/1

k bwc k3/2/kJ3/2wc

In Ar w3/1

p3/2

k bwc k312lk. J3/2w

In C6H6 w3/2

4

2

k bWc

k3/2/kJ3/2wc

04.9-33 = 4.9 x 10-33,

1.0+7 1.3 3.8-5 1.7-1 24.4

574

The Journal of Physical Chemistry, Vol. 94, No. 2, 1990

Pawlowska and Oref

TABLE IV: Threshold Energies, Number of Modes, and Temperatures at Which ( E ) / E , " = 0.5 quadricyclane cyclobutene cyclobutane c yclopropene Eo, kcal/mol 33.61 32.15 62.7 I 42.5 .sb 39 24 30 15 11.5 8 12 7. I sed T *.d K 7 30 1000 1300 1500

CH2DCH2CI 55.6 18 9.1 1550

C,H5Br 52.2 18 8.7 1500

a ( E ) , average internal energy; Eo, threshold energy. b s , a number of internal modes. cs,fhnumber of effective internal modes. dTemperature at which ( E ) / E o = 0.5.

TABLE V: Parameters of Linear Fit of the Quation log W , / Z T '= / ~aT + 1,for Strong Collider and a = 250, 500, and 1200 cm-I a quadricyclane cyclobutene cyclobutane cyclopropene CH2DCH2CI C2H5Br 250 (a)a 8.25 f 0.016 4.34 f 0.020 5.57 f 0.029 2.88 f 0.027 3.56 f 0.028 3.11 f 0.026 250 (b)" 1.93 f 0.012 6.82 f 0.021 2.85 f 0.034 8.24 f 0.039 4.99 f 0.037 4.11 f 0.038 500 (a) 7.93 f 0.022 4.12 f 0.016 5.49 f 0.023 2.77 f 0.023 3.49 f 0.021 3.04 f 0.021 500 (b) 1.71 f 0.016 6.58 f 0.016 2.55 f 0.017 7.95 f 0.033 4.67 f 0.028 5.80 f 0.030 I200 (a) 7.29 f 0.022 3.69 f 0.013 5.18 f 0.018 2.54 f 0.018 3.23 f 0.017 2.80 f 0.017 1200 (b) 1.69 f 0.017 6.50 f 0.014 2.41 f 0.021 7.78 f 0.026 4.51 f 0.022 5.64 f 0.024 2.56 f 0.018 2.17 f 0.020 4.37 f 0.034 1.99 f 0.019 2.98 f 0.010 sc (a) 6.12 f 0.013 4.36 f 0.024 5.62 f 0.028 2.45 f 0.039 7.66 f 0.027 1.78 f 0.010 6.40 f 0.010 sc (b) First number (a) is multiplied by IO' and second number (b) by IO5. 40

TABLE VI: Values of the Constants in Eq 7 for Strong and Weak Colliders a,cm-' In c E I R , K m Inc E I R , K m quadricyclane cyclobutene -3.86 2.15 0.83 394 -0.46 250 -0.14 -5.29 -98.6 -1.02 -2.63 251 -0.80 500 -1.16 -6.43 -187 -4.68 116 -1.07 I200 -1.41 -5.87 -301 -1.03 -7.32 -134 sc 250 500 1200 sc

-5.20 -5.13 -6.20 -6.63

cyclobutane 54.4 -1.17 58.8 -1.15 -24.4 -1.28 -1.29 -182

250 500 1200 sc

-1.81 -2.57 -3.57 -6.97

CH2DCHZCI 190 0.60 133 0.69 50.4 0.81 -290 -1.23

cyclopropene -1.87 90.0 -0.54 -2.76 23.0 -0.65 -3.77 -57.6 -0.77 -0.72 -3.63 -171 -3.16 -3.92 -4.98 -5.40

C2H5Br 17.6 -36.8 -122 -254

200

400

600

0

sc

0

000

o

a,=250 cm"

a2:500cm-' A

,

,

a3=1200cm-3 sc 1

1

I

400

1

1

'

600

"

'

800

0

'

.

1000

1

,

1200

1400

TEMPERATURE, K

Figure 12. J312vs temperature for cyclobutene isomerization in the presence of strong and weak colliders. Points denote values of J3/2given in Table 111 and lines are calculated by the Gaussian model (eq 8). TABLE VII: Values of tbe Constants in Eq 8 for Strong and Weak Colliders a,cm-' A, K-2 E T,,K A , K-2 E T I ,K cyclobutene quadricyclane -4.76 3.39 1.05 -12.8 4.54 0.774 250 3.27 1.09 -12.6 4.37 0.780 -4.41 500 -11.7 4.25 0.805 -3.76 3.16 1.16 1200 1.28 -1.66 2.70 sc -8.51 4.04 0.830

ci3 1200 cm-' i

Gaussian Model

200

-0.73 -0.82 -0.95 -0.97

eq 7

I

I

io00

250 500 1200 sc

cyclobutane -3.59 6.03 1.35 -3.75 5.71 1.31 -3.66 5.61 1.34 -2.48 5.25 1.43

cyclopropene -1.59 3.86 1.62 -1.59 3.72 1.64 3.60 1.70 -1.50 -0.99 2.99 1.68

250 500 1200 sc

CH2DCH2CI -2.30 4.74 1.53 -2.42 4.54 1.50 -2.23 4.41 1.56 -1.53 4.19 1.72

-1.81 -1.83 -1.81 -1.23

TEMPERATURE, K

Figure 11. J3,2 vs temperature for cyclobutene isomerization in the presence of strong and weak colliders. Points denote values of J312and lines were obtained from eq 7 .

peratures. An empirical Gaussian expression which yields good agreement with master equation calculations is

where A, T , and B are empirical parameters which define the function. The agreement is very good for all molecules and for all types of colliders used. We show a sample in Figure 12 where J3/* is given for strong colliders as well as for weak colliders with cy = 250, 500, and I200 cm-' interacting with cycobutene. The values of A , T, and B are given in Table VI1 for all six molecules studied. Examination of Tables VI and VI1 show a systematic behavior of the various constants in eq 7 and 8. However, instead

C2HSBr 4.13 3.93 3.83 3.29

1.59 1.59 1.62 1.69

TABLE VIII: Values of the Constants in Eq 11 and 12 for the Six Compounds Studied cyclobutane cyclobutene quadricyclane C2H5Br CH2DCH2CI cyclopropene

-0.064 -0.055 -0.00034 -0.0431 -0.029 -0.056

-0.139 -0.132 -0.023 -0.104 -0.074 -0.127

-0.159 1.245 -2.603 2.845 -0.182 -0.926 -2.10 2.752 -0.081 -0.640 -1.552 2.223 1.140 -2.475 2.923 -0.148 -0.1 12 -0.914 -2.01 1 2.492 -0.164 1.041 -2.269 2.733

of belaboring the systematics and trends of the constants as a function of temperature, it is important to realize that eq 7 and

The Journal of Physical Chemistry, Vol. 94, No. 2, 1990 575

Unimolecular Rate Coefficients I

i-

I

c =cyclobutane

~

d :C2HgBr

0-I-

e =cyclopropene

-2-

f = cyc lobutene

-3-

1

8

ZOO

He

400

1

1

1

600 (I

1

"

800 ( c d )

Figure 13. A ( a ) and B ( a ) defined by eq 9 vs

I

I

1

loo0

1200

-71, LY

b=500K

-4-5-6-

c=800K d =IOOOK ,

,

,

-4 - 3 - 2 - 1

for various reactants.

,

,

,

,

0 I 2 3 LOG P(Torr)

, 4

,

5

Figure 15. Rate coefficient vs pressure for cyclobutene isomerization in the presence of weak colliders at various temperatures. The full line was obtained from eq 1, using values of J,/2wcgiven in Table 111; the points were calculated from eq 1 by using J3/2wcgiven by eq 9-1 1. I

a1= 250 cm-l

He

a2=500 cm-'

200

400

800

600

1000

1200

TEMPERATURE , K

Figure 14. J,,2wc/J!lzw vs temperature for cyclobutane fission in the presence of weak colliders. Lines were calculated by using eq 9-1 1, while points denote the J3/2Wc/J3/2P values given in Table 111.

Ar

2

-O 1- t

2 -21 8 describe two effects, the dependence on a and on internal energy or temperature. For the presentation given here to be useful it is necessary to represent the weak collision effect, as embodied in the value of J312wc,as a function of two independent parameters. Thus J312scis found once and if the dependence is known, J312wc can be calculated for every combination of reactant and bath molecule at any desired temperature. One such function is

-J 3 /-2 w c

1000

(9)

J3/2sc

where A ( a ) and B ( a ) are given by

g -

-3

-41

-5

::

c6 H6

d = W K e =1203K

-5b I 2 3 4 5 -4-3-2-1

0

LOG P (Torr)

Figure 16. Rate coefficient vs pressure for cyclobutane fission in the presence of weak colliders at various temperatures. The full line was obtained from eq 1, using values of J3/2wcgiven in Table 111; the points were calculated from eq 1 by using J,12wc given by eq 9-1 1.

uirbi, and ci are constants evaluated in the following way. J312~' and J3$ are evaluated by solution of the master equation for rate coefficients for weak and strong colliders. These are used to find J3/2 from the J equation. The ratio J312wc/J312* is fit to eq 9 and the A ( a ) and B(a) are used in eq 10 and 11 to finally find ai, b , and ci. Figure 13 shows the dependence of A ( a ) and B(a) on a. The values of the constants are given in Table VIII. Figure 14 calculated from shows sample comparison of the ratio J312wc/J3/2x eq 9 as a function of a and T. Figures 15 and 16 demonstrate the tit for cyclobutane. A comparison is made among k calculated directly from a solution of a master equation, k calculated with the help of the J equation (eq 3 ) , and k calculated from the J equation when the J312wcwas evaluated by the use of eq 9-1 1. The agreement is as good as can be expected. The results for the other molecules studied are similar to those shown. The values of the constants in eq 9 and 10 are given in Table VIII. Equation 9 needs

further elaboration. 5312, a function of the temperature and number of internal vibrational-rotational modes,depends on some average internal energy ( E ) as discussed before in terms of the ratio ( E ) / E o .Division of ( E ) by Eo is required because similar molecules, e.g., cyclobutene and cyclobutane, have vastly different Eo and thus different k and 5312 values. For cyclobutane J3/2% at 1200 K is 170 while for cyclobutene at the same temperature it is 15-more than an order of magnitude difference. It is thus desirable to consider the correlation between J312"' and ( E ) / & . This is given as

It has the same form as eq 9 since T is related to the average energy. Calculation of J3/zWcby this equation yields very good results. Figure 17 shows the dependence of D ( a ) and F ( a ) on a for the six compounds studied. As can be seen, D ( a ) is inde-

J . Phys. Chem. 1990, 94, 576-581

576

I

C

I I

i-b,e I

'd

'

E -0004: I

~

t

-0010-

5

indicate that the contribution of the average energy and a to J3/2wc/J3/2sc are in most cases additive. Sample calculations which show the dependence of J312wc/J3/2s on ( E )are depicted in Figure 18 for cyclobutene isomerization. We did not pursue the utilization of eq 12 further since from a pragmatic point of view eq 9 is preferable because the temperature is a measurable parameter while ( E ) has to be calculated explicitly from the normal modes of the reactant molecule.

Conclusion

I Ob

o = cyclobutene b: cyclobutane

I

c= quodricyclane d * C2H,Br e: CHIDCHPCl

I

1

u

a (c m-'

Figure 17. D(a) and F ( a ) defined by eq 12 vs a for various reactants 26t

(I,

~ 2 5 Cm-l 0

a2 =500 cm-'

06 02

LU-_L_L__L---L-

007

017

027

037

047

(E)/Eo

Equations are given by which unimolecular rate coefficients can be calculated under a variety of temperature, pressure, and ( AE)dconditions. The results are compared with master equation calculations. The agreement is good and deviations are less than a few percent for the pressure and temperature ranges where the reaction order is larger than 3/2. The steps to follow in calculations of rate coefficients are as follows: (a) kfoand k , are calculated by using eq 5 and the Arrhenius expression, respectively. (b) q I 2is found from the relation ~ 3 1 2= k,/kfo. (c) k3j2is calculated from the solution of the master equation. (d) 5 3 1 2 is found from k , and k 3 / 2by eq 2. (e) Once J3/2is known at a few temperatures it can be calculated from the parameters of eq 7 or 8. (f) J312wc is found from the value of J3/2Scand the parameter of eq 9 which are found from eq 10 and 11. (g) The value of the rate coefficient at any temperature and in eq 3. Finally pressure is found by inserting ko, k,, and J312wc an estimate of the degree of falloff at any temperature can be JjlZXland k b are independent found with the help of eq 4. J312wc, of the molecular weight and cross section.

Figure 18. J312wc/J312g vs ( E ) / & for cyclobutene isomerization in the presence of weak colliders.

Acknowledgment. This work was supported by the US-Israel Binational Science Foundation and the Fund for Promotion of Research at the Technion.

pendent of CY except for cyclobutene and cyclopropene which are small molecules with low Eo. For the rest of the compounds studied D varies between 3 X IO4 and 1.4 X F(cY)does vary with CY by about 25% over the range of CY studied. These results

Registry No. CH2DCH2C1,23072-56-2; C2H5Br,74-96-4; quadricyclane, 278-06-8; cyclobutene, 822-35-5; cyclobutane, 287-23-0 cyclopropene, 2781-85-3.

Fluorescence Quenching Studies of Cyclodextrin Complexes of Pyrene and Naphthalene in the Presence of Alcohols Gregory Nelson7 and Isiah M. Warner* Department of Chemistry, Emory University, Atlanta. Georgia 30322 (Received: April 20, 1989)

Fluorescencequenching of naphthalene- and pyrene-cyclodextrin complexes using iodide ion is examined. A model describing the fluorescence intensity quenching of naphthalene:@cyclodextrin complexes is proposed and evaluated. This model assumes that cyclodextrin-complexed naphthalene is essentially unquenched by iodide ion. This was found to be true for naphtha1ene:cyclodextrin complexes in the presence of benzyl alcohol. Pyrene fluorescence lifetime data are presented which support the conclusion that cyclodextrin inclusion complexes in the presence of alcohols are quenched to a significantly lesser extent than in the absence of alcohols.

Introduction Previous studies have demonstrated the change in fluorescence properties of pyrene in the presence of alcohols upon cycl&xtrin (CD) c~mplexation.'-~Both fluorescence intensity and lifetime are significantly enhanced for pyrene:CD complexes in the presence of certain alcohols such as tert-butyl alcohol.2 Such *Author to whom correspondence should be addressed. 'Present address: Tennessee Eastman Co., Kingsport, TN 37662.

0022-3654/90/2094-0576$02.50/0

effects may be examined to provide information about the comPlexation Phenomena. The reduction of PYrene quenching by CD complexation has also been establi~hed.~The rate of quenching (1) Nelson,G.; Patonay, G.; Warner, 1. M. Anal. Chem. 1988.60, 274. (2) Nelson, G.; Patonay, G.; Warner, I. M. J . Incl. Phenom. 1988,6,277. (3) Patonay, G.; Fowler, K.; Nelson, G.; Warner, I. M. Anal. Chim. Acta 1988, 207, 25 1 . (4) Nelson, G.; Neal, S . L.; Warner, I. M. Spectroscopy 1988, 3, 24. (5) Edwards, H.; Thomas, J. K. Curbohydr. Res. 1978, 65, 173.

0 1990 American Chemical Society