1268
TERENCE K. KETTAND DONALD K. ANDERSON
Ternary Isothermal Diffusion and the Validity of the Onsager Reciprocal Relations in Nonassociating Systems by Terence K. Kett and Donald K. Anderson Department of Chemical Engineering, Michigan Stale Ungversity, East Lansing, Michigan
(Received July 8 9 . 1968)
Ternary liquid diffusion in the syatems dodeoane-hexadecane-hexane and toluene-chlorobenzene-bromobenzene is studied. Experimental diffusion data for both systems are used to verify the Onsager reciprocal relations within experimental error and to demonstrate the applicability and usefulness of hydrodynamic theory to multicomponent diffusion.
Introduction Miller1*2has derived a necessary and sufficient condition for which Onsager's reciprocal relation should be valid for liquid diffusion in a ternary system of constant volume. Using this condition as a basis of testing ~ Onsager's relation, Miller,1--3 Gosting, et ~ 1 . , ~ -and Toor, et ~ 1 . , ~ have - ~ studied various miscible ternary systems and found that within the limits of their experimental accuracy the condition has been satisfied. However, there has always been some doubt because of the limitations of the experimental methods employed. In the preceding paper,'o the hydrodynamic theory of Hartley and Crank" was applied to multicomponent diffusion and the results indicate that Onsager's reciprocal relation is valid for nonassociating systems. However, experimental evidence can be used to check both the hydrodynamic theory and the reciprocal relation. Such evidence is presented in this paper for the ternary systems dodecane-hexadecane-hexane and toluene-chlorobenzene-bromobenzene.
F2 a In a1
[C1C2(2 - ;)( x)
J2 = 9
c1
+ cl(
ax
u1
9
c1
(1 - VzC2) I
+ c2(
c~)(a;huz)cJ"
62
ax
By comparing eq 1 and 2, it can be seen that
Theory Equations for Lij and Dij. Baldwin, Dunlop, and GostingI2 have generalized Fick's law to include interaction effects. For a ternary system, these equations are J~ = - ~ l l ( a c l / a x ) - D , ( ~ c ~ / ~ x ) (1) J~ = -Dzx(acl/ax) - D ~ ~ ( ~ c ~ / ~ z ) where Dij is the diffusion coefficient and J i is the molar flux of species i with respect to the volume-fixed plane. Using the hydrodynamic approach, the following equations are obtained for the fluxes
(3)
(1) D. G. Miller, Chem. Rev., 60, 15 (1960). (2) D.G. Miller, J. Phys. Chem., 63, 570 (1959). (3) D.G. Miller, ibid., 69, 3374 (1965). (4) P. J. Dunlop and L. J. Gosting, J. Amer. Chem. Sac., 77, 5238 (1955). (5) H.Fujita and L. J. Gosting, ibdd., 78, 1099 (1956). (6) H.Fujita and L. J. Gosting, J. Phys. Chem., 64, 1256 (1960). (7) J. K. Burchard and H. L. Toor, ibid., 6 6 , 2015 (1962). (8) F. 0. Shuck and H. L. Toor, ibid., 67, 640 (1963). (9) H.T. Cullinan, Jr.. and H. L. Toor. ibid., 69, 3941 (1965). (10) T.K. Kett and D. K. Anderson, ibid., 73, 1262 (1969). (11) G.S. Hartley and J. Crank, Trans. Faraday Soc., 45, SO1 (1949). (12) R. L. Baldwin, P. L. Dunlop, and L. J. Gosting, J. Amer. Chem. Soc., 77, 5235 (1955). The Journal a j Physical Chemistry
TERNARY ISOTHERMAL DIFFUSION
(1 -
+ c2(
P2C2) +
C
1269
~
y
l
n
a2
has shown that when the two independent fluxes are written as Ji = LuYi L12Y2 J2
=:
+ L12Y1 + L2zY2
(4)
experimental phenomenological coefficients, Lij, one must know Pi, Ci, D i , and api/dC. In general, the last is the most difficult to obtain since multicomponent activity data are required. It is encouraging to note, yhowever, that more ternary activity data are becoming available. The chemical potential of the ith species can be written as p i = pio
+ RT In ai
(9)
where p+O is a function of T and P, R is the gas constant, and ai is the activity. Hence, at constant T and P
where the Lij are the phenomenological coefficients and the Yj are the independent forces defined by From the definition of activity, ai (11)
ai = y i X i
then L12 = Lzl. The expressions for the cross-phenomenological coefficients are L12
=
dD21 aD12 - cD11 ; LZl = ad - bc ad
- bDzz - bc
where yi is the activity coefficient and X i is the mole fraction, eq 10 becomes
(6) For a ternary system
where
and
x1 = Cl/(Cl + c2 + Cd C l B l + CZB2
(7)
(13)
+ C3V8 = 1
(14)
Using eq 13 and 14 in eq 12, one obtains
and c and d are the same, respectively, as a and b except that (d/aC1),, is replaced by (a/aC2) cl. Expressions for the Lij's in terms of ai)s can be obtained by substituting eq 3 and 7 in eq 6 or by the general approach outlined in the earlier paper.1° In both cases, the cross coefficients are identical and given by LIZ =
Lz1
=
clc2csvs~-~ VlClC2 US71
a171
+ +
where CT = C1 C2 C3. The term be calculated from the activity data. The experimental techniques available to obtain values of Dij, and thus Lij, have been described by Fujita and Gosting5ge and by Burchard and Toor.' Tracer techniques providing for the determination of the friction coefficients, a;, have been discussed elsewhere."JJS In addition a reasonable predictive method was presented. As a result, the hydrodynamic theory, which predicts that Onsager's reciprocal relations are valid, can be checked experimentally by comparing the predicted and experimental diffusion coefficients, DU, and the phenomenological coefficients,Lij. Expressions for aFi/dCj. In order to calculate the
a In yi/dCj
can
Experimental Section The isothermal diffusion for this study was followed with an optical diffusiometer. The diffusion took place in a glass-windowed cell immersed in a constanttemperature bath maintained at 25 f 0.03" by means of a mercury thermoregulator. The diffusion process was followed with a Mach-Zehnder type interferometer. This setup was very similar to the diffusiometer described by Caldwell, Hall, and Babb.14 (13) T. K. Kett, Ph.D. Thesis, Michigan State University, 1968. (14) C. 8. Caldweil, J. R. Hall, and A. L. Babb, Rev. Sci. Instrum., 28, 816 (1957).
Volume 75,Number 6 May 1060
TERENCE K . KETTAND DONALDK. ANDERSON
1270
the equation
Table I: Initial Concentration Differences for the Dodecane (1)-Hexadecane (2)-Hexan_e (3) System (T = 25'; = 1.615; = 1.464; Cs = 1.333 mol/l.) Run no.
AC,, mol/1.
ACZ,mol/l.
ffl
154 155 157 158 160 161 162
-0.1200 -0.0185 0.0168 0.0788 0.1558 0.1276 0.0716
0.1343 0.0571 0.0467 -0.0122 -0.1310 -0.0548 -0.0087
-1.134 -0.2-9 0.176 1.352 -2.416 3.599 1.257
In the theoretical derivations, it was assumed that the diffusion coefficients, Dij, were constant and in the calculations that the partial molar volumes were constant. To meet this condition experimentally, the initial concentration differences within the cell were chosen as small as possible. For the system studied here, these differences are listed in Table I. Two systems are considered in this analysis. The first consisted of the three hydrocarbons dodecane, hexadecane, and hexane and was studied experimentally in this laboratory. The other consisted of toluene, chlorobenzene, and bromobenzene and was studied by Burchard and T00r.~ The hydrocarbon chemicals were purchased in the purest forms available from Matheson Coleman and Bell Co. The hexane was spectroquality and the dodecane and hexadecane were 99+% (olefin free) pure. The purity of the chemicals was further confirmed by comparing their densities and refractivs indices with values given by Tirnmermans.l6 These are given in Table XI. I n determining concentrations, various predetermined amounts of each component were weighed out together, and the density of the resulting solution was measured with a pycnometer. Volumes calculated on the basis of the densities of the pure components agreed within 0.15% of the measured volumes but were lower than the measured values in all cases. The very small differences indicated the constancy of the molar volumes with concentration. For the system toluene-chlorobenzene-bromobenzene, the concentrations were calculated from the mole fractions given by Burchard and Toor7 according to Table 11: Comparison of Physical Properties with Previous Recorded Data Component Hexane Dodecane Hexadecane
--at
Density 2 5 O , g/cc-
0.6650 0.7450 0.7698
0.6549" 0.7451 0.7699
Bverage of several recorded data.
The Journal of Physical Chemistry
ni-d -ex,
Refractive masD--
1.3720 1.4193 1.4319
1.3723" 1.4195 1.4325
Because of the lack of ternary data, the viscosities of the solutions were determined according to the following relationship assumed by Yon and Toor.16 3
In r) =
i-1
~ i l Vni
(17)
The ternary diffusion coefficients,Dij, were calculated from measurements of the reduced height: area ratios and reduced second moments of the refractive index gradient curve (dn/dx us. x) according to the equations of Fujita and Gosting. A refractive index curve (n vs. x) in the form of a photograph was obtained directly by the diffusiometer at various times during a diffusion run. From this, the refractive index gradient curve was obtained. The reduced height: area ratios and reduced second moments were then obtained using numerical methods. It was assumed that the total refractive index change across the boundary, An, could be expressed by the equation hn = RlAC1 RzAC2 (18) where C; represents the concentration in moles per liter and Ri is the differential refractive index increment of component i. Division of eq 18 by AC1 gives
+
(AnlACl) = RI
+ Rz(AC2/AC1)
(19)
Thus values of R1 and Rz could be obtained from the intercept and slope of a plot of An/AC1 vs. ACZ/ACI. Values for R1and R2 are given in Table V. Experimental values of the friction coefficients for the hydrocarbon system were obtained from tracer diffusion runs made a t the same concentration at which the experimental ternary diffusion coefficients were determined. Values of the friction coefficients were also calculated from mutual and self-diffusion data available in the literat~re.l~-~O Thus a check on the predicted friction coefficients was available. For the toluene-chlorobenzene-bromobenzene system, only predicted values of the friction coefficients were available. The necessary diffusion data are provided in Tables XI1 and IV. From these friction coefficients, values for the ternary diffusion coefficients, Dii, and for the phenomenological coefficients, Lij, were calculated for both systems and compared to the experimentally deter(15) J. Timmermans, "Physico-Chemical Constants to Pure Organic Compounds," Elsevier Publishing Co., Inc., New York, N. Y., 1950. (16) C. M.Yon and H. L. Toor, private communication. (17) C . S. Caldwell and A. L. Babb, J. P h y s . Chem., 60, 51 (1956). (18) D. L. Bidlack and D. K. Anderson, { b i d . , 68, 206 (1964). (19) D. L. Bidlack and D. K. Anilerson, { b i d . . 68, 8790 (1964). (20) C. J. King, L. Hsueh, and K . Mao, J. Chem. Eng. Data, 10, NO.4, 348 (1965).
1271
TERNARY ISOTHERMAL DIFFUSION 1.9 L
355
Table 111: Binary Diffusion Coefficients at Infinite Dilution Temp.
DijoX 106, Dijo X qi X 107,
Binary system (i dilute in j )
OC
cma/sec
dyn
Hexane in dodecane Hexane in hexadecane Dodecane in hexane Dodecane in hexadecane Hexadecane in hexane Hexadecane in dodecane Chlorobenzene in bromobeneene Chlorobenzene in toluene Bromobeneene in chlorobenzene Bromobeneene in toluene Toluene in chlorobenzene Toluene in bromobenzene
25 25 25 25 25 25
1.42 0.85 2.74 0.49 2.21 0.67
1.91 2.58 0.81 1.47 0.66 0.90
30 30
1.36 2.36
1.34 1.22
30 30 30 30
1.76 2.27 1 .BO 1.41
1.26 1.17 1.28 1.39
mined values. Since the parameter RT/ai always appeared together in the calculations, this value was determined for each component rather than u i itself. The relationship used in this calculation was
RT/ai = XiDiqi
+ XjDijOvj + XkDikaqk
(i = 1, 2, 3) (20)
where Di is the self-diffusion coefficient of component i in cm2/sec, DijO is the mutual diffusion coefficient of the i-j binary at infinite dilution of component i in cm2/sec, q i is the viscosity of pure i in dyn sec/cm2, and X i is the mole fraction of component i at which the diffusion coefficients were measured. Activity data at 20" for the hydrocarbon binaries hexane-dodecane and hexane-hexadecane were obtained from Br@nsted and Koefoed.21 Since they concluded that the system exhibited regular solution behavior, the activity data for the ternary system were obtained by a Van Laar fit. The equations used were
T In y1 =
+ + +
( C z d Z CaAa~2/B7;;)~ (ciA12 C2 CaAaz)2
(21)
Figure 1. Linear relrttions of the reduced second moment, D ~ M , and of the reciprocal square root of reduced height :area rattio, I/& us. the refrttctive index fraction of dodecane, 011, for the system dodecane-hexadecane-hexane.
constants obtained from the binaries. The toluenechlorobenzene-bromobenzene system was assumed ideal. Values for the Aij and Bij are given in Table IX.
Results and Discussion Figure 1 shows the graphs of the experimental values us. cy1 and D ~ us. M cy1 for the system dodecane of 1/ ( l ) ,hexadecane (2), and hexane ( 3 ) . The curves are seen to be linear in accordance with the equations proposed by Fujita and Gosting. The best straight line through each set of points was determined by the least-squares technique and the slopes and intercepts for each are given in Table V. Using these values of the slopes and intercepts, the diffusion coefficients, Dij, were calculated according to the Fujita-Gosting equations and the phenomenological coefficients, Lij, according to Miller's equations (eq 6 ) .
v'z
~~~
~~
Table V: Slopes and Intercepts of Reduced Height:Area and Second-Moment Data for the System
Hexane-Dodecane-Hexadecanea
where Ci is the concentration of i and Aij and
Bij
are
292.30 -24.45 267.85 1.192 0.149 1.341
~~
Table IV: Self-Diffusion Coefficients~
Component Hexane Dodecane Hexadecane
Temp,
Dj X 106,
Di X q d X 107,
00
cma/sec
dyn
25 25 25
4.21 0.87 0.36
0.9390
0.01091 0.01803
1.25 1.20 1.09
Experimental data for toluene, chlorobenzene, and bromobeneene were not available. The friction coefficients for these components were estimated using the binary intercepts given in Table 111. (I
(21) J. N.
Br6nsted and J. Koefoed, Kgl. Danske Videnskab. Selskab
Mat. Fis. Medd., 2 2 , No. 17, 1 (1946). Volums 73,Number 6 May i9Sg
1272
TERENCE K. KETTAND DONALD K. ANDERSON
Table VI: Friction Coefficients Dodecane (1)-Hexadecane (2)-Hexane (3) (T = 25') X 107, dyn--
-RT/ul
-RT/ur X 107,dyn-Pred. (eq 2 0 ) Exptl
Xl
Xa
Pred. (eq 2 0 )
Exptl
0.350
0.317
1.089
1.118
0.887
X 107, dyn--
-RT/ua
Pred. (eq 20)
Exptl
1.899
1.873
0.848
Toluene (1)-Chlorobenzene (2)-Bromobenzene (3) (T = 30")
x1
R T I U ~x 107
xz
0.25 0.26 0.70 0.15 0.45 0.18
RT/ui X IO7
(pred. values)
R T / u s X 107
1.323 1.373 1.338 1.307 1.340 1.350
1.277 1.306 1,243 1.276 1.268 1.300
1.222 1,199 1.188 1.235 1.202 1.214
0.50 0.03 0.15 0.70 0.25 0.28
Table VII: Comparison of Experimental Diffusion and Phenomenological Coefficients with Those Calculated from Friction Coefficients for the System Dodecane (1)-Hexadecane (2)-Hexane (3) (T = 25"; XI= 0.350; Xx = 0.317) -----Du X
cma/sec-from-Pred. Exptl 105,
--Dlz
-Calcd
Exptl 0.968
uqs
u's
1.099
1.115
Exptl 0.266 -------Lis
Exptl
-0.453
X 105,cm2/sec--Calcd from-Pred. Exptl
u's
u's
0.366 X RT X
--Cacld Pred. 8'8
-0.469
0.386
Exptl 0.225 -LZI
105--
from-
Exptt
X 105,cm*/sec--Calcd from-Pred. Exptl
---
1 0 5 , cmz/sec-----Calcd from-Pred. Exptl
Dne X
0's
0's
Exptl
0.187
0.167
1.031
u's
4's
1.007
0.971
X RT X lOS--
-Calcd Pred.
u's
Exptl
u's
-0.465
-0.444
-0.469
The predicted values of these coefficients were also determined from the friction coefficients using eq 3 and 8, respectively. Both sets of values are recorded in Table VII. A similar procedure was carried out for the toluenechlorobenzene-bromobenzene system. The experimental values had already been determined by Burchard and Toor. These and the values determined from friction coefficients are recorded in Table VIII. In determining diffusion coefficients, Dzj, and phenomenological coefficients, Lij, based on the hydrodynamic model, experimental or predictive friction coefficients were required. For the hydrocarbon system, these were obtained both ways. This fulfilled two purposes: (1) to obtain accurate values of the friction coefficients, thus placing more confidence on the diffusion and phenomenological coefficients calculated from them, and (2) to check on the predictive method of calculating friction coefficients (eq 20). Values of the friction coeffcients calculated by both methods are given in Table VI. A comparison shows that the predicted friction coefficients compare quite favorably with those obtained experimentally by tracer methods. The fact that the molar volumes of the hydrocarbons differed considerably and that the friction coefficients for the same component calculated from the infinitely dilute The Journal of Physical Chemistry
-Dnl
from-Exptl u's
-0.465
binary diff usion coefficients and self-diffusion coefficients differed appreciably indicated that the effects of the other components were strong. Under these conditions, the predictive method for nonassociating systems would be the poorest. With this in mind, the agreement between the experimental and predicted friction coefficients under the least applicable conditions provides reasonable confidence in the predictive method. The diffusion and phenomenological coefficients calculated from these friction coefficients are given in Table VII. In comparing the values obtained by optical methods t o those calculated from hydrodynamic eq 3 and 8, it can be seen that there is good agreement. The hydrodynamically obtained values are considered well within the expected accuracy of the experimental values. The agreement of the experimental LIZ and Lzl and the agreement of these with the LIZ= Lzi obtained from the friction coefficients indicates that Onsager's reciprocal relations are valid. The toluene-chlorobenzene-bromobenzene system was assumed to be ideal. On this basis, phenomenological coefficients were calculated from the diffusion coefficients given by Burchard and Toor using Miller's equations. Friction coefficients were obtained by the predictive method. The fact that the molar volumes were almost equal and that the friction coefficients
TERNARY ISOTHERMAL DIFFUSION
1273
Table VI11 : Comparison of Experimental Diffusion and Phenomenological Coefficients with Those Calculated from Friction Coefficients for the System Toluene (1)-Chlorobenzene (2)-Bromobenzene (3) (T = 30') -Dii
XI
Xl
0.25 0.26 0.70 0.15 0.45 0.18
0.50 0.03 0.15 0.70 0.25 0.28
Exptl
X IO&----
1.848f0.066 1.570f0.088 2.132 f 0.098 1.853f0.108 2.006zt0.108 1.774f0.107
-DIZ
Calcd
Exptl
Calcd
x
Xi
Xs
Exptl
0.25 0.26 0.70 0.15 0.45 0.18
0.50 0.03 0.15 0.70 0.25 0.28
-2.384 -0.145 -2.048 -1.778 -2.212 -0.961
~
~~~
Table IX: Quantities Involved in Calculation of
RT
-Dz1
Exptl
X IO+-
-0.052f0.093 -0.012f0.033 -0.071 f0.026 -0.O68fO0.026 -0.198f0.163 0.003&0.162
-0,019 -0.033 -0.063 -0.009 -0.043 -0.019
-0.063f0.109 -0.077f0.100 0,051 f0.163 0.049f0.162 -0.020f0.026 -0.037f0.026
1.822 1.611 2.069 1.819 1.883 1.656
---L~~
~~
X lOs-
x io+
- L ~ ~ x RT
Calcd
Exptl
-2.265 -0.148 -2.175 -1.943 -2.502 -0.758
-2.248 -0.136 -2.123 -1.856 -2.119 -0.868
Calcd
-DM
X 1OS-
1.797&0.076 1.606f0.109 2.062f0.108 1.841 f O . 1 0 8 1.890f0.108 1.518f0.107
-0.073 -0.006 -0.039 -0.072 -0.052 -0.049
x
Exptl
Calcd 1.753 1.580 2.071 1.750 1.844 1.593
105-
Calcd
-2.248 -0.136 -2.123 -1.856 -2.119 -0.868
~~
D,j
and L,j
A. Dodecane (1)-Hexadecane (2)-Hexane (3) (T = 25") Xl
Xa
q, C P
a/RT
b/RT
c / RT
d/RT
0.350
0.317
1.250
2.519
2.415
2.415
3.781
R T / m X 107 R T / o n X 107 RT/ua X 107
1,089
0.887
1,899
Van Laar Constants Ai2
AN
Ala
A83
AI8
As1
fi
4%
1.0188
0.9815
0.9331
1.0717
0.9505
1.0521
0.0334
-0.0331
6
2/86;
4%
-5.6502
3.7336
-3.6066
dlz 3.6994
B. Toluene (1)-Chlorobenzene (2)-Bromobensene (3) (T = 24.6') x 1
XI
7, C P
a / RT
b/RT
c / RT
d/RT
0.25 0.26 0.70 0.15 0.45 0.18
0.50 0.03 0.15 0.70 0.25 0.28
0.713 0.824 0.596 0.713 0.679 0,801
0.883 0.560 0.927 1.474 0.619 0.792
0.455 0.158 0.765 0.759 0.379 0,208
0.455 0.158 0.765 0.759 0.379 0.208
0.660 3.626 1.465 0.900 0.797 0.579
calculated from the infinitely dilute binary diffusion coefficients were approximately equal indicated that reasonable values of the friction coefficients could be expected. These are given in Table VI. Table VI11 lists the diffusion and phenomenological coefficients. (Also see Table IX.) Excellent agreement between the hydrodynamically obtained values can be seen from a comparison. Except for a very few cases, the values obtained from the friction coefficients were well within the 95% confidence limits. Reasonable agreement between the experimental LIZ and Lzl is evident and these agree favorably with those obtained from friction coefficients.
Conclusions It can be concluded from this study, which includes the companion article, that hydrodynamic theory
RT/ur X IO7 RT/an X IO7 R T / a s X IO7
1.323 1.373 1.338 1.307 1.340
1.350
1.277 1.306 1,243 1.276 1.268 1.300
1,222 1.199 1* 188 1.235 1* 202 1.214
should play an important role in describing multicomponent diffusion. It predicts that Onsager's reciprocal relations should be valid for nonassociating systems of any number of components. Furthermore, it leads to generalized equations which enable the calculation of diffusion coefficients for quaternary and higher order nonassociating systems from friction coefficients, which can be obtained accurately from tracer measurements, and activity data. The number of tracer runs required for experimental determination of the a's is equal to the number of components in the system. Calculation of diffusion coefficients for nonassociating systems in this way would eliminate the laborious and time-consuming techniques now available and provide data for nonassociating systems of higher order than ternary. Such data, currently not available, would be extremely useful in considering mass flow problems. Volume Y3, Number 6 May 1888
1274
J. W. SIMONS AND 0.W. TAYLOR
Experimental evidence obtained in this study verifies reciprocal relations for two nonassociating ternary systems within the limits of experimental error. In addition, it demonstrates the applicability of hydrodynamic theory to multicomponent liquid diffusion. These conclusions are based on the fact that good agreement between the diffusion and phenomenological coefficients obtained by optical methods and those obtained from friction coefficients was found for the nonassociating systems studied. Hopefully experimental techniques will be developed in the future which will enable measurement of quater-
nary or higher diffusion coefficients, Dij, so that those calculated from friction coefficients,ui, can be checked. Since hydrodynamic theory requires fewer independent friction coefficients than independent diffusion coeficients to describe diffusion in systems of more than three components, such experimental techniques would be extremely valuable in checking the applicability of hydrodynamic theory to such systems. Acknowledgment. This work was supported by a grant from the Petroleum Research Fund, administered by the American Chemical Society. Grateful acknowledgment is hereby made to the donors of the fund.
Chemically Activated cis-1,2-Dimethylcyclopropane from Photolysis of
cis-2-Butene-DiazomethaneMixtures in the Presence of Oxygen'. by J. W. Simons and G.W. Taylorlb Chemistry Department, New Mexico State University, Las Cruces, New Mexico
88001
(Received July 3 1 , 1968)
A study of the geometric and structural isomerization rates of chemically activated cis-1,2-dimethylcyclopropane formed by singlet methylene radical addition to the double bond of cis-2-butene is reported. The singlet methylene radicals were produced by diazomethane photolysis a t 4358 and 3660 d in the presence of added oxygen. The structural isomerization rate was determined by an internal comparison method that eliminates uncertainties due to pentene decomposition. The rates are in excellent agreement with RRKM theory calculations at an adjusted energy for each photolysis wavelength. The energies carried into the activated molecule by the methylene radical were 113.0 and 116.8 kcal/mol at 4358 and 3660 A, respectively. These values suggest that the methylene radical carries into the addition reaction a relatively small fraction (0.30) of the total excess energy available from its formation reaction. Energies determined in this work, applied to related systems, give results that differ by at most -3 kcal/moI from earlier determinations.
Introduction It has been demonstrated for a few unimolecular chemical reactions that sensibly chosen activated complexes, which have been adjusted to fit the preexponential factors for the thermal reactions, give theoretical rates calculated from Rice-Ramsperger-KasselMarcus (RRKM) theory which are in remarkably good agreement with experiment for the corresponding chemically activated2 species at considerably higher energies than those in the thermal systems.*-b The success of RRKM theory in predicting rates in chemical activation systems should permit its use, combined with experimental rates, to determine the energy of a chemical activation system if this is an uncertain quantity. Setser and Rabinovitch have utilized this approach previously for a number of methylene radical reaction systems producing chemically activated cyclopropanes.e The Journal of Physical Chemislyl
Chemically activated cis-l,2-dimethylcyclopropane from the reaction of methylene radicals with cis-2butene is a particularly interesting chemically activated molecule for the determination of methylene radical energies, since reliable thermal data for both the geometric and the structural isomerizations are available' and activated complex structures have been (I) (a) This work was supported by NSF under Grant No. GP-6124: (b) NDEA Predoctoral Fellow. (2) B. S. Rabinovitch and M. 0. Flowers, Quart. Rev. (London), 122 (1964). (3) B. 9. Rabinovitch and D. W. Setser, Advan. Photochem., 3 , 53 (1964).and references therein. (4) J. W. Simons and B. S. Rabinovitch, J . Phys. Chem., 6 8 , 1322 (1964). (6) (a) J. C. Hassler, D. W. Setser, and R. L. Johnson, J . Chem. Phys., 45, 3231 (1966);(b) J. 0.Hassler and D. W. Setser, (bid., 45, 3287. 3246 (1966). (6) D. W. Setser and B. S. Rabinovitch, Can. J . Chem., 40, 1425 (1962). (7) M.0.Flowers and H. M. Frey, Proc. Roy. SOC.(London). A257, 122 (1960);A260,424 (1961).
