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The Journal of Pbysical Chemistry, Vol. 83, No. 12, 1979
M. Iwasaki and K. Toriyama
efficient formation of tert-butyl radicals, the selective formation of solute radicals was greatly suppressed a t 4.2 K as shown in Figure 7d and in Table I1 together with other cases. M. Iwasaki, K. Toriyama, H. Muto, and K. Nunome, J . Chem. Pbys., 65, 596 (1976);Cbem. Phys. Lett., 39, 90 (1976). K. Nunome, H. Muto, K. Toriyama, and M. Iwasaki, Cbem. Phys. Lett., 39. 542 (1976). M. Iwasaki, K. Toriyama, K. Nunome, M. Fukaya, and H. Muto, J . Pbys. Cbem., 81, 1410 (1977). K. Toriyama, K. Nunome, and M. Iwasaki, J. Am. Chem. SOC.,99,
5823 (1977).
25 G Figure 11. ESR spectral changes caused by the gradual lowering of the speclmen temperature. The spectra are obtained from neopentane containing 0.5 mol % of HI with 2 mol % of isobutane UV irradiated in a liquid helium dewar. Measurements were made (a) 22 min, (b) 60 min, and (c) 93 min after the UV light was turned off. The microwave power was 0.1 mW, field modulation width, 3.0 G.
temperature during UV irradiation. Kinugawa et al.15 also reported that the selective formation of solute radicals is not suppressed a t 4.2 K upon radiolysis of neopentane containing isobutane and HI. We have also carried out a similar experiment using neopentane containing 2 mol % of isobutane and 0.5 mol 7'0 of HI. Although the sample y-irradiated a t 77 K showed
M. Iwasaki and K. Torivama. J. Am. Cbem. SOC..100. 1964 11977). M. Iwasaki, K. Toripma, H. Muto, and K. Nunome, Cbem.'Phys. Lett., 56, 494 (1978). T. Miyazaki, K. Klnugawa, and J. Kasugai, Radiat. Phys. Chem., 10, 155 (1977h and references cited therein. R. W. Fessenden, J. Cbem. Pbys., 37, 747 (1962). M. Iwasaki, J. Magn. Reson., 16, 417 (1974). T. Ichikawa and P. K. Ludwig, J. Am. Cbem. SOC.,91, 1023 (1969). H. Enokido, T. Shinoda, and Y. Mashiko, Bull. Chem. SOC.Jpn., 42,
84 (1969). H. Muto, M. Iwasaki, and K. Nunome, The 21st Symposium on Radiation Chemistry, Tokyo, Japan, Sept 30, 1978;M. Iwasaki and K. Torlyama, unpublished work. L. Perkey and J. E. Willard, J . Chem. Phys., 60, 2732 (1974). K. Toriyama and M. Iwasaki, J . Phys. Chem., 82, 2056 (1978). K. Kinugawa, T. Miyazaki, and H. Hase, Radiat. Phys. Cbem., 10,
341 (1977). J. H. Freed, J. Cbem. Pbys., 43, 1710 (1965). W. L. Gamble, I. Miyagawa, and R. L. Hartman, Phys. Rev. Lett., 20, 415, 1221 (1968);R. B. Davidson and I.Miyagawa, J . Cbem. Phys., 52, 1727 (1970);J. W. Wells and H. C. Box, ibid., 51, 2542 (1968);S. Clough and F. Poldy, /bid., 51,2076 (1969);M. Iwasaki, K. Nunome, H. Muto, and K. Toriyama, /bid., 51, 1839 (1971). M. Iwasaki and K. Toriyama, J. Pbys. Chem., following paper in this issue.
A Simple Model for Hydrogen Atom Reactions in Neopentane-Cyclohexane Mixtures Irradiated at 4.2 K Machio Iwasaki" and Karumi Toriyama Government Industrial Research Institute, Nagoya, Hirate, Kita, Nagoya, Japan (Received October 18, 1978; Revised Manuscript Received February 12, 1979) Publication costs assisted by the Government Industrial Research Institute, Nagoya
In this paper, a simple model is given to explain why hydrogen atoms produced in crystalline neopentane at 77 K have a high probability of reacting with dilute solute molecules, whereas at 4.2 K they abstract hydrogen from neopentane matrix molecules. The model is based on the following assumptions: at 77 K hydrogen atoms which do not undergo short-range hot abstractions migrate until they encounter a solute molecule, and abstract hydrogen from it. At 4.2 K the hydrogen atoms which would be mobile at 77 K are immobilized at their place of birth and eventually abstract only from one of the molecules in the cage wall surrounding each hydrogen atom. The experimental data at 4.2 K are compared with estimates of the solute radical yields for different assumed values of (a) the fraction of hydrogen atoms which undergo short-range hot abstraction; (b) the ratio of the rate constants for the reaction of immobilized hydrogen atoms with a matrix and a solute molecule; and (c) the number of molecules in the cage wall. By this comparison plausible numerical values are obtained for each of these parameters.
Introduction When crystalline neopentane containing a small amount of cyclohexane is irradiated at 77 K, hydrogen atoms formed from the neopentane selectively react with solute molecules forming cyclohexyl radicals with much higher yield than that which would be proportional to the solute c~ncentration.~" On the other hand, when the same mixture is irradiated at 4.2 K, hydrogen atoms which would 0022-3654/79/2083-1596$0 1 .OO/O
react with solute molecules at 77 K react instead with solvent molecules.ly2 The detailed analysis of the ESR spectra suggests that solvent radicals are formed closely in pairs a t 4.2 K, while the solvent and solute radicals are more isolated when irradiated a t 77 K,l indicating that at 4.2 K hydrogen atoms formed from a homolysis of neopentane react with neighboring neopentane molecules, while a t 77 K they migrate a long distance t o react with 0 1979 American Chemical Society
Hydrogen Atom Reactions in Neopentane-Cyclohexane Mixtures
The Journal of Physical Chemistry, Vol. 83, No. 12, 1979
solute molecules without reacting with solvent molecules. If the cause is in the difference in the reactivity ratio of hydrogen atoms with solvent and solute molecules, we may have to assume that at 4.2 K most of the hydrogen atoms react with neighboring neopentane molecules before being thermalized. However, it is difficult to explain why short-range reactions by hot hydrogen atoms predominate at 4.2 K. For this reason, we have concluded that the hydrogen atoms cannot migrate a long distance to encounter solute molecules at 4.2 K.' In this case, however, we have to assume that hydrogen atoms can abstract a hydrogen atom from neopentane even at 4.2 K after being thermalized, because some of hydrogen atoms must not undergo hot abstraction and must be trapped a t 4.2 K if they are unreactive. However, our experiments show that hydrogen atoms are not trapped even at 1.5 K. Recently, we have obtained experimental evidence that thermal hydrogen atoms trapped at 4.2 K in irradiated CH4 containing 0.5 mol % of C2H6 are detrapped at 10-20 K and selectively abstract a hydrogen atom from C2H6 forming C2H5.4The results are satisfactorily explained by reactions which proceed by a tunneling process at low t e m p e r a t ~ r e The . ~ results strongly suggest that hydrogen atoms can abstract hydrogen atoms from neopentane by the tunneling process even at 4.2 K after being thermalized.6 In the present paper, it will be shown that the drastic difference in solute radical formations at 4.2 and 77 K reported in the preceding paper2 can be quantitatively explained in terms of the immobilization of hydrogen atoms a t their place of birth a t 4.2 K, if we admit the occurrence of abstraction reactions of thermal hydrogen atoms in solid hydrocarbons at cryogenic temperatures.
Results Simple Model for Hydrogen A t o m Reactions. The reactions involved in the radiolysis of neopentane containing cyclohexane are assumed to occur as follows: neo-C5H12* neo-C5Hl1*+ H-* (1) C-C&12* C-C6H11*+ H.* (2)
-+
+ neo-C5H12 kS H** + C-CgH12
H.*
k4
H.* H.
c-CsH11.
+ H2
-+H.
+ neo-C5H12 ke H. + C-CgH12 k7
H.
neo-C5Hl1- + H2
(4) (5)
neo-C5HIl. + H2 C-C&11*
(3)
+ H2
-
+ R. (R. = H., neo-C5HI1.,c-C6Hl1.)
(6) (7)
products (8)
where H.* represents a hot hydrogen atom while H. is a thermalized hydrogen atom a t 4.2 K and a mobile hydrogen atom at 77 K. Now, we assume that hydrogen atoms produced by radiolysis partly react with neighboring molecules before being thermalized and that hydrogen atoms, which do not undergo short-range hot abstraction, can migrate through the crystals a t 77 K while they are immobilized at their place of birth a t 4.2 K without migration. Then, if the immobilized hydrogen atoms find solute molecules in the cage wall surrounding each hydrogen atom, they are assumed to react with solutes with a reactivity ratio of k7/k6. However, if they do not find any solute molecules, they are assumed to react only with solvent molecules. For the sake of simplicity, it is assumed that the reactivity ratio of k 4 / k 3for hot reactions is unity even though
1597
this is not necessary. Under these assumptions, the fractional yield of solute radicals is given by the sum of the following three contributions from primary dissociation reaction 2, hot hydrogen atom reaction 4, and thermalized hydrogen atom reaction 7 : from (2) f / [ ( l - ~)GNP/GcH+ fl (9)
Nf (1 - N ) F
from (4) from (7)
(10) (11)
where the total yield is normalized to 2, f is the fraction of solutes, GNP/GCH,the ratio of the radical yield in pure neopentane and cyclohexane, N , the fraction of hot reactions, and F , the fraction undergoing thermal reaction with cyclohexane. Now, F is easily obtained for the reactions at 4.2 K in the following manner. The probability of finding i molecules of solutes among n surrounding molecules is given by f(1 (12)
f)n-i(nc,)
and the probability of reacting with solutes in the site having i molecules of solutes is given by
i / [ i + (n- i)k6/k7]
(13)
Therefore, the fraction of hydrogen atoms which react with cyclohexane in all kinds of sites is given by n
F = Cf(1- f)n-z(nCL)i/[i + (n- i)k6/k7]
(14)
2=1
It is noted that if k 6 / k 7 = 0, F becomes as follows: n
F = Cf(1 - f)n-2((,C,)= 1 - (1 - f)" 2=1
(15)
and if k 6 / k 7 = 1,F becomes as follows: i=l
(16)
as is expected from the simple considerations for these extreme cases. On the other hand, assuming a simple competition for the reactions of the mobile hydrogen atoms at 77 K, F is given by
F =
f/v + (1- m 6 / k 7 1
(17)
for the concentration range where the contribution from recombination reactions 8 is negligible. Comparisons with Experimental Data. Now, GNP/GCH is determined to be 0.69 from the radical yield measured at 77 K for the pure samples irradiated at 4.2 K.2 The ratio may be slightly different if the measurements are made at 4.2 K. However, the concentration dependence of solute radical yields was measured a t 77 K so that the value for GNP/GCH determined under the same conditions was adopted for the calculations. First, the fraction of hot hydrogen atoms is assumed to be the same as that of nonscavengeable hydrogen atoms by solutes at 77 K. For neopentane-cyclohexane mixtures, the fraction of nonscavengeable hydrogen atoms was estimated to be 0.36 in our previous study.l Using these values, the experimental data obtained at 77 K2p3were approximately fitted by the reactivity ratio k 6 / k 7 = 1/2000 as is shown in Figure la. Previously we estimated k 6 / k 7 < 1/87 from the experimental data obtained at a solute concentration of 2 mol % assuming that all hydrogen atoms undergo competition reactions with solvent and solute molecules. However, a considerably
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The Journal of Physical Chemistry, Vol. 83,
6oc
n
:
No. 12, 1979
M. Iwasaki and K. Toriyama
14
60 -
N 501.
-z 50 -
n=14 k ~ p / k c= ~1/2000 N =Fraction of H
-
N=O-
0.1
*
02030405-
-40-
-
06-
30 -
V
20
-
IO
-
Y
OW
0.05
I
I
I
010
015
020
Solute Fraction
Flgure 1. The yield of cyclohexyl radicals vs. concentration of cyclohexane in neopentane irradiated (a) at 77 K and (b) at 4.2 K. The circles represent the observed values and the solid curves are calculated assuming N = 0.36 and k 6 / k 7= 1/2000 at 77 K while N = 0.4 and n = 14 at 4.2 K. The reactivity ratio, k , / k 7 , is varied as a parameter in (b). The experimental data at 77 K in the solute concentration below 0.02 are cited from ref 3.
,,I
n
:
Numbers of neighbor molecules
Solute Fraction
Figure 2. The yield of cyclohexyl radicals vs. concentration of cyclohexane in neopentane irradiated at 4.2 K. The circles represent the observed values and the solid curves are calculated for various values of n assuming k s / k , = 1/2000 and N = 0.4.
higher ratio was obtained if the experimental data3 reported for the low solute concentration region are included and it is assumed that only scavengeable hydrogen atoms participate in the competition reaction. As is described in our previous paper,ll* the reactivity ratio does not change very much with lowering of the temperature below 77 K because the reaction rate by the tunneling process becomes nearly independent at such low temperatures. For this reason, we have tentatively assumed the same reactivity ratio ( k 6 / k 7= 1/2000) at 4.2 K to calculate the theoretical prediction for the solute concentration dependence a t 4.2 K. Assuming that the fraction of hot hydrogen atoms at 4.2 K is the same as that of the nonscavengeable hydrogen atoms a t 77 K, we have calculated the solute concentration dependence by changing the number of neighboring molecules, n, as a parameter. As shown in Figure 2, the experimental values are well fitted by the theoretical curve with n = 14. Since the crystal structure of neopentane is face-centered cubic, the number of surrounding molecules is 14 if the trapped hydrogen atom sits at the center of the unit cell. The numbers of the first and the second nearest molecules in this trapping site are six and eight, respectively. The first and the second nearest molecules are considered to consist of a cage wall for the trapped hydrogen atoms. Strictly speaking, we have to discriminate reactivities with the first and the second nearest molecules. However, it was not possible to obtain a good fit from the assumption that the first (n = 6) and the second (n’ = 8) nearest molecules have greatly different reactivities. The results suggest that the immobilized hydrogen atoms react with both first and
I
OV
005
I
010 Solute Fraction
I
015
020
Figure 3. The yield of cyclohexyl radicals vs. concentration of cyclohexane in neopentane irradiated at 4.2 K. The circles represent the observed values and the solid curves are calculated for various values of Nassuming k , / k , = 1/2000 and n = 14.
second nearest neighbor molecules with considerably higher reactivity when compared with that of neopentane. Using a fixed value of n = 14, the results shown in Figure 3 are obtained by changing the fraction, N , of hot hydrogen atoms as a parameter. As is seen, the curve is fairly sensitive to N giving a good fit to the experimental value when N = 0.4. There may not be any a priori reasons that the fraction of hot hydrogen atoms is largely different a t 4.2 K from that at 77 K. Figure l b is calculated by changing k 6 / k 7 with fixed values of N = 0.4 and n = 14. As is seen, the experimental data can be fitted by k6/k7over the range from 0 to 1/500. The theoretical curve is not as sensitive to k6/k7 when the reactivity ratio becomes as high as 1/500 and there is no appreciable difference between the curves with k6/h7 = 1/2000 and 0.
Discussion The remarkable difference in the solute radical yields at 77 and 4.2 K can thus be reproduced without introducing any drastic differences in the reactivity of hydrogen atoms and the fraction of hot hydrogen atoms. The cause of the drastic difference is in the immobilization of hydrogen atoms at their place of birth at 4.2 K in neopentane matrices. This merely prohibits competition reactions of hydrogen atoms with solvent and solute molecules. The reactivity of hydrogen atoms with neopentane having primary C-H bonds must be considerably low at such low temperatures, even though the tunneling effect allows this reaction to proceed. Therefore, a t 4.2 K hydrogen atoms may be thermalized and immobilized first, and then react rather slowly with surrounding molecules as is revealed by the simple calculations presented in this paper. As is reported in the preceding paper,2 suppression of solute radical formation becomes prominent with lowering of temperature below 77 K, indicating that hydrogen atom migration in neopentane matrices becomes more and more difficult a t lower temperature and that finally hydrogen atoms are immobilized at 4.2 K. If the immobilized hydrogen atoms were unreactive with the matrices, they would have been trapped at low temperature and become mobile at some elevated temperatures to react with solute molecules. These hypotheses may be supported by our recent experiments reported for the reaction of thermal hydrogen atoms in solid methane containing a small amount of ethane.4 The trapping of radical pairs between a hydrogen atom and a methyl radical in methane matrices at 4.2 K7pa demonstrates that the thermalized hydrogen atoms are trapped in the vicinity of each parent radical, if the matrices are sufficiently inert with hydrogen atoms. The fact that ethyl radicals are selectively formed as the
The Journal of Physical Chemistry, Vol. 83, No. 12, 1979
Thermodynamics of Electrolytes
trapped hydrogen atoms are detrapped clearly demonstrates what happens if hydrogen atoms are mobilized in the matrices. The reactions of mobile hydrogen atoms in such solid phases must be diffusion controlled especially at low temperature as is suggested from the suppression of hydrogen atom migration below 77 K and immobilization a t 4.2 K. It is likely that the reactions would be more or less diffusion controlled even at 77 K, so that simple competition may not be assumed even for mobile hydrogen atoms. The steric and orientational factors may be also important for the diffusion-controlled reactions of hydrogen atoms in such solid phase^.^ It is suggested that factors such as the efficiency of the hydrogen atom immobilization and the diffusion of hydrogen atoms in the matrices, as well as steric and orientational factors and the reactivities of hydrogen atoms with matrix and solute molecules are correlated in a complicated manner and play an important role in the selective formation of solute radicals at 77 K and its suppression at temperatures lower than 77 K. Unless all of these factors are properly taken into consideration, the sound understanding of the solid state reactions of hydrogen atoms may not be obtained. Note Added in Proof. Recently Katsumura et al.l0have reported that selective radical formation is not observed in the plastic crystal phase of neopentane. In the light of the present study, it may be suggested that the higher
1599
diffusion velocity of hydrogen atoms in the plastic crystal phase leads to a higher probability of recombination reaction 8 than that of abstraction reaction from solutes (reaction 7). They have also reported that the solute radical yield relative to the total yield in the crystal phase levels off at high irradiation doses. This may be also interpreted by the relative importance of recombination reaction 8 at higher doses. These results seem to suggest that the reactions of hydrogen atoms in crystalline neopentane are diffusion controlled. References and Notes (1) M. Iwasaki, K. Toriyama, K. Nunome, H. Fukaya, and H. Muto, J . Phys. Chem., 81, 1410 (1977). (2) M. Iwasaki, H. Muto, K. Toriyama, H. Fukaya, and K. Nunome, J. Phys. Chem., preceding paper in this issue. (3) T. Miyazaki, K. Kinugawa, and J. Kasugai, Radlat. Phys. Chem., 10, 155 (1977). (4) M. Iwasaki, K. Toriyama, H. Muto, and K. Nunome, Chem. Phys. Lett., 56, 494 (1978). (5) K. Toriyama and M. Iwasaki, J . Phys. Chem., 82, 2056 (1978). (6) K. Toriyama, K. Nunome, and M. Iwasaki, J. Am. Chem. Soc., 99, 5823 (1977). (7) W. Gordy and R. Morehouse, Phys. Rev., 151, 207 (1966). (8) K. Toriyama, M. Iwasaki, and K. Nunome, The 21st Symposium on Radiation Chemistry, Tokyo Japan, Sept 30, 1978. (9) See, for example, M. Iwasaki, MTP Int. Rev. Sci.: Phys. Chem., Ser. One, 4,339 (1972); M. Iwasaki and K. Toriyama, Chem. fhys. Lett., 41, 59 (1976). (10) Y. Katsumura, A. Fujita, H. Kadoi, K. Ishigure, and Y. Tabata, Radiat. Phys. Chem., 12, 69 (1978).
Thermodynamics of Electrolytes. 12. Dielectric Properties of Water and Debye-Huckel Parameters to 350 "C and 1 kbar Daniel J. Bradley and Kenneth S. Piker" Department of Chemistry and Lawrence Berkeley Laboratory, university of California, Berkeley, Callfornia 94 720 (Received December 13, 1978) Publication costs assisted by the University of California
In preparation for work with aqueous electrolytes at above saturation pressures and at temperatures to 350 "C, an equation was developed for the representation of the dielectric constant of water over this range. With this equation and an equation of state for water, the Debye-Huckel limiting law parameters for osmotic and activity coefficients, enthalpies, heat capacities, volumes, compressibilities,and expansibilities were calculated and are presented.
We are extending our investigations of the thermodynamic properties of aqueous electrolytes a t high temperatures to conditions at higher-than-saturation pressure. It is expected that at these higher pressures the solutions will retain to higher temperatures the properties associated with strong electrolytes. As the critical temperature is approached along the saturation curve, the dielectric constant decreases rapidly and various properties can be expected to show anomalies. However at somewhat higher pressures, properties show more gradual change and should be subject to more accurate interpretation. Also many cases of practical interest, such as equilibria with solid phases under geological conditions, occur a t above saturation pressure. In earlier work focused on the saturation pressure, Silvester and Pitzerl developed an equation for the dielectric constant of water extending to 300 "C.While this 0022-3654/79/2083-1599$0 1.OO/O
equation included terms for pressure or density dependence, the pressure derivative of the dielectric constant is not as accurately represented as is desirable or possible. Consequently we sought a new form of equation which will more naturally represent the volume or pressure dependence of the dielectric constant while still representing its temperature dependence. Dielectric Constant The dielectric constant of water has been measured as function of temperature and pressure by relatively few investigators.2-6 At temperatures above 70 "C there are only two sets of data, those of Akerlof and Oshry5 along the vapor saturation curve from 100 to 370 "C and those of Heger3 from 100 to 550 "C and saturation pressure to 5000 bar. The accuracy of most of the high temperature data is better than 1%.Below 70 "C, however, the data 0 1979 American
Chemical Society
