J. Phys. Chem. 1980, 84, 1991-1995
Back-reactions of the type represented by eq 11 and radical reactilons such as H+(solv) OH-(solv)
+ e-
+ h+
-
+
13.
(12)
OH-
(13) (14)
He + OH. H20(1) +
become especially important in the photosynthetic mode. A clear advantage of PEC cells in this regard lies in an efficient meains of separating the reaction products; Le., oxidation and reduction reactions ace carried out at different sites (at the photoanode and counterelectrode, respectively, for the case of n-type semiconducting electrodes) rather than on the same grain as in the case of the photosynthetic mode. Other loss mechanisms such as recombination of photogenerated carriers in the semiconductor bulk also become more predominant in the photosynthetic mode particularly because a field-assisted means of separating the carriers (e.g., by means of an external bias) does not readily exist unlike in the case of PEC cells. Partial metallization of the semiconducting grains would,, however, serve to reduce recombination losses in the bulk. ‘This approach has worked well with other photocatalytic and photosynthetic reactions on Ti02 and SrTi0314 Work is continuing in this laboratory on the effect of metallization of H202 and H2 yields utilizing various semiconducting materials. Acknowledgment. This research program was funded by the U.S.Department of Energy under Grant G-77-C02-4258.
References and Notes (1) (a) S. R. Morrlson and T. Freund, J . Chern. Phys., 47, 1543 (1967); (b) W. P. Oomes, T. Freund, and S. R. Morrison, Surf. Sci., 13, 201 (1968); (c) (3.Gerischer and K. Cammann, Ber. Bunsenges. Phys. Chem., 7 6 , 385 (1972); (d) H. Yoneynama, Y. Toyoguchi, and H. Tamura, J. Phys. Chem., 76, 3460 (19’72); (e) M. Miyake, H. Yoneyama, and H. Tamura, Electrochim. Acta, 21, 1065 (1976); (f) M. Miyake, H. Yoneyama, and H. Tamura, ibid., 22, 319 (1977); (9)
I991
S. N. Frank and A. J. Bard, J . Am. Chem. Soc., 99, 303 (1977); (h) M. Mjake, H. Yoneyama, and H. Tamura, Bull. Chem. SOC.Jpn., 50, 1492 (1977); (i) 8. Kraeutler and A. J. Bard, J. Am. Chem. Soc., 99, 7729 (1977); (i)ibM., 100,2239 (1978); (k) ibM., 100, 4317 (1978); (I) ibid., 100, 5985 (1978); (m) M. Miyake, H. Yoneyama, and H. Tamura, J. Catal., 58, 22 (1979); (n) H. Reiche, W. W. Dunn, and A. J. Bard, J. Phys. Chem., 63, 2248 (1979); (0)F. F. Fan and A. J. Bard, J . Am. Chem. Soc., 101, 6139 (1979). (a) A. J. Nozik, Annu. Rev. Phys. Chem., 29, 89 (1978); (b) W.A. Gerrard and L. M. Rouse, J . Vac. Sci. Techno/., 15, 1155 (1978); (c) L. A. Harris and R. H. Wilson, Annu. Rev. Mater. Sci., 8, 99 (1978); (d) K. Rajeshwar, P. Singh, and J. DuBow, Electrochim. Acta, 23, 1117 (1978); (e) H. P. Maruska and A. K. Ghosh, Sol. Energy, 20, 443 (1978); (f) M. Tomkiewicz and H. Fay, Appl. Phys., 18, 1 (1978). (a) M. C. Markam and K. J. Laldler, J. Phys. Chem., 57, 363 (1953) (seealso references cited in this article); (b) T. R. Rubin, J. G. Calvert, G. T. Rankin, and W. M. MacNevin, J. Am. Chem. SOC.,75, 2850 (1953); (c) J. 0. Calvert, K. Theurer, G. T. Rankin, and W. M. MacNevin, ibid., 76, 1575 (1954); (d) 0. A. Korsumovskii and Yu. S. Lebedev, Russ. J. Phys. Chem. (Engl. Trans/.),35, 528 (1961); (e) T. Freund and W. P. Gomes, Cafal. Rev., 3, 1 (1969); (f) D. R. Dixon and T. W. Healv. Aust. J. Chem.. 24. 1193 (1971): (a) M. D. Archer, J . Appl. Eiectrochem.,5, 17 (1975); (h) J. R. Ha6our and M. L. Hair, J . Phys. Chem., 83, 652 (1979). R. E. Stephens, B. Ke, and D. Trlvlch, J. Phys. Chem.,59,966 (1953). J. J. Rowlette, Sol. Energy, 7, 8 (1963); (a) M. S. Wrlghton, D.S. Ginley, A. B. Ellis, P. T. Wolcranskl, D. L. Morse, and A. Linz, Proc. Natl. Acad. Sci. U.S.A.,72, 1518 (1975); (b) P. Clechet, C. Martelet, J. R. Martin, and R. Olier, €lectroch/m. Acta, 24, 457 (1979). D. J. Savage, Analyst (London), 78, 224 (1951). 1. M. Koltholi and P. J. Eking, Eds., “Treatise on Analytical Chemistry", Vol. 7, Part 11, Intersclence, New York, p 7. L. Kuechler and H. Pick, Z . Phys. Chem. Abt. B, 45, 116 (1939). The equilibration time for the gas chromatograph was found to be -20 min from calibration experiments. Thls Induction period is shown as dotted lines in Figures 2a and 3. H, yields quoted in this paper are also uncorrected for the gas dissolved in the solution. The initial saturation of the evolved gas in the solution phase would account for part of the induction period. “Semiconductor Liquid-Junction Solar Cells”, The Electrochemical Society, Princeton, NJ, 1977, Chapter VII, p 272. (a) H. Gerlscher in “Solar Power and Fuels”, J. R. Bolton, Ed., Academic Press, New York, 1977, Chapter 4; (b) H. Gerischer, J . Vac. Sci. Techno/.,15, 1422 (1978); (c) W. M. Latimer, “Oxidation Potentials”, Prentice-Hall, Englewood Cliffs, NJ, 1952. H. W. Gundlach and K. E. Heisler, Z . Phys. Chem. (Wiesbaden), 112, 101 (1978). M. S. Wrighton, P. T. Wolczanskl, and A. B. Ellis, J. SolM State Chem., 22, 17 (1977).
Kinetics OF Phase Separatlon in Binary Liquid Mixtures J. Wenzel, U. Limbach,t G. Bresonik, and G. M. Schnelder” University of Bochum, Department of Chemistiy, Bochum, West Germany (Received November 13, 1979) Publicatlon costs assisted by the University of Bochum
The kinetics of liquid-liquid phase separation was studied in some binary liquid mixtures (2-butoxyethanol-H20; triethylamine-H20; nitrobenzene-2-methylbutane) by observing light scattering due to the formation of droplets of the new phase. Demixing was initiated by relaxation techniques (pressure-or temperature-jump);here pressure or temperature was changed within less than 0.1 ms to get very rapidly from the homogeneous into the heterogeneous region. After such a jump the intensity of the scattered light ran through a maximum value as a function of time, and the time t, between passing the coexistence curve and the occurrence of the intensity maximum varied between and lo1 s depending on how far the jump entered the miscibility gap. The dependence of t, on jurnp width was very strong for low supersaturations but became much smaller with increasing quench depth. The transition between these two types of behavior was found to occur at a definite superheating temperature T,which is assumed to be related to the so-called spinodal curve. These findings are supported by experiments with a1 different method using light absorption by dyes.
I. Introduction In the thermodynamic theory of liquid-liquid &mixing1 the limit between one-phase and two-phase regions in the +Deceased.
isobaric T-x diagram is called the coexistence or connodal curve where the chemical potentials of each component in the two coexisting phases are equal (see Figure 1); in the following the temperatures on this curve will be denoted by TE. The mutual solubility of liquids is also
0022-3654/80/2084-1991$01.00/00 1980 American Chemical Society
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The Journal of Physical Chemistry, Vol. 84, No. 15, 1980
Wenzel et at.
UCSl 110 o c
T
T
LCSl 5 0 OC
i 0.0
I
I
x2c
x2
1.0
Flgure 1. Isobaric T-xphase diagram of a binary liquid mixture showing liquid-liquid phase separation (schematic).
pressure dependent, and the miscibility gap may become smaller or larger with rising pressure.2 Inside the miscibility gap we find the spinodal curve where according to the classical theory the second derivative of the Gibbs energy with respect to composition becomes zero. The spinodal curve surrounds a region where one liquid phase is unstable. Between the connodal and spinodal curves the metastable region is found. In the metastable region the appearance of a new phase is known to proceed via “n~cleation”,~ the formation of small nuclei of the new phase in the “mother phase’’ (frequently showing a time lag) and their growth to macroscopic dimensions by transport of material from the surroundings. During the last two decades discussion has also been initiated concerning demixing phenomena in the unstable region (so-called spinodal decomp~sition).~-~ Especially interesting in this context is the behavior of the binary diffusion coefficient D12. According to the classical theory it is directly proportional to the second derivative of the Gibbs energy with respect to composition and thus positive in the stable and metastable regions, zero on the spinodal curve, and negative in the unstable region. For a discussion of the assumptions made, of the sign of DI2, and of its influence on demixing, see, e.g., Hilliarda5 The study of the formation of a new liquid phase in a liquid mixture was begun only a few years and the phenomenon of metastability has scarcely been observed in low molecular weight nonelectrolyte solutions up to now; the reason for this may be the (normally) high speed of liquid-liquid decomposition. For polymer solutions, however, metastable conditions could be realized by using the method of pulse-induced critical scattering (PICS).14 In the present work15 relaxation techniques developed for the investigation of fast reactions in solutions (such as temperature- and pressure-jump) are used to enter the two-phase region starting from a homogeneous mixture; then the intensity of scattered light is measured as a function of time. A similar method was used for selected polymer solutions by Smolders, van Aartsen, and Steenbergen.16 11. Experimental Section A . Pressure-Jump Measurements. Most of the measurements in this work were performed on the liquid-liquid system 2-butoxyethanol-water with a pressure-jump apparatus that has been developed by other authors17J8and will be described in detail e1~ewhere.l~ The system exhibits a closed immiscibility loop which vanishes with rising pressure,lgso that it is possible to get from the homogeneous to the heterogeneous region by a pressure drop or a temperature rise, respectively (see Figure 2). The
Flgure 2. p- T-x phase diagram of a binary liquid system with a closed miscibility gap at normal pressure that vanishes with increasing pressure, schematic representation of measuring technique used for P-butoxyethanol-H,O (temperature and pressure data given refer to this system).
pressure
d p - 1 3 8 bars (in co. 100 ,us1
I L
0
I
I
5
10
tlms
Flgure 3. Scattered light intensity and pressure vs. time tfor a typical pressure-jump experiment on 2-butoxyethanol-H20 (Ap = 138 bar, X(H,O) = 0,843; = 66.9 OC; r, = 61.5 OC).
r
pressure drop from about 140 bar down to 1 bar is performed within about 100 bs by a rupture diaphragm. During the experiments the intensity of scattered light under an angle of 90’ and the pressure change are observed simultaneously, and the signals stored with transient recorders. The information is then plotted on an x-y chart recorder and may be analyzed graphically (for details see ref 15). A characteristic experiment is represented in Figure 3. Similar curves have been obtained by Smolders, van Aartsen, and Steenbergen16for selected polymer solutions where the time evolution, however, was slower by a factor of about lo5. In Figure 3 the intensity of scattered light runs through a maximum as a function of time. According to the socalled Mie t h e ~ r this y ~ maximum ~ ~ ~ is attributed to a definite radius of a growing droplet. The time between passing the coexistence curve and the occurrence of the maximum of scattered light intensity corresponds to the time t , that a nucleus needs to be formed and to grow up to a sphere of about 0.2 pm (wavelength of incident light 600 nm). This maximum time t , is observed to be strongly dependent on how deeply the system is quenched into the miscibility gap. At a given composition of the mixture, t , has been found to vary up to five orders of magnitude by changing the depth of quenching only (see Figure 4). B. Temperature-Jump Measurements. Other measurements were performed in a temperature-jump apparatus described e l ~ e w h e r e . ~A~ ~small * ~ amount of salt, however, had to be added to the solution in order to obtain a sufficiently high electrical conductance to produce a fast
The Journal of Physical Chemistry, Vol. 84, No. 15, 1980
Kinetics of Phas,e Sepavatlon in Liquid Mixtures
'rc--
1
t
1
1
1
' 1 \
1993
',
9
1
i
3
1'1,
i
Flgure 4. Plot of log t, vs. T - T, for pressure-jump experiments on 2-butoxyethanol-H20 (x(H,O) = 0.908; T, = 49.1 OC).
temperature rise by passing the energy of a high-voltage capacitor through the solution. Here similar results have been found as with the pressure-jump technique. The Mie theory was used for the discussion of the intensity of scattered light as a function of time obtained in both types of experiments. It may be applied if the size distribution of growing droplets is sufficiently homogeneous. Then 11change in concentration of scatterers would only change the intensity of the maximum, whereas the radii of the scattering spheres would be the same as before. The problem becomes more complex; if the concentration of scatterers is; so high that multiple scattering occur^.^^^^ Our measurernents where scattered and transmitted light intensities were recorded simultaneously indicate that in the present experiments multiple scattering may be neglected.
111. Results A . Results of Pressure-J u m p Measurements. Figure 4 shows a typical logarithmic plot of the maximum time t , against the temperature T of the mixture before the pressure release. With TEbeing the static decomposition temperature at 1bar, T - TE= 0 K would then correspond to the temperature of decomposition at atmospheric pressure. The plot shows a strong dependence of t , on T - T E for low supersaturation. This dependence abruptly becomes much smaller at a definite quench depth, and at still higher supersaturation t, becomes essentially constant. Here as well as in Figures 3, 6 , and 7 the temperature T i s not yet corrected for adiabatic cooling (see below). It is also possible to measure the time t0.5, until the intensity of the Scattered light reaches half its maximum value which was already used by Smolders et a1.16 According to the Mie theory t0.5, corresponds to a definite but smaller radius (see Figure 5a). Another possibility is to perform quenches with the same depth but at different wavelengths (correspondingto different radii of the scatterers at ;maximum intensity, see Figure 5b). The interesting result is that the temperature Ts at which the supersaturation dependence changes abruptly remains essentially tht, same. It should be mentioned that the adiabatic expansion of the liquid mixture produces a temperature drop so that the temperature measured just before the pressure jump in the autoclave is not the same as that at which the 1
1
2
3
L
IT-TE)/K
5
,
1
I
2
3
L
5
(T-TE)/K
Figure 5. Pressure-jump experlments on the system P-butoxyethanol-H,0: (a) plot of log t, and log to.smvs. T- TE (x(H,O) = 0.974; TE = 47.8 OC); (b) pbt of log t, vs. T - TE at three different wavelengths (x(H2O) = 0.970; TE = 47.6 "C).
mixture undergoes phase separation. The temperature TS of the break in the logarithmic diagram must then be diminished by the temperature drop, the value of which is given by aT AT = -Ap PCP
where a = the cubic expansion coefficient, p = the density, and cp = the specific heat capacity. Data for a,p, and c were taken from the literature for and some binary mixtures;30the the pure actual parameters under experimental conditions were obtained from a computer program which used second- and third-order regression and Redlich-Kister fitting, respectively, to interpolate and extrapolate the literature data as a function of temperature and composition.16 The equation for AT used above only holds for a homogeneous sample. A shift in concentration which in principle begins already during the pressure jump adds another term to AT that was estimated to be less than 0.1 K and was neglected. C. Results of Temperature-Jump Measurements. For comparison one binary mixture with a small amount of electrolyte (KN03) added was studied in the pressure- and temperature-jump apparatus as well. The salt lowers TE, an effect that is frequently ~bserved.~' Additionally, the difference between TE and the break temperature Ts decreased. In the temperature-jump experiments the maximum of scattered light intensity was not as distinct as in the pressure-jump measurements. Thus a plot of log t0.5,,, against the square of capacitor voltage (which is proportional to the temperature change) was preferred. Unfortunately the actual break temperature Ts could not be determined exactly from such a plot since not all physical parameters which are necessary to calculate the temperature rise as a function of capacitor charge for the apparatus used were known. It could, however, be shown that the temperature-jump results qualitatively support the pressure-jump investigation^.^^ D. Effect of Composition and Phase Diagram. The measurements on liquid mixtures of different composition in Figure 6 show that the corrected temperature Ts at which the break in the logarithmic plot occurs lies higher above the static decomposition temperature TE the more the composition is different from critical; additionally, a striking analogy is found between Figures 1 and 6. This essential point will be discussed in section IV in more detail. E. Influence of Constituents of the System. In order to prove that the observed effect is not limited to special
substance^^^^^
1994
The Journal of Physical Chemistry, Vol. 84, No. 15, 1980
Wenzel et al.
I 090
085
095
x(H20)
100
Flgure 6. T, and T, vs. x(H,O) for 2-butoxyethanol-H20 (obtained from pressure-jump experiments; see text). T, is corrected for adiabatic cooling.
0
-m0 4
3tI 1
I
P -15
-10
-05
I
00 (T-TE
I
105
I +10
IIK
Figure 7. Plot of log t, vs. T- TE for pressure-jump experiments on nitrobenzene-2-methylbutane (x(nitrobenzene) = 0.63; TE = 21.5 OC; see Figure 4).
liquid-liquid systems (e.g., aqueous solutions showing lower critical solution temperatures (LCST)), a system that differed very much from 2-butoxyethanol-water was chosen for further investigations, namely, nitrobenzene-2methylbutane, which is a nonaqueous liquid system with an upper critical solution temperature (UCST). Indeed the results presented in Figure 7 show the same effect as before. It has to be taken into account that here, however, the adiabatic cooling increases the effect of the pressure quench (see Figure 7); as already mentioned above the T values are not corrected for this effect. Since the physical parameters of the system are somewhat unfavorable (Le., the UCST exceeds the boiling temperature of 2-methylbutane at normal pressure), it was unfortunately not possible to investigate the miscibility gap over the whole concentration range in the apparatus used for the present experiments. Additional measurements are underway. IV. Conclusions
The strong dependence of log t , (or log to.6m)on supersaturation resembles that of nucleation effects in other systems where similar large changes over several orders of magnitude are usually found in the metastable region. Add to this the close analogy between Figures 1 and 6. These findings suggest that the demixing phenomena in the range of rapidly changing t , (or values should happen in the metastable region and that the break in the logarithmic plot should mark the entrance into the unstable region, this break being situated near the spinodal itself. In each case the experiments show two different ranges to be present in the heterogeneous region of liquid
100
200
tlps
300
Flgure 8. Intensity of transmitted light It,ans vs. time t for a mixture of triethylamine (TEA)-H,O with small amount of PADA (w(H,O) = 0.919; PADA = 2 X mol dm3; capacitor voltage = 14.05 kV): (a) initial temperature, 18.8 O C ; (b) initial temperature, 19.5 OC.
binary mixtures that might be attributed to “metastable” or “unstable” states irrespective of how a “spinodal” will be defined. These results are confirmed by measurements obtained from a quite different method that has been developed by Jost, Limbach, and L i ~ h a r dit; ~will ~ be discussed shortly in the following. The calibration procedure of a relaxation apparatus with optical detection usually involves the use of solutions of acid-base indicators in a buffered system of given pH. Usually the reaction rates of these systems are high enough to follow the change of an external parameter (e.g., temperature) nearly simultaneously. Thus the temperature jump following the capacitor discharge produces a rapid change of transmitted light intensity that depends exponentially on time; consequently each intensity corresponds to a definite temperature. Triethylamine-water with an appropriate indicator (e.g., trans-pyridine-2-azodimethylaniline (PADA)) allows temperature measurements of this kind; in addition, it exhibits a miscibility gap with a lower critical solution temperature. Thus it should be suited to follow the change of temperature in a quench experiment. First investigations on this system gave a surprising result. For moderate quench depth the experimental curves have the expected shape (see Figure 8a). The intensity of transmitted light increases as a consequence of indicator absorbance change caused by the temperature rise, and then it decreases because of the beginning of light scattering from growing droplets of the new phase. If the jump is sufficiently large, however, the curves exhibit the anomaly shown in Figure 8b. First cU/dt shows the same behavior as before but then after a sudden break its absolute value becomes distinctly larger before transmission decreases again as a consequence of light scattering. BresonikZ4could show that the temperature associated with this phenomeon lies above the static demixing temperature. Some of her measurements are represented in Figure 9. Here the composition dependence resembles that of the light scattering experi-
J. Phys. Chem. 1980, 84, 1995-1997
30
T
Ik L
o 01
References and Notes
T jumplwith indicator)
-
o T-jump iscattisring 1
(1) J. S. Rowlinson, “Liquids and Liquid Mixtures”, 2nd ed., Butterworth, London, 1969. (2) G. M. Schneider, Spec. Period. Rep.: Chem. Thermodn.,2, (1978). (3) A. C. Zettlemoyer, Ed., “Nucleation”, Marcel Dekker, New York, 1969. (4) J. W. Cahn, Trans. Metall. SOC. AIM€, 242, 166 (1968). (5) J. E. Hilliard in “Phase Transformations”, American Society for Metals, Metals Park, Ohio, 1970, Chapter 12. (6) J. S.Langer, Ann. Phys. ( N . Y . ) , 85, 53 (1971). (7) K. Binder and D. Stauffer, Adv. Phys., 25, 343 (1976). (8) R. Reich and M. Kahlweit, Z.Phys. Chem. (Frankfurtam Main), 27, 80 (1961). (9) A. Jost and G. M. Schneider, J . Phys. Chem., 79, 858 (1975). (10) U. Limbach, A. Jost, and G. M. Schneider, J . Phys. Chem.,80, 1952
-1
OC
20
I
1995
l
(1976\ \.-.
-I.
a Figure 9, Concentration dependence of phase separation in the system triethylamine (TEA)-water according to the indicator method ( B r e s ~ n i k ) ~ ~ and the light-scattering method (Wenzel)” (see text).
ments shown in Figure 6. In order to check this resemblance a triethylaminewater mixture was studied in the same temperature-jump apparatus as used by B r e ~ o n i kthis , ~ ~time, however, without any addition of indicator and using the light-scattering method. The result is additionally given in Figure 9; it fits well into the other points obtained by the quite different indicator method. Thus it is shown that the indicator and the light-scattering methods (the latter being also used in the pressure-jump experiments) give consistent results. As can be seen from the time scale of Figures 4 and 8b, the indicator method, however, seems to be sensitive to a much earlier state of demixing, and the break in the transmittance curve in Figure 8b may be interpreted as the early beginning of phase separation. New experiments are underway to interease the data that heretofore have been rather sparse and inaccurate. Acknowledgment. The authors thank Professor A. Jost, Bielefeld, andl Professor G. Ilgenfritz, Koln, for their help in constructing the pressure-jump apparatus. Financial support of thle Fonds der Chemischen Industrie e.V. is gratefully acknowledged.
(11) R. Strey, Dissertation, Unlversity of Gottingen, Gottingen, West Germany, 1978. (12) J. S. Huang, W. I.Goldburg, and A. W. Bjerkaas, Phys. Rev. Lett., 32, 921 (1974). (13) N. C. Wong and C. M. Knobler, J . Chem. Phys. 89, 725 (1978). (14) K. Durham, J. Goldsbrough, and M. Gordon, Pure Appl. Chem., 38, 97 (1974). (15) J. Wenzel, Dissertation, University of Bochum, Bochum, West Germany, in preparation. (16) C. A. Smolders, J. J. van Aartsen, and A. Steenbergen, KolloidZ. Z . Polym., 243, 14 (1971). (17) W. Knoche and 0. Wiese, Rev. Sci. Instrum., 47, 220 (1976). (18) G. Ilgenfrltz, University of Cologne, Cologne, West Germany, prlvate communication. (19) G. Poppe, Bull. SOC. Chim. Belg., 44, 640 (1935). (20) G. Mie, Ann. Phys. (Leipzig), 330, 377 (1908). (21) M. Born and E. Wolf, “Principles of Optics“, Pergamon Press, Oxford, 1965, pp 633-64. (22) H. C. van de Hulst, “Light scattering by Small Particles”, Wiley, New York, 1957. (23) M. Kerker, “The Scattering of Light“, Academic Press, New York, 1969. (24) G. Bresonik, Dlplom thesis, University of Bochum, Bochum, West Germany, 1979. (25) S. W. Churchill, G. C. Clark, and C. M. Sliepcevich, Discuss. Faraday SOC.,30, 192 (1960). (26) D. H. Woodward. J . Oot. SOC.Am.. 54. 1325 (1964). (27) C. Smart, R. Jacobsen; M. Kerker, J. P . Kratohvil; and E. Matijevie, J . Opt. SOC.Am., 55, 947 (1965). (28) G. S. Kell and E. Whalley, Phi/os. Trans. R. SOC.London, 258, 565 (1965). (29) A. M. Sirota and P. E. Beljakov, Teploenergetika, 8, 67 (1959). (30) U. Onken, Z . Elektrochem., 83, 321 (1979). (31) W. A. P. Luck, Ber. Bunsenges. Phys. Chem., 89, 69 (1965). (32) A. Jost, U. Llmbach, and K. G. Llphard, University of Bcchum, Bochum, West Germany, private comrnunicatlon.
Improved Diffusion-Model Analysis of Cage Reactions of Chiral Radical Pairs John F. Garst Department of Chemistry, UniversiYy of Georgia, Athens, Georgia 30602 (Received:March 3, 1980)
Naqvi, Mork, and Waldenstrram have obtained recently the exact form of Noyes’ particle-pairreactivity function h(t). They suggest that serious errors can arise through the use of approximate forms proposed earlier by Noyes. An analysis of cage reactions of chiral radical pairs1 using one of Noyes’ approximate forms is improved here by using the exact form. The previous conclusions are generally supported, but the new results show that the predictions of diffusion and first-order models can differ somewhat more than was indicated by the earlier calculations.
Recently I pointed out that the distribution of cage products from decompositions of RMR, assumed to occur by simultaneous cleavage of both R-M bonds, is predicted differently by “first-order” and “diffusion” models of geminate radilcal pair dynamics when R is a chiral group bonded at its lchiral center (eq 1).l Here M contains the 0022-3654/80/2084-1995$01 .OO/O
elements of a small molecule (e.g., N2,COz),RR and R’R’ are enantiomers, and RR’ is meso. It is assumed that radical orientation at the moment of a collision that might give reaction is the determining factor for product identity, and it is assumed that interconversions of (R. Re), (R. R’.), and (R’. R’.) are first order, with the same rate constant 0 1980 American Chemical Society
