NOTES
747
(P,- Po)
=
Atn = P
(1)
where P, is any parameter whose changes with time (t) and temperature (T’K) are studied (for example: lattice parameter, resistivity, weight change, grain size, etc.), Pois its initial value, and A and n are constants a t any given temperature. Generally, A changes with temperature in some regular manner, while n is either a constant or shows some temperature dependence depending on the nature of the process. Examples of kinetic processes representable by eq 1 are well kn0wn.1-~ The activation energy (E,) for the process can be obtained by a simple Arrhenius plot. Often the constant A has an explonential temperature dependence similar to that expected for rate. Because of this there exists a temptation to obtain the activation energy using the temperature variation of A from the slope of In A vs. 1/T; for example, see Martens, et aLs Generally, the value obtained from the slope of In A vs. 1/T is not the correct activa,tion energy. As becomes evident, only under specific circumstances can one obtain the correct activation energy from such a graph. From eq 1 rate = dP/dt = nAtn-’ = nP/t In (rate) = In (nP)
- In t
(2)
(3)
From (1) and (2) rate = nAtn-1 = nAl/npl-l/n In (rate) = lni n
+ (l/n)In A + (1 - l/n)In P
(4) (5)
Often it is valid to assume that the temperature variation of A is given by
A = A&-Q/RT =
In. n
+ (l/n) In
A0
Q/nRT
P -
= Al/n pl-1/72
1
(9)
(10)
Thus, a graph of In (P/t) vs. In P yields a family of parallel lines for various temperatures (when n # f(T)). A plot of the intercept l / n In A vs. 1/T yields the correct activation energy Ea. Furthermore, for a chosen extent of transformation (P) the instantaneous rate at any temperature is proportional to the ordinate (P/t) in the graph In (P/t) os. In (P), and the proportionality constant (n) can be evaluated from the slope of the same graph. (1) P. Gordon, Trans. AIME, 203, 1043 (1955). (2) Creep and Recovery, ASM Publication, Cleveland, Ohio, 1957. (3) W. R. Price, S. J. Kennett, and J. Stringer, J . Less-Common Metals, 12, 318 (1967). (4) R. K. Linde, Trans. AIME, 236, 58 (1966). (5)H.E. Martens, L. D. Jaffe, and D. D. Button, Jet Propulsion Laboratory Progress Report No. 20-373, 1958. (6).If n # f(T), then In P vs. In t yields a family of parallel lines for various temperatures and the amount of “shift” necessary to make these parallel lines coincide can be used to determine the activation energy. Furthermore, if n is independent of temperature, it implies a single valued activation energy for the process in the temperature ranges studied, while a temperature-dependent n implies a process having a spectrum of activation energies. (7) W. V. Kotlensky, Carbon, 4, 212 (1966).
Motion : Photodiffusiophoresis
(7) by J. R. Brock
From the Arrhenius equation ln (rate) = In Ro
pl-l/n
t
Some New Modes of Aerosol Particle
-
+ (1 - l/n)In P
n__ P = nAl/n
(6)
This being the case, we have from ( 5 ) and (6)
In (rate)
for n # f(T), such a graph yields the correct Ea, it is to be noted that In (l/t) does not give the actual value of the rate. The above problems can be overcome simply by the following procedure. From (2) and (4)
- E./RT
(8)
To obtain the activation energy Ea, a graph of In (rate) against 1/T for a given P is used. If n is independent of temperatu~re,~ a plot of In (rate) vs. 1/T yields a family of parallel lines for various values of P , and Ea is obtained from the slope.’ Using (7) and (8) it can be shown that when n # f(T), E . = Q/n. Hence, Ea is equal to Q as obtained from the slope of In A vs. 1/T only when the kinetic relationship given in eq 1 is linear (when n = 1, Ea = Q). It is also worth noting that even though rate = f(P), if n # f(T), then E . # f(P) and E , =: Q/n. However, if n = f(T) then E , = f (P)and E. arid Q are not related in any simple manner. Often to obtain Ea, In ( l / t ) (where t is the time necessary to attain. a given extent of transformation P) is plotted against 1/T instead of In (rate). Although
Department of Chemical Engineering, The University of Texas, Austin, Texas 78711 (Received September 11, 1967)
The purpose of this note is to point out some new modes of aerosol particle motion which do not appear to have been previously observed. These new modes, which we term “photodiffusiophoresis,” arise from molecular diffusion velocities produced by electromagnetic radiation incident on a particle. The electromagnetic radiation produces the diffusion velocities by altering the chemical or physical equilibrium between an aerosol particle and a surrounding gas mixture. Photodiff usiophoresis therefore is quite different from what has usually been termed “photophoresis,”l which (1) N. A. Fuchs, “The Mechanics of Aerosols,” Pergamon Press, New York, N. Y., 1964.
Volume 71, Number B February 1968
NOTES
748 occurs even in a pure gas and which in mixtures involves no diffusive motion of the gas molecules. Photophoresis describes the particle motion owing to unequal heating of the particle surface by incident electromagnetic radiation; this unequal heating causes gas molecules which rebound from the particle surface to impart more momentum from the hot portions of the surface than from the cold portions with D resultant force on the particle. I n photophoresis, unlike photodiffusiophoresis, the molecular fluxes to and from the particle surface are equal. The term “photophoresis,” as it has been applied in the past, should be changed more properly to “photothermophoresis,” and both photothermophoresis and photodiffusiophoresis could be described by the term photophoresis. We shall consider in some detail an example of photodiff usiophoresis which arises as a result of photocatalytic reaction at a particle surface. Brief mention will be given to two other examples of photodiffusiophoresis which arise from radiation-induced variations over the particle surface of surface reactions and of vapor pressure.
Some Illustrations of Pbtodiffusiophoresis Here we shall consider only effectively spherical particles with radius R much less than the mean free path, L, of surrounding gas molecules. For R 0, and f i - is the function for molecules hitting the surface, dS. v i < 0. For simplicity in the analysis which follows, it will be assumed that all the fi- are equilibrium distribution functions
-
minated in a non-uniform manner cause photodiffusiophoresis. We select a coordinate system fixed in a stationary particle with the z axis in the direction of the electromagnetic radiation energy flux, Is,impinging on the particle. It is supposed that a single elementary reaction is photocatalysed at the particle surface
Also, the reactant species are similarly related.
-
= ni-(rn,/27rkT)”/” exp( -m,vi2/2kTf
(2)
For the reflected molecules from the surface, assumptions concerning the distribution function will be made appropriate for the physical process occurring. We proceed now to a detailed examination of photodiflusiophoresis produced by a photocatalytic reaction at a surface. It appears to be well established experimentally3 that photocatalytic reactions occur on various metallic oxides, sulfides, etc. The photocatalytic oxidation of CO on ZnO and CuO has received much attention.3 It is easy to show in the free-molecule region that such photocatalytic reactions on particles illuThe Journal of Physical Chemistry
All the Oi are some complicated functions of the state of the particle surface, the state of the molecules taking part in the reaction, and the intensity, wavelength, polarization, etc., of the incident radiation. We make the simplifying assumption that the Oi are independent of the intensity of the incident radiation above some minimum level of intensity. This assumption does not appear unreasonable for at least one system.* With these various assumptions and the application of eq 1, we obtain the following expression for the force F,, giving rise to photodiffusiophoresis (2) L. Waldmann, 2.Naturforsch., 14a, 689 (1969). (3) T. S. Nagarjunan and J. G. Calvert, J . Phys. Cfiem., 6 8 , 17 (1964).
NOTES
749 Intramolecular Comparison of the Insertion into the C-H Bonds of Alkanes by Singlet Methylene Radicals la
where reactant R1has been selected as a reference species and T is the temperature of the gas surrounding the particle. The sum over Pr,the products, is understood to include only those species which leave the particle surface. It is also possible to imagine the particle being comsumed in a photosensitized reaction. A similar expression for Fp would result. Approximate calculations, using eq 8 and the results of experiments for the photocatalytic oxidation of CO on ZnO given in ref 3, show that for 10% CO in air and 0.25-p particles of density of the order of 10 g/cc \PPIis of the order of the gravitational force on the particle. Speculation o a the possible effect of this particular example of photodiff usiophoresis on the distribution of atmospheric aerosols is not profitable without more information than is now available on aerosol chemistry in the atmosphere. We examine briefly now photodiffusiophoresis arising from radiation-induced variations over the particle surface of surfaae reactions and of vapor pressure. In these two examples, unlike the example we have just discussed, incident radiation plays only an indirect role in differentially heating a particle. Inasmuch as surface reaction rates and vapor pressure are both temperaturedependent properties, the differential heating by radiation would produce variations in either of these properties over the surface of the particle. In general, a force producing photodiffusiophoresis would arise from either of these physical property variations. Photodiff usiophoresis in all the examples discussed could occur either in the same or opposed to the direction of the incident radiation. Just as in photothermophoresis, one speaks then of positive and negative photodiffusiophoresis. Experimental study of photodiffusiophoresis would be of value. A very simple example for study would be the particle motion or force produced by a photocatalytic reaction. The observation of the force on ZnO particles participating in the photocatalytic oxidation of CO would be of particular interest inasmuch as the photocatalytic reaction for this sytem has been investigated already in some detail.a It is interesting to note that according to eq 8, in the free molecule region, the force producing photodiffusiophoresis can be used to obtain detailled information on the reaction through determinatioin of the phenomenological parameter Of.
Acknowledgment. This study was supported by the National Center for Air Pollution Control, Bureau of Disease Prevention and Environmental Control, U. S. Public Health Service, through Grant APOO479-02.
by J. W. Simons, C. J. Mazac,lb and G. W. Taylor10 Chemistry Department, New Mexico State Univeraiiy, La8 Cruces, New Mexico (Received September 19,1967)
It has become evident that only in the presence of small percentages of radical scavengers such as oxygen ~
~-
Table I :a Insertion Ratios with n-Butane CHpSiHa-n-Butane-DM-Oa Mixtures (3660A) 180-
PCH88iHi
P(n.butane)
9.70 9.00 23.80 27.10 24.00 23.80 0.00 99.70 302.70 278.00 301.80
9.90 9.10 23.40 13.90 24.00 23.80 40.20 101.00 100.40 94.60 94.30
PDM
1.70 2.30 10.10 4.60 4.70 1.90 4.40 22.80 37.80 36.50 38.80
POa
11.70 15.60 21.10 19.50 22.20 17.20 9.00 20.30 9.40 42.90 108.90
pentane/ n-pentane
0.94 0.89 0.96 0.92 0.91 0.95 0.93 0.89 0.90 0.89 0.88
cis-Butene-2-n-Butane-Diazomethane-Oxygen Mixtures (3660A) 18G-
P(butene)
P(n.butane)
PDM
Poi
pentane/ n-pentane
31.90 2.48 26.90 16.82 0.88 0.27 6.72 0.56 36.90 0.18 0.36 0.21
31.60 0.56 8.25 4.69 0.29 0.10 2.05 0.19 10.90 0.06 0.11 0.07
7.50 0.34 4.14 3.50 0.13 0.06 0.93 0.13 3.90 0.04 0.07 0.05
12.00 0.48 6.00 6.00 0.21 0.07 10.00 0.18 8.80 0.05 0.08 0.07
0.84 0.93 0.86 0.87 0.87 0.87 0.91 0.95 0.98 0.88 0.87 0.88
0.17 0.21 5.52 0.56 0.37 2.47 60.30 0.16 36.38
0.05 0.07 1.59 0.18 0.11 0.72 18.00 0.05 11.05
(4358A) 0.03 0.05 0.89 0.11 0.05 0.32 8.77 0.05 9.73
0.05 0.07 8.80 0.16 0.07 0.36 7.60 0.07 12.40
0.87 0.92 0.92 0.89 0.93 0.94 0.89 0.91 0.83
' All pressures are in centimeters. Volume 76,Number 9 February 1968
