NOTES
746
474cm-1
\
434cm-1
SLIT WIDTH
251.5~m-~
Figure 2.
Raman spectrum of rhombic sulfur (Stokes region).
separate experiment involving direct passage of the attenuated laser beam into the monochromator. Under such conditions grating ghosts were observed a t 14 and 19 crn-l. Raman peaks centered at 86 cm-1 (E2), 218 cm-l (A1), and 437 cm-' (E3)were each cleanly resolved, in both Stokes and anti-Stokes spectra, as shown in Figure 2, into doublets with components separated by 4.5, 4, and 6.5 cm-', respectively. The peaks centered at 152 cm-' (E2),248 cm-' (E3), and 475 cm-I (A1 and Ez)showed incipient splittings of the same order (4.5 1+ 0.5 cm-1). The assignments used here are those of Scott, McCullough, and Kruse.12 According to these assignments, there is an accidental degeneracy of an AI and an Ez mode at 475 cm-l. This should split into four components under suitably high resolution. In addition, the Raman-forbidden fundamental centered at 191 cm-l (El) appeared weakly in the high-resolution spectra. Using a slit width of 1.8 cm-', the forbidden line appeared with appreciable intensity as an asymmetric peak with components at 189 and 194 cm-l. Group theory also predicts three Raman-active rotational modes (A2 2Ea), and three infrared-active translational modes (Bz 2E1), for the isolated SS molecule. Intermolecular forces will cause the rot& tions to be somewhat hindered in the crystalline state and may be strong enough to cause transgression of the selection rules forbidding Raman activity of the translational modes. However, since the interaction is weak, the principal effect expected should be the splitting of each rotational fundamental into two components, in the same way as explained for the vibrational fundamentals. Translational modes should not appear in the Raman spectrum. Experimentally, four of the six expected rotational fundamentals were observed. These occurred at 26, 42.5, 49.5, and 61, respectively, with the peak at 61 em-' appearing as a shoulder to the intense line at 49.5 cm-l. Inspection of the disposition of the molecules in the unit cell indicates that only rotations, R,, about the x molecular axis (perpendicular to the nodal plane) will experience little hindrance from neighboring molecules. It is reasonable
+
+
The Journal of Physical Chemistry
to assume, therefore, that R, rotations should have lower energy than either R, or R, (rotations about the orthogonal axes in the nodal plane), which will be highly hindered. On this basis, the Raman peak of moderate intensity centered at 26-om-' is assigned to the unresolved components of the At (R,) fundamental, and the triplet comprising the peaks at 42.5, 49.5, and 61 cm-l is assigned to the incompletely resolved quartet expected for the two Ea rotational modes R,, R,. The results are to be compared with those of Chantry, et aL2 These workers were able to resolve the splittings of the two infrared-active vibrational fundamentals Bz and El lying below 400 cm-' using an interferometer with an effective slit width of 4 cm-l. The Ba mode, which they observed at 186 cm-1 in spectra of saturated solutions of sulfur in carbon disulfide, was resolved into components at 186 and 197 cm-', in the spectrum of the solid. The El mode, observed by them at 239 cm-I in the spectra of the carbon disulfide solutions, was also split into a doublet, but with much smaller spacing. In addition, two infrared-forbidden fundamentals, 152 cm-' (Ez)and 216 cm-' (A1),appeared with moderate intensity in the rhombic sulfur spectrum. It is apparent that the results of the infrared and Raman investigations are complementary. The splitting of fundamentals by from 4 to 11 cm-', observed by both methods, is considered to arise as a consequence of a small perturbation of the molecular force field produced by van der Waals forces between oppositely oriented nearest neighbors. The number and activity of fundamentals predicted according to the site-symmetry approach is not in accord with the spectral data obtained here from a sample at room temperature. Better agreement with the site-symmetry predictions might be obtained at lower temperatures, where additional lines might become resolvable due to a decrease in thermal line broadening. Acknowledgment. Helpful discussions with G. Lucovsky and R. Zallen of this laboratory are gratefully acknowledged.
The Determination of Activation Energies in Solid-State Kinetic Processes
by H. N. Murty, D. L. Biederman, and E. A. Heintz Research Laboratory, 8peer Carbon Division, Air Reduction Company, Inc., Niagara Falls, New Yorlc l@O9 (Received August 30,1967)
The isothermal kinetic changes of a given property in many solid-state processes can be expressed by an empirical equation of the form
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
