1123
DIFFUSIONCOEFFICIENTS FOR GASESIN LIQUIDS
(b) rapid recombination, within the solvent cage, of the free radicals produced by dissociation. However, since the quantum yield in the photolysis of liquid water (1470 A) is close to unity,29processes (a) and (b) do not appear t,o be important, at least for water excited photolytically. This suggests either that the excited states produced by photolysis of liquid water differ markedly from those produced radiolytically or that the radiolytic excitation process responsible for decomposition in the vapor phase does not occur in the liquid phase. Molecular Yield of Hydrogen. The value of G(D2)w = 0.64 f 0.06 obtained in D20-rich mixtures (Figure 3) agrees with values for the molecular yield of hydrogen reportbedp r e v i o u ~ l y . ~G(D& ~ ~ from DzO
+
C3Hs decreased with increasing CaHs concentration , suggesting that CSHS may react with some precursor of Dz. In the systems D2O N20 and DzO SFs, on the other hand, G(D2)wincreased as the concentration of N20 or SFe increased. This indicates that the energy absorbed by NZO and by SFs can, to some extent, contribute to the formation of Dz. Further investigation of these effects would be of interest.
+
+
Acknowledgments. We thank Professor J. J. Weiss for his interest and Dr. G. Scholes for helpful discussions. The work was supported financially by the Atomic Energy Research Establishment, Harwell, England. (29) U. Sokolov and G . Stein, J . Chem. Phys., 44, 2189 (1966).
Diffusion Coefficientsfor Gases in Liquids from the Rates of Solution of Small Gas Bubbles
by Irvin M. Krieger, George W. Mulholland, and Charles S. Dickey Department of Chemistry, Case Institute of Technology, Cleveland, Ohw
(Received August 31, 1966)
A new technique is described for suspending a small gas bubble in a liquid and measuring its size as a function of time while it dissolves. From these data and the gas solubility, diffusion coefficients for the gas in the liquid can be calculated. The experimental technique involves catching a bubble on a fine horizontal fiber and photographing its projected image. To analyze the data, Fick’s law is integrated for unsteady, spherically symmetrical conditions; the resultant equation is fitted to the data by an iterative least-squares technique. Results are presented for 0 2 , N2, and He in HzO and in organic liquids.
Introduction The study of the liquid state encompasses both equilibrium and nonequilibrium phenomena. Diffusion, which is the transport of matter under the influence of a concentration gradient, is an important nonequilibrium process for which an adequate general theory is not yet available. Diffusion in dilute gases has been treated successfully by the kinetic theory, as has the diffusion of colloidal particles in a continuous
medium. The diffusion of ordinary molecules in a dense gas or a liquid has shown itself to be much less tractable, however. To stimulate and guide the development of theory, it would be desirable to have on hand a large body of accurate diffusion coefficient measurements on many diversified systems. Unfortunately, the data available are meager and, judging from the discrepancies among different techniques and observers, they are Volume 71, Number 4 March 1967
1124
frequently inaccurate. Part of this lack is due to difficulty in making accurate measurements. Of the various experiments suitable for quantitative study of diffusion of gases in liquids, the dissolution of a stationary bubble into an essentially infinite liquid is particularly simple, since the spherical symmetry and the absence of troublesome boundaries greatly facilitate the theoretical analysis. The rate of solution of a stationary gas bubble in a liquid is governed by both the solubility and the diffusion coefficient of the gas in the liquid. The liquid at the bubble surface maintains :t gas concentration equal to the saturated solution concentration a t the local pressure. If the surrounding fluid has a lower concentration, then the gas molecules diffuse outward, thereby depleting the solution a t the bubble surface and permitting more gas to dissolve. By considering the diffusion to occur radially outward into an essentially infinite solution and by applying Fick's law with a concentrationindependent diffusion coefficient, an equation can be obtained giving the bubble radius as a function of time. The parameters of this equation are the solubility and the diffusion coefficient; if the solubility is known, then the diffusion coefficient may be determined by fitting the equation to experimental radius vs. time data. The theoretical analysis of the bubble solution rate problem was developed by Epstein and Plessetl in 1950. Because the analysis is restricted to a stationary bubble, it could only be applied to experiments in very viscous fluids, where the upward motion of the bubble due to buoyancy could be neglected. In 1957, Lieberman2 developed the technique of catching the bubble under a microscope slide and measuring the radius a as a function of the time t. He assumed steadystate behavior and introduced a correction factor for the plane boundary calculated from the analogous electrostatic problem. Because of the steady-state assumption, a linear graph of a2US. t was predicted. Lieberman used his technique to study the rate of solution of air in water and found a slight upward curvature in the a2 us. t graph, which he attributed to the formation of a film of organic impurities a t the bubble surface. hlanley3 obtained similar results. Houghton4 and co-workers performed extensive studies and concluded that the initial linear portion of the a2 vs. t curve could be used with Lieberman's analysis to give diffusion coefficients which agreed with literature values to within f 10%. The present study differs from those described above in both the experimental method and the data analysis. An experimental technique was developed in which the bubble was caught and held on a fine horizontal The Journal of Physical Chemistry
I. M. KREIGER,G. W. MULHOLLAND, AND C. S. DICKEY
fiber, thereby approximating closely the assumed condition of a spherical bubble in an infinite solution. In the analysis of the data, the steady-state assumption was discarded, permitting the use of radius vs. time data extending all the way to complete disappearance of the bubble. The curvature in the a2 vs. t graph appears as a natural result of the departure from a steady-state concentration gradient.
Theory Consider a gas bubble of radius a at the center of a large volume of solution whose initial gas concentration is Cog/cm3. If the liquid a t the bubble's surface is saturated, then the gas concentration C in the solution should vary with the radial coordinate r from C,, the saturated solution concentration at r = a, to Coat r = a. Assuming a constant diffusion coefficient D, Fick's second law for this spherically symmetrical geometry is
The solution of eq 1 subject to the boundary conditions at r = a and r = is expressible as an error integral
C
r-a __
a
- Co = -(Cs- Co)[l - 2 r
2(Dt)'/2
x " ' ~
e-"dg]
(2)
The rate at which the mass m of the bubble decreases is determined by the outward flux Jaof gas at the bubble surface dm - = -4na2J, dt
(3)
Now if p is the density of the gas bubble, then m = 4 / 3 ~ a a pand dm da - = 47ra2pz dt
(4)
Thus the rate of decrease of the bubble radius is da -= dt
Ja --
P
We may obtain J, from Fick's first law J = -DVC =
bC -D-br
(1) P. S. Epstein and M. S. Plesset, J . Chem. Phys., 18, 1507 (1950). (2) L. Lieberman, J . A p p l . Phys., 28, 207 (1957). (3) M. Manley, Brit. J. Appl. Phys., 11, 41 (1960).
(4) G. Houghton, P. D. Ritchie, and J. A. Thornson, C h . Eng. Sci., 17, 224 (1962).
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DIFFUSION COEFFICIENTS FOR GASESIN LIQUIDS
The concentration gradient a t eq 2
T
=
a is available from
P(nDt)’/’ I p7rDt) In- 1+ao2 a2 a a
(
2P (4p
We thus obtain Epstein and Plesset’s differential equation for the bubble radius
+ +J
da dt = - D(ca Pa-
(8)
At sufficiently long times, a / d a t
