are then substituted into the impedance equations (Equations 24, 27, 34, 53, and 54). These calculations illustrate, for example, the type of results predicted for an a.c. chronopotentiometric experiment a t a hanging mercury drop. Deviations between faradaic impedances obtained with the two modes of control of the d.c. parameter obviously are influenced by the fact that time is constant in one case and variant in the other, but this is not important for the present discussion. The relevant point illustrated in Figures 1 and 2 is that differences in impedances are predicted by the spherical diffusion model which are not predicted by the planar diffusion model, regardless of the value of time (Equation 57). Rationalization of the effect of spherical diffusion given elsewhere (reference 7 and references therein) is applicable here. Briefly, spherical diffusion influences the magnitudes of the surface concentrations of the electroactive species which play a most significant role in determining the faradaic impedance ( I , 3, 6, 7, IS). The magnitude of this effect depends on the direct current density leading to a corresponding dependence of the faradaic impedance on direct current. APPENDIX
A Di Ci
electrode area = diffusion coefficient of species i = concentration of species i =
Cis
Eo
= initial concentration of species i = surface concentration of species i = standard redox potential in
European convention EA) = instantaneous value of potential = amplitude of applied alternating potential Ed.c. = d.c. component of potential .ptlz= reversible half-wave potential (planar diffusion theory) = fundamental harmonic faradaic alternating potential due to applied alternating current R = Faraday’s constant ideal gas constant absolute tem erature number of efxtrons transferred in heterogeneous charge transfer step total faradaic current (cathodic current positive) fundamental harmonic faradaic alternating current due to applied alternating potential amplitude of applied alternating current d.c. faradaic current component phase angle of fundamental harmonic alternating current relative to fundamental harmonic alternating potential angular frequency time auxiliary variable of integration apparent heterogeneous rate constant for charge transfer a t E” charge transfer coefficient distance from center of spherical electrode spherical electrode radius faradaic impedance equivalent series resistive component of faradaic impedance equivalent series capacitive component of faradaic impedance
LITERATURE CITED
(1) Biegler, T., Laitinen, H. A., ANAL. CHEM.37, 572 (1965). (2) Breyer, B., Hacobictn, S., Australian J . Chem. 7, 225 (1954). (3) Delahay, P., in “Advances in Electro-
che+try and Electrochemical Engineering,” Vol. 1, P. Delahay and C. W. Tobias, e&., Chap. 5, Interscience, 1961. New York. ~-(4) Delahay, P., Adam, T. J., J . Am. Chem. SOC.74, 5740 (1952). (5) Delahay, P., Mamantov, G., Ibid., 76, I
5323 (1954). (6) Delmastro, J. R., Smith, D. E., ANAL. CHEM.38. 169 (1966).
(7) Delmastro, J: R.,’Smith, D. E., J. Electroanul. Chem. 9, 192 (1965). (8) Gerischer, H., 2. Physik. Chem. Lezptig 198, 286 (1951). (9) Glasstone; S., Laidler, K. J., Eyring, H., “The Theory of Rate Processes,” pp. 575-87, McGraw-Hill, New York,
1941. ~. (10) Hurwitz, H. D., J. Electroanal. Chem. 7, 368 (1964). (11) Matsuda, H., 2.Elektrochem. 61, 489 (1957). (12) Ibid., 62,977 (1958). (13) Smith, D. E., ANAL. CHEM.36, 962 (1964). (14) Smith, D. E., Dept. of Chemistry,
Northwestern University, Evanston, Ill., un ublished data, 1965. (15) Tack, I., Senda, M., Bull. Chem. SOC. Japan 28, 632 (1955). DONALD E. SMITH
Department of Chemistry Northwestern University Evanston, Ill. RESEARCHsupported by the National Science Foundation.
Reverse Osmosis as a Concentration Technique SIR: I n recent years there has developed a wide interest in the use of synthetic membranes as a tool for separating a variety of materials, as well as for simulating biological membranes (2, 6). Of particular interest is the development of cellulose acetate membranes by Loeb and Sourirajan for the purpose of desalination by reverse osmosis (6). I n this technique pressure usually greater than the osmotic presstwe is applied to the saline solution, resulting in the flow of water through the membrane with a high degree of retention of the salt. Using this process these membranes have also been shown to be capable of retaining large percentages of low molecular weight organics in aqueous solution a t concentrations in the range of 0.05 to 1.5M (9). Although reverse osmosis or ultrafiltration (the latter term referring to membranes separating only because of their sieve properties) have long been used to separate high molecular weight materials such as proteins from smaller molecules (1) and to concentrate them (S), until recently these techniques could not be used with similar efficiency to separate or concentrate low molecular
weight organics because of their low retention by the available membranes. In the procedure described here, dilute aqueous solutions of sucrose were concentrated by reverse osmosis in a batch process. It is shown how the per cent retention of the solute may be predicted and related to the retained solute concentration and its solution volume. EXPERIMENTAL
Membranes. Bac-T-Flex B20 cellulose acetate membranes were supplied by Carl Schleicher and Schuell Co., Keene, N. H. They were treated by immersion in water a t 84’ C. for 10 minutes, resulting in 157, linear shrinkage, following the method of Loeb and Sourirajan (6). Field Apparatus. The 316 stainless steel high pressure filter cell used was Model MFH-P60, Microcell Filter Co., Grosse Point Farms, Mich. It has a capacity for batch filtering 150 mi. of solution with an effective filter area of 21.2 sq. cm. Pressure was applied to the solution being filtered by tank nitrogen through a high pressure stainless steel system.
Analytical Method. The concentrations of the aqueous sucrose solutions were determined by absorption spectrophotometry a t 620 mp using the anthrone procedure (8). Reagents. All chemicals were reagent grade. Procedure. After the solution to be concentrated was placed on top of the membrane in the cell, the solution port was closed and nitrogen pressure applied t o result in reverse osmosis through the membrane. Fractions of the effluent solution were collected and analyzed, as was the cell concentrate at the end of the experiment. All procedures were carried out a t 24-25’ C. RESULTS AND DISCUSSION
A summary of the results of the sucrose concentration experiments is given in Table I. I n both experiments the applied pressure was 600 p.s.i. The heading “Final Solution” refers to the concentrate. The fraction of the solute retained in the concentrate, F , may be calculated by dividing the concentration factor obtained by that of the volume ratio. VOL 38, NO. 2, FEBRUARY 1966
351
I
II
1 1
LL
1 1
bb
II
LL
II
LI
II
//
/
INITW. SUCROSE COmENTRATIOM MOLAR
0
-
log
VOLUME
yq
RATIO
f loq y/v.
- YlVr
Figure 1. Volume-concentration relationship for concentrates of aqueous sucrose solutions in batch reverse osmosis experiments
To relate the concentration factors, volume ratios, and F values obtained in these experiments, it is convenient to consider the reverse osmosis process on a differential basis-that is, the instantaneous flow of solute and solvent across the membrane as they are related to the solution from which they are flowing. For such a process the differential fractional retention,f, of the solute is defined by (dn/dV)/C = 1 - f (1) with dn/dV as the number of moles of solute per volume of solvent passing across the membrane from the solution being concentrated, and C as the concentration of solute in that solution. For dilute solutions the solute volume may be neglected and V taken as the solution volume. In the batch reverse osmosis concentration process the relationship between the solute concentration and the solution volume of the concentrate may be obtained by integrating Equation 1 between the limits of n1, V Iand n2,V 2 with , subscripts 1 and 2 referring to the initial and final values, respectively, for the solution being concentrated. For this purpose f will be taken as constant throughout the con-
FRACTIONAL RECOVERY I N CONCENTRATE
Figure 2. Fractional recovery in concentrate vs.volume and concentration ratio
centration process and C = n / V . The integral form of Equation 1 is, then
352
7.8 3.9
8.1 2.6
ANALYTICAL CHEMISTRY
33.6 97.8
concentrate, F , is of importance and is defined as
F = CzVz/CiVi (5) From Equations 4 and 5 it may be shown that and In n2/n, = (1 - f)ln V ~ V I (3) Again using C expressed as
=
n/V this may be
log C,/Cl = f l o g
v,/vz
(4)
As a test of Equation 4 and the assumption that f is constant over the range of sucrose concentrations encountered in the above experiments, the concentration ratio or factor, CZ/Ci, vs. V l / V 2is plotted on a log-log scale in Figure 1 for both the final and intermediate points. The linearity of the plot indicates a reasonably constant value of f over the concentration range of 3.9 X 10-6 to 9.8 lO-4M sucrose encountered in these experiments. The reference line for f = 1 corresponds to complete retention of the solute in the concentrate. In this batch concentration process the fraction of the solute retained in the
Table 1. Reverse Osmosis Experiments with Aqueous Sucrose Solutions Initial soln. Av. Duration Concn., Final soln. fl& - Of Volume, M x Volume, Concn., Concn. Volume rate, expt., 106 ml. M X 106 factor ratio F ml./hr. hr. ml .
50 140
- F.100
4.3 25.0
6.2 53
0.70 0.47
4.4 3.5
11 30
log F = (1
- l/f) log Cz/Ci
(6)
log F = (f
- 1) log Vi/V,
(7)
and These relationships are plotted in Figure 2 for several values off, including that of 0.814 obtained in the above sucrose experiments. These plots are useful for determining the F values and concentration ratios obtained for given volume ratios. For example, using the plots for f = 0.814, for a volume ratio of 53, point a, one may expect an F value of 0.47, line ab, and a concentration ratio of 25, point b. Also, for a given f one may readily determine the volume ratio needed to obtain a given concentration factor. Even for a relatively low f value of 0.5 one may obtain reasonably high concentration factors, although the volume ratios required may be substantially higher. Although these sucrose concentration experiments indicate a reasonably constant value for f in the concentration range encountered, this is not a necessary condition to concentrate efficiently by this technique. The relationships between F , C2/C, and V I / V Zmay be calculated for a variable f , provided that it may be functionally related to C over the complete range of concentrations encountered. In the concentration range of 0.05 to 1.5X1f tends to increase with decreasing C, although for sucrose it was shown for one cellulose acetate
membrane that between 0.05 and 0.25M it became constant a t 0.87 (9). I n a recent review of membrane transport phenomena ( I ) ,both constant and varying f values were reported for glucose and sucrose in the concentration range of 10-5 to 10-4M, depending on the membrane being utilized. These were obtained, however, with dialysis tubing and gels, where the f values were less than 0.2. When varying, these values did increase with decreasing solute concentration. The practicability of the technique is indicated by the fact that f values greater than 0.5 have been obtained for aqueous solutions of organics such as iso-, sec- and tert-butyl alcohol, isopropyl alcohol, glycerol, and sucrose (9). -41though the flow rates of 0.2 ml./sq. cm./hr. a t a pressure of 600 p.s.i. reported here are relatively low and would necessitate inconveniently long times to concentrate large volumes of solutions (10 to 20 liters) down to a few milliliters, with similar membranes a t 750 p s i . flow rates five times as large have been ob-
tained (9). Recently polyion complex membranes have been prepared with water permeabilities of 14 to 17 ml./sq. cm./hr. a t a pressure of 100 p.s.i. with anfvalue of 0.9 for raffinose (7). I n the same study one type of membrane had f values of 1.0 for sucrose and raffinose and 0.25 for sodium chloride. This low value for salt is particularly advantageous when concentrating organics in a salt solution, in that lower osmotic pressures would have to be overcome as the solution becomes concentrated, as compared to membranes with high f values for inorganic salts. The advantage of reverse osmosis for concentrating dilute solutions of organics is that it avoids phase changes, high temperature, or transfer techniques in the concentration process. Although the interest of the authors lies primarily in the concentration of dilute aqueous solutions of organics in natural and municipal waters so as to facilitate their identification and analysis, there are numerous other potential applications for this technique.
LITERATURE CITED
(1) Cummins, A. B., Hutto, F. B., in “Separation and Purification,” A. Weissberger, ed., 2nd ed., p. 711, Interscience, New York, 1956. (2) Friedlander, H. Z., Rickles, R. N., ANAL.CHEM.37, No. 8, 27A (1965). (3) Jesting, E., J. Lipid Res. 5, 135 (1964). (4) Lakshminarayanaiah, N., Chem. Rev. 65. 491 (1965). (5) Li, N. “:,-Long, R. B., Henley, E. J., Ind. Eng. Chem.. 57, No. 3, 18 (1965). (6) Loeb, S., Sourirajan, S., Aduan. Chem. Ser. 38, 117 (1963). (7) Michaels, A. S., Ind. Eng. Chem. 57,
No. 10, 32 (1965).
(8) Snell, F. D., Snell, C. T., “Colorimetric Methods of Analysis,” Vol. 111,
3rd ed., p. 215, Van Nostrand, New York, 1953. (9) Sourirajan, S., Ind. Eng. Chem. Product Research Develop. 4, 201 (1965).
JULIAN B. ANDELMAN MICHAEL J. SUESS
Graduate School of Public Health University of Pittsburgh Pittsburgh, Pennsylvania 15213
Repeated Gas Chromatographic Sampling Technique in Gas Phase Chemical Kinetics Photolysis of Acetone SIR: Pratt and Purnell (9, 10) recently described a convenient technique for the study of the kinetics of the thermal decomposition of acetaldoxime using a specially designed rotary valve (11) to sample repeatedly from the reaction cell directly into a gas chromatograph. Since then, the method has been successfully employed to study other gas phase reactions (2,4-?‘, 12, IS). With this technique, concentration us. time curves for the various products are quickly and easily obtained without the necessity of tedious and often inadequate preliminary fractionations. However, the finite dead volumes unavoidably present in the valve and connecting lines and the finite sensitivity of the gas chromatograph impose a practical lower limit to the size of the sample volume, and consequently the changes in concentration of products and reactants due to repeated sampling are often quite significant. Thus, in the process of adapting this technique to the study of the reaction of methyl radicals with aromatic molecules (S), it became essential to correct for the changes in concentration due to sampling, since the rates of production of methane and ethane from acetone
photolysis have different functional dependences on concentration. The main purpose of this paper is to outline the procedure used in the mathematical analysis of our data, inasmuch as the effect is quite general and appears not to have been considered before. We also describe in some detail the important aspects of our experimental technique. T o illustrate the application of the method and show its v.L
Figure 1. paratus
Schematic diagram of ap-
V, sampling valve;
A, sampling stopcock; B, metal valve; F, furnace; R.C., reaction cell; L, lamp; V.L., vacuum line; G.C., gas chromatograph; P, vacuum pump. Valve shown in “push” position. Dotted lines indicate the positionsof the O-rings in the “pull” position
reliability, we report the results of our measurements of the rate constant for the hydrogen abstraction reaction of methyl radicals with acetone and compare them with the results of earlier workers using more conventional techniques. These results will be used in a later publication to interpret the data in the aromatic systems. EXPERIMENTAL
A schematic diagram of the apparatus is shown in Figure 1. Instead of a Pratt-Purnell valve ( I I ) , we used a commercially available stainless-steel gas-chromatograph sampling valve of the push-pull type (Loe Engineering, Model L-208-6, with Viton “A” 0rings). Connections to the valve were made with 1/8-inch copper tubing using Swagelock fittings. The sample loop was a piece of 1/8-inch copper tubing about 50 em. long; its internal volume was ca. 1.1 cc. The cylindrical borosilicate glass reaction cell (50-mm. id., 176 mm. long) was provided with a flat borosilicate glass window a t one end and a thermocouple well which extended axially nearly the whole length of the cell. It was placed inside an electrically heated aluminum block furnace, the temperature of which was manually controlled. VOL. 38, NO. 2, FEBRUARY 1966
353
