A Method for Obtaining a Characteristic Permeation Function for Membranes SIR: The diffusion experiments with KCl in an open cell reported earlier (3) can be interpreted in terms of a permeation coefficient average, defined in the present paper. Consider an open dialysis cell in which the volume of solution is negligibly small compared with that of external fluid, so that the solute concentration in the latter may be taken as zero. Let the effective membrane area of a single "channel" be given by AIAj, where A is the actual total membrane area, and the effective distance of a point within the membrane from the surface of the unstirred layer on the side
in analogy to Fick's law, where z is the amount of solute "inside" the dialysis cell up to the surface just defined and D is the diffusion coefficient of free diffusion. If the change in the amount of solute present within the membrane and unstirred layer in any time interval is negligibly small compared with the amount of solute passing through in the same time interval, one can consider dx/dt to be independent of and to represent the rate of loss of solute from the inside of the dialysis cell. This rate of loss may be found by integrating Equation 1 a t any time 1:
of the solution be given by [ f . i d a , where a is the actual vertical distance. Both f A i and f a i are functions of (Y and of the concentration ci(a). Accepting for the sake of discussion the model of spherical particles passing through meandering channels of varying cross section, one would think of f A j as the ratio of the cross section available for the centers of the particles in the j t h chaanel a t the position where the concentration in the channel is c j ( a ) , over the membrane area A , and of f u j as the ratio of the differential length of the j t h channel, over the distance da. Thus, jAi and f u i are related to the porosity of the membrane and the tortuosity of the channels, respectively. The amount of solute passing a surface which goes through all channel cross sections a t positions where the concentration is c, per unit time in the direction from the higher to the lower concentration is given by
+
+ 1.21080 + 2.58880
E i = (0.00322
where a. is the thickness of the membrane including the unstirred layers on both sides, and c i is the solute concentration inside the cell. Defining a permeation coefficient average, F,, by
- dx -
x x 108," Moles
1450
ANALYTICAL CHEMISTRY
From these by differentiation and integration we obtained: = (-o,oo4o458
e-maiGit
e-0,W1633f)
mole/sec.
JZi fi dc = (1.9958Ei - 0 . 4 0 7 1 0 ~ i ' + ~~~' Values of
106,
Ci," Ci, Moles Moles/L. Moles/L. Sec. 8.209 3.8050 0 3993 3.8028 4.550 1.9666 2410 2.0241 262 3,691 1.5900 1.5721 360 1949 360 1950 1.5734 600 1302 1.0080 0.9886 2.265 610 1217 0.9402 1265 432 0.3123 0.3063 0,6?27 0.1598 0.07076 2145 99 0.07010 0.0266 0.01270 19 0.01305 3300 From Hoch and Williams ( 3 ) . * Derived from Gosting (1) and Harned and Nuttall (2).
0
= (1.9958 - 0.56302 ~ 0 . 3 8 3 fL 0.54158 c ~ -. 0.080063 ~ ~ ~c1.149 l o + sq. cm./sec.
clt - 0.0041649
Diffusion of KCI through Cellophane
A
moles/liter
0.30667Ei'.766- 0.0376752~"$)10-8 niole/1000 cm. sec
dt
t,'
e--0.W525L
b
'2
we have from Equations 1 and 2
- d? Sec.
+
2 = (-0.00045 1.27750 e--0*w3167f 2.71624 e - 0 , W 1 6 a 3 f ) mole
Thus, F , multiplied by cy0 is interpreted to be the escape rate expressed as the
Table 1.
fraction of the hypothetical rate which would be observed in the absence of the effects of the membrane, if the concentrations a t the surfaces were maintained a8 in the membrane experiment. Table I shows the pertinent data which were used for computing F, as a function of concentration in a case of diffusion of KC1 through cellophane. In these experiments the volume increased, as a consequence of osmotic flow, from 1.05 to 1.44 ml. Empirically selecting approprbte algebraic forms and using the method of least squares (6),the following estimation equations were obtained :
ti
h d c X lo8,*
fi,,
Moles 1000 Cm. Sec.
Cm.+
7.5920 3.7390 2,9937
14.42 16.23 16.44
1.8361
16:45
0.'5'871 0.1404 0,0245
15.18
:
15 82
14.48
&, d&/dt, andf
bdc have
been computed for the given data and are presented in Table I. Estimates, were found by dividing the column 5 values by the product of the column 6 values and A = 7.5 sq. cm. F, reflects the effects of osmotic and hydraulic flows, of surface charges, and of any other restrictive membrane propertie:. While the range of the values for F , was within 13% of the average, E, was low both a t high concentrations as a consequence of osmotic flow (increase of effective channel length) and a t low concentrations because the permeability of cellophane for electrolytes decreases as the surface charges in the channels become effective. Under the conditions of the present experiment the hydraulic flow was estimated to be negligibly small. I t is of interest that for a. = lo-* cm. (a reasonable value for the cellophane membrane used), F,ae was equal to 0.16, or that the diffusion through the membrane was 16% of free dif-
P,,
fusion under the hypothetical conditions stated above. F,cq is a simultaneous measure of porosity of the membrane and tortuosity of the channels in the membrane. Its numerical value can be interpreted as the minimum value fer the fraction of membrane area available for diffusion. The magnitude of this value for cellophane, 0.16, confirms the high degree of porosity, unity being the upper possible limit a t zero restriction. The method of treating the data as given by Hoch and Williams (9) is incorrect, since the coefficients in the equation on which the method was based are not activity coefficients as stated there. For a similar reason, Equation 28 given by Laidler and Shuler (4) is open to question. The estimation equation for D reproduces the literature data for concentrations between 0.00125 and 3.9M, including the one point which Gosting (1) believed to be out of line, accurately t o within +0.47 and -0.28%. The estimated standard deviation of D was k0.0044 x 10-6 sq. cm. per second. At concentrations below 0.005M the
equation is systematically off the theoretical by more than O.l%, but this is not unexpected because the limiting law alc0.5 and not: requires: D = Q
+
+
D = Q alc0,383. A cubic equation with exponent 0.5 did not give an acceptable fit. A higher powered polynomial with an exponent of 0.5 could be made to fit but would introduce two disadvantageous effects: A larger number of coefficients would have to be estimated, thereby producing indeterminacy in the system, and there would be undesirable oscillations in the fitted equation. After this paper had been prepared for press, a note appeared (6) giving a theoretical discussion of the same problem as ours, in which it is shown that the use of an equation of the form:
ACKNOWLEDGMENT
The authors thank L. G. Longsworth for making available to them his lecture notes containing a revealing discussion of free diffusion, and to David Yphantis for very valuable criticism. LITERATURE CITED
(1) Gosting, L. J., J. Am. Chem. SOC. 72,4418(1950). (2) Harned, H. S., Nuttall, R. L., Zbid., 71, 1460 (1949). (3) Hoch, H., Williams, R. C., ANAL. CHEM.30,1258(1958). (4) Laidler, K. J., Shuler, K. E., J. Chem. Phys. 17,851 (1949). (5) Toor, H. L., J. Phys. Chcm. 64, 1580 ( 1960). E. J., “Regression Anal(6) W~~liams, ysis, Wiley, New York, 1959.
HANSH O C H ~ MALCOLM E. TURNE~R Medical College of Virginia Richmond, Va.
does not depend upon the assumption of unidirectional diffusion. This equation differs from ours in that it implies that A,/L. is not a function of C.
Present address, Geriatrics Reaearch Laboratory, VA Center, Martinsburg, W. Va. WORKsupported by U. S. Public Health Grant C-3977.
Differential Thermal Analysis of Explosives and Propellants under Controlled Atmospheres Robert
L. Bohon, Central Research Department, Minnesota Mining and Manufacturing Co., St. Paul 19, Minn.
TECHNIQUE of differential T:fermal analysis (DTA) has been successfully used for studying the physical chemistry of systems capable of spontaneous thermal ignition, such as rocket propellants (4, 16), oxidizers (6, 7, 14), and pyrotechnic mixtures (8, 9). These studies have been carried out under a pressure of 1 atm. of air or in a flowing stream of inert gas. Since the thermal decomposition and exothermic reactions are pressure dependent for most materials of interest in the rocket industry, a rugged, variable-pressure DTA system was designed and constructed for specific use with experimental rocket fuels. Particular emphasis was placed on versatility, ruggedness, chemical inertness to fluorine-containing samples, and easy replacement of thermocouples damaged by detonations. Only sparse reference was found to DTA investigations a t elevated pressures ( I , 12, 17, 20). The apparatus used was either too fragile for explosives (12, 17) and with too limited a pressure range (1, 17) or too massive
because of extremely high pressure requirements (20). I n the present apparatus, the differential thermocouple was isolated from the sample to avoid destruction from explosions or chemical reaction with the sample. This arrangement is superior for measuring heats of reaction (& 10) and avoiding spurious heat effects (13). The over-all cell dimensions (Figure 1) were chosen to fit a Kanthal R E H 4-30 furnace element. The thermocouples are commercial swaged units with 24-gage Chromel-Alumel elements encased in an Inconel sheath (Conax IKC 12K-UT2-MPGT-10) and are pressuresealed to the base with a Teflon gasket (Tru-Seal). A Monel pressure manifold system containing pressure gages, vents, rupture disk, vacuum pump, traps, and manometer is used to feed inert gas to the cell through an inlet port in the base, or to trap evolved gases. The pressure-vacuum DTA apparatus of Lodding and Hammell (11, 19) has several of the same design features
described here, but their Mullite pressure vessel would be too fragile for many propellent studies. The sample and reference cups (Figure 2) fit snugly on the thermocouple protection tubes and are readily removed for filling, weighing, or cleaning. A high-speed grinder fitted with steel brush is especially useful for removing debris from the cup interior after a run. Cup f is actually a tiny constanb volume bomb similar to that described by Vold (18) but much easier to construct, load, and sed. I t is particularly useful for studying volatile materials or obtaining heats of explosion. The dome may be omitted when using these containers. With cup g the dome atmosphere can readily penetrate the porous metal cap (Micro Metallic Corp., Glen Cove, N. Y.) and the product gases escape, but the solid sample cannot expel itself no matter how violently i t may be buffeted by rapidly-evolved gases. This cup allows simple and accurate control of the pressure exerted on the VOL 33, NO. 10, SEPTEMBER 1961
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