J. Phys. Chem. 1981, 85,1741-1746
Figure 11. Comparison of the steady-state modo1 (plotted as log (k-'/k2)on the abscissa) and the true electrochemical reversibility (plotted as log [ fr/(l - f , ) ] ) on the ordinate. (See Table VI (supplementary material) with D = 1.0 X cm2 s-', v := 259 mV s-', p = 0.5, lo2 > k , > lo-' cm s-', C K C lo5 s-'). The line is arbitrarily drawn with a slope of unity to emphasize the identity.
E:
-
E;
,
mV
1741
The steady-state model is of practical value for the measurement of reaction rates by cyclic voltammetry. For example, standard electrochemicaltheory provides a means of measuring rate constants for chemical reactions from the analysis of the CV sweep dependence of the peak potential E,.13326 However, this theory assumes a complete Nernstian behavior, Le., f, = 1. Such an assumption has been addressed previously, by defining regions in a twodimensional zone diagram in which electrochemical reversibility appears to be sufficientlyhigh to apply standard theory.26 In contrast to this qualitative approach, the results in Figure 11 provide a quantitative measure of electrochemical reversibility by a rapid, direct procedure. Thus, the resultant chemical rate constants K , obtained by standard cyclic voltammetric technique^,^^^^^ may be used in conjunction with eq 18 and 19 to readily calculate the reversibility f, of the electrochemical system. Cyclic Voltammetric Determination of the Standard Rate Constant k,. Comparison of E q 26 and 27. According to Nicholson,37 the standard rate constant k , is related to the peak separation EPa - E; through the function in eq 27, which is numerically evaluated with the aid of tabular data. The relationship between the peak separation and the standard rate constant, reformulated in the dimensionless form k , [ ~ D n F v / ( R 5 " ) ] - is ~ / illus~, trated in Figure 12. For comparison, the dependence of this dimensionless rate constant on the peak separation evaluated by eq 26 is also included in Figure 12. The rapid convergence of the two plots demonstrates the equivalence of the two methods, especially at peak separations greater than 150 mV.
Flgure 12. Comparison of the standard rate constant k, evaluated by equations 26 (e)and 27 (0)at p = 0.5.
Acknowledgment. We thank the National Science Foundation for financial support of this research.
simple evaluation of the rate-constant ratio k-,/k2, is an excellent measure of the true electrochemical reversibility, fJ(1- f,), over the entire reasonable range of rate parameters k , ranging from I O 4 to I O 2 cm s-l and K from to 105 s-1.
Supplementary Material Available: Listing of the input parameters A, A, and P for the calculation of the current-time profile according to eq 28 and 29 (Table VI) (4 pages). Ordering information is available on any current masthead page.
Second Surface Vlrial Coefflclenit for Argon Adsorbed on Graphite S. Sokolowski Department of Theoretical Chemistry, Institute of Chemistry UMCS, 2003 1 Lublin, Nowotki 12, Poland
and J. Stecki" Institute of Physical Chemistry, Polis19Academy of Sciences, 0 1224 Warszawa, Kasprzaka 44, Poland (Received: July 3, 1980; In Final Form: February 17, 198 1)
Physical adsorption data for iugon adsorbed on graphite are analyzed by means of the three-dimensional virial theory of adsorption. The second surface virial coefficients are calculated by taking the intermolecular pair potential to have the Lennard-Jones 12-6 form and accepting the summed Lennard-Jones 10-4 potential or the Lennard-Jones 9-3 potential for gas-solid interactions. The calculated virial coefficients are fitted to experimental data. Over the available temperature range the fit is good, and the evaluated values of the depth of the gas-gas potential are in moderate agreement with estimates based on the two-dimensional model of adsorption.
Introduction In our previous papers122 we have developed the virial expansions in a three-dimensional model of adsorption. It (1) S. Sokoyowski and J. Stecki, Acta Phys. Pol. A , 55, 705 (1979). (2) J. Stecki and S. Sokoyowski, Mol. Phys., 39, 343 (1980).
has been shown that, in the case of adsorption on completely impenetrable solids, the virial expansion for the adsorption isotherm can be written as r = kHn + \ k ( n ) ( a [ p / ( k T ) ] / d n ] - l (1) where r is the Gibbs adsorption, n is the bulk density, k
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Sokolowski and Stecki
The Journal of Physical Chemistry, Vol. 85, No. 12, 1981
is the Boltzmann constant, p is the pressure, and h~ is the Henry constant kH = x m [ g ( z )- 13 dz
g(z) = exp[-v(z)/(kT)I
T A B L E I: Experimental Values of KH and 2 W, Evaluated f r o m the Adsorption Isotherms of Argon on Graphite P33( 2700)”
(2)
where v(z) is the gas-solid potential. The new function @(n)when expressed in terms of the bulk density takes the form @(n) = 2w2n2 3w3n3+ 4w4n4+ ... (3)
+
where wm values are the cluster integrals of a new e.g.
where f(r) = exp[-u(r)/(kV] - 1 and u(r) is the interparticle potential. Equation 1gives the adsorption isotherm in a new form and differs from the virial expansion resulting from the so-called gas-solid virial theory.44 We recall that according to the latter theory the amount of adsorbed gas is obtained as the difference between the average number of molecules in the adsorption system and the average number of molecules in the hypothetical calibration system of equal volume (V), temperature, and activity, but with the gas-solid energy equal to zero inside V and equal to infinity on and inside the solid. In other words, the calibration system is a certain gas-hard-wall system; l the theory relates to a shift in adsorption associated with attractive gas-solid interaction. While these may well be acceptable low-temperature approximations, it is of interest to have an accurate description; moreover both the adsorption and the Henry constant depend on the assumed position of the Gibbs dividing surface, whereas the difference F - kHn and hence the virial series given by eq 3 do not. Thus eq 1 provides a possibility of working with quantities independent of the arbitrary position of the Gibbs dividing surface. The three-dimensional theory1i2p7was inspired by the 1962 papers by bell ern an^,^^ who then developed a May-
s.
(3) J. Stecki and SokoYowski, Phys. Rev. A , 18, 2361 (1979). (4) R. A. Pierotti and H. E. Thomas, J . Chem. SOC.,Faraday Trans. 2, 70, 1725 (1974). (5) J. R. Sams, Prog. Surf. Membr. Sci., 8, 1-46 (1974). (6) R. A. Pierotti and H. E. Thomas, Surf. Colloid Sci., 4, 93 (1971). (7) S. SokoYowski and J. Stecki, J. Chem. SOC.,Faraday Trans. 2, in
press. (8) S. Sokoyowski, J. Colloid Interface Sci., 74, 26 (1980). (9) J. R. Sams, G. Constabaris, and G. D. Halsey, J. Phys. Chem., 64, 1689 (1960). (IO) J. R. Sams, G. Constabaris, and G. D. Halsey,J. Chem. Phys., 36, 1344 (1962). (11) R. Wolfe and J. R. Sams, J. Chem. Phys., 44, 2181 (1966). (12) M. D. Johnson and M. L. Klein, J. Chem. Soc., Faraday Trans. 1, 60, 1964 (1964). (13) D. H. Everett in “Surface Area Determination”, Butterworths, London, 1970, p 181. (14) J. A. Barker and D. H. Everett, J. Chem. SOC.,Faraday Trans. 1, 58, 1608 (1962). (15) J. E. Kirzan, J . Chem. Phys., 42, 2933 (1965). (16) T. R. Rybolt and R. A. Pierotti, J. Chem. Phys., 70,4413 (1979). (17) M. P. Freeman, J . Chem. Phys., 62, 723, 729 (1958). (18) D. H. Everett, Discuss. Faraday Soc., 40, 177 (1965). (19) R. Wolfe and J. R. Sams, J. Phys. Chem., 69, 1129 (1965). (20) A. L. Gostma, R. D. McCarthy, and J. G. Hust, Natl. Stand. Ref. Data Ser. (U.S. Natl. Bur. Stand.), 27 (1969). (21) W. A. Steele, J . Phys. Chem., 82,817 (1978); “The Interaction of Gases with Solid Surfaces”, Pergamon Press, Elmsford, NY, 1974. (22) F. A. Putnam and T. Fort, J. Phys. Chem., 81, 2164 (1977). (23) F. A. Putnam and T. Fort, J . Phys. Chem., 79, 459 (1975). (24) 0. Sinanoglu and K. S. Pitzer, J. Chem. Phys., 41, 1322 (1964). (25) J. A. Hirschfelder, C. F. Curtis, and R. B. Bird, “Molecular Theory of Gases and Liquids”, Wiley, New York, 1954, p 1110. (26) I. D. Morrison and S. Ross, Surf. Sci., 39, 21 (1973). (27) A. Bellemans, Physica (Utrecht),28, 493, 617 (1962).
10-5 2 w, , m z A‘‘/
KH, cm3/g
T,K 240.019 220.393 207.713 175.082 166.135 158.087 150.140 145.114 140.607
this w o r k
ref 1 0
(gmol)
0.0812 0.1158 0.1508 0.3650 0.4933 0.6720 0.9421 1.1850 1.4802
0.0801 0.1162 0.1502 0.3654 k 0.4932 t 0.6731 t 0.9425 t 1.1856 5 1.4816 &
-15.27 -21.05 -23.74 0.0005 -74.79 0.0006 -78.86 0.0008 -50.31 0.001 88.16 0.001 308.8 0.0015 807.2 a See ref 9 a n d 10. T h e d a t a were corrected f o r the thermal expansion of the sample a n d sample cell 9.
er-cluster theory of a system confined to a volume V and showed a way of extracting the surface terms proportional to the surface area. With new “virial coefficients” w2 and w 3 and with new expressions for them, it is most interesting to seek comparison with experiment. The new features, i.e., the diffusivity of the boundary between “bulk” and “adsorbed”, would be best revealed at somewhat elevated pressures and not-too-low temperatures. Unfortunately we found no data of high precision other than the well-known data on the Ar-graphite system at subatmospheric p r e s s u r e ~ . ~The J~ advantage of analyzing this particular set of data is that we can compare our results with numerous earlier papers,4dJ+19especially those of Everett and Barker14J8and Wolfe and Sams,lgwho used the two-dimensional theory. In the next section we extract the Henry constant and w2, then we describe the gas-solid potential used and compute the Henry constant and the gas-gas potential, and finally we calculate w2 from eq 4 and compare it with the experimental w2.
Analysis of Data The only data available in the literaturegJOare the coefficients in an expansion of the adsorption isotherm in power series of p/(k!T) which are usually obtained by least-squares fits of the adsorption isotherm to a polynomial in p/(k!T). As can be seen by expanding eq 1, the successive terms contain, besides irreducible clusters wm, also contributions from the gas imperfection. It is therefore preferable to use polynomials of the bulk gas density in place of the Thus, the experimental values of Henry constants KH = AkH and of W2= Aw2 (A denotes here the surface area) were determined by us directly from experimental adsorption isotherms of argon on graphite P33(2700)9810according to the following relations (which follow from eq 1) KH = lim (N,/n) n-0
2W2 = lim [(N,/n - K~)/nl(d[p/(k!i“)l/dnl (5) n-0
where N , = A r denotes the amount adsorbed. Because the experimental adsorption isotherms are given in the form of the volume of adsorbed gas at STP vs. pressure, we calculate the densities n correspondingto the pressure p and compressibilities d[p/(lzr)]/dn with the aid of the NBS equation of state of argonOaThe experimentalvalues of Henry constants and W2are given in Table I as found by us along with the values of the Henry constants as given in Sams, Constabaris, and HalseyagOur values of KH were obtained by graphical extrapolation of N / n vs. n, as indicated by eq 5. The value of the second virial coefficient
Second Surface Virial Coefficient for Argon
The Journal of Physical Chemistty, Vol. 85,No. 12, 1981
1743
K, =1.4801
K, = l L - z ,
I
I
2
4
I
lo4 n
6
mole/[
( N J n - K H )vs. n for three indicated Figure 1. Plot of F ( n ) = values of the Henry constant in cm3/g.
W2is very sensitive to the value of the Henry constant, just as the bulk virial coefficient is sensitive to the value of RT, i.e., of the first term in the polynomial series. Fortunately, the graph of F(n) = (d[p/(kT)l/dn)(N'K H n ) / nshould go through the origin if the existence of strong sites may be neglected and should be linear at low density-and that provides a certain test. From the slope of such a graph, or from a polynomial fit, w 2 can be obtained. To give an indication of the accuracies and numbers involved, a graph is presented in Figure 1 for a low temperature of 140.607 K for three values of the Henry constant-the one accepted as the best, KH = 1.4816 cm3/g, and two values limiting the interval of possible Henry constants from above and below, KH f A, with A equal to the estimated uncertainty, A = 0.0015 cm3/g (see Table I). As can be seen from Figure 1, a shift of the assumed value of the Henry constant does change the slope as well as shift the curve away from the origin. The new quality W2,which is an irreducible cluster integral independent of any assumption about the Gibbs dividing surface,' is also plotted in Figure 2. The estimated uncertainty in W 2is indicated by the error bars.
I
I
160
I
200
T
Figure 2. Comparison of experimental and calculated values of 2 W,. All labels refer to the appropriate parameters in Tables I1 and IV.
where d is the distance between basal planes in graphite.21 One of the long-standing problems in fitting the Henry constant to a theory is that the parameter determined is the product Azo rather than individual A or zo values. In the works of Sams et al."J9 and Pierotti et aL4J6a substitution of reasonable choices for zo into the experimentally determined products yields surface areas that differed considerably from the BET area of the graphite. Putnam have given a detailed discussion of the results and Fort22*23 of various types of surface area determination of graphite and have argued that the measurement of lattice spacing in diffraction experiments can be combined with isotherm data to give more estimates of surface area than by other techniques. Putnam and Fort estimated the surface area of their graphite to be 11.4 f 0.4 m2/g. They also estimate that the ratio of the area of P333 used by Halsey et al.gJO Gas-Solid Potential Energy and the Henry to that for their sample was 1.04, which gave A = 12.0 f Constant 0.5 m2/g. In order to have some indication of the sensiA variety of Lennard-Jones-type potentials have been tivity of the results to the value chosen for substrate area, used in previous works to model the gas-solid interactions the presented eq 7 in the rare gas-graphite s y ~ t e m Putnam . ~ ~ and~ ~ ~ calculations ~ ~ out ~ for ~ both ~ ~A =here ~11.5~for ~the ~potential ~ ~Thein~value and 12 m2/g. were carried Fort,z1 considering adsorption of krypton on graphite, Azo obtained for the potential in eq 6 from the analysis conclude that the Lennard-Jones m-3 functions give a of the Henry constant is9J023.877 X lozoA3;this value for somewhat better fit of the adsorption data than the LenA = 11.5 m2/g gives zo = 2.076 A, a value unreasonably nard-Jones 10-4 function and that variations in m from small in comparison with that resulting from the Lor9 to 16 give a marginally best fit at m = 12. Halsey et al.9Jo enz-Bertholet rule.21 Potential (7) not only has the best and Pierotti et a14J6tried several models for the interaction theoretical justification and fits the available data as well potentials of argon, including the Lennard-Jones 9-3,12-3, as the Lennard-Jones 9-3 function,21 but also gives a and m-3 functions. However, they were unable to make reasonable value of zo when the best estimate of the area a choice of the best potential model as a result of their is used (cf. Table 11, where all values of the parameters of analysis. Here we also use the Lennard-Jones 9-3 potential potentials (6) and (7) used in our calculations are collected). V(z) = ~ I ~ ( E , , / ~ ) [ ( Z O / Z -) ~( Z O / Z ) ~ I (6) where E,, is the well depth, along with the summed Lennard-Jones 10-4 potential ~ ( z =) E
@
jto
y3[
(+
Lz ) o j d
-
(_ie)] + z
jd
(7)
Gas-Gas Potential The intermolecular potential energy between two inert gas molecules is considerably altered when these molecules are next to a solid surface. A formula for the two-molecule surface potential was derived from quantum mechanical
1744
The Journal of Physical Chemistry, Vol. 85, No. 12, 1981
TABLE 11: Parameters of Potentials ( 6 ) and ( 7 ) and the Surface Area
Sokolowski and Stecki
itative behavior just as expected from earlier calculations. The data do not extend to temperatures high enough to reveal the existence of two Boyle temperatures2p7found for eq +la eq 6b square-well potentials. Moreover, the agreement of the a b C d experimental w 2 with that calculated from eq 4 can be 1107 1107 968 E,,/k, K 959 made quantitative by varying the parameters of the gas2.77 2.076 3.21 20, A 3.38 gas potential, Table IV compares the results of such an 8.62 11.5 11.5 12 A, m2/g analysis with some of the previous investigations, whereas a See ref 21. b See ref 10. Figure 2 displays the differences between computed and experimental w 2values. The agreement is satisfactory, and the parameters E,, determined by all variants of the perturbation theory by Sinanoglu and P i t ~ e r .In ~ ~the least-squares procedure agree very closely with those espresent paper, however, we take, along with earlier authors, timated previously. This agreement is indeed satisfactory the empirical Lennard-Jones 12-6 potential since the equations of the three-dimensional theory and of the two-dimensional theory are quite different. u(r) = 4E,,[(a/r)12 - (a/r6)] (8) Although the depth of the adsorbate-adsorbate potential (see Table IV) is not strongly coupled to the accepted with an effective parameter Egr gas-solid potential, in the case of potential (c) (see Table 11) the minimization of all three sums I, 11, and I11 leads Results and Discussion to practically the same value of E,,, whereas in the case The numerical calculations of the cluster functions w 2 of potentials (a) and (b) the values of E obtained by as required by eq 4 for gas-solid potentials (6) and (7) and minimizing SI differ from those obtained t y minimizing for the gas-gas potential (8) were performed by using a SIIIby -1 K. The latter fact is probably due to the difmultiple gaussian method. Table I11 presents the calcuferent method of determination of the parameter zo in the lated values of w 2for different values of EB/k and different potential functions (a), (b), and (c). In the case of potential gassolid potentials. The experimentalw 2values have been (c) the parameter zo was computed independently,g'O and fitted as a function of temperature to each of the four A was determined from the experimental product Azo. In gas-solid potentials (see Table 11). The fitting was done the case of potentials (a) and (b), however, we accept for by a scanning process. A range of E /k values was chosen, A the best estimate of the surface area A = 11.5 or 12 m2/g and the value of E /k was found wfich minimizes (I) the and compute zo from the product Azo. where P denotes the average value Our analysis was performed along lines similar to those sum SI = Ci(Piof Pi= W2(Ti)expt1/~2(Ti)cdcd,5~9~15~17-19 (11) the sum SII= described in ref 4-6, 9, 10, and 12-19; Le., we analyzed C;[wz(TJexPt1 - AwZ(Ti)Cdcd]2, and (111) the sum SIII= independently Henry constants and virial coefficients w2. However, the method of analysis of adsorption of krypton Ci([WZU'J~'~' The parameter Q, however, was not optimized and was assumed to be on graphite applied by Putnam and Fortz2was quite difequal to 3.405 A.25 Thus, while the minimization of the ferent. Starting with the two-dimensionalvirial expansion sum SI leads to reevaluation of the surface areas A ( A = for the adsorption isotherm P for the best value of E g g / k )minimizing , the sum SI1 and In [ N a / ( p K ~ )=l 2 B 2 ~ r4- 3/2c2Drz (9) Smwe have set A = 11.5 and 12 mz/g for potential (7) and A = 8.62 and 11.5 m2/g for potential (6) (cf. Table 11). where BzDand Czn are the second and third two-dimenAlso, owing to the rather large uncertainties in experisional virial coefficients, Putnam and Fort determined the mental W2values at the highest three temperatures (the potential energy of functions (6) and (8) by performing a measured adsorption isotherms consist of two experimental direct multiparameter fit of eq 9 to the experimental data. points), only the seven lowest temperatures were used in Such a multiparameter fit may often be preferable, and our fits. indeed it would be interesting to repeat the calculations Of course, the use of the surface area as an adjustable by Putnam and Fort but by using the three-dimensional parameter may be seriously questioned.21p22The miniequations based on eq 1. We felt it would be more profmization of the sum SIwas performed in order to compare itable to extract kH and w2 by graphical procedures, and the well depths E,, determined by us with previous estiseparately. Incidentally, it is regrettable that w 3could not mations; in previous studies of Halsey et al.,9v10Pierotti be extracted from those low-pressure data. and tho ma^,^,^ Sams et al.,5911,1gand Everett and BarkConclusions er,9J4J8the surface area was treated as a best-fit parameter. In the adsorption isotherm of eq 1,the quantities w, are The extraction of the new second surface virial coeffiindependent of the position of the Gibbs dividing surface, cient w zfrom experimental data is quite possible even with whereas N , = AI' and kH are both dependent on the aslow-pressure data and in low- and medium-temperature sumed position of this surface. At the low temperature ranges. This second virial coefficient can be very well and at low pressures this uncertainty is negligible. In our described, at least in the very limited temperature range calculations we have located the Gibbs dividing surface at studied, by the known gas-solid and gas-gas potentials, z = 0, Le., at the divergence of the potential v(z). although the full potential of the three-dimensionaltheory The Henry constant extracted with the aid of eq 1 of adsorption should be revealed by high-pressure data also agrees, of course, with earlier investigations, except for a extending from the lowest to the highest temperatures. small and inconclusive shift. Access to really raw data Appendix would be necessary to draw any conclusions or discriminate In adsorption measurements the amount adsorbed is between different values extracted by different procedures. defined as Na = No - N,, where No is the known initial Also, the existence of any strong sites has been neglected number of moles of the gas and N E is the calculated without any direct evidence. More importantly, the new amount in the volume available to the gas including the quantity w2 which we call the second virial coefficient volume of the adsorbed layer. The latter volume is never (although it entails the interaction of at least three bodies), known exactly since its definition entails the introduction when extracted from experimental data, shows the qual-
fit2,
Second Surface Virial Coefficient for Argon
The Journal of Physical Chemistry, Vol. 85, No. 12, 1981 1745
a
PA
P
J. Phys. Chem. 1981, 85, 1746-1750
1746
TABLE IV: Best-Fit Values of E , J k for Argon Adsorbed on Graphite P33a a I
I1
b
I11
I
C
I11
I
I1
I11
d. I11
2D
E
96.6 96.2 97.2 95.2 95.2 96.8 99.4 94.6 95.1 96.0 97.1 3.405 3.405 3.405 3.405 3.405 3.405 3.405 3.405 3.46 3.48 Zor a 3.405 11.5 11.5 17.45 12.0 9.07 8.62 8.62 11.5 10.08 8.52 A , mz/g 15.15 a The symbols a, b, c, and d refer to the appropriate gas-solid potentials (cf. Table 11), whereas the symbols I, 11, and I11 denote the method of calculation (see text). The two last columns include the results of previous estimations: the data 2D were obtained from the two-dimensional models5whereas the symbol E denotes the results obtained by Everett.Io
E,,/k, K
object; the solid being impenetrable, n(l)(z)= 0 with arbitrarily good accuracy for z sufficiently small and n(l)(z) = n with arbitrarily good accuracy for z sufficiently large. The integral of n(l)(z)over the whole volume of the gas yields N,. The difficulty starts with the subtraction of N g = Vn since n # 0 in the region where n(l)(z)= 0 so that one has to fix an arbitrary lower limit of integration zl, defining
A
We have
Figure 3. Schematic representation of the density profile as a function of distance normal to the surface. The amount N,‘ = N o / Ais independent of the choice of the Gibbs surface z , and is given by the shaded area. The amount N,‘ = N,/A depends upon the choice of the Glbbs surface, and the difference N~z,‘- N ~ Z , ’is’given by the dotted area. of the Gibbs dividing surface, which in this case limits the gas volume. Suppose, for simplicity, that one has a homogeneous solid surface; the density profile (see Figure 3 ) n(l)(z)dr = n(l)(z)dz dA is a perfectly well-defined
provided z1is so chosen that always the entire adsorption layer is included in the integration region. Clearly, the definition of the Henry constant suffers from the same drawback; however, we have found1 that the difference AI’* = A r - AkHnis independent of z1and consequently all three-dimensional ,virial coefficients w, also are. Incidentally, any experimental error which is proportional to the gas density n will be cancelled exactly in the subtraction procedure if AkH will be obtained by fitting or extrapolation of the same data.
Thermodynamics of Osmotic Flow with Cavitation in Semipermeable Porous Membranes Hugo A. Massaldi*+ and Carlos H. B o d Instituto Mu/tidiscipiinario de Bioloda Celular, Casilla de Correo 403, 1900 La Plata, Argentina (Received: August 21, 1980; In Flnai Form: February 17, 1981)
An alternative (with respect to a simple viscous mechanism) for osmotic flow through a strictly semipermeable porous membrane is presented to show that internal phase transitions (cavitation) can take place. In this way, the response to arbitrary high values of the osmotic pressure is explained without resorting to postulate negative pressures within the pores. Experimental results taken from the literature are discussed in connection with the present model to show that this is plausible. The theoretical finding that the osmotic permeability is a function of the osmotic pressure is emphasized, since it precludes the general equivalence of hydraulic and osmotic pressures as driving forces.
Introduction The question of membrane structure and function has been intensively studied during the last 50 years, both in the biological and the technological fields. Permeability determinations under different conditions has been one
of the methods used to gain more insight into membrane characteristics. The determinations have been carried out in cellular, synthetic bilipid, and artificial membranes (Stein’), the main objective in the latter case being the attempt to understand the physical phenomena involved
‘Member of Carrera del Investigador, CONICET, Argentina. Under Fellowship from CONICET, Argentina.
(1) Stein, W. “The Movement of Molecules Across Cell Membranes”; Academic Press: New York, 1967; Chapters 3 and 4.
*
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