J . Chem. SOC., Faraday Trans. I , 1982, 18, 2329-2336 Application of the Elovich Equation to the Kinetics of Occlusion Part 3 .-Heterogeneous Microporosity BY CHAIM AHARONI* AND YAAKOV SUZIN Department of Chemical Engineering, Technion-Israel Institute of Technology, Haifa, Israel Received 29th May, 1981 A model is considered in which occlusion takes place in parallel in an array of micropores with different coefficients of diffusion. The rate equation for occlusion in a pore is approximated by a parabolic or by an exponential equation, and the rate for the overall process is obtained by summing the rates in the pores. The plot of &/ V , against In t, where & and V, are the amounts sorbed at times t and infinity, respectively, is sigmoid and has an intermediate part with greatest slope.The slope of this intermediate part is related to the heterogeneity of the system. If uE(E) is constant, where E is the energy of activation for diffusion and uE the pore volume for an energy between E and E+dE, the reciprocal of the slope is equal to the difference between the highest and lowest E divided by RT. If E is constant the slope is close to 0.2. The kinetics of occlusion by solids with a homogeneous network of micropores with the same diffusion coefficient and particles of the same shape and size have been discussed in Part 1.' It was shown that the plot of the reciprocal of the rate z = (dK/dt)-l against the time, t , is S-shaped and has an inflexion point at t = t,. Around t = t,, the rate equation for &(t) can be approximated by an Elovich equation (1) & = A + (1 / b ) In (IllI, + I,) where the constants A, b and t, are determined by the shape of the particles.In the present paper it is shown that Elovichian kinetics are also applicable to occlusion by sorbents possessing micropores with different lengths and different coefficients of diffusion. Occlusion is defined in this case also as a sorption process in which the fluid undergoes a change of phase when entering a micropore, and all the sorbate in a micropore is in the same phase. MODEL FOR THE OCCLUSION PROCESS We consider specifically the following model. The micropores open into a network of wide macropores that communicate with the surrounding atmosphere. There is no resistance to the flow of fluid and no adsorption in the macropores, and the fluid reaches the entrances of all the micropores rapidly and at the same time.The penetration of the fluid into the micropores and the accompanying change of phase are rapid. The movement of the condensed sorbate in the micropore is by activated surface diffusion, and this determines the rate of the overall occlusion process. The volume v; sorbed at time t in a micropore of length I with a coefficient of diffusion D is given by n--00 ui/ul, = 1 - (8/7r2) X [ 1 /(2n+ 1)2] exp [ - (2n + l)"t/z)] (2) n-0 23292330 KINETICS OF OCCLUSION where u; is the volume sorbed at t = o(> and is considered as the ‘available pore volume’ for the pressure and temperature at which occlusion is performed. z is a parameter related to I and D by z = 412/n2D.(3) If 1 and D are the same in all micropores, the volumes adsorbed in all micropores at t = t and t = co, & and Vm, are related to the volumes adsorbed in a single micropore, ( 4 ) ul and v;, by </Vm = u;/v; and eqn ( 2 ) applies to the overall system. The treatment for ‘slabs’ in ref. (1) is valid, and in the region t z t, = 0.4742 one can apply eqn (1) with A/Vw = 0.238, bVm = 2.281 and t, = 0.793. If the coefficients of diffusion or the lengths of the micropores vary, it is necessary to use a distribution function in order to characterize the system. It is convenient to use the function v,(z), where v,dz is the available pore volume for the micropores with z ranging from z to z+dz. The available pore volume is as defined above, the maximum volume of occluded adsorbate at the pressure and temperature considered.It is implicitly assumed that z can vary from micropore to micropore, but it does not vary in the same micropore. The total occlusion in all the micropores is given by where vTt is the volume of adsorbate in the micropores characterized by z at time t, and zi and z, are, respectively, the lowest and highest values of z in the system. In order to solve eqn (9, one has to introduce the appropriate distribution function v,(z) and replace vrt/v, = v;/ul, by its value according to eqn (2). The calculations are greatly simplified if one uses a suitable approximation instead of eqn (2). At small t eqn ( 2 ) reduces to the parabolic expression (6) u;/u:, = V,t/V, = (4/n9(t/z)4 and at large t it reduces to the exponential expression v;/v; = uTt/uT = 1 - [(8/n2) exp (- t / z ) ] .( 7 ) It has been shown1 that the plot of vl/v& against t calculated according to eqn (6) is practically congruent to the plot calculated according to eqn ( 2 ) for u ; / u k < 0.5, and similarly the plot according to eqn (7) is practically congruent to the one according to eqn ( 2 ) for u;/& > 0.5. A more strict test is to compare differential plots. Plots of z’vl,/z = v’,/z(du;/dt) against t / z calculated according to eqn (2), (6) and ( 7 ) are depicted in fig. 1. The figure shows that eqn (6) is valid with reasonable accuracy when 0 < t < t, and eqn (7) is valid in the range t, < t c co. In the system under consideration occlusion takes place in parallel in pores with different z, and there are at any moment t some pores in which t , is just being reached; these are the pores for which z = zt = t/0.474.In the pores with z < z~, z is beyond the inflexion point and the kinetics obey eqn (7). In the pores with z > zt the inflexion point is not reached and eqn (6) is still valid. Eqn (5) can thus be rewritten asC. AHARONI AND Y. SUZIN 233 1 fh FIG. 1.-Plots of Z’V&/T against t / t . The plots are calculated according to eqn (2) (solid line), eqn (6) (dots) and eqn (7) (squares). MICROPORES WITH vE CONSTANT Eqn (8) can be solved analytically for some simple distributions v,(z). An interesting case is the one in which the available pore volumes, correspondingto equal increments of the activation energy for diffusion, are equal.Denoting by vE(E)dE the available pore volume corresponding to an activation energy for diffusion between E and E+dE, we have u,(E) constant. As D = D,exp(-E/RT) (9) z is related to E by z = z, exp (E/RT) (10) where T o = 4l2/n2DO. (1 1) (12) Therefore v, = c / z (13) If Do is the same for all pores and the lengths I are equal, the distribution v,(z) is related to the distribution v,(E) by u, = U,RT/z. where C is a constant. The total available pore volume V, is given by V, = jrrm v,dz = Cln (zm/zi). (14) The volume adsorbed at t is obtained by combining eqn (13) with eqn (8) and normalising by using eqn (14)2332 KINETICS OF OCCLUSION Differentiating according to t and multiplying both sides by t gives [In (zrn/7i)] [(d &/ V,)/d In t] = 1 -(4/d) (t/tm)'-(8/n2) exp (- t/zi)* (16) The plot of VJV, against In t corresponding to eqn (16) has the following characteristics (see fig.2). 4 3 I 2 Int FIG. 2.-Plots of VJVm against In t : (1) Ei = 400, Em = 10 000; (2) Ei = 5400, Em = 15000; (3) Ei = 400, Em = 6000; (4) Ei = 400, Em = 2000 ( E in arbitrary units, RT = 600 arbitrary units, z, = 1). The points corresponding to t = ti and t = tm and the tangents at these points are shown for curve 1. The dotted line cutting curve 4 is a line with slope l/ln (zm/ri) = 0.375. (1) If zi is sufficiently small and z, sufficiently large, i.e. if the range of activation energies is sufficiently wide, there is a range of t for which the two last terms in eqn (16) are negligible, and eqn (16) then becomes (17) The plot of &/ V, against In t is linear in that range, and its slope is l/ln(zm/zi).This means that when heterogeneity increases (a) the range at which the curve is linear increases and (b) the slope of the linear part decreases. d (&/ V,)/d In t = 1 /In (zm/zi). (2) At small t eqn (16) is approximated by d( &/ V,)/d In t = [ 1 /In (zm/zi)] [ 1 - (8/n2) exp ( - t/zi)]. (18) The plot &/ V, against In t is concave towards the Q/ Vm axis in that range since d2 (&/ V,)/d (In t)2 = [ 1 /In (zm/zi)] (8t/n2zi) exp ( - t/zi) > 0. (19) When t = zi eqn (1 8) becomes d (&/ VJ/d In t = 0.702/ln (zrn/zi) i.e. the point at which t = zi is the point in the concave region for which the slopeC . AHARONI AND Y. SUZIN 2333 is 0.702 times the slope of the linear part.When t = 0 eqn (18) becomes d ( G / Vm)/d In t = 0.189/1n (zm/zi) (21) i.e. the minimum possible value of the slope at the beginning of a run is 0.189 its value in the linear region. (3) At large t eqn (16) is approximated by d ( &/ V,)/d In t = [ 1 /in (z,/zi)] [ 1 - (4/$) (t/zm)']. dz (&/ V,)/d (In t)2 = [ - 2/& In (z,/zi)] [t/zm]' < 0. d (Q'V,)/d In t = 0.282/1n (z,/zi) (22) (23) (24) The plot of &/ V, against In t is convex towards the VJ Vm axis since When t = z, eqn (22) becomes i.e. the point at which t = r , is the point in the convex region for which the slope is 0.282 times the slope of the linear region. Plots corresponding to eqn (1 6) were computed numerically and are depicted in fig. 2. Introducing uE(E) instead of u,(z) in eqn (8), replacing the integration by a summation and normalising by introducing one obtains wheref,(z) is uTt/u, as given by eqn (6),f2(z) is vJu, as given by eqn (7), Ei and Em are, respectively, the lowest and highest values of E in the system and Et = RTln ( z t / z o ) = RTln (t/0.474 zo).For the specific distribution uE constant eqn (25) becomes Curve 1 in fig. 2 refers to an array of 100 pores with vE constant and E varying progressively and by equal increments between Ei = 400 and Em = 10000; RT = 600 and z, = 1. (Ei, Em and RT are in the same arbitrary units.) The curve is calculated by introducing in eqn (26) the appropriate values of Ei, Em and AE. Curves 2, 3 and 4 refer to similar arrays of pores with other ranges of E and they are computed in a similar way: E ranges between 5400 and 15000 units for curve 2, between 400 and 6000 units for curve 3 and between 400 and 2000 units for curve 4.Curve 1 is practically linear over a wide range of &/Vm, between 0.15 and 0.85. The slope of this linear part is l/ln (z,/zi) = RT/(E,/Ei) = 0.0625 in agreement with eqn (17). The point corresponding to t = zi = 1 . 9 5 ~ ~ is on the concave part of the plot and the slope of the tangent at that point is 0.7/ln(z,/zi) = 0.044, as required by eqn (20). The point corresponding to t = z, = 1.73 x lo7 q, is on the convex part of the plot and the slope of the tangent at that point is 0.28/ln(z,/zi) = 0.018, as required by eqn (24). In curve 2 the ratio2334 KINETICS OF OCCLUSION z,/zi is the same as in curve 1, and the linear parts of these two curves are parallel.In curve 3 the ratio zm/ti is smaller, the linear part is shorter and steeper and its slope is l/ln(zm/zi) = 0.107. In curve 4 z, and zi are too close and it is not possible to discern a linear part with a slope of l/ln(zm/zi) = 0.375; if the region around the inflexion point is approximated by a straight line, the slope of this line is ca. 0.22. PLOT OF VJV, AGAINST lnt FOR A N ARRAY OF PORES WITH THE SAME z It is useful to compare the kinetics implied by eqn (16) with the kinetics for a homogeneous system in which the micropores have the same value of z. The kinetic equation for the homogeneous system, eqn (2), approaches eqn (1) at t = t, and eqn (7) at large t. The following expressions for d(&/V&J/dlnt are obtained by differentiating these equations with respect to In t and taking into consideration eqn (27) (4) : valid at intermediate t, and (28) valid at large t .Eqn (27) and (28) show that d(VJV,)/dlnt increases with t at intermediate t and decreases at large t . It follows that the plot of &/ V, against In t is sigmoid and has an inflexion point with maximum slope at a point t = tinf in the transition region between the regions described by eqn (27) and (28). This is illustrated by curve 1 in fig. 3 which depicts a plot of K/ V, against In t calculated according to eqn (2) for E = 5200, RT = 600 arid z, = 1. ( E and RTin the same arbitrary units.) d (K/ G ) / d In t = (l/b Gl)/(l+ tJ,/t) d (K/ Vm)/d In t = (8/n2z) t exp (- t / z ) In t In t In t In t In t FIG. 3.-Plots of &/ Vagainst In t for various distributions: curve (1) Econstant, (2) Gaussian distribution [eqn (30)], (3) triangular-shaped distribution [eqn (29)], (4) Rayleigh distribution [eqn (3 l)], (5) uE constant.The dotted curves refer to distributions with uE constant adjusted to fit the curves referring to the other distributions. The slope of the plot of against In t is known at the point t = tp. Introducing in eqn (27) t , / t = 1, bV, = 2.281 and tr = 0.793 [see ref. (l)], one obtains d(l$/V,)/dlnt = 0.244 for that point. At the point t = tinf the slope is > 0.244. If the plot &/ V, against In t is approximated by a straight line in the region aroundC. AHARONI A N D Y. SUZIN 2335 tinf this line will have a slope > 0.24.This property gives the possibility to distinguish between a strongly heterogeneous system with a small slope given by 1 /In (zm/zi) and a homogeneous or weakly heterogeneous system with a slope of the order of 0.24. The slopes for curves 1 , 2 and 3 in fig. 2 are 0.06, 0.06 and 0. 1, respectively, whereas the slopes for curve 4 in fig. 2 and curve 1 in fig. 3 are of the order of 0.24. OTHER DISTRIBUTIONS Eqn (16) is a solution of eqn (8) for the particular distribution u,(E) constant. For other distributions it is difficult or impossible to solve eqn (8) analytically, however plots analogous to those in fig. 2 can be computed by introducing in eqn (25) the appropriate function v,(@ and the assumed Ei, Em and AE. Three different distributions were examined, plots of v,/V, against E for these distributions are depicted in fig.4 , and the corresponding plots of &/Vm against In t are given in fig. 3. In all the cases a system comprising 100 pores is assumed; E is assumed to increase progressively and by equal increments between Ei = 400 and Em = 10000 arbitrary units, and therefore AE = (Em - Ei)/ 100 = 96 units; RT is 600 units and z, is 1. The distributions chosen are such that uE is maximum at the energy E = (Em - Ei)/2 = 5200 units. 2 3 4 5 2 5 5 7 5 'U' 2 5 5 7 5 E/ 103 E/ 103 ~1103 ~1103 EI 103 FIG. 4.-Plots of uE against E for the distributions given in fig. 3 (E in arbitrary units). Curves 2-4 have a maximum at E = 5200 units. Curve 3 in fig. 3 and 4 represents a triangular-shaped distribution vE(E) = 4(E- Ei)/(Em - Ei)2 for E < E and u E ( E ) = 4(Em-E)/(Em - EJ2 for E > E.The summation Curve 2 represents a Gaussian distribution uE(E) = (kG/n)4 exp [ - kG(E- IT)'] (30) k , = 3 x lo-' (energy units)-' was chosen; with this value vE at Ei and at Em are small as compared with uE at E Em Ei uE(Ei)/uE(E) = ~ E ( E ~ ) / V E ( E ) = uE(E)AE2336 KINETICS OF OCCLUSION can be taken as 1. Curve 4 represents a Rayleigh distribution, and distribution on which the Dubinin-Radushkevich equation is based2 (31) u,(E) = - 2k,(E- Ei) exp [ - k,(E- This distribution is unsymmetrical: uE is maximum at which differs from the average energy defined by Eav. = L?%,/Cu,. k , = (1 /2) (E- Ei)2 = 2.17 x (energy units)-2 was chosen giving U, maximum at Ev(max) = = 5200 units, and the resulting value for Eav.is 5500 units. v,(Ei) is zero, uE(Em)#O, EizEmVE(E)L\E = 0.86. Plots representing the systems E constant and vE constant are also depicted in fig. 3 and 4. With all the distributions examined the plot of K/ V against In t is sigmoid. If the intermediate section with greatest slope is approximated by a straight line, the slope of this straight line appears to depend on the distribution function u,(E) : it is greatest for E constant and decreases along the sequence v,(E) given by eqn (31), u,(E) given by eqn (29), u,(E> given by eqn (30) and finally uE(E) constant. This means that the distributions in which the pores with extreme values of E have a greater weight give plots of &/V, against In t with a lesser slope, a result consistent with the behaviour noted above: d (K/ Vm)/d In t decreasing with increasing heterogeneity.As all the plots in fig. 3 seem to be of the same form except for the variation in slope, it would be of interest to find out if they are significantly affected by the form of the distribution uE(E) or only by the implied heterogeneity. The dotted curves in fig. 3 represent plots for V , constant adjusted in order to give the best fit with the solid curves representing the. other distributions. The plots for u, constant were calculated by using a value of Eequal to that of the original distribution and a value of (Em - Ei)/RT equal to the reciprocal of the slope in the linear part for the original distribution [see eqn (17)]. These uE constant distributions are also depicted in fig. 4. In the three cases the plots of Q/Vm against In t for uE constant are congruent with the plots for the other distributions during most of the run. However, K/ Vm is slightly smaller at small t and slightly larger at large t, and the intermediate part of the curve is therefore closer to a straight line. It is unlikely that one could be able to assess the actual distribution by examining kinetic data obtained experimentally, although it should be possible in many cases to distinguish between more heterogeneous systems and less heterogeneous ones. CONCLUSION In conclusion, a simple test is suggested in order to assess the heterogeneity of a system of pores. K/ V, is plotted against In t, and if at the intermediate region with maximum slope the slope is ca. 0.24, the system is homogeneous or slightly heterogeneous, if it is + 0.24, the system is significantly heterogeneous. The reciprocal of the slope can conveniently be used as a parameter for expressing the degree of heterogeneity. This parameter increases as the heterogeneity increases and is indepen- dent of the form of the distribution uE(E). It is exactly equal to (Em-Ei)/RT for the distribution uE constant. C. Aharoni and Y . Suzin, J. Chem. Soc., Faraday Trans. I , 1982, 78, 2313. H. Marsh and B. Rand, J. Colloid Interface Sci., 1970, 33, 101. (PAPER 1 /872)