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Accepted Manuscript Modified GAB model for correlating multilayer adsorption equilibrium data Longhui zou, Linghui Gong, Peng Xu, Guochao Feng, Huiming Liu PII: DOI: Reference: S1383-5866(16)30026-0 http://dx.doi.org/10.1016/j.seppur.2016.01.026 SEPPUR 12808 To appear in: Separation and Purification Technology Received Date: Revised Date: Accepted Date: 16 November 2015 14 January 2016 18 January 2016 Please cite this article as: L. zou, L. Gong, P. Xu, G. Feng, H. Liu, Modified GAB model for correlating multilayer adsorption equilibrium data, Separation and Purification Technology (2016), doi: http://dx.doi.org/10.1016/ j.seppur.2016.01.026 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Modified GAB model for correlating multilayer adsorption equilibrium data Longhui zoua,b, Linghui Gonga,*, Peng Xua , Guochao Fenga,b ,Huiming Liua a).Key Laboratory of Cryogenics, Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, Beijing 100190,PR China b).University of Chinese Academy of Sciences, Beijing 100049, PR China *Corresponding author. Tel.: +86-010-82543527 E-mail address: [email protected] (L. H.Gong) Abstract A new multilayer gas adsorption model is built based on the GAB (Guggenheim-Anderson-de Boer) model and L-F isotherm (Langmuir-Freundlich isotherm). Accounting for the heterogeneity of the adsorption system, the adsorption rate is assumed to beαth power of the surface area as demonstrated in L-F isotherm; the modified GAB model has the same form as that of the GAB equation with the difference lying on the relative pressure form; the nominal relative pressure is the α th power of the relative pressure. Different adsorption isotherm models are applied to correlate the adsorption data of microporous materials. The modified GAB model has the best conformity with the experimental data and can get almost the same so called “BET monolayer capacity” with that of the original BET equation using consistence criteria. The parameters in the modified GAB model were calculated as well; the non-unity of α indicated that the heterogeneity of the adsorption system is un-negligible even on the highly homogeneous surface. Keywords: Multilayer adsorption; Heterogeneity; BET; Microporous material 1. Introduction Adsorption is a common technology to study the characteristic of porous material and adsorption isotherm is mostly used to describe the adsorption process. There are many famous models available to describe the pure component adsorption, such as the Langmuir model[1], Freundlich, BET[2], Langmuir-Freundlich isotherm[3], Toth model[4] and DR (DA) equation[5], etc. The BET model is usually used to evaluate the specific area of the non-porous and mesoporous material and it’s applicable when the relative pressure is 0.05-0.35. Because the existence of micropores can lead to non-linear BET plot[6] and capillary condensation may occur in high pressure. A modified BET equation is built by Anderson [7] and Brunauer et al[8] respectively by employing a third parameter k, which is a measurement of the attractive force field of the adsorbent. And it greatly extends the applicability of the BET model with P P 0 or c c 0 up to at least 0.9[9] while the BET areas are almost the same for the two models. Timmermann[10] did more research about the parameters in GAB (Guggenheim-Anderson-de Boer) isotherms(that is the modified BET model derived by Anderson and Brunauer) and found that although the GAB isotherm can reproduce the results of the BET equation, the GAB monolayer value is about 15% higher than the BET value and the GAB energy constant C reduces 35%-40% or more than the BET constant. That is huge difference. And these two models are unable to explain and represent the sorption at very low activities (relative pressure P P0 0. 1). Bashiri and Orouji[11] derived a new multilayer adsorption model based on the BET model and accounting for the heterogeneity by assuming the first layer being heterogeneous and the lateral layers being homogeneous. And it shows it has a good conformity with experimental data in some system. BET equation is traditionally used to evaluate the specific surface area of non-microporous materials. But still, it’s routinely used in microporous system. There’re several problems applying the BET model to microporous materials[12];(1)The BET monolayer capacity is doubtful,(2) monolayer structure is different for different materials,(3) Localized monolayer adsorption or micropore filling could occur in low relative pressure. Rouquerol et al[13]did further study about the applicability of BET equation in microporous adsorbent. They found that two other criteria (consistency criteria) besides the relative pressure range being 0.05-0.35 should be included if the BET equation was used to microporous adsorbent and the monolayer capacity was more like “BET strong retention capacity”. Bae et al[12] and Walton, Snurr[14]did molecular simulation to predict nitrogen adsorption in MOFs and zeolite with micropores and even ultra-micropores .The BET surface area calculated with the consistency criteria agreed well with the accessible surfaces areas obtained from crystal structures, indicating that BET theory can be applied to microporous materials on heterogeneous surface. In this work, a new multilayer adsorption isotherm is built based on the GAB model and the BET model; the adsorption surface is heterogeneous, not just for the first layer. And it shows that the new modified GAB model can be applied to describe the adsorption process in microporous material and can get the almost the same so called “BET monolayer capacity” with original BET model in the full relative pressure range. 2. Theory Assume that one molecule occupies sites when being adsorbed, then the rates of adsorption and desorption rate are proportional to 1 and respectively ( is the surface coverage); Similar to the derivation of Langmuir model, from adsorption and desorption kinetics: d kaP(1 ) kd dt When adsorption equilibrium is arrived, (1) d 0 dt Thus: 1 n nma x bP 1 bP This is the L-F isotherm[3], b ka kd 1 , V V n , Vm V m n ma x (2) . ka is the adsorption rate while kd is the desorption rate. n is the adsorption amount when adsorption equilibrium is reached. nmax is the saturation capacity. When is unity, the Langmuir-Freundlich isotherm reduces to Langmuir model. Thus can be a symbol of heterogeneity. Based on the concept of multisite occupation of one molecule when being adsorbed, the adsorption rate is proportional to surface coverage. And the new multilayer adsorption model is built. There are several assumptions need to make before the derivation. ⑴ One molecule occupies sites when being adsorbed. ⑵ The adsorption occurs at second and further layers before the completion of the first one[15]. ⑶ The heat of adsorption differs from that of liquefaction by a constant amount d from the second layer and above[7]. ⑷ The number of adsorbed layer is infinite when adsorption equilibrium is reached. Accounting for the heterogeneity of the system whether it is the heterogeneous surface or the force between adsorbed molecules, all is displayed in the parameter . When adsorption equilibrium is reached, the adsorption rate on the free surface (not occupied by molecules) is equal to the desorption rate of the first layer, that is[11]: 1 1 E a1PS0 b1S1 exp 1 RT (3) Here a1 and b1 are constants. P is the equilibrium pressure, E1 is the heat of adsorption. Si is the number of adsorption sites on each layer of the multilayer adsorption. For the second layer: 1 1 E d a2PS1 b2S2 exp 2 RT (4) The quantity d will be added to the adsorption heat of the first layer as well for convenience [7]. For the i th layer: a a Si 1 2 b b 2 1 E +d E d ai exp 1 2 bi RT RT E d i RT We know from the derivation of BET equation, E2 , E3 , i P S0 (5) Ei is assumed to be equal to the heat of liquefaction. From the second layers and above, the adsorption is mainly due to the attracting force among adsorbed molecules, the interaction among adsorbed molecules is similar (not equal) to the interactions in a pure liquid, so it is expected that d Ei and ai bi is constant. a2 a 3 b2 b3 ai bi a b (6) Define: x E +d a2 exp 2 P b2 RT (7) y E +d a1 exp 1 P b1 RT (8) C1 y x (9) The total amount of adsorption is: V i S (10) i i 0 Si is the monolayer adsorption capacity: Vm S (11) i i 0 Surface coverage is defined: V n θ Vm nmax i Si i 0 Si i 0 (12) x 1 , we get: θ C1 x 1 x 1 x C1x (13) When adsorption saturation is reached, the adsorption layers is infinite and the adsorption amount is infinite at the liquefaction pressure of the adsorbate[7], seen from Eq.(7), x must be infinitely close to unity and d=0 [7]. That is: E a exp P0 =1 b RT (14) P d Combine Eq.(7) and Eq.(14), we get: x exp . P R T 0 d C=C1 RT , Define: k exp P Vm C k P0 V P P P 1 k 1 k C k P0 P0 P0 (15) There’s a variant form of Eq.(16), that is: Vm C K P V 1 K P 1 K P C K P (16) There’re four adjustable parameters in the modified GAB model, that is nmax ,C, k, . The monolayer capacity nmax and energy constants C vary with the temperature, adsorbate and adsorbent. Anderson[7] and [8] found that by multiplying a constant less than unity with the relative pressure can greatly improve the applicability of BET model, situation is the same here. The parameter k accounts for the adsorption heat deviation from that of liquefaction. The parameter suggests the heterogeneity of the adsorption system. The only parameter which can be compared to that of other adsorption isotherms is the monolayer capacity. It will differ from different adsorption isotherms, but the difference is small enough to be neglected. That’s also a standard self-checking principle when we use different adsorption isotherm in the fitting process. In the next section, adsorption data from already published literature [16, 17]is applied to the modified GAB isotherm model. It shows that the new modified GAB model has good conformity with experimental data. 3. Application of Eq.(15) and Eq.(16) to Microporous material 3.1 Physisorption of ethane on Template carbon The first system is the ethane adsorption on the Template carbon, as the literature described[16]; the Templated carbon has a good microporous volume and a relatively homogeneous microporous structure with the pore sizes distributed around 1.0 nm[16]. To fit the experimental data of ethane adsorption with the modified GAB model, nonlinear curve fitting method is used. The adsorption data of ethane on Template carbon at 263K, 273K and 303K are also correlated by Eq.(16), displayed in Fig. 1 and it shows a good match. Fig. 1 Adsorption of ethane on a Template carbon respectively at 263K, 273K and 303K. Solid black Squares, solid red circles and solid blue triangles are respectively experimental data at 263K,273K and 303K from[16]. Red solid line, magenta solid line and olive solid line are correspondingly predictive data using modified GAB model derived in this work. Black dash line, blue dot line and pink dash dot line are correspondingly predictive data using GAB model derived by Anderson[7]. From Tab. 1, we can see that parameter varies from 0.4-0.6 at different temperature and the saturation capacity decreases as temperature increases since the adsorption process is exothermal. The terms and nmax calculated by Eq.(15) and Eq.(16) stay put but the parameter k(or K) differ. The parameter K in Eq.(16) is always reduced an order of magnitude compared to that of the parameter k in Eq.(15).We can see from Eq.(15) and Eq.(16), the parameter K in Eq.(16) is k P0 . Tab. 1 Correlated Parameters of Eq.(15) and Eq.(16) with data from [16] C k(or K) nmax/(mol·kg-1) Adj.R-Square 263K Eq.(15) 263K Eq.(16) 273K Eq.(15) 273K Eq.(16) 303K Eq.(15) 303K Eq.(16) 0.43169 215.1713 0.00156 55.01412 0.998287 0.43067 216 0.0004358 55.7216 0.998286 0.5674 176.6126 0.00941 18.1814 0.99947 0.5674 176.5703 0.00156 18.1797 0.99947 0.59841 105.5281 0.024629 13.5659 0.999965 0.5984 105.52798 0.002478 13.5659 0.999965 In order to investigate the accuracy of the modified GAB model, Langmuir-Freundlich isotherm, the GAB model are also used to correlate the experiment data and it’s showed in Fig. 1. The Langmuir-Freundlich isotherm has the same effect as the modified GAB model did. And the two lines overlap with each other into one line, and you couldn’t find the difference. So it’s not displayed in Fig. 1.The monolayer capacities obtained from the two models are almost identical. It’s showed in Tab. 2. Although the GAB model can fit the experimental data well, the “BET monolayer capacity” calculated from the GAB model is much different from the modified GAB model and the L-F isotherm. From the Fig. 1 and Tab. 2, we can see the modified GAB model built in this work has the best conformity with the experimental data. Tab. 2 Comparison of L-F isotherm, GAB and Modified GAB model in correlating adsorption data Langmuir-Freundlich Modified GAB model GAB isotherm (This work) (Anderson) Temp nmax/(mol·kg-1) Adj.R-square nmax/(mol·kg-1) Adj.R-square nmax/(mol·kg-1) Adj.R-square 263K 56.39186 0.99786 55.01412 0.99755 10.01543 0.97493 273K 18.66808 0.99934 18.1814 0.99923 9.0874 0.98391 303K 14.23082 0.99995 13.5659 0.99994 6.76205 0.99124 The Clausius–Clapeyron equation[18] is usually used to calculate the isosteric enthalpy. In order to look into more details about the modified GAB model, isosteric enthalpy of ethane adsorption is calculated with the modified GAB model and the Langmiur-Freundlich model. d l n P Qst RT 2 dT N (17) As show in Fig. 2, the isosteric enthalpy of ethane adsorption on Templated Carbon calculated with the Clausius–Clapeyron equation is very high at low coverage and decrease gradually until the loading reaches 6mol/kg, it then increases. It shows that the Templated carbon has energetically heterogeneous surface. This phenomenon is due to the two assumptions that an ideal bulk gas phase and a negligible adsorbed phase molar volume when using the Clausius-Clapeyron equation. As Pan, Ritter et al[19] indicated in their work, the largest effect comes from the ideal gas assumption at high loadings. Choi, Lee et al[20]did calculation of isosteric heat of ethane in the temperature range of 293.15K-313.15K and found the same phenomenon. The heterogeneity of the surface is the dominant effect at low surface loading, and then the interaction between adsorbate and adsorbate becomes the main effect, resulting in the decrease of isosteric enthalpy at low surface loading and then increase. The isosteric enthalpy of ethane is about 60kJ/mol at low surface loading, the same as what is indicated in Fig. 2 with the calculation using adsorption data at 263K and 273K. Other calculations using the adsorption data at temperature 263K and 303K or using the combination of adsorption data at 273K and 303K is misleading. (a) (b) Fig. 2 The isosteric enthalpy of ethane adsorption on Templated Carbon (a) isosteric enthalpy plotted as lnP vs 1/T. (b) isosteric enthalpy plotted as isosteric enthalpy vs loading. The solid lines with black square, red circle and blue triangle are the isosteric enthalpy obtained by Langmuir-Freundlich model using adsorption data respectively at 263K and 273K, 263K and 303K, 273K and 303K. While the solid lines with magenta up-down triangle, olive diamond and navy blue left triangle are obtained by the Modified GAB model. 3.2 Physisorption of Carbon Dioxide adsorption on Graphitized Carbon Black The second system comes from literature[17]. It measured the adsorption data of dioxide and ethane on STH-2 graphitized carbon black using volumetric method and the relationship between the relative pressure P P0 and the adsorption equilibrium amount at different temperature displays a characteristic curve as the potential theory predicts. We correlated the adsorption equilibrium data with several multilayer adsorption models. Through Fig. 3, we can see that the model built by Anderson[7] and the modified GAB all have good conformity with the experimental data and almost the same effect. But seen from the enlarged drawing, the modified GAB model built in this article fits experimental data the best. The model derived by Anderson is a special case of the model derived in this paper in the case that alpha( ) reaches unity. The Graphitized Carbon Black surface is regarded to be homogeneous[17], the term is approaching unity ranging from 1 to 1.3, displaying some heterogeneity of the surface. (a) (b) (c) Fig. 3 Comparison of model prediction with experimental data for CO2 adsorption on graphitized carbon black. Fig. (a), (b) and (c) represents adsorption at 263.2K, 273.2K 303.2K respectively. Black squares represent experimental data. Blue dash line represents the data predicted by BET model[2]; black dot line represents the data predicted by Bashiri[11]; Red solid line represents the data predicted by the modified GAB model built in this work; Oliva dash dot line represents the data predicted by the GAB model[7]. Tab. 3 Correlation Parameters of Eq.(15) and Eq.(16)with adsorption data at different temperature from [17] C k nmax/ (mol·kg-1) Adj.R-Square 263.2K 273.2K 303.2K 1.31775 16.06288 0.7871 0.59695 0.99985 1.25905 14.13714 0.84528 0.544432 0.99935 1.04088 5.14199 0.85644 0.69365 0.9999 One of the usages of BET isotherm is to calculate the specific surface area of an adsorbent, but when it comes to microporous material, Rouquerol. et.al[13] did a further study about how we can get the so called “monolayer content” in use of BET isotherm in microporous material. Including the condition that the relative pressure range should be 0.05-0.35, two other conditions should be considered. The value of C should be positive and the term of n P0 P should increase with the relative pressure. We took the adsorption data of carbon dioxide on graphitized carbon at 263.2K to demonstrate how the modified GAB model can applied to calculate the so called BET monolayer capacity in the full relative pressure range. The saturation capacity was calculated by using the BET model with the two added conditions and by using the modified GAB model (Eq.(15)). Fig.4 Plot of the term n P0 P vs. P P0 From Fig.4 we can see, in the relative pressure range of 0.05-0.35, the term n P0 P increases with the relative pressure, so we choose the data from relative pressure range 0.05-0.35, if the linear criterion of the BET plot is not satisfied, the relative pressure range should be narrowed down until the determination coefficient of the non-linear fitting process is larger than 0.99. The saturation capacity was calculated using the original BET equation and the modified GAB model in this work. Adsorption data of other temperature was also analyzed, the comparison is as follows: Tab. 4 the saturation capacity of carbon dioxide adsorption on graphitized carbon at 263K calculated by BET model and Eq.(15) adsorption model nmax /(mol·kg-1) Adj.R-square 263.2K BET Eq.(15) 0.54359 0.59695 0.9926 0.99985 273.2K BET Eq.(15) 0.56213 0.54443 0.99376 0.99935 303.2K BET Eq.(15) 0.65409 0.69365 0.99979 0.9999 We can see from Tab. 2 that the saturation retention capacity (“so called BET monolayer capacity”) in microporous material, calculated from the modified GAB model and the original BET equation is almost the same, the error was less than 9%.Once the saturation capacity nmax was known, the specific surface area can be calculated from Eq.(18) A = nmax NA am (18) NA is the Avogadro number and am is the molecular projected area[11, 21] which in this 2 circumstances is the projection area of carbon dioxide of 20 A / mol ecul e . 4. Conclusions The paper built a new multilayer adsorption model based on the original BET model and the GAB model; the modified GAB model has the same form with the original BET model, but the so called relative pressure is different, the nominal relative pressure is αth power of the relative pressure and it can improve the applicability of the BET model with the relative pressure being up to at least 0.9. BET model and the models supposed by Anderson[7] and Bashiri[11] are special cases of the model in this work. Microporous material system adsorption data are applied to the modified GAB model; it shows the model has good conformity with the experimental data. Comparison between several adsorption models is provided, and the model supposed by Anderson[7] and the model derived here has almost the same effect correlating experimental data, but the modified BET model derived in this work is the most accurate regarding the coefficient of determination. The term alpha ( ) in the modified GAB model is larger or smaller than unity accounting for the heterogeneity of the adsorption system. The so called “BET monolayer capacity “or more like the “BET strong retention capacity “of microporous materials calculated by the modified GAB model in the full relative pressure range and the original BET model with linearity criterion and two other criteria proposed by Rouquerol et al[13] is almost the same. It’s a new way to characterize the microporous materials. Nomenclature ai BET constant ( i 1, 2, , ) am projected area occupied by one molecule( nm molecule ) A the specific surface area( m kg ) b equilibrium constant(1/Pa) bi evaporation coefficients in BET equation ( i 1, 2, c solute concentration in the liquid phase (mole/kg) 2 2 , ) c0 solute saturation concentration in the liquid phase(mole/kg) d C adsorption enthalpy deviation from that of liquefaction(J/mole) energy constant Ei adsorption enthalpy in the i th layer(J/mole, i 1, 2, GAC k Guggenheim-Anderson-de Boer GAB model constant ka the adsorption rate(1/(s Pa)) kd the desorption rate(1/s) K GAB model constant ( k P0 ) L- F Langmuir-Freundlich NA Avogadro number n adsorption equilibrium amount(mole/kg) nmax monolayer saturation capacity(mole/kg) V adsorption equilibrium amount(m3/kg) Vm monolayer saturation capacity(m3/kg) P gas pressure (Pa) P0 saturated vapor pressure (Pa) Qst adsorption enthalpy(kJ/mole) R universal gas constant( J mole K ) Si number of adsorption sites in the i th layer( i 1, 2, T temperature (K) , ) 1 1 , ) Greek symbol display of the heterogeneity of the adsorption system surface coverage Subscripts N equivalent of surface loading when calculating the isosteric heat Notes The authors declare no competing financial interest. ACKNOWLEDGEMENTS Project supported by the National Natural Science Foundation of China (Grant No. 51406217) and by the National Key foundation (from National Ministry of Finance of the People’s Republic of China) for developing Major Scientific Instruments under Grant No.ZDYZ2014-1. [1] I. Langmuir, THE ADSORPTION OF GASES ON PLANE SURFACES OF GLASS, MICA AND PLATINUM, J. Am. Chem. Soc., 40 (1918) 1361-1403. [2] S. Brunauer, P.H. Emmett, E. Teller, Adsorption of Gases in Multimolecular Layers, J. Am. Chem. Soc., 60 (1938) 309-319. [3] R. Sips, On the structure of a catalyst surface, The Journal of Chemical Physics, 16 (1948) 490-495. [4] Toth.J, State equations of the solid-gas interface layers, Acta Chim Acad Sci Hungar, 69 (1971) 311-328. [5] M.M. Dubinin, The Potential Theory of Adsorption of Gases and Vapors for Adsorbents with Energetically Nonuniform Surfaces, Chem. Rev., 60 (1960) 235-241. [6] S. Thomas, P. Schaetzel, Multilayer adsorption equilibrium model for gas adsorption on solids, Adsorption-journal of the International Adsorption Society, 19 (2013) 121-129. [7] R.B. Anderson, Modifications of the Brunauer, Emmett and Teller Equation1, J. Am. Chem. Soc., 68 (1946) 686-691. [8] S. Brunauer, J. Skalny, E.E. Bodor, Adsorption on nonporous solids, Journal of Colloid & Interface Science, 30 (1969) 546–552. [9] T. Gu, The application of the modified BET equation to multimolecular adsorption from solution, Journal of Colloid & Interface Science, 85 (1982) 601–603. [10] E.O. Timmermann, Multilayer sorption parameters: BET or GAB values?, Colloids & Surfaces A Physicochemical & Engineering Aspects, 220 (2003) 235–260. [11] H. Bashiri, S. Orouji, A new isotherm for multilayer gas adsorption on heterogeneous solid surfaces, Theor. Chem. Acc., 134 (2015) 1-7. [12] B. Youn-Sang, Y. A Ozgür, R.Q. Snurr, Evaluation of the BET method for determining surface areas of MOFs and zeolites that contain ultra-micropores, Langmuir the Acs Journal of Surfaces & Colloids, 26 (2010) 5475-5483. [13] J. Rouquerol, P. Llewellyn, F. Rouquerol, Is the BET equation applicable to microporous adsorbents?, Studies in Surface Science & Catalysis, 160 (2007) 49–56. [14] K.S. Walton, R.Q. Snurr, Applicability of the BET method for determining surface areas of microporous metal-organic frameworks, J. Am. Chem. Soc., 129 (2007) 8552-8556. [15] S. Thomas, P. Schaetzel, Multilayer adsorption equilibrium model for gas adsorption on solids, Adsorption, 19 (2013) 121-129. [16] J. Chen, L.S. Loo, K. Wang, An Ideal Absorbed Solution Theory (IAST) Study of Adsorption Equilibria of Binary Mixtures of Methane and Ethane on a Templated Carbon, J.chem.eng.data, 56 (2011) 1209-1212. [17] M. Li, J. Liu, T.L. Wang, Adsorption Equilibria of Carbon Dioxide and Ethane on Graphitized Carbon Black †, J.chem.eng.data, (2010). [18] T.L. Hill, Statistical Mechanics of Adsorption. V. Thermodynamics and Heat of Adsorption, J. Chem. Phys., 17 (1949) 520-535. [19] H. Pan, J.A. Ritter, P.B. Balbuena, Examination of the Approximations Used in Determining the Isosteric Heat of Adsorption from the Clausius−Clapeyron Equation, Langmuir, 14 (1998) 6323-6327. [20] B.U. Choi, D.K. Choi, Y.W. Lee, B.K. Lee, S.H. Kim, Adsorption Equilibria of Methane, Ethane, Ethylene, Nitrogen, and Hydrogen onto Activated Carbon, Journal of Chemical & Engineering Data, 48 (2003) 603-607. [21] D. Do, Adsorption Analysis: Equilibria and Kinetics. 1998, in, Imperial College Press, London. Highlights A new multilayer adsorption isotherm is built The new model is found to correlate adsorption data in microporous material well The model extends the applicability of the BET equation