Derivation and constants determination of the Freundlich and (fractal) Langmuir adsorption isotherms from kinetics

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$IOP onference Series: Materials Science and Engineering PAPER OPEN AESS Derivation and constants determination of the Freundlich and (fractal) Langmuir adsorption isotherms from kinetics To cite this$

1 IOP onference Series: Materials Science and Engineering PAPER OPEN AESS Derivation and constants determination of the Freundlich and (fractal) Langmuir adsorption isotherms from kinetics To cite this article: Patiha et al 208 IOP onf. Ser.: Mater. Sci. Eng Related content - Kinetic and Isotherm Studies for the Adsorption of Phanerochaete hrysosporium on Lignites Yu Niu, Xian Niu, Shiyue Wu et al. - Isotherm adsorption studies of Ni(II) ion removal from aqueous solutions y modified caroxymethyl cellulose hydrogel L Anah and N Astrini - Potential adsorption of methylene lue from aqueous solution using green macroalgaeposidonia oceanica. F-N Allouche and N Yassaa View the article online for updates and enhancements. This content was downloaded from IP address on 5/09/208 at 23:00

2 Derivation and constants determination of the Freundlich and (fractal) Langmuir adsorption isotherms from kinetics Patiha, M Firdaus, S Wahyuningsih, K D Nugrahaningtyas and Y Hidayat Department of hemistry, Seelas Maret University, Surakarta 5726, Indonesia Tel./Fax patiha3@yahoo.co.id Astract. The Freundlich adsorption isotherm is mostly presented as an empirical equation and the (fractal) Langmuir adsorption isotherm is derived from kinetics. However, their constants are determined y thermodynamic approach. As yet, oth are used as an independent tool for determining the type of adsorption, just physical or chemical adsorption. This study aims to introduce an alternative way of driving Freundlich adsorption isotherm from kinetics, to elaorate its relation with the (fractal) Langmuir adsorption isotherm, and to determine their constants simultaneously y kinetics approach. An alternative way of determining the constants in oth isotherm equations y kinetics approach was introduced. The results were then compared with that otained y the conventional method. The study is theoretical ut the validity of determination technique is ased on the statistical tests on data adapted from literature. The Freundlich adsorption isotherm is introduced; the isotherm is the (fractal) Langmuir adsorption isotherm at relatively low concentration. It is irrelevant to use oth adsorption isotherm separately. The new method is more reliale and its represent the reaction.. Introduction One of the first isotherm equations introduced to descrie the adsorption phenomenon is the Freundlich Adsorption Isotherm (FAI): S = K () where S is the mass adsored per adsorent mass; is the concentration in solution; K and are fitting constants []. The value of K is otained from the intercept and from the slope of the doulelogarithmic plot of log S versus log of equation (Eq.): log S log K log (2) The other is the Langmuir Adsorption Isotherm (LAI): S Kads k (3) where K ads, and k are fitting constants [2,3]. The curve pattern of S versus of (Eq.) (3) is a rectangular hyperola. Thus, it is difficult to determine the values of K ads and k directly from Eq. (3). By taking the reciprocal of oth sides of Eq. (3), however, is transformed into its linear form ontent from this work may e used under the terms of the reative ommons Attriution 3.0 licence. Any further distriution of this work must maintain attriution to the author(s) and the title of the work, journal citation and DOI. Pulished under licence y Ltd

3 / S / K ( k / K ) (4) ads ads / The K ads value is otained from the intercept and k from the slope of the curve (/S ) versus. (/). However, for some reason, Eq. (4) cannot e used to get the exact values of K ads dan k [4]. In many textooks, the FAI is presented as an empirical equation with nearly no, or limited in its usefulness for its aility to fit data or to interpret the physical significance of the coefficient [5]. The kinetics approach has also een introduced [6]. It is ased on equation: n2 d / dt k k2s (5) At equilirium the derivative is equal to zero. Then, after some steps Eq. (5) ecomes: S ( k / n / k2) 2 n2 (6) where n and k are the order and fractal rate constant for adsorption and n 2 and k 2 for desorption. Eq. (6) is identical with Eq. () if (k /k 2) /n 2 = K and (n /n 2) =. The modified or fractal LAI has also een introduced y the same author. It is ased on equation: d / dt k ( S S) k S o 2 (7) At equilirium the derivative is equal to zero. Then, after some steps Eq. (7) ecomes: S k 2 / S o k n (8) where S 0 is the maximum adsorption capacity and k and k 2 are the fractal rate constants. Or, if S o = K ads ; n =, and k 2/k = k ecomes S K ads k (9) Eq. (9) is the fractal LAI and identical with Eq. (3) if =. Another form of equation has also een reported earlier [7]. The reciprocal of oth sides of Eq. (9) is / S / / Kads ( k / Kads) (0) Eq. (0) is identical with Eq. (4) if =. The value of K ads is otained from the intercept of the linear plot of /S versus / and then used to determine k from the slope of the line. It is interesting to compare Eq. (0) with the equation otained from Eq. (9) on the condition of which is relatively higher and lower than k. If is higher than k, then +k, Eq. (9) ecomes S K ads () The reciprocal of Eq. [] is 2

4 / S / (2) K ads which is differ with Eq. (0). Inserting Eq. (2) into Eq. (0) gives / Kads / Kads ( k / Kads)/ or k = 0 (3) That is to say, Eq. (0) does not apply if is relatively higher than k. If is lower than k, and so +k k, Eq. (9) ecomes S ) ( Kads/ k (4) The reciprocal of Eq. (4) is / S ( k / K ads )/ (5) which is also differ with Eq. (0). By assumption, S o = K ads and therefore, if all surfaces occupied (or S = S o), then S = K ads and Eq. (5) ecomes k = (6) That is to say, Eq. (0) also does not apply if is relatively lower than k. Thus, it can e concluded that, theoretically, Eq. (0) is amiguous and therefore it cannot e used to determine oth K ads and k. There are some prolems arises. First, Eq. (6) does not always apply to every reaction. The fitting constant has a value of 0 [8]. But, the reaction orders are usually integers (, 2,..,) or halfintegers (/2, 3/2,...), positive or negative [9]. If, n is higher than n 2, then. Furthermore, unlike Eq. (7), in Eq. (5), the rate of adsorption is assumed to e independent on the vacant sites, S (= S o S). This goes against theory. Normally, the amount of vacant sites and the types of adsorent always affect the rate. Thus, the rate of adsorption in Eq. (5), must depend on S. Moreover, the order with respect to S is n 2 ut in Eq. (7) is the first-order. Second, usually the FAI and LAI equations are treated separately. For example in determining the type of ond formed in the adsorption process; chemical or just physical ond [0]. However, the FAI can e derived from LAI []. Allegedly, it is suspected that this statement can e confirmed y using Eq. (9) and Eq. (). If so, then it is possile to determine the constants of oth FAI and LAI simultaneously. Third, as yet, despite eing derived from kinetics, the constants in oth isotherms are still determined y the thermodynamic approach. The data used are always otained from equilirium condition at the same contact time. In practice, y using a certain concentration, the first step is to find the time needed to achieve the equilirium condition. Then y using the time otained to find the S change from some other concentrations. For some reasons, this treatment is not recommended. It is possile that the data are just taken from a certain condition and only valid to Eq. () i.e is higher than k or Eq. (4) i.e is lower than k. This is not representing the sample as a whole. Basically, the S concept is resemle to the rate concept; the changes of the amount of reactant or product at a given time interval. To e more specific, it is similar to the differential method in kinetics. The order otained y differential method is fixed ut the value of the constant must e checked with that otained y integral method [2]. In addition, the only criterion used to determine the validity of the calculation result is ased on the 3

5 regression coefficient. The result is always claimed to e valid if the regression coefficient is close to ±.000. It is never questioned whether the data used have met the adsorption isotherm conditions for FAI and / or LAI or not. Based on the aove discussion, the ojectives of this study are to derive FAI y kinetics approach, to elaorate the relation of FAI and LAI, and to determine the FAI and LAI constants simultaneously y kinetics approach. 2. Experimental 2.. Theory This research is a literature study. The first goal is to drive Eq. () from Eq. (7). The second goal is to drive Eq. () from Eq. (9). The third goal is to find an easier and reliale technique for the simultaneous determination of, K, K ads, and k y kinetics approach and will e achieved y the following steps. i. By Eq. (4), the order () is otained from the slope of the curve of S versus for = and 0.5. This ecause, y kinetics, to get, the possile orders are or 0.5. The curve should cross the y- axis at y = 0 and x-axis at x= 0 with regression coefficient closer to ± and intercept closer to y = The k value is determined using the slope and K ads otained y Eq. (2). The more exact value (at the conditions of higher and lower than k) will e otained y using integral equations: [ A o ] [ A] akt for n = 0 (7) [ A o ] [ A] 2 akt for n = 0.5 (8) ln[ A]/[ A0 ] akt for n = (9) The equation is selected with regression coefficient closer to ±.000. ii. The value of K ads is determined simply, y making the curve of S versus at the conditions needed for Eq. (0) is valid to e used. This is possile ecause y Eq. (0) the reaction is zeroth-order with respect to. The curve should cross the x-axis at x = 0. The intercept is the value of K ads. The curve chosen is that with slope closer to x = 0.00 and regression coefficient closer to ± Model This study uses data, not from adsorption experiment, ut taken from the romination of acetone: (H 3) 2O + Br 2 H 3OH 2Br + Br - + H +. Br 2 is assumed to e the adsorate and acetone, A, (which is in excess) is the adsorent, H + is the catalyst (which is also in excess) to activate acetone. The mechanism of the reaction has een given [3] and from that, the rate law of ( kk d[ P]/ dt ( k 2 2 ) / k [ A][ H ][ Br2 ] / k )[ H ] [ Br ] 3 2 (20) If in the eginning, [Br 2] o, is higher than (k -2/k 3)[H + ] (actually higher than [H + ] released from acetone), at the early-stage, the reaction is zeroth-order with respect to [Br 2] ut at the near-end is firstorder. During the reaction process, [Br 2] is decreased while [H + ] is almost constant (in this reaction H + acts as a catalyst and is added excessively). At this condition, this reaction is resemle to adsorption reaction. But, if in the eginning, [Br 2] o is lower than (k -2/k 3)[H + ], for the same reason, the reaction is always first-order [4]; [5]. 4

6 As seen in Tale, the data are not presented in concentration unit ut in asorance of Br 2 in the solution, measured at =400 nm (asorance index 60 M - ) and room temperature. The data were otained from the experiment where, originally, [Br 2] o is higher than [H + ] released from acetone. Tale. Asorance, A, of Br 2 in Bromination of Acetone: 0 ml Acetone 4.0 M, 5.0 ml Hl.0 M, 25 ml H 2O, 0 ml Br M [8] Early stage Near end No. t / s A t / s A In order to make it more similar to the experimental technique on adsorption, each oservation which was taken at the same time interval and the oservation resulting at the end of one time interval eing considered as the initial value for a new time interval. Each direct oservation is taken as o and the following as. The difference etween each pair is taken as S (thus the amount of S is always taken at the same time interval). The setting of the data is presented in Tale 2. The value of is determined y integral method using Eq. (7), Eq. (8), and Eq. (9). It was ased on the value of regression coefficient (closest to ± ) and intercept (closer to 0.00). This step is of special important. This is ecause the reaction proceeds in different order at the early-stage and nearend. Tale 2. Setting the data of Tale. for n = Early-stage Near-end No. t/s t/s o o = S o o = S Result and Discussion 3.. Derivation of the FAI Derivation of the Freundlich isotherm is ased on the equation: d / dt k ( S S) k S o 2 In Eq. (5), the order with respect to S (= S o S) and S are assumed to e first-order ecause there is no change in the structure of the adsorent and the adsorption is assumed to e monolayer. It is just a process of ending and releasing of an adsorate. It is differing with. Here, the order is n (which is 5

7 can e an integer or non-integer). The mechanism of the ending process of may e differing. A change in the structure of an adsorate could happen. As an example, for a given adsorent, an acid HA will ionize and eing adsored as its ion. By ionization, HA H + + A -, then A - = K a If, S S O, Eq. (5) ecomes d / dt k S k S o 2 (2) At equilirium the derivative is equal to zero. The equation can then e solved for S: S k / k ) S o (22) ( 2 or, if k /k 2= /k and S o=k ads, then Eq. (22) ecomes S ( K / k) (23) ads Eq. (23) is identical to the standard Freundlich isotherm, Eq. (), if K ads / k = K and n = Relation of FAI and LAI At high concentration or is relatively higher than k, Eq. (9) ecomes: S () K ads If S approaches the limiting value or S = S o, all surfaces are covered and S o = K ads. From kinetics point of view, the reaction is of the zeroth-order with respect to. At low concentration or is relatively lower than k, Eq.(9) ecomes S ) ( Kads/ k (4) Eq. (4) is identical with Eq. () if K ads / k = K. From kinetics point of view, the reaction is of the -thorder with respect to. This means, the FAI can e derived from the fractal Langmuir equation; the same as discussed aove. The FAI is the LAI at low concentration Simultaneous determination of LAI and FAI from kinetics Based on aove discussion, it is important to determine the order of the reaction at oth conditions; at the early-stage (to determine K ads) and at the near-end (to otain K ads/ k, and then used to determine k ). This treatment is then called the New Method (NM). For easier calculation, the A o for the early- stage is and for the near-end is assumed to e for n = and for the early-stage is and for the near-end is assumed to e for n = 0.5. The results of the calculation y integral method for n = = 0, 0.5, and are presented in Tale 3. Tale 3. Order, n =, of the Reaction at the Early-stage and Near-end Position Early-stage Near-end n = A B r A B R * * 6

8 Based on Tale 3, the new method confirms that at the early-stage, the process is zeroth-order (r = ) and first-order at the near-end (r = ). Thus, y the NM, the value of K ads is determined from the intercept of the curve of S versus of the data at the early-stage (Eq. ) and k from the slope of the curve of S versus of the data at the near-end (Eq. 4). The results are then compared with the values of K ads and k otained y conventional method using Eq. (0) and K ads / k (otained from Eq. (4) with K otained from Eq. (2). The results of the determination of FAI y conventional (Eq. 2) are presented in Tale 4. Tale 4. The Results of Statistical Analysis for the Determination of FAI onstants Data Source A B r Early-stage Near-end The slope B in Tale 4, is used to determine (of Eq. (2)). In this case its value is positive. In Tale 4, the slope B at the early-stage is negative. This means that the Freundlich constants of the reaction, cannot e determined from the data at the early-stage. Then, the Freundlich constants of the reaction, can only e determined from the data at the near-end. The slope B in Tale 5. is used to determine k /K ads. Both intercepts have positive value. But, the slope B at the early-stage is negative. The slope can not e used to determine k /K ads of Eq. (4) or Eq. Tale 5. The Results of Statistical Analysis for the Determination of LAI onstants Data Source A B R Early-stage Near-end (0) for =. This means that the Langmuir constants of the reaction, can only e determined from the data at the near-end. Tale 6. The Results of Statistical Analysis for the Determination of K ads and k y the NM Data Source A B R Early-stage Near-end The slope B at the early-stage in Tale 6. is close to to zero: the curve is parallel to x-axis and so, the intercept can e used to determine K ads(of Eq. ()). The intercept A at the near-end is close to zero. Then, its slope can e used to determine K ads / k (of Eq. (4)). Tale 7. The values of K ads and k y the onventional and the NM Method Freundlich Langmuir New Method B K r K ads k r K ads K Earlystage Near-end

9 There are five interesting facts reveal in Tale 7. that need to e discussed in their relation with the aims of this study.. The FAI and LAI constants could not e determined using the conventional method from the data at the early-stage ( in the condition of k). This is consistent with Eq. (). This means, the constants of FAI and LAI counted y conventional method do not represent the reaction as a whole. 2. By the NM, K ads is determined from the data at the early-stage and k from the data at the nearend.these mean, the values of K ads and k determined y the NM represent the reaction as a whole. 3. The value of, which is the same as that y integral method, indicates that the FAI is the fractal LAI at low concentration. And, that, relatively, the K value (of FAI = ) K ads / k (of LAI = / = ). This again confirms the fact that the FAI is a part of fractal LAI. (This is ecause, FAI is determine y differential while NM y integral method) 4. The fact that, at low concentration, FAI has regression coefficient (r = ) higher than LAI (r = 0.940) not just mean, FAI works etter than LAI ut, simultaneously, confirms that FAI is the fractal LAI at that condition. So, it is irrelevant to distinguish oth isotherms. 5. At first sight, the value of K ads and k otained from the reciprocal of Langmuir equation, Eq. (4), are the real values. Both are higher (than that otained y NM.) However, the data used for the calculation is the data at the near-end (fact.) At this condition, is relatively lower than k and the reaction is first-order; the curve of S versus cross x-axis at x = 0 and y-axis at y = 0. There is no maximum value. K ads can not e determined. Thus, the NM is est used to determine K ads and k. The values given are exact and the process is simple; directly using oservation data. It is interesting to note that the K ads and k values derived from the reciprocal of LAI (Eq. (4)) is approximately ten times higher than that from NM. Is it ecause, unlike NM, in Eq. (4), the time factor (in this case 0 seconds) is not included in the calculation? It is unlikely; the value of K (of FAI = ) otained from Eq. (2) is relatively the same with K ads / k (of LAI = / = ) otained y using NM. 4. onclusion In summary, the FAI equation has een derived from kinetics. The equation is the fractal LAI at low concentration. By NM, K ads and k of the fractal LAI should e determined separately: K ads from the intercept of the linear plot of Eq. (7): (relatively high ) and k from the calculated-value of K ads and the slope of the linear plot of Eq. (9): (relatively low.) References [] Freundlich HMF 906 Z. Phys. hem [2] Langmuir I 96 Part I The Research Laoratory of The General Electric ompany 222 [3] Langmuir I 98 Part II The Research Laoratory of The General Electric ompany 848 [4] Patiha, Heraldy E, Hidayat Y, Firdaus M 206 IOP onf. Ser. : Mat. Sci Eng. 07 () [5] Spark D L 2003 Environmental Soil hemistry 2 nd (San Diego : Academic Press) pp. 5 [6] Skopp J 2009 J. hem. Ed [7] Sposito G 980 Soil Sci. Soc. Am. J [8] Gregory T, Karns L, Shimizu K D 2005 Anal. him. Acta [9] Levine I N 2003 Physical hemistry 6 th Edition (New York : McGraw-Hill ompanies, Inc) pp. 57 [0] astellan G W 983 Physical hemistry 3 rd Edition (Massachusetts : Addison-Wesly Pulishing ompany, Inc) pp. 428 [] Sawyer N, Mcarty P L, Parking G N 2003 hemistry for Environmental Engineering and Science 5 th Edition (New York : McGraw-Hill ompanies, Inc) pp [2] Laidler K J 987 hemical Kinetics 3 rd Edition (New York : Harper ollins Pulisher, Inc) pp. 28 8

10 [3] Daniels F, Mathews J H, Williams J W 970 Experimental Physical hemistry, 7 th Edition (Tokyo : Kogakusha ompany, Ltd) pp [4] Patiha 203 Kajian Kritis terhadap Persamaan-persamaan dan Teknik untuk Penentuan Tetapan Michaelis-Menten Penelitian Fundamental FMIPA UNS (Surakarta) [5] Patiha, Heraldy E, Hidayat Y, Firdaus M. 206 IOP onf. Ser.: Mat. Sci. Eng. 07 () 9

The langmuir isotherm adsorption equation: The monolayer approach

The langmuir isotherm adsorption equation: The monolayer approach IOP Conference eries: Materials cience and Engineering PAPER OPEN ACCE The langmuir isotherm orption equation: The monolayer approach To cite this article: Patiha et al 06 IOP Conf. er.: Mater. ci. Eng.