Langmuir 1991, 7, 1400-1408
1400
Thermodynamic Theory of Stoichiometric Adsorption A. M.Tolmachev Chemical Faculty of Moscow State University, 119899, Moscow, Lenimkie Gory, USSR Received December 19, 1989. In Final Form: October 16, 1990 On the basis of a model in which the adsorption solution is treated as the phase having a limited capacity with respect to the system components,a general thermodynamictheory of adsorption has been developed, which enables one, by proceeding from uniform principles, with the same choice of standard states of the components(and, in particular, of the adsorbent) in the phases, to describe equilibria of physical (localized and nonlocalized) adsorption and chemisorption of individual fluids and their mixtures from bulk phases of any aggregate state (gases, vapors, liquids). Within the framework of the method of activities, a mathematicaltool of the theory has been developed, applicable to the problemsof thermodynamictreatment of experimental data related to the adsorption of one- and two-componentfluids on adsorbents of different structure. It is demonstratedthat, owing to ita general character,this theory may provide the thermodynamic basis for the development of methods of a priori calculation of adsorption equilibria.
Introduction In order to solve one of the most important problems of the contemporary theory of adsorption, which consists in calculating the equilibrium characteristics of adsorption in one-component and multicomponent systems, it is necessary to construct, as the starting basis, a maximum general thermodynamic theory, which will enable one to describe adsorption equilibria in such systems, by resorting to a unified thermodynamic language and by using the same standard and reference states. The requirements of adsorption technology and the solution of the fundamental problem of describing the properties of substances in adsorbed state generate a need for subdividing the system into two phases: a bulk phase and an adsorption phase, insofar as the latter must be characterized by the corresponding parameters of state and, first of all, by the value of adsorption capacity with respect to the components of the bulk phase. In this connection the use of the Gibbs "method of excess which allows the obtaining of information on changes in the thermodynamic properties of the adsorption system in the whole, proves to hold little promise, and one has to use methods of "total content", based on this or that model of the adsorption phase.3 The techniques of thermodynamically correct choice of the dimensions of the adsorption phase in the adsorption of gases, vapors, and liquid mixtures have been discussed in detail in refs 3-10. It turned out that in the case of (1) Gibbs, J. W. Termodinamisheskie Raboty (Thermodynamic Works);Moscow-Leningrad, 1950; p 492 (in Russian). (2) Lopatkin, A. A. Teoreticheskie Osnovy Fizicheskoi Adsorbtsii (Theoretical Fundamentals ofPhysica1Adsorption);MoscowUniversity Publishers: Moscow, 1983; p 344 (in Russian). (3) Tolmachev, A. M. Usp. Khim. 1981,50, 769 (in Russian). (4) Larionov, 0. G. in Rasshirennye Tezisy Dokl. IV Vsesoyuzn.Konf. P O Teor. Voprosam Adsorbtsii (Extended Summaries ofpapers Read at the Fourth All-Union Conference on Theoretical Problems of Adsorption); Moscow, 1973; Issue 1, p 111 (in Russian). (5) Larionov, 0. G.; Yakubov, E. S.Zh. Fiz. Khim. 1982,56,1523 (in Russian). (6) Avramenko, V..A.; G!uschenko, V. Yu.; Zhidachevskaya, V. Ya.; Madaras, I. N. Zh. Fiz. Khim. 1984,58,2119 (in Rusaian). (7) Fomkin, A. A.; Seliverstova, I. I.; Serpinsky, V. V. Synth., Struct., Technol. and Appl. Proc. Int. Symp., Pertorose, 1984; Amsterdam, 1985, p 443. (8) Serpinsky,V.V.;Yakubov,T.S.Izv.Akad. NaukSSSR,Ser. Khim. 1985, 12 (in Russian). (9) Fomkin, A. A.; Seliverstova, I. I.;Serpinsky, V. V. Izv. Akad. Nauk SSSR, Ser. Khim. 1986, 280 (in Russian). (10) Chkhaidae, E. V.; Fomkin, A. A.; Serpinsky, V. V.; Tsitaishvili, G. V. Izv. Akad. Nauk SSSR, Ser. Khim. 1986,299 (in Russian).
0743-7463f 91f 2407-14OO802.5Qf 0
microporous adsorbents, the volume of the adsorption phase in any system is approximated fairly well by the volume of the micropores and, in the case of macroporous adsorbents, by the monolayer (gases and liquids), and in some systems by two or three layers (liquids). In all the cases mentioned above the adsorption phase has a limited capacity with respect to the system components. Only the adsorption of vapors in the macropores is complicated by the processes of polymolecular adsorption and capillary condensation and a special thermodynamic analysis is therefore required. When one develops a model of the adsorption phase, the problem resides in that the choice of components of the adsorption solution, ensuring the simplest way of obtaining the necessary thermodynamic relations for different onecomponent and multicomponent adsorption ~ y s t e m , ~ should be made within the thermodynamically permissible limits. With due regard for the requirements set forth, the author of the present paper suggested a sufficiently general model of the adsorption solution within the framework of the method of "total content" in 1973." From this model, a thermodynamic theory of stoichiometric adsorption (STA)was developed later.3J2-15 In the present paper we shall give for the first time a complete exposition of the contemporary (substantially refined and revised) version of this theory. This theory is built upon a large amount of literary material concerning the thermodynamics of adsorption and, as a matter of fact, uses the achievements obtained in this field by a large number of researchers. In this context the most important original and review works are menti0ned.l~*~~J6-~~ (11) Tolmachev, A. M. Zh. Fiz. Khim. 1973,47, 2629 (in Russian). (12) Tolmachev, A. M. Zh. Fiz. Khim. 1978,52, 1050 (inRussian). (13) Tolmachev, A. M. Zh. Fir. Khim. 1986, 69, 2764 (in Ruesian). (14) Tolmachev, A. M.; Belousova, M. E. Vestn. Mosk. Univ., Ser. 2 Khim. 1988,29 (4), 257 (in Russian). (15) Artywhina, G. G., Tolmachev, A. M. Vestn. Mosk. Univ. Ser. 2 Khim. 1988,29 (6), 571 (inRussian). (16) Butler, J. A. V. Proc. R. SOC.London, A 1932,135A (2), 348. (17) Hill, T. L. Adv. Catal. Related Subj. 1952, 4, 211. (18) Haneen, R. S. J. Phys. Chem. 1962,66,410. (19) Everett, D. H. Trans. Faraday SOC.1964,60,1083 and 1965; 61. 2478. (20) Shay, J.; Nagy, L. G. J. Colloid Interface Sci. 1972, 38, 302. (21) Shay, J. Pure Appl. Chem. 1976,48, 393. (22) Sircar, S.; Myers, A. L. Adsorpt. Sci. Technol. 1985, 2, 69. (23) Sircar, S.; Myers, A. L. Sep. Sci. Technol. 1986,21,536. (24) Everett, D. H.; Podoll, R. T. Colloid Sci. 1979,3, 63. (25) Davis, J.; Everett, D. H. Colloid Sci. 1983, 4, 84.
0 1991 American Chemical Society
Langmuir, Vol. 7, No. 7, 1991 1401
Thermodynamic Theory of Stoichiometric Adsorption Unlike these and other works in which rather stringent thermodynamic ratios were obtained, as a rule, in relation to separately examined categories of adsorption systems (adsorption of gases, of vapors, of gas mixtures, of liquid solutions, etc.),this paper makes an effort to build a rather general thermodynamic theory enabling one, as noted above, to carry out a thermodynamically harmonized processing of experimental adsorption-related data in any adsorption (and chemisorption) system.
Choice of Model. General Relationships This paper uses the following principal designations: pi, ni, Ci, and Ti, chemical potentials, number of moles, concentrations (m0l-L-1)~and coefficients of component activities of a volumetric phase (if vapor and liquid volumetric phases are being examined simultaneously, then the latter relevant values are noted with a tilde as Ei and so on); pi^, piR, n,R, c a , and ya,complete and “internal” chemical potentials, number of moles, concentrations (mol-kg-l), and coefficients of component activities of adsorption solution (index R means an adsorbent, as CR, (PR, and so on); cia and Yis, concentrations (mol-L-l) and coefficients of activities of saturated vapor of single liquid adsorptives; Eio and CiRo, concentrations of single liquids (mol-L-’) and “standard” concentrations (molskg-l) of adsorbates in the adsorption phase; CiR”, the highest attainable concentrations of adsorbates in the adsorption phase under ci, Eio m; k < I, normalization factor determining the choice of standard condition of an adsorbent; W, surface (volume) of macro- (micro-)porous adsorbents; 9,a portion to fill the adsorption volume with a sum of adsorbantes with respect to filling under standard conditions (9= Iunder c a = cao;it is obvious that values (9” = I/k)may be attained with the adsorption of liquids under an additional hydrostatic pressure); NiR, a volume portion i, component from the adsorption volume occupied )~~, by a sum of adsorbates; (ciyi)st and ( c ~ R Y ~ Rcomponent activities of volumetric and adsorption solutions under selected standard conditions; Zi = Ci/ (CiYiIBt and ZiR = Clx/ (CiRYa)Bt, nondimensional concentratioqs of components of volumetric and adsorption solutions; sa and sa, molar areas or component volumes of an adsorption solution; D i O , molar volumes of single liquid adsorptives; fli and fl, stoichiometric coefficients of quasi-chemical equations of adsorption; Ki*, Ki(i)*, R*, Ki*, Ki(j)*,and K*, thermodynamic constants and coefficients of an adsorption equilibrium i component, i component of a binary mixture and competitive adsorption binary mixtures of substances; u, UiR,st, and q a t , surface tension or internal pressure in the adsorption phase in equilibrium and in selected standard conditions of adsorption solution components. Under a simultaneous analysis of adsorption of a singleor two-component fluids, all designations for a singlecomponent adsorption are supplemented with a prime, as c’i, ~ ’ a and , SO on. In the STA,to extend Langmuir’sideas, as components of the adsorption solution, which is the phase having a limited capacity with respect to the system components, choice is made of complexes of the molecules of adsorbates (A, B, ...) with adsorption sites (R)of adsorbent (ARl/@,, BR1pB,...) and free adsorption sites of adsorbent (R), the entire mass of which is subdivided into a definite number of “moles” (nRO),and to each mole there corresponds a
-
(26) Bering, B. P.; Maiers, A. L.; Serpinsky, V. V. Dokl. Akad. Nauk SSSR 1970, 193,119 (in R w i a n ) . (27) Rusanov, A. I. Farouye Raunouesiya i Pouerkhnostnye Yauleniya {PhaseEquilibria and Surface Phenomena);Khimiya Publishers: Leningrad, 1967; p 388 (in Russian).
definite area or volume of micropores (SR) respectively. Adsorbates with effective (i.e. those taking into account the number of layers in the adsorption phase in the case of macroporous adsorbents) partial mole areas or volumes (in the case of adsorption in micropores) may occupy a certain part of the surface or volume of the adsorption phase (sm,QR, ...) with the formation of corresponding complexes in accordance with simultaneously proceeding quasi-chemical reactions of addition
and competitive adsorption
(3) wherein Binding energy in adsorption complexes may vary within a wide range: from energies corresponding to chemisorption processes to very small energies of dispersion interactions, while adsorption may be either localized or nonlocalized. In the latter case, stoichiometry of the complex is defined by the statistical average surface (or volume) per mole of the given component of the adsorption phase. As concerns the process of competitive adsorption (3), the following comments are deemed to be appropriate. In accordance with (1) or (2), complete saturation of the adsorbent may take place only when the pressure (concentration) of the component in the bulk phase is infinitely high. Therefore, in real adsorption processes there is always CR # 0 and the transition from the state c m , ~CBRJ, , CR,1 to the state CAR,^, CBR,p, CR.2 may be regarded as a sum of two processes: a transition from CR,1 to csz (joint adsorption-desorption of the components (A,B) and then a transition to cm,2,CBRgat CR,2 = constant, i.e. there occurs a competitive mutual displacement of the components (A, B) from the adsorption solution. Since equilibrium concentrations of all the Components in the phases satisfy simultaneously the system of eq 1-3, in the case of adsorption of a two-component mixture of adsorbates, complete information of the system may be obtained in the analysis of any two of the three equations. Very comprehensive and valuable information in such a case may be obtained from the analysis of eq 3; it is most expedient that this analysis should be carried out along the section CR = constant. Such an approach is basically more general as compared with the examination of processes of a competitive adsorption of liquid solutions when the adsorption phase is regarded as “saturated”with a sum of a d s o r b a t e ~ . ~ J ~ ~ ” * ~ ~ The value SR is chosen arbitrarily (see below), but it remains constant for the given adsorbent, irrespective of the nature of the adsorbates and of the corresponding values SIR. This important condition of the theory allows one, when considering the adsorption of any adsorbates or their mixtures, to deal with an unambiguously thermodynamically defined adsorbent, mutual recalculations of adsorption equilibria becoming thus substantially simplified. In contrast to other, frequently encountered versions of the choice of components of the adsorption ~ o l u t i o n , 3 * ~ Jin 6 Jthe ~ *STA ~ ~ the adsorbent is a component of the adsorption phase, and by this virtue the latter is always saturated by the sum of components
1402 Langmuir, Vol. 7, No. 7, 1991
Tolmacheu (5)
E n i R dviR+ nR d‘pR + W du = I
where W is the surface or volume and n, and nR are the numbers of moles of the adsorption phase components, respectively. Insorfar as in the model adopted in the STA, the adsorption phase comprises at least two components (ARlIp, and R), its properties may be analyzed by using the adequately well developed tool of chemical thermodynamics (see (1-3)) and by resorting to the recent advances made in the theory of solutions with molecules having different dimensions. The volume W is assumed to be the volume of the adsorption solution, and in spite of the condition W = constant, the values of partial mole values (s‘R,SR) are constant (and different from zero) within the framework of the adopted model of “rigid” solution, insofar as, for instance, derivatives S‘R = (d W/dniR)p,nR must be taken at nR = constant, that is, upon adding to the system of the adsorbent volume (d W = SR dnR) corresponding to its dnR moles, which have reacted with dniR moles of the adsorbate according to (1) or (2). Here it is necessary to mention two assumptions in the model of adsorption solution, adopted in the STA, which simplify the development of the tool of the theory: the adsorption phase inhomogeneous with respect to the coordinates is regarded as homogeneous, with the values of corresponding parameters averaged (over the layers or over the volume); the values of partial mole areas or volumes (SIR, SR) are assumed to be independent of the composition of the adsorption solution both in the case of “rigid” adsorbents (carbon blacks, silica gels, activated carbons, zeolites, etc.), which practically do not change their surface of volume of pores in adsorption, and in the case of “swelling” adsorbents (insoluble polymeric adsorbenta, etc.), whose volume changes appreciably in adsorption. In the latter case energy expenditures for changes in the adsorption volume are included into the corresponding energy characteristics of the processes (1-3). These assumptions, inevitable in the present-day state of our knowledge of the physical state of adsorbed substances, do not interfere with the thermodynamic strictness of the consideration, since all deviations from the adopted model (probable changes of s a , SR, W) are formally taken into account within the framework of the method of activities, since the corresponding coefficients of activity (y,, YR) comprise the above-stated deviations from the adopted model of “rigid adsorption solution”. As experimental data become accumulated, the contribution of these deviations to the values YiR, YR may be studied specially. In the thermodynamic analysis of adsorption systems, distinction is sometimes drawn between adsorption on the surface and that in the volume of micropores.l6 Such an approach may be useful, if the adsorbent is not regarded as a system component. Within the framework of ageneral quasi-chemical model, such distinction becomes unnecessary, because in all cases one should take into account additional work against the forces of surface tension (u, J.cm-2) or the forces of internal pressure ( 0 , J.cm-3), respectively. Within the framework of quasi-chemical model of the STA the fundamental equation and the equation for the adsorption solution of Gibbs may be, at P, T = constant, represented in two equivalent forms dG =
& dni + cvlRdn, i
Since “complete” (v,) and ”internal” (PiR) chemical potentials of the adsorption solution components (similarly to (PR and PR) are interrelated1J2127
+
v ~ R= qipiRo RT In z ~ R Y ~-RsjR(u - aiR,&) s ~ R u ~ R , RT ~ ~ In z ~ R Y ~-RS ~ R U= p a o
+
= ptR0 +
+ RT In Z ~ R Y B -
S ~ R U= piR
- S ~ R U(8)
and, consequently, in view of ( 5 )
Here and hereafter: Pi = pio + RT In ZiYi are chemical potentials of the_bulk phase components (in the case of liquid SOlUtiOnSki, etc.); zi = Ci/(CiYi)at, ztR = CiR/(CiRY,)&, and ZR = cR/ (cRYR)st are nondimensional concentrations of the components of the bulk and adsorption solutions; ci, tip, CiR, CiR,st,SR, SR,st, Yi, Ti,&,YiR, YiR& YR, and YRrt are concentrations (moLL-l, molakg-l) and the corresponding coefficients of activity of the components in the equilibrium phases and in the chosen standard states (in the case of simultaneous description of adsorption from onecomponent and multicomponent bulk phases, all the symbols for one-component systems are “primed”, for example, zil, YR(i)’, etc.); ua,st and m,stare surface tension or internal pressure in the adsorption phase in the standard states chosen for the given components. For closed systems under consideration, at any constant P, T, W (6) is equal to zero, and from ( 5 ) we obtain
Simultaneous solution of (6) and (11) leads to the initial conditions of interphase equilibrium in the STA3J1J2 ~ A R - P A--‘PBR-PB - = -‘4r -... SAR
SBR
(12)
SR
and, correspondingly PAR-PA -SAR
PBR-PB --- ... - PR
-5-
SBR
SR
(13)
Equations 12 and 13 confirm the possibility of regarding adsorption processes as corresponding quasi-chemical reactions, make it possible to exclude for finite equations the values u that are not measured experimentally, and, in combination with (a), lead to equations of the law of mass action type, which differ from the classical version of the latter only in that &, 6 in the general case are temperature-dependent (see below). Thus, in the case of onecomponent adsorption
In the case of adsorption from two-component bulk phases, two equations (14) are written down for components A and B and, in addition, the condition of equilibrium of the process (3)
+ ‘ p dn, ~ + u dW
i
In the case of multicomponentadsorption, the correspond-
Langmuir, Vol. 7, No. 7, 1991 1403
Thermodynamic Theory of Stoichiometric Adsorption ing eqs 14 and 15 are added. Here and hereafter Ki*(T),
R*(T),Ki*(T,ziR),K*(T , z ~ , z B are R ) thermodynamic con-
stants and coefficients of equilibrium of the processes (l), (2), and (3), respectively. It is apparent that the numerical values of Ki*, Ki*, R*, and K* and of the corresponding changes of thermodynamic functions, characterizing the processes (1-3), depend on the method of choosing the values Of cg,Bi, and /3 and the standard states of components in each of the phases. Constancy of such choice for all the above-quoted values ensures thermodynamic consistence of the data on the one-component and multicomponent adsorption of different adsorbates on a given adsorbent. We shall consider these methods of standardizing as applied to different absorption systems. Imagine an adsorption isotherm at T < T,, of some substance (A) on a microporous ad~0rbent.l~ When the concentration of saturated vapor (CAJ is reached or upon contact with pure liquid ( E A O ) , the value of adsorption reaches the values cmo,which is often regarded to be "ultimate". In this case, however, the adsorption phase is two-component (ARIlp, with the concentration cmo and R with the concentration CR*). It is practically impossible to measure experimentally the values CAR,-, i.e. (CR = 0), necessary for the determination of /3i from (4); therefore, we shall introduce a certain coefficient k < 1 defined by the relation CAR,^^ = CAR'. It is apparent that the choice of the numeric value of k (for instance, k = 0,9) will determine the value of initial concentration of the adsorption sites of the adsorbent under vacuum (cRO) or the value of SR and further of pi and CR. Indeed, if M is a certain adsorbate, adopted as "standard", for which PM = 1 at the temperature TO,then (16) In accordance with (16) the value of CRO (and further of CR) is determined arbitrarily, but unambiguously for the given adsorbent, through the choice of a standard adsorbate and by the values of cmoand k~ at TO.Within the framework of rigorous theory it is necessary to take into account that k~ = f (T) and k~ # k A # kg # ki. The precision experimental data available indicates, however, that on the present-day level of adsorption measurements it may be assumed with fairly good approximation that
k, = k, = k,
=
... = ki # f ( T )
(17)
Thus, proceeding from the choice of standard vapor, the , is, as a matter of fact, of the value of TOand k ~ that standard state of the adsorbent, it is possible to determine unambiguously all the necessary parameters in the adsorption of any one-component or multicomponent systems on the given adsorbent, since from (4), (16), and (17) it follows
The values C ~ R O = f ( T )are determined experimentally in adsorption of pure liquids or as parameters of corresponding equations of the theory of space filling of micropores
(TSFM).28 Experimental studies demonstrated that CBO = f (T)is described with sufficient accuracy by the linear equation28 In cBo(T) = In clRo(To) - a,(T - To) (22) Insofar as (see (18))the temperature dependence of SR (i.e. of CRO) may be neglected, the variation of pi = f (27 is defined by the relation C ~ R O= f (T)and may be calculated from (22). Since for many adsorbates the values of thermal coefficients of expansion ((YiR) are close, the corresponding dependences 0 = f(T)are expressed very weakly (13% at AT = looo)and in many cases it may be assumed that 0 # f (V3. Furthermore, in view of the fact that Tcr for pure adsorbates is considerably smaller than T, for the binary adsorbate-adsorbent system, (22) may be extrapolated linearly into the region T > Tcrto the values T* E Tcr+ 100-150°.29 This makes it possible to calculate the values of CiRo in the supercritical region and, consequently, to determine all the necessary parameters from (18) to (21) for the case of adsorption of gases on microporous adsorbents. When choosing standard states of components in equilibrium phases, it is possible to introduce "hypothetical" (actually nonrealizable) standard states, traditional for the chemical thermodynamics 1 # f (T); (CRYR)st = 1 # f (T); (CiYi)st = 1 z f ( T ) (23) In this case K i * = Kj',Ki* = K [ , K* = K', K* = K', z l=~ CB, ZR = CR, and zi = ci (all the concentrations and constants are nondimensional!). Such choice of the standard states is convenient in the analysis of the temperature dependence of adsorption, since adsorption enthalpies,-calculated from the temperature dependence of Kj',Kj', K', and K', may be compared with the corresponding values of adsorption heats, determined calorimetrically (see below). Nevertheless, as shown in refs 13 and 29, in the analysis of adsorption equilibria and, especially,in the development of methods for a priori calculations of the latter, it is convenient to use equilibrium-corresponding and, by this virtue, temperature-dependent standard states of the components in the phases (CiRo is equilibrium with respect t o q S(or In contradistinction to (23),such conditions are actually attainable in the experiment (CiRYiR)st =
= f (T); (CRYR)st = CRoYR,st = f(T); (ciYi)st = CirYi, = f ( T ) (24) It is natural to assume additionally that the reference states for Y ~ and R YR coincide with the chosen standard states of the components, i.e. YiR& = 1, YR,st = 1. It should be emphasized that, in contradistinction to the components of the bulk phase and R, for which standard states correspond to pure substances at ci = cia ( E i = E i O ) and CR = CRO, the standard state for adsorption complexes corresponds to a two-component adsorption solution, in which CB = caoLandt_he concentration c~ = CR* (see (21)). In this case Ki* = Ki, Ki* = Ki, K* = K , and K* = K and on the basis of their temperature dependence enthalpies corresponding to the "pure" heats of the processes (1-3) may be calculated. The use of the equilibrium-corresponding standard states of the components poses the question concerning the method of determining Cip = Ci,.* at temperatures exceeding critical ones (T,) for pure individual (ciRYiR)st = CiRoYiR,st
(28) Dubinin, N. N. h o g . Surf. Membr. Sci. 1976, 9, 1. (29) Tolmachev, A. M.; Dubinin, M. M.; Belousova,M. E.;Fomkin, A. A. Zzu. Akad. Nauk SSSR 1982,56, 2022 (in Russian).
1404 Langmuir, Vol. 7,No. 7, 1991
Tolmachev
adsorbates. A detailed analysis of this question, carried out in ref 29 on the basis of precision experimental data, demonstrated that the values of CiRo and Cis* prove to be equilibrium-corresponding,if c i ~ O( T > T,) are found from (22) and the values of Cis* at T > T,, are calculated from the relation In cis* = A - B / T
(25)
where the constants A and B are determined from the relation In Cis = f (1/T) in the range of temperatures Tcr > T > Tmrlt. The entire tool of the TSFMB may be extended to the range T > Tcrand ci < Cis*." The determination of all the necessary parameters for the case of adsorption of gases on macroporous adsorbents may also be carried out in a similar manner. First, proceeding from the data on polymolecular adsorption of corresponding vapor ( T < it is necessary to determine the temperature dependenceof the capacity of the monolayer ( c a 0 ) from (22), using the BET or Aranovich theories,81 then it is necessary to find the values of C~R' = f (29,Cist = f ( T ) from (22) and (25) for P > T > Tcr, and, further, all the necessary values (cRO, pi (T),@(T),etc.) from equations (16-21). The determination of the upper possible boundary (P) for such extrapolation requires in this case additional experimental verification. The problem that may arise in such systems for the molecules having a complicated geometric and energy structure resides in taking into account different orientation of the molecules at the surface and, as a result of it, changes occurring in the values of pi. Such taking into account provided that the necessary special information is available and may be carried out by introducing the dependence pi = f (cB), but, within the spirit of the adopted model, it will be more consistent to preserve pi = constant and to take the corresponding changes into account in the values of y,~. In the case of adsorbents with limited swelling,the values of ki may differ appreciably because of energy-hindered penetration. of the molecules of adsorbates into the adsorption phase, associated with the "creation" of adsorption space. Nevertheless, the densities of the adsorbates (within the framework of the model discussed) are constant and close to the densities of pure liquids (Tcr> T > Tmelt). Within the scope of this assumption, which requires additional precision experimental verification, it may be adopted that @i( r ) = sR/siR
= sR/B:(T); B ( T ) = ~ i 0 ( ' 1 1 / ~ ~ ( ' 1 1(26)
(Dio(T) being mole volumes of pure liquid adsorbates), and the values Of cR0 and SR may be chosen arbitrarily, but in such a way that for any adsorbates there should be CROSR > C~ROS~R. Then extrapolation to higher Tis carried out as described above. In this case, if the necessary data are available, it is better to calculate C ~ Rin moles per liter of the adsorption solution. If such data are absent, the variation of W will be taken into account in the values of y i ~YR. , A more detailed analysis of such systems requires accumulation of experimental data, especially in the case of adsorption of two- and multicomponent fluids. It is necessary to stress that in the case of adsorption on rigid microporous adsorbents or on adsorbents with limited "swelling", molecular-sieve ("geometric" or "energy") effects may become manifest. No doubt, it is clear that the consideration carried out above relates to the case when such effects are absent. But if, for instance, in the adsorption of substances (A and B) the first of them (30)Brunauer, S. J. Am. Chem. SOC.1938,60, 309. (31)Aranovich, G. L.Zh. Fix. Khim. 1988,62, 3000 (in Russian).
may fill the entire adsorption space (W), while for B only the volume W Bis available, the adsorption of the mixture must be considered within the volume Wg,whereas within the volume W - W Bonly the one-component adsorption of A should be considered. Accordingly, subdivision of CRO into CRO ( WB) and CRO ( W - WB) etc. must be taken into account. We shall now consider in greater detail the application of STA to the description of equilibrium of adsorption from one-component and multicomponentbulk phases.
Adsorption of One-Component Fluids on "Rigid" Adsorbents Proceeding from eqs 14 and 20 or from the relations that follow from (20) kzm + zg = 1;ciRo k@$!Ro (27) it is easy to derive, with due account for the difference in the choice of standard states (23) or (24), equations
which represent in a generalized form the equations of adsorption isotherm. Transforming (7) to the form
and taking into account that dCiR = -pi dCR, it is possible to present (28) (and, similarly, (29)) in a different form
Ri =
Ki' (kgi)l/aic
iR
0(1-(1/5i))
-CiR ciYi(CiRo - kciRle
(31)
It is of interest to consider some particular cases. 1. In "ideal" adsorption, Le. YiR, YR, Yi 1 and, consequently, pi = 1, eq 28 reduces to an equation of Langmuir type (when passing from the STA model to the Langmuir model (k 1). 2. Solution constituted by molecules of different dimensions (& # 1) cannot be ideal (YiR = 1and, at the same time YR = 1 at all c , ~ ) however, ; if YiR = 1 and YR = f (C~R) changes in accordance with (301, then (31) also formally reduces to an equation of Langmuir type (at Yi = l),but for a nonideal adsorption solution. 3. In adsorption on microporous adsorbents the relations YiR = f (CiR) and y~ = f (CR) may be represented for a large group of adsorbates in the form3
-
YiR = cp-1; .yR= cRgR-l (32) in this case gi, g R retain constant values within a wide range of variation c i ~ O> CiR > 0, whereas at CiR ---cCiROgi 1 (at CjR 0,i.e. CR CRO, g R 1). Since from (30) it follows thatgR = pigi, upon substitution of (32) into (28) (at Yi = 1) relations may be obtained, similar in form to the equations of Kisarov-Cohen and of the osmotictheory of adsorption3
-
-+
- -
which adequately describe isotherms of adsorption on mi-
Langmuir, Vol. 7, No. 7, 1991 1405
Thermodynamic Theory of Stoichiometric Adsorption croporous adsorbants. It should be emphasized that the equation of the form (33) was derived by the authors of ref 3 for the particular case of @i = 1. The presented conclusion shows that the Kisarov-Cohen equation corresponds to the general case of adsorption solutions with molecules of different dimensions (@i # 1). With the choice of other forms of dependence of YiR and YR on the composition of adsorption solution, the equations of Hill, Fowler-Guggenheim, and others may be ~ b t a i n e d . ~ 4. If the adsorption solution is constituted by noninteracting molecules of different dimensions? (28) may be represented in h e a r form (Ti = 11, since in this case YiR # constant, YR # constant, but y,/y~l/@i = constant CiR 1 In Rill = In -+ 7 In (cZRo - kc,)
Ci
01
+ In (constant)
(34)
which makes it possible to find j3i from experimental data. It is of interest to note that the coordinates of eq 34 are more sensitive than the usually employed coordinates of the linear form of the Langmuir equation, and processing of the literature data, which were believed to be in fair agreement with the Langmuir isotherm (Bi = l ) , in that for all systems j3i accordance with (34), has # 1, and, in conformitywith STA,in the case of adsorption on one and the same adsorbent
...
= CARo:CBRo:CiRo (35) this being an additional confirmation of the stoichiometric quasi-chemical model adopted in the STA. If none of the particular forms of (28) (and, correspondingly (29)) can describe adequately the experimental data, it is necessary to carry our calculations of y , ~ YR, , and Ki*. Correspondingrelations may be obtained in simultaneous solution of (14) and (30)14 BA:
Bi
In Ki* = kJ1ln Ki* d ziR+ (1 - k) In Ki*(ziR= 1) lk(1-
Pi) (38)
Pi
Certainly, it is clear that the values of ygx,YR do not depend on the choice of standard states of components in the phases (23) or (24), whereas the value of Ki* is defined by this choice. Apart from the strict relations (36-38) the values of YiR, YR, and Ki* may be assessed in the case of microporous adsorbents from experimental data for a certain “standard” (M) vapor with the use of the property of “similarity” of the characteristic curves (CC) of adsorption, substantiated in the TSFM.B In this case it is convenient to use equilibrium-corresponding standard states (24) Ki =
ZiR li(mZMTM(1 - kziR)1/8i
In ygR= li(mIn ym
+ (lim - 1) In ziR
adsorbate with respect of the CC of the standard vapor M. For the exposition that follows, it is important to note that, as a rule, li(M)Bi/oM does not differ strongly from unity and, consequently, YR(i), 7 ~ 0YR(M), , etc. have close values. Certainly, eqs 39-41 hold true only within that interval of Z~Rvariation, in which the condition lam = constant # f ( z , ~ is) fulfilled. From the values of K,.’(T) and& ( T )one may further calculate variations of standard thermodynamic functions of processes 1 or 2 for two sets of standard states of components in the phases
(39)
(40)
Relations 42 give “full” values of variations of thermodynamic functions, whereas relations 45 give “net”values, which differ from the “full” ones by the corresponding variations of the thermodynamic functions in the process of condensation of vapor into liquid (of “supercritical” vapor (gas) at T > T,, into “supercritical” liquid.%). It is useful to note that with the use of (24) AGiO = AGR(~)O; i.e. it is equal to the standard variation of the isobaric potential of Gibbs for the adsorbent, corresponding to ita transition from pure state into solution with CR = cR*.l3 Indeed, from (13) for adsorption at Ci = Ci,g it follows (44)
where p~(i)* is the chemical potential of the adsorbent in the adsorption solution at CR = CR* (see (21)). Using (44) we obtain
Insofar as the values of M i o , as a rule, are high, and changes in the entropy of the adsorbent (rigid adsorbents!) are small, it may be assumed with good approximation that RT In Ki = -Mio# f (T) (46) More complete information on processes 1 or 2 may be obtained from analysis of dependences of ’)’iR and YR or Ki* and differential mole thermodynamic functions of processes 1 or 2 on Z ~ Rand on the temperature. The corresponding relations are easily derivable from (14), taking into account that in the two-component adsorption solution (AR1/@,and R) R T In Y,R = ~ l i R- p,(id); RT In YR = P R - pR(id);Pi = hi Tsi; hiR(id) = hiR,st = hiRo;hR(id) = hR,st = hRo;SiR,(id) S,Ro - R In z~R;sR(id) = S R o - R In Z R (47) Substituting (47) into (14), we obtain 1 RTln Ki = RTln Ri -RTln yiR+-In yR = -mio + Pi
TASio - (pa- p,,(id))
1
1
+ & (FR - pR,(id)) = hio + -hR Bi
+ R T In 1/~i ZiR = - mil0 + ZR
(41) Here li(m is the similarity factor of the CC of the ith (32) Tolmachev, A. M.; Denisova, L. V. Zh. Fiz. Khim. 1974,47,2634 (in Rueelan).
+
ZiR TASiso’ R T In = zRllBi
+ T ASiso
= -AGiso
Tolmachev
1406 Langmuir, Vol. 7,No. 7,1991 S R O are partial enthalpies and entropies of components in solutions of composition z a and the values of these functions in standard state (24),AHieO, A5'ico, AGiaO, are differential mole changesof the thermodynamic functions of process 1or 2. From 48 it follows (in a similar manner the relations may be obtained also for the case of (23),see, for example, (49)
of precision experimental datam show that appreciable differences (to 30-50%) between qc and qco may be manifest only in the region of c a ~ ca~O, whereas the relation (57) (without taking into consideration the contribution (apparently insignificant) of the nonequilibrium character of the calorimetric experiment) accurately takes into account differences of the enthalpies of real gas in the standard (ciyi = 1)and equilibrium states. Certainly, it is clear that at relatively small pressures Ah 0 and q d = qco.
-
-
'k(1-
4
AH? = J1AHie0 dz,
/3JR (51)
+ (1 - k)AHiso(z, = 1) (52)
It is important to emphasize that in the calculation of Ki*,
K{,Ki, etc. at different temperatures it is neeessary to
take into account the relation /3i = f (T). It is apparent that the form of eq 50-52 reflects the fact that the standard state for AR1/@,(ZAR = 1)is a two-componentsolution, the composition of which (cR*) depends on the choice of k. It is of interest to point out that from (48)it also follows that
Adsorption of Binary Mixtures of Fluids on YRigidnAdsorbents First of all, it should be emphasized that eq 15 describes equilibrium of the process of competitive adsorption (3) of any pair (A, B)from a multicomponent mixture of fluids at any overall values of adsorption of the given components and in this sense it is more general than similar equations derived by different authors when describing the adsorption of binary liquid solution^.^^^^^^^^^^^^ Approximation of y , = ~ f (ZaR, Z,R) with the help of different equations that follow from the theory of liquid solutions allows the obtaining of different particular forms of (15). Two cases are of a certain interest. 1. If YAR/YBR@ = constant within a sufficiently broad range of variation of zm, ZBR, or along the line ZAR + ZBR = constant, then, by representing (15)in linear form (TA, YB = 1)
In A
Thus, thermodynamic substantiation of the ideas of Polanyi-Dubinin concerning the temperature invariance of CC, i.e. RT In ziyi = f )2,(
#
f (T)
-
(54)
resides in that the part of the entropy ASi,Zo' 0; i.e. adsorption solutions must be regular. In conclusion of this section, using (49),we shall compare the values of A H i , e o = -qco with isothermal adsorption heats obtained calorimetrically ( q d ) and also with those calculated approximately from thermodynamic data in accordance with the relation (55)
which holds true at relatively low pressures and adsorption values. From comparison of (14),(49),and (55)it follows that
Further, taking into account that in the calorimetric experiment, in principle, a nonequilibrium process is realized and the heat being measured is the heat of transition into the adsorption solution not from the standard state of the adsorbate but from the corresponding equilibrium pressure, we obtain qd I qeo-
I:'@(2)
T
dci = qco - Ah = qc - Ah + Aq (57)
Corresponding assessments and verification on the basis
= In K* - @ In
- In (constant)
(58)
ZB
it is possible to determine /3 from experimental data and to compare it with /3 calculated from (19). In all possible cases the values of /?from (58) and from (19)coincided practically precisely. 2. If, similar to the case of one-component adsorption, Y~R/YR(A,B)'~@~ = const, then the general equation of partial adsorption isotherm of any component (A) with the use of the conditions (23)
becomes transformed into an equation similar to the Langmuir equation, but nonlocalized adsorption of a mixture of fluids with molecules of different dimensions CAR
(60) cA'YA(c,o/k@, - cM/@A- cBR//3B)1/BA and exactly corresponds to it, if @A, BB,YA = 1and, certainly, k 1. As has been already pointed out, it is reasonable to carry out thermodynamic analysis of the processes of competitive adsorption along the section zm + ZBR = constant, i.e. CR = constant. In this case experimental values of ci, C,R or zi, z a are ~ used to calculate K', K , Nax, 8,and the dependence of K', K on NAR(1 1 NAR1 0)are analyzed along different constant sections l / k > 8 > 0. From balance consideration it follows that ZR + kz, + kzBR = 1; ZR = 1 - k 8 ; B c ~ O= cBRO;8 = KA(B 1 (8, = l/k) may be reached in the case of adsorption of liquids at an additional hydrostatic pressure) N a is the volume fraction of the ith component from the volume occupied by the mixture of adsorbates (A + B). For constructing a series of relations In K* = f (NAR), 8 = constant, T = constant), it is necessary to have a very large experimental ensemble of data on the values of K* a t different 1 1 NAR1 0; 11 8 > 0, T Z> T > T I . In this connection we shall cite here two empirical equations substantiated by this author with co-workers,l6which make it possible, at 0.95 1 NAR1 0.05 and 1 1 8 1 0 . 2 , to obtain the required ensemble of data from a very limited experimental information with an accuracy not inferior to that of contemporary experimental research (f5-8% in the values of K*). Thus, at Nm = constant In K*(8,) = In K*(e,) + In MAB(8,)- In MAB(eZ) (62) in the case of Nm, 8 = constant and /3 # f (T) T In K*(T) = T I In K*(T1)- ( T , - T ) X
" + (1 - 0)In 8 + In 4
(1 - Nm)'
The values of In MAB(8)# ~ ( N A R are) calculated from the isotherms of individual adsorption of the components A and B (see below), while 4 = 1 a t K* = K and 4 = C A R O . (CB$yB,s)8/CAgyA,sCeRo'at K* = K'. Thus, the dependence of In K* on NARobtained in a usual experiment at TI,with the values of 8 slightly varying with the growth of NAR (for example, from 81 to 8z), may be easily transformed in accordance with (62) (provided that adsorption isotherms of the components A and B are present) into the relation In K* = f (Nm,T I ,8 ) at the constant value of 8 2 > 8 > 81, and then such dependences may be calculated at other 8 and T from (62) and (63) (if necessary, the dependence of 0 on F 6 is taken into account in (62)). This, in its turn, makes it possible to carry out a sufficiently complete thermodynamic analysis of the system studied. The values of K*,ym, Y B R , and ~ R ( A , Bin ) the case of adsorption of two-component fluids at TImay be calculated from ratios (64-67) which are obtained in this paper for the first time proceeding from the joint resolution of two equations (14) recorded for adsorption of each of the binary mixture components and from eq 30. Thus, if there are experimental relationships In KA(B)*,In KB(A)*and accordingly In K* from Nm at 8 = constant and respective isotherms of component adsorptions A and B then In ym(Nm8) = In Ym'(8A=. 8 ) - ke[(l -Nu) x In K*(Nm,e)- Jjmln K* CW,, - ( 1 - NAR)(I - 0)1 (1 - ke)[ln KA(B)*(NAR,e) -In K A ( B ) * ( N A R = ~(64) ,~)]
B In yBR(NAR,e) = 0In yBR'(eB=e) +
x
In K*(N,,e) - J 9 n K* W, - N A R ( I - B ) ] P(1 - k8)[1n KB(A)*(Nm8)- lfiB(A)*(NAR=o,e)l (65) In yR(A,B)(Nm,e)= ln T R ( A ) ( @ A = e ) + k,dA8 x [NARIn KA(B)*(Nm,e)- In KA(B)*(Nm=1,8)+ l d m l n KA(B,W A R
+ (1- N
~ ) -OP)I + k&e x
[ ( I - Nm) In KB(A)*(Nm,e)- s i m l n KB(A)*"1
(66)
In I?* = L l l n K* WAR + 1- ,d + In M,(8) In KA(B)*= In KA*;In I?B(A)* = In KB*
(67) (68)
In MAB(8)= In ym'(eA=e) - 0In T B R ' ( 8 B - 8 ) l-ke [In yR(B)'(8B=8) - In yR(A)'(eA=e)l (69) k,dAO
It is clear, of course, that values In M u do not depend on NAR and may be calculated from the isotherms of component adsorptions A and B for any values of 6. The same method of calculation may be extended further to adsorption of multicomponent mixtures. It is important to emphasize that as a result of the same choice of the adsorbent as the system component in the adsorption ofpne-component and multicomponent fluids, the values of K* may be calculated only from the adsorption isotherms of the components (A and B), since from (1)to (3) it follows that In K* = In KA* - 0In KB* (70) Comparison of the values of K* (68) and K* (70) serves as a good criterion of thermodynamic consistence (Le. of experimental correctness) of the data on the adsorption from one-component and two-component fluids. As in the case of one-component adsorption, the use of (23) and (24) leads to two sets ("full" and "net") of the values of variations of standard thermodynamic functions, characterizing the process (3)
More complete information is obtained in the analysis of relations, obtained similarly to (47-52) with the use of (24) -RT In KN,e = AGN,,' = ",eo - TASN,eo' -
R T [ l - 0+ In MAB(8)](74) Since the state of the adsorption phase in the micropores practically does not depend on the aggregate state of the fluid phase (liquid or vapor), with the use of (24)
K (liquid) = K (vapor)
(75) and in the presence of the relation In K = f (Fm)in_the adsorption of liquid solutions and of data on Ti and yb it is possible to calculate (yi, yj = l), using (621, similar relations for the adsorption from vapor phase at any 8 1 0.2. It is of interest to note that from (73) it follows
RT In 2 ,
= RT(P In z B- In z A ) =
+ TASN,,"
(76) Comparison with (53) shows that if the relation RT In zm = f (Nm,8 = constant) is regarded as the characteristic curve of mutual displacement (CCMD) of components from the adsorption solution in the course of the process of competitive adsorption (3) at any 8 = const, then for its being temperature-invariable it is necessary that (asin
1408 Langmuir, Vol. 7, No. 7, 1991
-
the case of (53)) ASN,~'' 0, i.e. that the adsorption solution (ARI/B,,BRl/h, R) should be regular. This very circumstance provides substantiation for the ideas of Polanyi-Dubinin and allows their extension to the adsorption of two-component and, probably, multicomponent fluids.
Tolmachev
-
Experimental substantiation of the condition A S N , ~ O ' 0 in (76) allowed the author with co-workers to develop on the basis of STA method for a priori calculation of adsorption equilibria of two-component fluids on microporous adsorbents.ls
