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Description of Adsorption and Absorption Phenomena from a Single Viewpoint
JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.
179, 335–340 (1996)
0224
Description of Adsorption and Absorption Phenomena from a Single Viewpoint A. V. TVARDOVSKI Laboratory of Adsorption, Institute of Physical Chemistry of the Russian Academy of Sciences, Leninski Prospect, 31, Moscow, 117915, Russia Received March 28, 1995; accepted August 1, 1995
On the basis of phenomenological thermodynamics, an equation has been derived that represents one of the interphase equilibrium equations and provides a description of both adsorption and absorption phenomena from a single point of view. It is shown that the well-known Henry, Langmuir, Fowler–Guggenheim, Temkin, and BET adsorption equations follow directly from the equation proposed with constants of clear physical meaning. q 1996 Academic Press, Inc.
Key Words: equilibrium ad- and absorption thermodynamics; theory of ad- and absorption; particular equations of adsorption.
INTRODUCTION
Up to the present time a great number of adsorption and absorption equations that reflect various properties of systems have been derived by a great variety of methods. Moreover, the number of such equations is growing continuously and seems to be even limitless, since the systems under study are so different and multiform in their nature. The most valuable are those works (e.g., (1–10)) that try to construct a general theory of adsorption. The above works present equations that reflect, in different forms, a condition of phase equilibrium. Their forms of expression may be diverse. Application of the results of any of the above works depends on the intricacy of the systems at issue, and on a researcher’s taste. This work derives, on the basis of phenomenological thermodynamics, an equation that is one of the expressions for the general condition of interphase equilibrium. The equation presented provides descriptions of both adsorption and absorption phenomena from a single viewpoint.
sider only those cases in which E can be completely determined from S, V, na0nt , and nads . Such an approach is clear when the sorbent under study has negligible surface effects compared to volume effects. However, if a sorbent has noticeable surface effects, this implies that an na0nt value change by d n a0nt means addition of a pure sorbent of the same dispersity, specific surface, etc. as that of the initial pure sorbent in all processes under study. Thus, in cases where an expression such as ‘‘specific surface’’ is meaningful, this quantity is proportional to na0nt rather than representing an additional independent variable ( 11 ) . Using conventional methods of solution thermodynamics we may, for instance, write dE Å TrdS 0 PrdV / madsrdnads / ma0ntrdna0nt , dmads Å 0sV adsrdT / £V adsrdP / ( Ìmads / Ìa)T ,Prda, where T is temperature, a Å nads /na0nt , sV ads Å ( ÌS/ Ìnads ) n a0nt,T ,P , etc., P is hydrostatic pressure, and m is chemical potential. For the gas, the well-known relationship may be given dmgas Å 0sI gasrdT / £I gasrdp, where s˜gas Å Sgas /ngas and £˜ gas Å Vgas /ngas are molar entropy and equilibrium gas phase volume, respectively. Equilibrium between the gas and the condensed phase implies that dmgas Å d mads and, consequently, that
METHOD, RESULTS, AND DISCUSSION
0sV adsrdT / £V adsrdP / ( Ìmads / Ìa)T ,Prda
Consider a two-component (sorbent and sorbed gas) condensed phase in equilibrium with a gas, where the sorbent is assumed to be nonvolatile. Such a condensed phase has energy E, entropy S, volume V, etc., and contains na0nt and nads moles of the two substances, respectively. We shall con-
Å 0sI gasrdT / £I gasrdp.
Under ordinary experimental conditions P Å p, so, for a constant ‘‘a,’’ we shall have
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( Ìp/ ÌT )a Å
sI gas 0 sV ads . £I gas 0 £V ads
The preexponential multiplier may be obtained as follows. Consider a certain sorption system at the temperature of T. The sorption value is a. The gas phase equilibrium pressure p over the sorbed substance is determined by [6]. Then, for p Å ps , we get
Suppose that £˜ gas @ £V ads ; then ( Ìp/ ÌT )a Å
sI gas 0 sV ads qst Å , £I gas T r£I gas
[1]
where qst is the isosteric heat of sorption. For an ideal gas phase differentiation of the left-hand side of [1] gives ( Ìp/ ÌT )a Å nrk / krT r( Ìn/ ÌT )a Å nrk / krT r(dTn/dT ).
qst 0 RrT rdT, RrT 2
[3]
ln(n) Å 0 wi ( DTi )/RrT / Ci ( DTi ). Similar equations correspond to each such DTi interval (which may be as small as possible). It is obvious that we shall have a continuous sequence of w and C changes as functions of temperature for a certain constant a value in a continuous transition from one interval DTi to another interval DTi /1 . In a general form we may write ln(n) Å 0 (qst (T ) 0 RrT )/RrT / C(T )
[4]
n Å M(T )exp[ 0 (qst (T ) 0 RrT )/RrT],
[5]
or
where M, thus, has the dimension of concentration. Note again that qst , C, and M in [4] and [5] appear to be certain functions of not only temperature, but also sorption a. Multiplying both sides of [5] by krT yields
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Now, if we divide [7] by [6], we can take logarithms of both sides and multiply them by RrT. Then, using the thermodynamical identity DGV ads Å DHV ads 0 TrDSV ads , where G, H, and S are Gibbs energy, enthalpy, and entropy, respectively (here, changes with respect to liquid state are taken), we can derive DSU ads Å R ln(Ms /M).
[6]
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[8]
It is well known (12) that a molar entropy change in the transition of a substance from one state to another can be defined as DS Å S(V, T ) 0 S(V0 , T 0 ) Å
where R is the universal gas constant. As follows from [1], Eq. [3] holds for a certain a Å const. Therefore, qst 0 RrT Å w(T ) is a function of the temperature and sorption, which makes integration of [3] difficult. If we integrate [3] for a certain a Å const. within very small DTi intervals, where w may be considered as temperature independent for the given DTi , we can obtain
p Å krTrM exp[ 0 (qst 0 RrT )/RrT].
[7]
[2]
In the last expression, dTn Å ( Ìn/ ÌT )ardT is a so-called differential of n with respect to T, n is the molecular concentration in the gas phase, and k is the Boltzmann constant. Substituting [2] into [1] and transforming gives d ln(n) Å
ps Å krTrMs exp[ 0 (qL 0 RrT )/RrT].
*
V 0, T
(CVrdT )/T /
V 0,T0
*
T,V
( ÌS/ ÌV )TrdV,
T ,V0
where C£ is the molar heat capacity of the substance at V Å const., V0 and V are molar volumes of the substance in its initial and final states, and T 0 and T are initial and final temperatures, respectively. It is clear that entropy change depends only on volume characteristics at T Å const. The following formula (12) can be used to calculate an ideal gas molar entropy change at T Å const., DS201 Å R ln(n1 /n2 ),
where n1 and n2 are molecular concentrations in two states, respectively. In those states the entire volume provided for the molecules is available for them. That means that the entire volume is free for them. In our problem, assume that the entropy change at T Å const. depends on the ratio of the concentrations with respect to the free volume. That is, we shall apply the free volume concept from the theories of liquids proposed by Eyring, Lennard-Jones, and Devonshire and present the development of their ideas detailed in (13). Then [8] can be transformed to DSU ads Å R ln[NA /(VH liq 0 Fliq ):NA /(Vads 0 Fads )], [9]
where NA is the Avogadro number, V˜ liq and Vads are, respectively, a liquid molar volume and the sorbed substance molar volume corresponding to isosteric heat qst for certain T and a (naturally, Vads and qst define a ‘‘differential level’’ of the substance sorbed rather than average characteristics), and
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ADSORPTION THEORY
Fliq and Fads are volumes unavailable to the motion of the mole of molecules in the liquid and sorbed substance, respectively (those characteristics are certain functions of T and p). Note that the Avogadro number NA is deliberately left in [9] as M, and Ms should have the dimension of concentration in [8]. Thus, in accordance with [8] and [9], formula [6] takes its final form: p Å [RrT/(Vads 0 Fads )]exp(1 0 qst /RrT ).
[10]
If there is a nonideal gas over the sorbed substance, the latter may be written as
In the last expression V mon /m Å srt, where s is the adsorbent specific surface, t is the monolayer thickness, m is adsorbate molar mass, and K is Henry’s constant. Formula [14] was derived from [12] using [13]. 2. Langmuir Equation (15) This model assumes that z Å 1. The adsorbent surface is also assumed to be homogeneous. In contrast with the Henry model, it takes account of the dimensions of the adsorbed molecules. Therefore, it follows from [11] that p Å [RrT/(crVads )]exp(1 0 qst /RT ) Å [NArkrT/(crVads )]exp(1 0 qst /RT ),
p Å zr[RrT/(Vads 0 Fads )]exp(1 0 qst /RrT ), [11] where z is the gas phase compressibility factor. Note that qst , in [11] as well as in [10], has the meaning of the equilibrium heat for one mole of molecules transferred from the sorbed state to the gas phase (intermolecular interactions in the latter are ignored) for a given p, T, and a. Thus, we have presented some general phenomenological approaches to describing both sorptive and adsorptive equilibria. The general equation [11] has been derived where entropy and energy characteristics are certain functions of a and T. Using models, these functions can be calculated to yield particular ad- and absorption equations. Let us now show some particular cases of Eq. [11] obtained with the help of system information. 1. Henry Equation (14)
where c Å (Vads 0 Fads )/Vads . Since the molecular lateral interactions are ignored and it is assumed that qst Å const., Eq. [15] with the help of [13] gives mon /amon )r krT r[exp(1 0 qst /RrT )]/c p Å ar(n ads
Å (Kra)/c Å Kramonru /(1 0 u ) Å Karu /(1 0 u ),
p Å (NA /Vads )r krT exp(1 0 qst /RrT ),
[12]
where NA /Vads Å nads is the current value of the adsorbed molecules concentration. Here concentration nads can be expressed linearly as mon )/amon , nads Å (arn ads
[13]
mon where a is the current value of adsorption, and n ads and amon are concentration and adsorption limit values, respectively, when forming a monolayer. Since this simple adsorption model ignores lateral interactions of adsorbed molecules and deals with only adsorbate– adsorbent bonds for qst Å const. (homogeneous surface),
p Å Kra,
where u is the adsorbent surface coverage. It is evident that [16] amounts to Ka Å Kramon Å const., and (1 0 u ) is nothing but c Å (Vads 0 Fads )/Vads . Expression [16] is known as the Langmuir equation.
The Langmuir equation assumes that the molecules adsorbed interact with adsorption centers with a certain energy and at the same time do not interact with each other. Such lateral interaction is provided by the Fowler–Guggenheim model. The molecule adsorption probability for a given center is u (coverage level). If each adsorption center has z * neighbor centers, the probability of the presence of a molecule at one of the neighboring centers is z *ru. Thus, the proportion of adsorbed molecules which interact with each other is (z *ru )/2. (The multiplier 12 is introduced so that no molecules should be considered twice.) If the two-molecule lateral interaction energy is v, the average and differential adsorption energies are additionally increased by (z *rvru )/2 and z *rvru, respectively (16). In fact, taking account of the average and differential quantities ˜ )/ Ì N Å UV , we get relationship, Ì (U Ì[((z *rvru )/2)r N]/ Ì N Å Ì[((z *rvr N/N*)/2)r N]/ Ì N
[14]
Å (z *rvr N/N*)/2 / (z *rvr N/N*)/2
mon /amon )r krT rexp(1 0 qst /RrT ) Å [RrT/ where K Å (n ads mon [(V /m)rm]]exp(1 0 qst /RrT ).
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3. Fowler–Guggenheim Equation (16)
The following assumptions hold for this type of adsorption: z Å 1 and Fads Å 0. Then [11] is
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Å z *rvr N/N* Å z *rvru.
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In the above expression N and N* are the numbers of adsorbed molecules and adsorption centers, respectively. Thus, if we have the same assumptions as the Langmuir model and introduce the additional term z *rvru (lateral interaction of molecules) into [11], then the Fowler–Guggenheim equation is derived, p Å [Karu /(1 0 u )]exp[ 0 (z *rvru )/(RrT )], where v is two-molecule lateral interaction energy per mole.
surface coverage by all the complexes, V1 is the volume occupied by single complexes (in this case this is a monolayer volume), and Fads is the volume unavailable to them. It is clear that V1 depends on the adsorbent surface area and adsorbed molecule diameter (or linear size). It should be emphasized that n1 concentration includes only the molecules which belong to the single complexes present in the V1 volume. Now, since there are no lateral interactions, a uniform molecular distribution on the adsorbent surface may be assumed. Therefore,
4. Temkin Equation (17) mon n1 (a) Å (a1rn ads )/amon ,
This model simulates monomolecular adsorption on an adsorbent heterogeneous surface when qst is linear with u, qst Å q0r(1 0 aru ), where q0 and a are constants. If we substitute the last expression into [11], assuming z Å 1 as above, we derive the Temkin equation:
As is known, this model considers polymolecular adsorption of vapor onto a homogeneous adsorbent surface. The vapor adsorption features a transition to volume condensation at a limiting pressure equal to the liquid-saturated vapor pressure for the liquid, p Å ps . The BET model implies that the vapor adsorption of a liquid wetting the solid body becomes infinite. In the derivation of the equation, the lateral interactions of the adsorbed molecules were ignored, but interlayer bonds were implied. There was also the assumption that adsorption heat was equal to molar condensation heat within all the layers except the first. According to the BET model, the adsorption pattern is a set of different-multiplicity noninteracting complexes of adsorbed molecules at any adsorption value. Consideration of a complex gas-phase equilibrium on the basis of [11] followed by corresponding summation yields a BET-type polymolecular adsorption equation with constants having clear physical meaning. Thus, first of all, let us use [11] to consider the equilibrium between single complexes and the gas phase. This yields the equation (see [15]) [17]
Here and below the gas-phase compressibility factor is assumed to be unity. In [17], qst,1 is the first-layer molecule adsorption heat, n1 is the concentration of adsorbed molecules belonging to the single complexes, and c1 Å (V1 0 Fads )/V1 Å 1 0 u, where u is the adsorbent homogeneous
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mon p Å a1r(n ads /amon )r krT[exp(1 0 qst,1 /RrT )]/c1
Å (Karu1 )/(1 0 u ) Å u1 /[K1r(1 0 u )],
5. Brunauer–Emmett–Teller (BET) Equation (18)
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where a1 is the current adsorption associated with the single mon complex, and amon and n ads are limit adsorption and concentration values, respectively, to forming a dense monolayer. Using Eq. [18], expression [17] can be presented as
Å Kra1r1/c1 Å (Kru1ramon )/(1 0 u )
p Å [Karu /(1 0 u )]exp[( ar q0ru )/(RrT )].
p Å (n1 /c1 )r krT rexp(1 0 qst,1 /RrT ).
[18]
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[19]
where K Å Ka /amon is the Henry constant, and 1/K1 Å Ka Å Kramon .
[20]
In Eq. [19], the difference (1 0 u ) may be substituted with u0 , characterizing an adsorbent free surface proportion. Thus, we have the Langmuir-type adsorption corrected for the presence of other-multiplicity complexes on the adsorbent surface. Then, consider the same Langmuir-type adsorption on the molecules that directly interact with the surface. Here, p Å u2 /(K2ru1 ), where u1 and u2 are surface fractions covered with the single and double complexes, respectively; the 1/K2 constant, taking account of the model assumptions, can be determined as 1/K2 Å nliqrkrT rexp(1 0 qL /RrT ).
[21]
In [21], qL is a liquid evaporation heat (per mole) at a given temperature, and nliq is a molecule’s concentration in the liquid. Further, adsorption on the molecules of the second, third, etc., layers is being considered. In the general case we get p Å ui /1 /(Ki /1rui ).
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Taking the BET model assumptions into consideration, we can write
or a/amon Å (Crp/ps )/[(1 0 p/ps )r(1 / (C 0 1)r p/ps )],
K2 É K3 É rrr É KL .
[23]
[30]
Then, as follows from general considerations, the surface fractions covered with different-type complexes will tend to become the same with pressure, i.e., ui /1 / ui r 1 for p/ps r 1. Therefore, it follows from [22] that
where C Å K1 /KL . Using [19] – [21] and [23] we can get
ps Å 1/KL . The further derivation can be conducted by the conventional procedure. In this connection the adsorption can be calculated as a Å amonr( u1 / 2ru2 / 3ru3 / rrr) Å amonr( u1 / 2r KLrpru1 / 3r(KLrp) 2ru1 / rrr) Å amonrK1rpru0r(1 / 2r(p/ps ) / 3r(p/ps ) 2 / rrr).
[24] It is absolutely clear that the series in parentheses in [24] is a derivative of the following series with respect to p/ps : 1 / p/ps / (p/ps ) 2 / rrr.
[26]
The adsorbent free surface fraction u0 can be found from the obvious formula u0 Å 1 0 ( u1 / u2 / u3 / rrr)
[31]
As is known, Eqs. [29] and [30] correspond to the polymolecular adsorption BET equation. Moreover, in contrast to the classical variant, the preexponential multiplier or entropy factor in [31] has a clear physical definition. 6. BET Equation for a Finite Number of Layers, n (19) It is absolutely clear that the present approach provides a derivation of the BET equation for a finite number of layers, n. All that is required is to sum with respect to n layers in expressions [24] and [27]. In this case we get another wellknown equation: a Å [(amonrK1rp)r[1 0 (p/ps ) nr(1 / n 0 nrp/ps )]]: [(1 0 p/ps )r[1 / K1rpr(1 0 (p/ps ) n ) 0 p/ps ]]. Here, n designates the maximum number of adsorbed layers.
[25]
Expression [25] is a descending geometric series with its sum equal to 1/(1 0 p/ps ). Thus, when the derivative is taken, Eq. [24] takes the form a Å (amonrK1rpru0 )/(1 0 p/ps ) 2 .
mon C Å (nliq /n ads )rexp[(qst,1 0 qL )/RrT].
SUMMARY
An equation has been derived on the basis of phenomenological thermodynamics that represents one of the expressions for interphase equilibrium and provides description of both adsorption and absorption phenomena from a single viewpoint. It has been shown that well-known Henry, Langmuir, Fowler–Guggenheim, Temkin, and BET adsorption equations directly follow from it with their constants having a clear physical meaning. New adsorption and absorption equations can be derived using the equation proposed and corresponding new model concepts.
Å 1 0 K1rpru0 (1 / p/ps / (p/ps ) 2 / rrr) Å 1 0 (K1rpru0 )/(1 0 p/ps ).
In the derivation of [27], the descending geometric series [25] was also replaced with its sum. It follows from [27] that u0 Å (1 0 p/ps )/(1 / K1rp 0 p/ps ).
[28]
Substitution of [28] into [26] gives a Å (amonrK1rp)/[(1 0 p/ps )r(1 / K1rp 0 p/ps )] [29]
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REFERENCES
[27]
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1. Bering, B. P., and Serpinski, V. V., Izv. Akad. Nauk SSSR, Ser. Khim. 11, 2427 (1974). [In Russian] 2. Tolmachyov, A. M., Usp. Khim. 50(5), 769 (1982). [In Russian] 3. Nicholson, D., and Parsonage, N. G., ‘‘Computer Simulation and the Statistical Mechanics of Adsorption.’’ Academic Press, New York, 1982. 4. Kiselyov, A. V., ‘‘Mezhmolekulyarnyye vzaimodeistviya v adsorbtsii i khromatografii (Intermolecular interactions in adsorption and chromatography).’’ Vysshaya shkola, Moscow, 1986. [In Russian] 5. Bojan, M. J., and Steele, W. A., Surface Sci. 199, 395 (1988). 6. Sircar, S., and Gupta, R., AIChE J. 27(5), 806 (1981). 7. Sircar, S., and Myers, A. L., Surface Sci. 205, 353 (1988). 8. Bakaev, V. A., Surface Sci. 198, 571 (1988).
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9. Tovbin, Yu. K., ‘‘Teoriya fiziko-khimicheskikh protsessov na granitse gaz-tverdoye telo (Theory of Physicochemical Processes on Gas– Solid Body Interface).’’ Nauka, Moscow, 1990. [In Russian] 10. Neimark, A. V., J. Colloid Interface Sci. 165, 91 (1994). 11. Hill, T. L., in ‘‘Kataliz. Voprosy teorii i metody issledovaniya (Catalysis. Theoretical Problems and Research Methods)’’ (transl. from English), p. 276. Inostr. literatura, Moscow, 1955. [In Russian] 12. Kikoin, A. K., and Kikoin, I. K., ‘‘Molekulyarnaya fizika (Molecular Physics).’’ Nauka, Moscow, 1976. [In Russian] 13. Smirnova, N. A., ‘‘Molekulyarnyye teorii rastvorov (Molecular Theories of Solutions).’’ Khimiya, Leningrad, 1987. [In Russian]
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14. Henry, D. C., Philos. Mag. 44(262), 689 (1922). 15. Langmuir, I., J. Am. Chem. Soc. 9, 1361 (1918). 16. Fowler, R., and Guggenheim, E. A., ‘‘Statisticheskaya termodinamika (Statistical Thermodynamics)’’ (Transl. from English). Inostr. literatura, Moscow, 1949. [In Russian] 17. Temkin, M. I., Zh. Fiz. Khim. 15(3), 296 (1941). [In Russian] 18. Brunauer, S., Emmett, P. H., and Teller, E., J. Am. Chem. Soc. 60(2), 309 (1938). 19. Emmett, P. H., in ‘‘Kataliz. Voprosy teorii i metody issledovaniya (Catalysis. Theoretical Problems and Research Methods)’’ (Transl. from English), p. 328. Inostr. literatura, Moscow, 1955. [In Russian]
coida
AP: Colloid
179, 335–340 (1996)
0224
Description of Adsorption and Absorption Phenomena from a Single Viewpoint A. V. TVARDOVSKI Laboratory of Adsorption, Institute of Physical Chemistry of the Russian Academy of Sciences, Leninski Prospect, 31, Moscow, 117915, Russia Received March 28, 1995; accepted August 1, 1995
On the basis of phenomenological thermodynamics, an equation has been derived that represents one of the interphase equilibrium equations and provides a description of both adsorption and absorption phenomena from a single point of view. It is shown that the well-known Henry, Langmuir, Fowler–Guggenheim, Temkin, and BET adsorption equations follow directly from the equation proposed with constants of clear physical meaning. q 1996 Academic Press, Inc.
Key Words: equilibrium ad- and absorption thermodynamics; theory of ad- and absorption; particular equations of adsorption.
INTRODUCTION
Up to the present time a great number of adsorption and absorption equations that reflect various properties of systems have been derived by a great variety of methods. Moreover, the number of such equations is growing continuously and seems to be even limitless, since the systems under study are so different and multiform in their nature. The most valuable are those works (e.g., (1–10)) that try to construct a general theory of adsorption. The above works present equations that reflect, in different forms, a condition of phase equilibrium. Their forms of expression may be diverse. Application of the results of any of the above works depends on the intricacy of the systems at issue, and on a researcher’s taste. This work derives, on the basis of phenomenological thermodynamics, an equation that is one of the expressions for the general condition of interphase equilibrium. The equation presented provides descriptions of both adsorption and absorption phenomena from a single viewpoint.
sider only those cases in which E can be completely determined from S, V, na0nt , and nads . Such an approach is clear when the sorbent under study has negligible surface effects compared to volume effects. However, if a sorbent has noticeable surface effects, this implies that an na0nt value change by d n a0nt means addition of a pure sorbent of the same dispersity, specific surface, etc. as that of the initial pure sorbent in all processes under study. Thus, in cases where an expression such as ‘‘specific surface’’ is meaningful, this quantity is proportional to na0nt rather than representing an additional independent variable ( 11 ) . Using conventional methods of solution thermodynamics we may, for instance, write dE Å TrdS 0 PrdV / madsrdnads / ma0ntrdna0nt , dmads Å 0sV adsrdT / £V adsrdP / ( Ìmads / Ìa)T ,Prda, where T is temperature, a Å nads /na0nt , sV ads Å ( ÌS/ Ìnads ) n a0nt,T ,P , etc., P is hydrostatic pressure, and m is chemical potential. For the gas, the well-known relationship may be given dmgas Å 0sI gasrdT / £I gasrdp, where s˜gas Å Sgas /ngas and £˜ gas Å Vgas /ngas are molar entropy and equilibrium gas phase volume, respectively. Equilibrium between the gas and the condensed phase implies that dmgas Å d mads and, consequently, that
METHOD, RESULTS, AND DISCUSSION
0sV adsrdT / £V adsrdP / ( Ìmads / Ìa)T ,Prda
Consider a two-component (sorbent and sorbed gas) condensed phase in equilibrium with a gas, where the sorbent is assumed to be nonvolatile. Such a condensed phase has energy E, entropy S, volume V, etc., and contains na0nt and nads moles of the two substances, respectively. We shall con-
Å 0sI gasrdT / £I gasrdp.
Under ordinary experimental conditions P Å p, so, for a constant ‘‘a,’’ we shall have
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( Ìp/ ÌT )a Å
sI gas 0 sV ads . £I gas 0 £V ads
The preexponential multiplier may be obtained as follows. Consider a certain sorption system at the temperature of T. The sorption value is a. The gas phase equilibrium pressure p over the sorbed substance is determined by [6]. Then, for p Å ps , we get
Suppose that £˜ gas @ £V ads ; then ( Ìp/ ÌT )a Å
sI gas 0 sV ads qst Å , £I gas T r£I gas
[1]
where qst is the isosteric heat of sorption. For an ideal gas phase differentiation of the left-hand side of [1] gives ( Ìp/ ÌT )a Å nrk / krT r( Ìn/ ÌT )a Å nrk / krT r(dTn/dT ).
qst 0 RrT rdT, RrT 2
[3]
ln(n) Å 0 wi ( DTi )/RrT / Ci ( DTi ). Similar equations correspond to each such DTi interval (which may be as small as possible). It is obvious that we shall have a continuous sequence of w and C changes as functions of temperature for a certain constant a value in a continuous transition from one interval DTi to another interval DTi /1 . In a general form we may write ln(n) Å 0 (qst (T ) 0 RrT )/RrT / C(T )
[4]
n Å M(T )exp[ 0 (qst (T ) 0 RrT )/RrT],
[5]
or
where M, thus, has the dimension of concentration. Note again that qst , C, and M in [4] and [5] appear to be certain functions of not only temperature, but also sorption a. Multiplying both sides of [5] by krT yields
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Now, if we divide [7] by [6], we can take logarithms of both sides and multiply them by RrT. Then, using the thermodynamical identity DGV ads Å DHV ads 0 TrDSV ads , where G, H, and S are Gibbs energy, enthalpy, and entropy, respectively (here, changes with respect to liquid state are taken), we can derive DSU ads Å R ln(Ms /M).
[6]
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[8]
It is well known (12) that a molar entropy change in the transition of a substance from one state to another can be defined as DS Å S(V, T ) 0 S(V0 , T 0 ) Å
where R is the universal gas constant. As follows from [1], Eq. [3] holds for a certain a Å const. Therefore, qst 0 RrT Å w(T ) is a function of the temperature and sorption, which makes integration of [3] difficult. If we integrate [3] for a certain a Å const. within very small DTi intervals, where w may be considered as temperature independent for the given DTi , we can obtain
p Å krTrM exp[ 0 (qst 0 RrT )/RrT].
[7]
[2]
In the last expression, dTn Å ( Ìn/ ÌT )ardT is a so-called differential of n with respect to T, n is the molecular concentration in the gas phase, and k is the Boltzmann constant. Substituting [2] into [1] and transforming gives d ln(n) Å
ps Å krTrMs exp[ 0 (qL 0 RrT )/RrT].
*
V 0, T
(CVrdT )/T /
V 0,T0
*
T,V
( ÌS/ ÌV )TrdV,
T ,V0
where C£ is the molar heat capacity of the substance at V Å const., V0 and V are molar volumes of the substance in its initial and final states, and T 0 and T are initial and final temperatures, respectively. It is clear that entropy change depends only on volume characteristics at T Å const. The following formula (12) can be used to calculate an ideal gas molar entropy change at T Å const., DS201 Å R ln(n1 /n2 ),
where n1 and n2 are molecular concentrations in two states, respectively. In those states the entire volume provided for the molecules is available for them. That means that the entire volume is free for them. In our problem, assume that the entropy change at T Å const. depends on the ratio of the concentrations with respect to the free volume. That is, we shall apply the free volume concept from the theories of liquids proposed by Eyring, Lennard-Jones, and Devonshire and present the development of their ideas detailed in (13). Then [8] can be transformed to DSU ads Å R ln[NA /(VH liq 0 Fliq ):NA /(Vads 0 Fads )], [9]
where NA is the Avogadro number, V˜ liq and Vads are, respectively, a liquid molar volume and the sorbed substance molar volume corresponding to isosteric heat qst for certain T and a (naturally, Vads and qst define a ‘‘differential level’’ of the substance sorbed rather than average characteristics), and
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Fliq and Fads are volumes unavailable to the motion of the mole of molecules in the liquid and sorbed substance, respectively (those characteristics are certain functions of T and p). Note that the Avogadro number NA is deliberately left in [9] as M, and Ms should have the dimension of concentration in [8]. Thus, in accordance with [8] and [9], formula [6] takes its final form: p Å [RrT/(Vads 0 Fads )]exp(1 0 qst /RrT ).
[10]
If there is a nonideal gas over the sorbed substance, the latter may be written as
In the last expression V mon /m Å srt, where s is the adsorbent specific surface, t is the monolayer thickness, m is adsorbate molar mass, and K is Henry’s constant. Formula [14] was derived from [12] using [13]. 2. Langmuir Equation (15) This model assumes that z Å 1. The adsorbent surface is also assumed to be homogeneous. In contrast with the Henry model, it takes account of the dimensions of the adsorbed molecules. Therefore, it follows from [11] that p Å [RrT/(crVads )]exp(1 0 qst /RT ) Å [NArkrT/(crVads )]exp(1 0 qst /RT ),
p Å zr[RrT/(Vads 0 Fads )]exp(1 0 qst /RrT ), [11] where z is the gas phase compressibility factor. Note that qst , in [11] as well as in [10], has the meaning of the equilibrium heat for one mole of molecules transferred from the sorbed state to the gas phase (intermolecular interactions in the latter are ignored) for a given p, T, and a. Thus, we have presented some general phenomenological approaches to describing both sorptive and adsorptive equilibria. The general equation [11] has been derived where entropy and energy characteristics are certain functions of a and T. Using models, these functions can be calculated to yield particular ad- and absorption equations. Let us now show some particular cases of Eq. [11] obtained with the help of system information. 1. Henry Equation (14)
where c Å (Vads 0 Fads )/Vads . Since the molecular lateral interactions are ignored and it is assumed that qst Å const., Eq. [15] with the help of [13] gives mon /amon )r krT r[exp(1 0 qst /RrT )]/c p Å ar(n ads
Å (Kra)/c Å Kramonru /(1 0 u ) Å Karu /(1 0 u ),
p Å (NA /Vads )r krT exp(1 0 qst /RrT ),
[12]
where NA /Vads Å nads is the current value of the adsorbed molecules concentration. Here concentration nads can be expressed linearly as mon )/amon , nads Å (arn ads
[13]
mon where a is the current value of adsorption, and n ads and amon are concentration and adsorption limit values, respectively, when forming a monolayer. Since this simple adsorption model ignores lateral interactions of adsorbed molecules and deals with only adsorbate– adsorbent bonds for qst Å const. (homogeneous surface),
p Å Kra,
where u is the adsorbent surface coverage. It is evident that [16] amounts to Ka Å Kramon Å const., and (1 0 u ) is nothing but c Å (Vads 0 Fads )/Vads . Expression [16] is known as the Langmuir equation.
The Langmuir equation assumes that the molecules adsorbed interact with adsorption centers with a certain energy and at the same time do not interact with each other. Such lateral interaction is provided by the Fowler–Guggenheim model. The molecule adsorption probability for a given center is u (coverage level). If each adsorption center has z * neighbor centers, the probability of the presence of a molecule at one of the neighboring centers is z *ru. Thus, the proportion of adsorbed molecules which interact with each other is (z *ru )/2. (The multiplier 12 is introduced so that no molecules should be considered twice.) If the two-molecule lateral interaction energy is v, the average and differential adsorption energies are additionally increased by (z *rvru )/2 and z *rvru, respectively (16). In fact, taking account of the average and differential quantities ˜ )/ Ì N Å UV , we get relationship, Ì (U Ì[((z *rvru )/2)r N]/ Ì N Å Ì[((z *rvr N/N*)/2)r N]/ Ì N
[14]
Å (z *rvr N/N*)/2 / (z *rvr N/N*)/2
mon /amon )r krT rexp(1 0 qst /RrT ) Å [RrT/ where K Å (n ads mon [(V /m)rm]]exp(1 0 qst /RrT ).
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3. Fowler–Guggenheim Equation (16)
The following assumptions hold for this type of adsorption: z Å 1 and Fads Å 0. Then [11] is
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Å z *rvr N/N* Å z *rvru.
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In the above expression N and N* are the numbers of adsorbed molecules and adsorption centers, respectively. Thus, if we have the same assumptions as the Langmuir model and introduce the additional term z *rvru (lateral interaction of molecules) into [11], then the Fowler–Guggenheim equation is derived, p Å [Karu /(1 0 u )]exp[ 0 (z *rvru )/(RrT )], where v is two-molecule lateral interaction energy per mole.
surface coverage by all the complexes, V1 is the volume occupied by single complexes (in this case this is a monolayer volume), and Fads is the volume unavailable to them. It is clear that V1 depends on the adsorbent surface area and adsorbed molecule diameter (or linear size). It should be emphasized that n1 concentration includes only the molecules which belong to the single complexes present in the V1 volume. Now, since there are no lateral interactions, a uniform molecular distribution on the adsorbent surface may be assumed. Therefore,
4. Temkin Equation (17) mon n1 (a) Å (a1rn ads )/amon ,
This model simulates monomolecular adsorption on an adsorbent heterogeneous surface when qst is linear with u, qst Å q0r(1 0 aru ), where q0 and a are constants. If we substitute the last expression into [11], assuming z Å 1 as above, we derive the Temkin equation:
As is known, this model considers polymolecular adsorption of vapor onto a homogeneous adsorbent surface. The vapor adsorption features a transition to volume condensation at a limiting pressure equal to the liquid-saturated vapor pressure for the liquid, p Å ps . The BET model implies that the vapor adsorption of a liquid wetting the solid body becomes infinite. In the derivation of the equation, the lateral interactions of the adsorbed molecules were ignored, but interlayer bonds were implied. There was also the assumption that adsorption heat was equal to molar condensation heat within all the layers except the first. According to the BET model, the adsorption pattern is a set of different-multiplicity noninteracting complexes of adsorbed molecules at any adsorption value. Consideration of a complex gas-phase equilibrium on the basis of [11] followed by corresponding summation yields a BET-type polymolecular adsorption equation with constants having clear physical meaning. Thus, first of all, let us use [11] to consider the equilibrium between single complexes and the gas phase. This yields the equation (see [15]) [17]
Here and below the gas-phase compressibility factor is assumed to be unity. In [17], qst,1 is the first-layer molecule adsorption heat, n1 is the concentration of adsorbed molecules belonging to the single complexes, and c1 Å (V1 0 Fads )/V1 Å 1 0 u, where u is the adsorbent homogeneous
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mon p Å a1r(n ads /amon )r krT[exp(1 0 qst,1 /RrT )]/c1
Å (Karu1 )/(1 0 u ) Å u1 /[K1r(1 0 u )],
5. Brunauer–Emmett–Teller (BET) Equation (18)
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where a1 is the current adsorption associated with the single mon complex, and amon and n ads are limit adsorption and concentration values, respectively, to forming a dense monolayer. Using Eq. [18], expression [17] can be presented as
Å Kra1r1/c1 Å (Kru1ramon )/(1 0 u )
p Å [Karu /(1 0 u )]exp[( ar q0ru )/(RrT )].
p Å (n1 /c1 )r krT rexp(1 0 qst,1 /RrT ).
[18]
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[19]
where K Å Ka /amon is the Henry constant, and 1/K1 Å Ka Å Kramon .
[20]
In Eq. [19], the difference (1 0 u ) may be substituted with u0 , characterizing an adsorbent free surface proportion. Thus, we have the Langmuir-type adsorption corrected for the presence of other-multiplicity complexes on the adsorbent surface. Then, consider the same Langmuir-type adsorption on the molecules that directly interact with the surface. Here, p Å u2 /(K2ru1 ), where u1 and u2 are surface fractions covered with the single and double complexes, respectively; the 1/K2 constant, taking account of the model assumptions, can be determined as 1/K2 Å nliqrkrT rexp(1 0 qL /RrT ).
[21]
In [21], qL is a liquid evaporation heat (per mole) at a given temperature, and nliq is a molecule’s concentration in the liquid. Further, adsorption on the molecules of the second, third, etc., layers is being considered. In the general case we get p Å ui /1 /(Ki /1rui ).
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Taking the BET model assumptions into consideration, we can write
or a/amon Å (Crp/ps )/[(1 0 p/ps )r(1 / (C 0 1)r p/ps )],
K2 É K3 É rrr É KL .
[23]
[30]
Then, as follows from general considerations, the surface fractions covered with different-type complexes will tend to become the same with pressure, i.e., ui /1 / ui r 1 for p/ps r 1. Therefore, it follows from [22] that
where C Å K1 /KL . Using [19] – [21] and [23] we can get
ps Å 1/KL . The further derivation can be conducted by the conventional procedure. In this connection the adsorption can be calculated as a Å amonr( u1 / 2ru2 / 3ru3 / rrr) Å amonr( u1 / 2r KLrpru1 / 3r(KLrp) 2ru1 / rrr) Å amonrK1rpru0r(1 / 2r(p/ps ) / 3r(p/ps ) 2 / rrr).
[24] It is absolutely clear that the series in parentheses in [24] is a derivative of the following series with respect to p/ps : 1 / p/ps / (p/ps ) 2 / rrr.
[26]
The adsorbent free surface fraction u0 can be found from the obvious formula u0 Å 1 0 ( u1 / u2 / u3 / rrr)
[31]
As is known, Eqs. [29] and [30] correspond to the polymolecular adsorption BET equation. Moreover, in contrast to the classical variant, the preexponential multiplier or entropy factor in [31] has a clear physical definition. 6. BET Equation for a Finite Number of Layers, n (19) It is absolutely clear that the present approach provides a derivation of the BET equation for a finite number of layers, n. All that is required is to sum with respect to n layers in expressions [24] and [27]. In this case we get another wellknown equation: a Å [(amonrK1rp)r[1 0 (p/ps ) nr(1 / n 0 nrp/ps )]]: [(1 0 p/ps )r[1 / K1rpr(1 0 (p/ps ) n ) 0 p/ps ]]. Here, n designates the maximum number of adsorbed layers.
[25]
Expression [25] is a descending geometric series with its sum equal to 1/(1 0 p/ps ). Thus, when the derivative is taken, Eq. [24] takes the form a Å (amonrK1rpru0 )/(1 0 p/ps ) 2 .
mon C Å (nliq /n ads )rexp[(qst,1 0 qL )/RrT].
SUMMARY
An equation has been derived on the basis of phenomenological thermodynamics that represents one of the expressions for interphase equilibrium and provides description of both adsorption and absorption phenomena from a single viewpoint. It has been shown that well-known Henry, Langmuir, Fowler–Guggenheim, Temkin, and BET adsorption equations directly follow from it with their constants having a clear physical meaning. New adsorption and absorption equations can be derived using the equation proposed and corresponding new model concepts.
Å 1 0 K1rpru0 (1 / p/ps / (p/ps ) 2 / rrr) Å 1 0 (K1rpru0 )/(1 0 p/ps ).
In the derivation of [27], the descending geometric series [25] was also replaced with its sum. It follows from [27] that u0 Å (1 0 p/ps )/(1 / K1rp 0 p/ps ).
[28]
Substitution of [28] into [26] gives a Å (amonrK1rp)/[(1 0 p/ps )r(1 / K1rp 0 p/ps )] [29]
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REFERENCES
[27]
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1. Bering, B. P., and Serpinski, V. V., Izv. Akad. Nauk SSSR, Ser. Khim. 11, 2427 (1974). [In Russian] 2. Tolmachyov, A. M., Usp. Khim. 50(5), 769 (1982). [In Russian] 3. Nicholson, D., and Parsonage, N. G., ‘‘Computer Simulation and the Statistical Mechanics of Adsorption.’’ Academic Press, New York, 1982. 4. Kiselyov, A. V., ‘‘Mezhmolekulyarnyye vzaimodeistviya v adsorbtsii i khromatografii (Intermolecular interactions in adsorption and chromatography).’’ Vysshaya shkola, Moscow, 1986. [In Russian] 5. Bojan, M. J., and Steele, W. A., Surface Sci. 199, 395 (1988). 6. Sircar, S., and Gupta, R., AIChE J. 27(5), 806 (1981). 7. Sircar, S., and Myers, A. L., Surface Sci. 205, 353 (1988). 8. Bakaev, V. A., Surface Sci. 198, 571 (1988).
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9. Tovbin, Yu. K., ‘‘Teoriya fiziko-khimicheskikh protsessov na granitse gaz-tverdoye telo (Theory of Physicochemical Processes on Gas– Solid Body Interface).’’ Nauka, Moscow, 1990. [In Russian] 10. Neimark, A. V., J. Colloid Interface Sci. 165, 91 (1994). 11. Hill, T. L., in ‘‘Kataliz. Voprosy teorii i metody issledovaniya (Catalysis. Theoretical Problems and Research Methods)’’ (transl. from English), p. 276. Inostr. literatura, Moscow, 1955. [In Russian] 12. Kikoin, A. K., and Kikoin, I. K., ‘‘Molekulyarnaya fizika (Molecular Physics).’’ Nauka, Moscow, 1976. [In Russian] 13. Smirnova, N. A., ‘‘Molekulyarnyye teorii rastvorov (Molecular Theories of Solutions).’’ Khimiya, Leningrad, 1987. [In Russian]
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14. Henry, D. C., Philos. Mag. 44(262), 689 (1922). 15. Langmuir, I., J. Am. Chem. Soc. 9, 1361 (1918). 16. Fowler, R., and Guggenheim, E. A., ‘‘Statisticheskaya termodinamika (Statistical Thermodynamics)’’ (Transl. from English). Inostr. literatura, Moscow, 1949. [In Russian] 17. Temkin, M. I., Zh. Fiz. Khim. 15(3), 296 (1941). [In Russian] 18. Brunauer, S., Emmett, P. H., and Teller, E., J. Am. Chem. Soc. 60(2), 309 (1938). 19. Emmett, P. H., in ‘‘Kataliz. Voprosy teorii i metody issledovaniya (Catalysis. Theoretical Problems and Research Methods)’’ (Transl. from English), p. 328. Inostr. literatura, Moscow, 1955. [In Russian]
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