J. Phys. Chem. 1993,97, 7097-7101
7097
Statistical Mechanical Aspects of Adsorption Systems Obeying the Temkin Isotherm Chung-hai Yang Department of Applied Chemistry, Nanjing Institute of Chemical Technology, Nanjing, Jiangsu 21 0009, People's Republic of China Received: November 12, 1992; In Final Form: March 26, I993
A model was set up for adsorption systems obeying the Temkin isotherm, in which both the lateral interaction between the adsorbed molecules and the energetic surface heterogeneity in adsorption were quantitatively described and then characterized by parameters, respectively. On the basis of this model, by using the ensemble theory in statistical mechanics as a theoretical tool, was derived the Temkin isotherm, in the form of 6 = ( R T / a ) ln(AP), its validity being examined both theoretically and experimentally. The derived statistical mechanical expression for the Temkin isotherm established the relation between the constants A and (Y and the microscopic properties of the adsorbed molecules, including the parameter of lateral interaction and the parameter of surface heterogeneity in adsorption, as proposed in the model. It was with this relation that these parameters were evaluated from the experimental data of the chemisorption of hydrogen on iron films. From the evaluated parameter of lateral interaction was further evaluated the molar lateral interaction energy between the adsorbed hydrogen molecules on iron films, the value of which was compared with that estimated from a Lennard-Jones (12, 6 ) potential. Introduction It is generally accepted that the correlation between the microscopic properties and the macroscopic behavior for adsorption systems is of theoretical significance in surface science. This paper is an attempt to correlate the macroscopic behavior of adsorption systems with the microscopic properties of the adsorbed molecule, of which the adsorption system is composed, by using statistical mechanics as a theoretical tool. In order to realize the above attempt, we try to derive the adsorption isotherm equation for an adsorption system by treating the model for the same adsorption system with statistical mechanics. Considering that the Temkin isotherm' is more likely to be obeyed by adsorption systems, we restrict ourselves to the derivation of the Temkin isotherm. Hayward et alS2and Young et al.3 have summarized the derivation of the Temkin isotherm. However, all of these derivations are done by inserting into the Langmuir isotherm4 the condition of a linear decrease in the differential heat of adsorption with increasing coverage or certain types of energy distribution function of adsorption sites, with the lateral interaction between the adsorbed molecules not being considered. In this paper, the Temkin isotherm is derived by using the ensemble theory in statistical mechanics as a theoretical tool, on the basis of the model for an adsorption system, in which the lateral interaction between the adsorbed molecules and the energetic surface heterogeneity in adsorption, which are widely accepted as the factors playing an important role in adsorption,5 are taken into account.
Model Consider an adsorption system composed of a gaseous phase and an adsorption phase, i.e. monolayer. The gaseous phase is assumed to be an ideal gas. Besides the assumption of a localized monolayer as made in Langmuir's theory for his isotherm, the following assumptions are made for the adsorption phase. ( 1 ) The lateral interaction between the adsorbed molecules is assumed to occur only in the nearest-neighbor pair. Therefore, the total lateral interaction energy between the adsorbed molecules (U) can be expressed as the sum of the lateral interaction energy of all possible nearest-neighbor pairs (w): namely U = Eiwi, where wi is the lateral interaction energy of ith nearest-neighbor 0022-3654/93/2097-7097%04.00/0
pair. The number of the nearest-neighbor pairs between the NA adsorbed molecules on the NSadsorption sites ( N U ) , is evaluated by Bragg-Williams approximation, namely6 NU = ( N A / N ~ ) ~ ( N s z / where 2 ) , z is the coordination number, characteristic of the adsorption system. The lateral interaction energy of a nearest-neighbor pair ( w ) is assumed to be of repulsive nature, and independent of coverage (e) in the middle range of coverage (denoted by 8' to 02) where the Temkin isotherm is valid.' Thus, w is characteristic of the adsorption system only. On the basis of the above statement, we have the expression for the lateral interaction energy between the N Aadsorbed molecules on the NS adsorption sites
u = N f i w = ( N A / N s ) 2 ( N s z / 2 ) w= NA(zw/2)8 (1) where e = NA/Ns. The significance of z w / 2 can be understood as follows. Rearranging eq 1, we have U/NA = (zw/2)8 ( z , w-independent of coverage, as their characteristics imply), leading to the conclusion that the total lateral interaction energy contributed by an adsorbed molecule (UINA) is proportional to coverage (e) with a proportionality constant of zw/2. For the different adsorption systems which obey the Temkin isotherm, at the same coverage, the larger the proportionality constant z w / 2 is, the stronger the lateral interaction between the adsorbed molecules appears. In the sense of the above statement, z w / 2 may be regarded as a parameter characterizing the lateral interaction between the adsorbed molecules. Thus, z w / 2 is, in this paper, defined as the parameter of lateral interaction. ( 2 ) On adsorption, the three degrees of translational freedom of a molecule in the gaseous phase can be assumed to appear as two degrees of translational motion around the adsorption site within the confine of area u,,, plus one degree of vibration with frequency u in adsorption bond normal to the surfaces8 On the above assumption and the assumption that kT >> hu (k,the Boltzmann constant; h, Planck's constant), as usually is the case, the partition function of an adsorbed molecule (Q) can be written as
Q" = (2.lrmkT/h2)ao(kT/hu)f(T) exp(-&kT)
=
QS(kT/hv) ( 2 ) where m is the mass of a molecule, P(T ) is the internal partition function of an adsorbed molecule, taking the energy of the lowest 0 1993 American Chemical Society
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7098 The Journal of Physical Chemistry, Vol. 97, No. 27, I993 energy state of an adsorbed molecule (et) as energy zero for the adsorbed molecules, and QF = (2~mkT/hZ)uoJS(T) exp( ft/kT) (3) As for vibrational frequency Y,the further assumption is made below. Experimental evidence shows that the surface of real solids is energetically heterogeneous? being described in terms of the type of distribution of adsorption sites according to energy, as proposed by previous workers.lO Experimental evidence further shows that the more energetic site will tend to be covered first with the adsorbed molecule? forming an adsorption bond with higher vibrational frequency. Consider a monolayer of NA molecules adsorbed on NS adsorption sites, thus the coverage being 8 = NAJNs. Denote by ul, ~ 2 ..., , V N the ~ vibrational frequencies in the adsorption bonds formed in sequence of adsorption during the formation of the above monolayer. Reasoning from the above experimental evidence, we are led to the conclusion that, for the above monolayer, the mean of the vibrational frequencies VI,29, ...,vNAmustdecrease with increasing coverage. It is convenient to use the geometric mean (i)in view of the statistical mechanical treatment of the proposed model, which will be discussed in the next section. For an adsorption system obeying the Temkin isotherm, the decrease in the geometric mean of vibrational frequencies (i)with increasing coverage (0) is assumed to be
applicable to the monolayer of molecules adsorbed on the heterogeneous surface. Correspondingly, the canonical partition funtion for the monolayer of molecules adsorbed on the heterogeneous surface should be written as
i a exp(-ae) i = V, exp(-a8) (e, I6 Ie2, a > 0) (3) where i = ( ~ Z V ~ ) and ‘ / ~Y, ^and “a” are constants characteristic
where xNMg(NA,NAA) = Ns!/NA!(Ns - NA)! and N U is supposed to be calculated by the Bragg-Williams approximation, namely = (NA/Ns)2(Nsz/2). Substituting eqs 3 and 6 into eq 5 , we have
of the adsorption system. Eq 3 indicates that, for the different adsorption systems which obey the Temkin isotherm, at the same coverage, the larger the value of “a” is, the larger is the rate a t which the geometric mean of vibrational frequencies (i)decreases with increasing coverage (e), and hence the more heterogeneous the surface covered in adsorption appears. From the above discussion, the “a” may be regarded as a parameter characterizing the surface heterogeneity in adsorption. Thus, the “a” is, in this paper, defined as the parameter of surface heterogeneity in adsorption.
N.
where N.
N.
NA
[Q7(kT/h;)lNA
(a:
;= ( ~ Y J ’ / ~ ^ ) ia 1
We define the quantity xg(NA,NAA)
by the equation6
exp(-NAAw/kT)
=
NAA
-
z = [Ns!/NA!(Ns
- NA)!]
x
exp(-N~zw/2NskT){@[ kT/ hv, exp(-ae)]
IN^
(7)
Substituting eq I into pSJkT = -(a In Z/aNA),, we obtain the chemical potential of an adsorbed molecule (ps) ps
= kT{ln[B/(l - e)]
+ (0zw/kT) - ln[Q~(kT/hv,)] - 2aBJ (8)
Derivation We now consider a monolayer composed of NA molecules adsorbed on NS adsorption sites with NAApairs of the nearestneighbor in a particular configuration. On the basis of the assumption of localized monolayer and assumption 1 made in the Model section, using the ensemble theory in statistical mechanics, we have the canonical partition function for the monolayer ( 2 ) 6
where g(NA,NAA) denotes the number of distinguishable configurations with NAApairs of the nearest-neighbor between NA molecules adsorbed on Ns adsorption sites. Strictly speaking, eq 4 is only applicable to the monolayer of molecules adsorbed on the homogeneous surface. However, in this paper, the monolayer is composed of molecules adsorbed on the heterogeneous surface, as indicated in the proposed model. In order to extend eq 4 to be applicable to the monolayer of molecules adsorbed on the heterogeneous surface, some modification should be made. In thecaseof adsorption on the heterogeneous surface, themolecules are adsorbed on the adsorption sites with different energies, resulting in the different vibrational frequencies in the formed adsorption bonds. On the contrary, in the case of adsorption on the homogeneous surface, the molecules are adsorbed on the adsorption sites with the same energy, resulting in the same vibrational frequency in the formed adsorption bonds. From the above comparison and eq 2, it is concluded that if the factor ( Q S ) N ^ is replaced by @[Q~(kT/hv,)], then eq 4 can be
As for the gaseous phase, the chemical potential of a gaseous molecule (pc) is, according to statistical mechanics, given by pG
= k T ln((P/kT) [ h3/(2.1rmkT)3/2][ 1/p( T)] X exP(c:/kT)I
(9)
where P is the pressure of the gas, and F(T) is the internal partition function of a gaseous molecule, taking the energy of the lowest energy state o f a gaseous molecule (e,“) as energy zero for the gaseous molecules. Substituting eqs 8 and 9 into p s = pG, the condition of equilibrium between monolayer and gaseous phase, and rearranging, we have In P = -1n~[u,/(2mnk~)’/~v,1 [JS(T)/JG(T)I exp[-(e, S -
t : ) / k T ] J + ln[e/(l -e)]
+ [(zw/kT) - 2a]6
(10)
Eq 10 is only applicable to the middle range of coverage, because eqs 1 and 3, on which eq 10is derived, are assumed to be applicable to that range of coverage. In the middle range of coverage, we have Iln [O/(l -e)][ 1, as illustrated in Appendix 11. In this case, the second term, In [e/(l - e)], may be neglected as compared with the third term, [(zw/kT) - 2~10,without serious error, and then eq 10 reduces to
6 = ( R T / a ) ln(AP)
(11)
Statistical Mechanical Derivation of the Temkin Isotherm
The Journal of Physical Chemistry, Vol. 97, No. 27, I993 7099
TABLE I: Slopes of 8-log P Isotherm for Chemisorption of H)on Fe Films’ T,K 103p [ ~ i ~ px e103] ~ S~ODCx 103 1 /slope
&
14.4 69.4 18.0 55.6 50.0 20.0 22.7 44.1 26.0 38.5 28.5 35.1 38.2 26.2 a See Appendix I11 for an illustration on [ ~ l o p e ] ~and , , ~slope.
147.5 177.8 195.3 217.4 250.3 273.1 306
6.78 5.62 5.12 4.60 4.00 3.66 3.27
2.57 3.20 3.57 4.03 4.63 5.07 6.80
where
and
a = zwL - 2 a R T (13) with L being Avogadro’s number and R being the gas constant. Being characterized by a linear variation of 6 with In P a t constant temperature, eq 1 1 is a Temkin isotherm. Equation 11 is referred to as the statistical mechanical expression for the Temkin isotherm, and is to be distinguished from the forms of the Temkin isotherm derived by using the other methods.2 Discussion (1) Testing the Validity of the Derived Statistical Mechanical Expression for theTemkin Isotherm. To be certain that a derived expression is truly identical with a particular adsorption isotherm equation, the following conditions must be satisfied. The first is that the P-8 relationship indicated by the derived expression must be shown to be the form characterized by the particular adsorption isotherm equation. The second is that the manner in which the differentialheat of adsorption (q) varies with coverage (e), deduced from the derived expression, must be shown to be the form implied by the particular adsorption isotherm equation. As shown in the Derivation section, the first condition has been satisfied. However, the second condition remains to be tested. Substituting eq 11 into the Clapeyron-Clausius equation, in the form of q = -AH(adsorption) = -R[d In P/d(l/T)]e, we have
4=qo-@
(14)
where qo = (RT/2) - L(c: - ):e and j3 = Lzw. For a given adsorption system, e,: ,:e z, and w are constants. Therefore, eq 14 indicates that, for a given adsorption system at constant temperature, the differential heat of adsorption (q) decreases linearly with increasing coverage (e), which is the form implied by the Temkin isotherm? thus leading to the verification of the derived statistical mechanical expression for the Temkin isotherm. Besides the above theoretical verification, the derived statistical mechanical expression for the Temkin isotherm can also beverified experimentally as follows. The expression for the slope of &log P plot is, according to eqs 11 and 13, given by slope = 2.303RT/ (zwL - 2aRT). Rearranging the above equation, we have l/slope = (zwL/2.303 X 103R)(103/T) - (2a/2.303) (15) Due to the temperature independence of z,w,and 4,as implied by their characteristics, eq 15 predicts that, for an adsorption system obeying theTemkin isotherm, the (1 /slope)-( 103/T) plot should be a straight line with a slope11 of (zwL/2.303 X lO3R) and an intercept of -(2a/2.303). On the basis of the experimental data due to Porter and Tompkins,12Hayward and TrapnellZfound that the chemisorption of hydrogen on iron films obeys the Temkin isotherm. Treating the above experimental data further, we found that, for the same adsorption system, l/slope decreases with increasing temperature (0,as shown in Table I. When l/slope
10’/T Figure 1. (l/s10pe)-(103/T)relation for chemisorption of hydrogen on
iron films.
is plotted against lo3/ T, the points are shown to have a tendency to fall on a straight line with a slopell of 11.0 and an intercept of -5.0, as shown in Figure 1. This linearity confirms the prediction fromeq 15, thus leading to the experimentalverification of eq 15, and hence the derived statistical mechanical expression for the Temkin isotherm. (2) Establishing the Relationship between Microscopic hop d e s and Macroscopic Behavior for an Adsorption System Obeying the Temkin Isotherm. m, et, ,e: .P(T), and P(T) are the properties of a molecule itself. w,a, and uo are the parameters characterizing the lateral interaction between the adsorbed molecules, the energetical surface heterogeneity in adsorption, and the two-dimensional translation of the adsorbed molecule around the adsorption site, respectively. Therefore, all of these are the microscopic properties of an adsorbed molecule, essential to the microscopic understanding of adsorption. The constants A and a,which can be determined experimentally, characterize the adsorption capacity of an adsorption system. Therefore, both of these may, to some extent, be regarded as the macroscopic behavior of an adsorption system obeying the Temkin isotherm. In the sense of the above statement, the derived statistical mechanical expression leads to the establishment of the relationship between the macroscopic behavior of a certain adsorption system, and the microscopic properties of the adsorbed molecule, of which the adsorption system is composed. (3) Evaluating the Microscopic Properties of the Adsorbed Molecule from the Experimental Adsorption Data. As a correlation of macroscopic behavior with microscopic properties, the derived statistical mechanical expression for the Temkin isotherm permits the evaluation of the microscopic properties of the adsorbed molecule from the experimental data of adsorption, which is illustrated by the following example. The slopell of the (l/slope)-( lo3/ T ) plot according to eq 15 and that according to the experimental adsorption data from Porter and Tompkinslz are (zwL/2.303 X 103R)and 11.0,respectively, as indicated in the previous paragraph. Comparing the two slopes, we obtain the numerical value of the parameter of lateral interaction (zw/ 2) between the adsorbed hydrogen molecules on iron films z w ~ / 2 . 3 0 3x 1 0 3 =~ 11.0 z ~ / 2= 12.7 x 103k (16) The intercept of the (1 /slope)-( lo’/ T ) plot according to eq 15 and that according to Porter and Tompkins’lz adsorption data are-(2a/2.303) and-5.0, respectively,as indicated in the previous paragraph. Comparing the two intercepts, we obtain the
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7100 The Journal of Physical Chemistry, Vol. 97, No. 27, 1993
numerical value of the parameter of surface heterogeneity in adsorption ( a ) for the chemisorption of hydrogen on iron films a = 5.8 -2a/2.303 = -5.0 The degree of validity of the evaluated value of the parameter of lateral interaction (zw/2) is examined as follows. Substituting eq 16 into eq 1, taking NA as L (Avogadro's number) and 8 as 0.5 (around which the Temkin isotherm is valid), and noting that Lk = R, we obtain the molar lateral interaction energy between the adsorbed hydrogen molecules on iron films (U)
U = L(12.7 X 103k)0.5 = 6.35 X 103R The above evaluated value is on the same order of magnitude as the value 4.12 X lO3R, estimated from a Lennard-Jones (12,6) potential.13 The estimation is presented in Appendix IV. Conclusion In the model set up for an adsorption system obeying the Temkin isotherm, two parameters have been proposed for characterizing the lateral interaction between the adsorbed molecules and the energetic surface heterogeneity in adsorption, respectively. By treating this model with the ensemble theory in statistical mechanics as a theoretical tool, we have derived the Temkin isotherm, namely the statistical mechanical expression for the Temkin isotherm. Substituting the derived expression into the Clapeyron-Clausius equation, we are led to the linear decrease in the differential heat of adsorption with increasing coverage, thus confirming the validity of the derived expression. The derived expression correlates the constants A and (Y with the microscopic properties of the adsorbed molecule zw/2, a, yo, uo, m, E:, COG, P(T), and P(T). In the sense of statistical mechanics, this correlation means the establishment of the relationship between the macroscopic behavior of a certain adsorption system and the microscopic properties of the adsorbed molecule of which the adsorption system is composed. With this relationship, from the experimental data of the chemisorption of hydrogen on iron films have been evaluated the parameter of lateral interaction (zw/2) and the parameter of surface heterogeneity in adsorption (a), their values being 12.7 X 1O3k and 5.8, respectively. The former leads to a molar lateral interaction energy of 6.35 X 103R,which is in agreement (order of magnitude) with the value 4.12 X lO'R, estimated from a Lennard-Jones (12, 6) potential. Appendix I. Comparison of the Absolute Value of the Term In[O/(l - e)] with the Value of Coverage (e) Some values of the term ln[e/(l - e)] at different coverages are given below.
0.401 0.410 0.450 0.500
-0.401 4.364 -0.201 0
0.550 0.600 0.650 0.659
0.201 0.406 0.619 0.659
From the above table, it is shown that the absolute value of In[@/(1 - e)] is smaller than or equal to the value of coverage ( 8 ) as 0.401 I 8 I0.659 (approximately). Furthermore, as 8 approaches 0.5, the smaller ln[e/(l -e)] is than 8, leading to the conclusion that, in the middle range of coverage, the absolute value of ln[e/(l - e)] is smaller than the value of coverage (e). This conclusion can also be drawn from the ln[8/( 1 -e)] - 8 plot, which is omitted here due to space considerations.
Appendix 11. Value of the Term [(zw/kZ')- 2 4 at Ordinary Temperatures As evaluated in section 3 of Discussion, the value of the parameter of lateral interaction (zw/2) is 12.7 X 103k, and that
of the parameter of surface heterogeneity in adsorption (a) is 5.8 for the chemisorption of hydrogen on iron films. On the basis of the above results, at ordinary temperatures (e.g. T = 298 K), we have (zwlkT) - 2a = (2 X 12.7 X 103k/k X 298) - (2 X 5 . 8 ) = 73.6 From the above evaluation, it is concluded that the term [(zw/ k7') - 2a] is much larger than 1, namely [(zw/kT) - 2a] >> 1, a t ordinary temperatures. This conclusion, together with the one obtained in Appendix I, leads to the conclusion that the term ln[O/(l - e)] is much smaller than the term [(zw/kT) - 2~18, namely ln[e/(l - e)]
