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Statistical Mechanical Study on Variation in Energy of the Lowest Energy State of Molecules during Adsorption
Journal of Colloid and Interface Science 243, 267–272 (2001) doi:10.1006/jcis.2001.7808, available online at http://www.idealibrary.com on
Statistical Mechanical Study on Variation in Energy of the Lowest Energy State of Molecules during Adsorption Chung-hai Yang Department of Applied Chemistry, Nanjing Institute of Chemical Technology, 5 New Model Road, Nanjing, Jiangsu 210009, People’s Republic of China Received June 22, 2000; accepted June 29, 2001; published online October 5, 2001
The variation in energy of the lowest energy state of molecules during adsorption was evaluated for hydrogen chemisorption on iron films by treating the adsorption data with statistical mechanics as a theoretical tool. The validity of the evaluated value of this variation was examined. The application of this evaluated value to the heat of adsorption led to the microscopic understanding of the heat of adsorption on a quantitative level, and the interpretation of the observed temperature independence of the heat of adsorption. °C 2001 Academic Press Key Words: statistical mechanics; energy of the lowest energy state; Temkin isotherm equation; heat of adsorption; microscopic properties of adsorbed molecules.
1. INTRODUCTION
From theoretical consideration, some knowledge can be obtained from the heat of adsorption and its variation with temperature and coverage. The heat of adsorption reflects the strength of the adsorption bond (1, 2), which is implicitly related to the structure of surfaces. As Kirchhoff’s law implies, for adsorption systems, the rate of the increase in the heat of adsorption with temperature gives the difference in the heat capacity between the gaseous phase and the adsorption phase, and hence, in a microscopic sence, the change in the state of motion of the molecule during adsorption (1, 2). The variation in the heat of adsorption with coverage reflects the lateral interaction between the adsorbed molecules, and the energetic surface heterogeneity (3, 4), factors which play important roles in adsorption (5). The above statement indicates that, theoretically at least, the study on the heat of adsorption, particularly its variation with temperature and coverage, is of significance in the surface science. Theoretically, the variation in energy of the lowest energy state of the molecule during adsorption (denoted by −(²oS − ²oG ) where ²oS and ²oG are the energies of the lowest energy state of an adsorbed molecule and a gaseous molecule, respectively) invariably contributes to the heat of adsorption, thus being of fundamental importance in studying the heat of adsorption. In this paper, an attempt is made to evaluate −(²oS − ²oG ) from adsorption data, so as to gain a microscopic understanding of the heat of adsorption on a quantitative level, and hence to interpret the observed phenomenon.
In a broad sense, −(²oS − ²oG ) is one of the microscopic properties of adsorption systems. Measured at equilibrium, the adsorption data can be regarded as one of the macroscopic properties of adsorption system. From the above statement, the evaluation of −(²oS − ²oG ) from adsorption data, which is to be done in this paper, reduces to a problem of correlating the microscopic properties with the macroscopic properties for the adsorption system. Theoretically at least, this correlation can be obtained with statistical mechanics as a theoretical tool (6). Actually, for an adsorption system obeying a certain adsorption isotherm equation (in its empirical form or derived theoretically), this correlation is implicitly given by the statistical mechanical expression for this adsorption isotherm equation (7). Therefore, the present study is confined to those adsorption systems which obey a certain adsorption isotherm equation, with its statistical mechanical expression being worked out. More precisely, this paper is an attempt to evaluate −(²oS − ²oG ) from adsorption data, on the basis of the statistical mechanical expression for the adsorption isotherm equation obeyed by these adsorption data. 2. PRINCIPLE
The principle is to be illustrated by adsorption systems obeying the Temkin isotherm equation (8). Having been already derived in Ref. (7), the statistical mechanical expression for this isotherm equation is directly given below (7), θ = (RT/α) ln(AP)
(θ1 % θ % θ2 ),
[1]
where £ ¡ ¤ ¢± ¤ £ A = σ/(2πmkT )1/2 ν0 [J S /J G ] exp − ²oS − ²oG kT
[2]
and α = zwL − 2a RT
[3]
(θ , coverage; namely the fraction of sites covered by adsorbed molecules at pressure P and temperature T ; R, gas constant; α and A, constants in the Temkin isotherm equation; θ1 % θ % θ2 , middle range of coverage where the Temkin isotherm equation
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is valid (9, 10); σ, area within which an adsorbed molecule translates around the adsorption site; m, mass of a molecule; k, Boltzmann constant; ν0 , constant characteristic of the adsorption system; J S, internal partition function of an adsorbed molecule, taking ²oS as energy zero for the adsorbed molecules; J G, internal partition function of a gaseous molecule, taking ²oG as energy zero for the gaseous molecules; zw, parameter of lateral interaction;1 L, Avogadro number; a, parameter of surface heterogeneity in adsorption). Following is an attempt to derive an expression for evaluating −(²oS − ²oG ) from the adsorption data, using the above cited statistical mechanical expression for the Temkin isotherm equation, actually that for the constant A, namely Eq. [2]. Taking logarithms of both sides of Eq. [2], assuming that J S = J G , and rearranging, we have £ ¤ £ ¡ ¢± log(AT 1/2 ) = log σ/(2π mk)1/2 ν0 + − ²oS − ²oG 2.303 ¤ [4] ×103 k (103 /T ). Due to the temperature independence of σ, ν0 , ²oS , and ²oG as implied by their characteristics, it is predicted from Eq. [4] that, for a given adsorption system obeying the Temkin isotherm equation at different temperatures, the plot of log(AT 1/2 ) against 103 /T should be a straight line with a slope of −(²oS − ²oG )/2.303 × 103 k. From the above prediction, we have the expression for evaluation of −(²oS − ²oG ), ¢ ¡ − ²oS − ²oG = [slope] × 2.303 × 103 k ¢ ¡ −L ²oS − ²oG = [slope] × 2.303 × 103 R,
[5] [6]
where [slope] is the slope of the log( AT 1/2 )–103 /T plot. From the adsorption data can be obtained the value of [slope] (the details of which will be given in the next section, Evaluation,) and hence that of −(²oS − ²oG ). The above method of evaluating −(²oS − ²oG ) is also, in principle, applicable to an adsorption system obeying other adsorption isotherm equations with its statistical mechanical expression being worked out. 3. EVALUATION
Adsorption data were recorded by Porter and Tompkins on chemisorption of hydrogen on evaporated iron films, covering a wide range of temperature (147 to 306 K) and pressure (10−5 to 10−1 mmHg) (11). The amounts adsorbed are given in millimoles per gram of iron, and denoted as sorption (mmol/g
1
Briefly, the parameter of lateral interaction (zw) characterizes the lateral interaction between the adsorbed molecules, the details of which have been given in Ref. (7), and hence is not repeated here. Intuitively, the larger zw is, the stronger the lateral interaction appears, and vice versa. As a limit case, zw = 0 means that there is no lateral interaction between the adsorbed molecules, which will be used in Section 4.2.
iron) in the figure and tables. The pressures P are expressed as logP(mmHg). When plotted as sorption(mmol/g iron)–logP (mmHg) isotherms, these adsorption data give straight lines, as shown in Fig. 1 in Ref. (11). The linearity of these isotherms indicates that this chemisorption obeys the Temkin isotherm equation (3). As stated in the previous section, Principle, an adsorption system obeying this isotherm equation is taken as an illustration of the principle of evaluating −(²oS − ²oG ). Therefore, an attempt is made to evaluate, from these adsorption data, the −(²oS − ²oG ) of this chemisorption as a sample evaluation. To evaluate the −(²oS − ²oG ), the above adsorption data are treated below. From the above sorption(mmol/g iron)– logP(mmHg) isotherms are taken sorptions at P = 1 mmHg and different temperatures. These sorptions are denoted by sorption(P = 1 mmHg), given in column 3 in Table 1, and, for the convenience of evaluation, are converted to θ(P = 1 mmHg), defined as the coverages at P = 1 mmHg and given in column 5, by the equation θ (P = 1 mmHg) = sorption(P = 1 mmHg)/saturation sorption.2 From the above isotherms are also obtained the slopes of isotherms at different temperatures, according to its definition, namely slope = 1 sorption/1 log P. These slopes are denoted by (slope)s , given in column 4, and, for the convenience of evaluation, are converted to (slope)θ , defined as the slopes of θ –log P(mmHg) isotherms and given in column 6, by the equation (slope)θ = (slope)s /saturation sorption. When applied to P = 1 mmHg, the Temkin isotherm equation, in the form of θ = (RT /α) ln(A P), reduces to θ (P = 1 mmHg) = (2.303RT /α) log A, where 2.303RT /α corresponds to the slope of the θ –log P(mmHg) isotherm (the graphical presentation of this isotherm equation), and hence the above (slope)θ . Substituting the above obtained θ(P = 1 mmHg) and (slope)θ into this reduced isotherm equation, we have the values of log A at different temperatures. These values are given in column 7. Now we evaluate −(²oS − ²oG ), using Eqs. [5] and [6], from the treated adsorption data given in Table 1. When log(AT 1/2 ) is plotted against 103 /T (Fig. 1), the points are shown to have a tendency to fall on a straight line with a slope of 11.6. Note that the slope of a log( AT 1/2 )–(103 /T ) plot is denoted by [slope], as stated in the previous section, Principle. Therefore, we have [slope] = 11.6. Substituting [slope] = 11.6 into Eqs. [5] and [6], we have the variation in energy of the lowest energy state of hydrogen molecules during its adsorption on iron films, ¢ ¡ − ²oS − ²oG = [slope] × 2.303 × 103 k = 26.7 × 103 k = 3.69 × 10−12 erg/molecule
¢ ¡ −L ²oS − ²oG = [slope] × 2.303 × 103 R = 26.7 × 103 R = 222.1 kJ/mol ∼ = 222 kJ/mol. 2 The saturation sorption is regarded as the Y -coordinate of the intercept of the extrapolated lines of sorption(mmol/g iron)–log P(mmHg) isotherms at different temperatures, which were plotted in Fig. 1 in Ref. (11), its value being 0.178 mmol/g iron.
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TABLE 1 Adsorption Data of Hydrogen Chemisorption on Iron Films Adsorption dataa (original) T (K)
103 /T
Sorption (P = 1 mmHg) (mmol/g iron)
147.5 177.8 195.3 217.4 250.3 273.2 306
6.78 5.62 5.12 4.60 4.00 3.66 3.27
0.167 0.165 0.162 0.159 0.155 0.150 0.148
a
Adsorption data (treated) (slope)s
× 103
θ (P = 1 mmHg)
(slope)θ × 103
log A
log (AT 1/2 )
0.938 0.927 0.910 0.893 0.871 0.843 0.832
14.4 18.0 20.1 22.6 26.0 28.5 38.2
65.1 51.5 45.3 39.5 33.5 29.6 21.8
66.2 52.6 46.5 40.7 34.7 30.8 23.0
2.57 3.20 3.57 4.03 4.63 5.07 6.80
From Ref. (11).
(It is necessary to emphasize the difference among [slope], (slope)s , and (slope)θ , so as to avoid confusion among their meanings. Their definitions are clearly presented in the above paragraphs.) 4. RESULTS AND DISCUSSION
4.1. Testing the Validity of the Evaluated Value of −(²oS − ²oG ) The validity is tested by comparing the heat of adsorption obtained from the evaluated value of −(²oS − ²oG ) with that confirmed by calorimetric measurement. Applying the Clapeyron– Clausius equation to isotherms at different temperatures (shown in Fig. 1 in Ref. (11)), Porter and Tompkins evaluated the heats of adsorption (−1H ) at different amounts adsorbed for chemisorption of hydrogen on iron films (11). The above results were presented in Fig. 3 in Ref. (11) as a −1H (kcal/mol)–amount adsorbed (mmol/g iron) plot. It was reported that the evaluated
values of −1H are confirmed by being in satisfactory agreement with those obtained by Beeck who used a calorimetric method (11). Of the values of −1H , only those in or near the middle range of coverage are, in this paper, taken as the standards for comparison, consistent with the Temkin isotherm equation being applicable in that range of coverage (9, 10). These values of −1H are given in Table 2. Besides, the heat of adsorption can be obtained from the value of −(²oS − ²oG ) as follows. Substituting Eq. [1] into the Clapeyron–Clausius equation, as q = −R[∂ ln P/∂(1/T )]θ , and assuming that J S = J G , we have the statistical mechanical expression for the heat of adsorption (q), ¡ ¢ q = (1/2)RT − L ²oS − ²oG − L[zw − T (dzw/dT ) (θ1 % θ % θ2 ),
+ 2kT 2 (da/dT )]θ
[7]
where dzw/dT is the derivative of zw with respect to T , and da/dT is the derivative of a with respect to T , both being constant for a given adsorption system at a given temperature. Physically, in Eq. [7], the terms Lzwθ and −L T (dzw/dT )θ can be regarded as contributions to the decrease in the heat of adsorption (q) by the lateral interaction between the adsorbed molecules, and the term 2RT 2 (da/dT )θ can be regarded as that by the surface heterogeneity in adsorption. It is shown in Eq. [7] TABLE 2 Comparison of Heats of Adsorption (q) with Heats of Adsorption (−∆H) Amount adsorbed
FIG. 1. log(AT 1/2 )–103 /T relationship for chemisorption of hydrogen on iron films.
Heats of adsorption, −1H a
Sorption (mmol/g iron)
Coverage θ
(Kcal/mole)
(kJ/mol)
Heats of adsorption, q (kJ/mol)
0.120 0.122 0.124 0.126 0.128 0.130
0.674 0.685 0.697 0.708 0.719 0.730
20.2 19.4 18.6 17.9 17.1 16.3
84.5 81.2 77.8 74.9 71.6 68.2
80.7 78.4 75.8 73.5 71.2 68.8
a
From Ref. (11).
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that, for a given adsorption system at constant temperature, q decreases linearly with increasing coverage, which is the form implied by the Temkin isotherm equation, thus leading to the verification of Eq. [7]. To evaluate the heat of adsorption from the value of −(²oS − G ²o ) using Eq. [7], some simplifying assumptions are made below. Assuming that dzw/dT and da/dT are small, and noting that the value of zw is 25.4 × 103 k (as evaluated in Ref. (7)), we would expect that, in the brackets in Eq. [7], dzw/dT and da/dT may numerically be neglected as compared with zw at not very high temperatures (as is usually the case for adsorption) without serious error. Based on the above assumptions, we have the simplified statistical mechanical expression for the heat of adsorption (q), ¡ ¢ q = (1/2)RT − L ²oS − ²oG − Lzwθ
(θ1 % θ % θ2 ). [8]
The verification of Eq. [8] is similar to that of Eq. [7], and hence is omitted here. Equation [8] is an approximate one, with the neglect of minute contributions to the decrease in the heat of adsorption (q) by the terms −L T (dzw/dT )θ and 2RT 2 (da/dT )θ. Due to its simplicity, yet retaining the essential feature of Eq. [7], Eq. [8] can give results of both theoretical and practical interest when employed to treat the adsorption data, even limited. Specifically, Eq. [8] will be employed not only to evaluate the heat of adsorption from the value of −(²oS − ²oG ) (as shown below), but also to examine the microscopic aspects of the heat of adsorption and to interpret the observed temperature independence of the heat of adsorption (as shown in Sections 4.2 and 4.3, respectively). Substituting T = 223.9 K,3 −(²oS − ²oG ) = 26.7 × 103 k (as evaluated in the section, Evaluation), and zw = 25.4 × 103 k (as evaluated in Ref. (7)), we have the values of the heat of adsorption (q) at different coverages (θ). (These coverages correspond to the amounts adsorbed, the heats of adsorption (−1H ) at which are taken as the standards for comparison.) The evaluated values of q at these coverages are given in Table 2. From Table 2, some results can be given as follows. In view of the probable error on the heat of adsorption obtained by Porter and Tompkins (−1H ) being 0.5–1.0 kcal/g iron (or 2.1–4.2 kJ/g iron) (11), the heats of adsorption obtained from the evaluated value of −L(²oS − ²oG ) (q) are close to the −1H confirmed by calorimetric measurement, thus confirming the q, and hence, to some extent, the evaluated value of −L(²oS − ²oG ).
3 The temperature is taken as 223.9 K, which is the average of 147.5, 177.8, 195.3, 217.4, 250.3, 273.2, and 306 K. From isotherms at these temperatures, which were plotted in Fig. 1 in Ref. (11), were obtained, by Porter and Tompkins, the heats of adsorption (−1H ), some of which are, in this paper, taken as the standards for comparison. The temperature is taken as such for the purpose of making a comparison between the heats of adsorption obtained by different methods (namely q and −1H ) under conditions of temperature as similar as possible.
4.2. Microscopic Understanding of Heat of Adsorption on a Quantitative Basis Following is an examination, with Eq. [8], of the microscopic aspects of the heat of adsorption. In Eq. [8], the first term (1/2)RT and the second term −L(²oS − ²oG ) can be regarded as contributions to the heat of adsorption (q) by the differences in translational energy and energy of the lowest energy state between the gaseous molecules and the adsorbed molecules, respectively. The third term −Lzwθ gives a linear decrease in q with increasing coverage (θ ) due to the lateral interaction actually existing in the practical adsorption system. It is shown from the above statement that, in a statistical mechanical sense, Eq. [8] leads, to some extent, to the microscopic understanding of the heat of adsorption. However, this microscopic understanding is of a qualitative nature. Therefore, an attempt is made to discuss the microscopic aspects of the heat of adsorption on a quantitative basis, using the evaluated value of −(²oS − ²oG ), the details of which are given below. On the basis of the significance of the parameter of lateral interaction zw, an adsorption systems with zw = 0 means that these adsorption systems are taken as those with no lateral interaction between the adsorbed molecules. (See footnote 1.) Substituting zw = 0 into Eq. [8], we have the statistical mechanical expression for the heat of adsorption for adsorption systems with no lateral interaction (qo ), ¡ ¢ qo = (1/2)RT − L ²oS − ²oG .
[9]
From Eq. [9], qo can theoretically be considered to be contributed by the variation in translational energy of molecules during adsorption, (1/2)RT , and that in energy of the lowest energy state of molecules during adsorption, −L(²oS − ²oG ). For chemisorption of hydrogen on iron films, the values of these contributions,4 together with that of qo , are given in Table 3. Substituting Eq. [9] into Eq. [8], we have another form of the simplified statistical mechanical expression for the heat of adsorption (q), q = qo − Lzwθ
(θ1 % θ % θ2 ).
[10]
From Eq. [10], the heat of adsorption for a practical adsorption system (obeying the Temkin isotherm equation) (q) can theoretically be considered to be the heat of adsorption for the adsorption system taken as the one with no lateral interaction (qo ) minus the decrease in the qo at a coverage of θ due to lateral interaction (Lzwθ ). For chemisorption of hydrogen on iron films, the values of Lzwθ , together with those of q, at the same coverages as those shown in Table 2, are also given in Table 3.
4 The value of temperature (T ) for evaluation of the contribution (1/2)RT is taken as 223.9 K, consistent with that for evaluation of the heats of adsorption (q) shown in Table 2.
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TABLE 3 Microscopic Understanding of Heat of Adsorption for H2 on Fe (Films) Contribution to qo by (1/2)RT
Contribution to qo by −L(²oS − ²oG )
Decrease in qo , Lzw θ
Heat of adsorption, qo (kJ/mol)
(kJ/mol)
(1/2)RT /qo
(kJ/mol)
−L(²oS − ²oG )/qo
Coverage, θ
(kJ/mol)
Lzwθ/qo
Heats of adsorption, q (kJ/mol)
223.0
0.9
0.4%
222.1
99.6%
0.674 0.685 0.697 0.708 0.719 0.730
142.3 144.7 147.2 149.5 151.8 154.2
63.8% 64.9% 66.0% 67.1% 68.1% 69.1%
80.7 78.4 75.8 73.5 71.2 68.8
From Table 3 and the above theoretical considerations, some quantitative results can be given for chemisorption of hydrogen on iron films. (1) The heat of adsorption (qo ) is invariably contributed by the variation in translational energy of molecules during adsorption (1/2)RT , and that in energy of the lowest energy state of molecules during adsorption −L(²oS − ²oG ). The relative importance of these contributions to the qo is as follows. The value of −L(²oS − ²oG ) is about 200 times larger than (1/2)RT at a temperature of 223.9 K, thus being the main contribution to the qo even at ordinary temperatures. (2) Due to lateral interaction between the adsorbed molecules, qo decreases by 63.8%–69.1% in the range of coverage θ = 0.674–0.730. The above results lead, to some extent, to the microscopic understanding of the heat of adsorption on a quantitative level. 4.3. Interpreting the Temperature Independence of the Heat of Adsorption Porter and Tompkins evaluated the heats of adsorption (−1H ) for hydrogen chemisorption on iron films, by applying the Clapeyron–Clausius equation to the isotherms at 147.5, 177.8, 195.3, 217.4, 250.3, 273.2, and 306 K, and then reported the fact that −1H is found to be independent of temperature (11). An attempt is made to interpret, mainly with the evaluated value of −L(²oS − ²oG ), this temperature independence of −1H as follows. Theoretically, for a given adsorption system (obeying the Temkin isotherm equation) at a given coverage, the increase in the heat of adsorption due to temperature increase can be evaluated, using the simplified statistical mechanical expression for the heat of adsorption (Eq. [8]), as follows, Increase in heat of adsorption = 1(1/2)RT = (1/2)R1T, [11] where 1T is the temperature increase. It is shown from Fig. 3 in Ref. (11) that the heat of adsorption for chemisorption of hydrogen on iron films obtained by Porter and Tompkins (−1H ) is dependent on the amount adsorbed. This implies that their above findings of temperature independence of −1H is made on the
condition of the given coverage. Therefore, the method of evaluating the increase in the heat of adsorption due to the temperature increase (namely Eq. [11]) can be applicable to this chemisorption under this condition. The range of temperature in which the values of −1H were obtained is 147.5 to 306 K, the temperature increase being 158.5 K. Substituting this temperature increase into Eq. [11], we have the increase in the heat of adsorption for this chemisorption due to this temperature increase, 1(1/2)RT = (1/2)R 1T = (1/2)R(158.5) = 0.659 kJ/mol, whereas the contribution to the heat of adsorption (expressed by Eq. [8]) by −L(²oS − ²oG ) is invariably 222 kJ/mol as evaluated in the section, Evaluation. The former (0.659 kJ/mol) is about 1/340 of the latter (222 kJ/mol), and hence can, in evaluation of the heat of adsorption with Eq. [8], be neglected as compared with the latter without serious error. The above comparison leads to the interpretation of the temperature independence of the heat of adsorption (−1H ), reported by Porter and Tompkins. (It is necessary to point out that the above interpretation is only valid for chemisorption without change in its mechanism, such as dissociation, in the range of temperature observed (3).) 5. SUMMARY AND CONCLUSION
The expression for evaluating the variation in energy of the lowest energy state of molecules during adsorption (−L(²oS − ²oG )) from the adsorption data was derived using the statistical mechanical expression for the constant A of the Temkin isotherm equation, namely Eq. [2]. The evaluated value of −L(²oS − ²oG ) is 26.7 × 103 R (or 222 kJ/mol) for chemisorption of hydrogen on iron films. The validity of this evaluated value was confirmed by the fact that the heats of adsorption obtained from this evaluated value are close to those confirmed by the calorimetric measurement. As one of its applications to the heat of adsorption, the evaluated value of −L(²oS − ²oG ) leads, to some extent, to the microscopic understanding of the heat of adsorption on a quantitative level, the main part of which is that, for chemisorption of hydrogen on iron films, the heat of adsorption decreases, due to the lateral interaction between the adsorbed molecules,
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by 63.8%–69.1% in the range of coverage θ = 0.674–0.730. Besides, it is shown from this evaluated value that, for this chemisorption, the contribution to the heat of adsorption (expressed by Eq. [8]) by −L(²oS − ²oG ) is numerically much larger than the theoretically evaluated increase in the heat of adsorption due to the temperature increase observed, thus suggesting the temperature independence of the heat of adsorption, as was found by Porter and Tompkins in this chemisorption. REFERENCES 1. Cerny, S., in “The Chemical Physics of Solid Surfaces and Heterogeneous Catalysis” (D. A. King and D. P. Woodroff, Eds.), Vol. 2, p. 1, Elsevier, Amsterdam, 1983.
2. Thomas, J. M., and Thomas, W. J., “Introduction to the Principles of Heterogeneous Catalysis,” Academic Press, London, 1967. 3. Hayward, D. O., and Trapnell, B. M. W., “Chemisorption,” 2nd ed., Butterworth, London, 1964. 4. Brunauer, S., “The Adsorption of Gases and Vapors,” Vol. 1, University Press, Princeton, 1945. 5. Jaroneic, M., and Madey, R., “Physical Adsorption on Heterogeneous Solids,” Elsevier, Amsterdam, 1988. 6. Kubo, R., “Statistical Mechanics,” North-Holland, Amsterdam, 1965. 7. Chung-hai Yang, J. Phys. Chem. 97, 7097 (1993). 8. Frumkin, A., and Slygin, A., Acta Physicochim. U.R.S.S. 3, 791 (1935). 9. Oscik, J., “Adsorption,” Ellis Horwood, Chichester, 1982. 10. Brunauer, S., Love, K. S., and Keenan, R. G., J. Am. Chem. Soc. 64, 751 (1942). 11. Porter, A. S., and Tompkins, F. C., Proc. R. Soc. A 217, 544 (1953).
Statistical Mechanical Study on Variation in Energy of the Lowest Energy State of Molecules during Adsorption Chung-hai Yang Department of Applied Chemistry, Nanjing Institute of Chemical Technology, 5 New Model Road, Nanjing, Jiangsu 210009, People’s Republic of China Received June 22, 2000; accepted June 29, 2001; published online October 5, 2001
The variation in energy of the lowest energy state of molecules during adsorption was evaluated for hydrogen chemisorption on iron films by treating the adsorption data with statistical mechanics as a theoretical tool. The validity of the evaluated value of this variation was examined. The application of this evaluated value to the heat of adsorption led to the microscopic understanding of the heat of adsorption on a quantitative level, and the interpretation of the observed temperature independence of the heat of adsorption. °C 2001 Academic Press Key Words: statistical mechanics; energy of the lowest energy state; Temkin isotherm equation; heat of adsorption; microscopic properties of adsorbed molecules.
1. INTRODUCTION
From theoretical consideration, some knowledge can be obtained from the heat of adsorption and its variation with temperature and coverage. The heat of adsorption reflects the strength of the adsorption bond (1, 2), which is implicitly related to the structure of surfaces. As Kirchhoff’s law implies, for adsorption systems, the rate of the increase in the heat of adsorption with temperature gives the difference in the heat capacity between the gaseous phase and the adsorption phase, and hence, in a microscopic sence, the change in the state of motion of the molecule during adsorption (1, 2). The variation in the heat of adsorption with coverage reflects the lateral interaction between the adsorbed molecules, and the energetic surface heterogeneity (3, 4), factors which play important roles in adsorption (5). The above statement indicates that, theoretically at least, the study on the heat of adsorption, particularly its variation with temperature and coverage, is of significance in the surface science. Theoretically, the variation in energy of the lowest energy state of the molecule during adsorption (denoted by −(²oS − ²oG ) where ²oS and ²oG are the energies of the lowest energy state of an adsorbed molecule and a gaseous molecule, respectively) invariably contributes to the heat of adsorption, thus being of fundamental importance in studying the heat of adsorption. In this paper, an attempt is made to evaluate −(²oS − ²oG ) from adsorption data, so as to gain a microscopic understanding of the heat of adsorption on a quantitative level, and hence to interpret the observed phenomenon.
In a broad sense, −(²oS − ²oG ) is one of the microscopic properties of adsorption systems. Measured at equilibrium, the adsorption data can be regarded as one of the macroscopic properties of adsorption system. From the above statement, the evaluation of −(²oS − ²oG ) from adsorption data, which is to be done in this paper, reduces to a problem of correlating the microscopic properties with the macroscopic properties for the adsorption system. Theoretically at least, this correlation can be obtained with statistical mechanics as a theoretical tool (6). Actually, for an adsorption system obeying a certain adsorption isotherm equation (in its empirical form or derived theoretically), this correlation is implicitly given by the statistical mechanical expression for this adsorption isotherm equation (7). Therefore, the present study is confined to those adsorption systems which obey a certain adsorption isotherm equation, with its statistical mechanical expression being worked out. More precisely, this paper is an attempt to evaluate −(²oS − ²oG ) from adsorption data, on the basis of the statistical mechanical expression for the adsorption isotherm equation obeyed by these adsorption data. 2. PRINCIPLE
The principle is to be illustrated by adsorption systems obeying the Temkin isotherm equation (8). Having been already derived in Ref. (7), the statistical mechanical expression for this isotherm equation is directly given below (7), θ = (RT/α) ln(AP)
(θ1 % θ % θ2 ),
[1]
where £ ¡ ¤ ¢± ¤ £ A = σ/(2πmkT )1/2 ν0 [J S /J G ] exp − ²oS − ²oG kT
[2]
and α = zwL − 2a RT
[3]
(θ , coverage; namely the fraction of sites covered by adsorbed molecules at pressure P and temperature T ; R, gas constant; α and A, constants in the Temkin isotherm equation; θ1 % θ % θ2 , middle range of coverage where the Temkin isotherm equation
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0021-9797/01 $35.00
C 2001 by Academic Press Copyright ° All rights of reproduction in any form reserved.
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is valid (9, 10); σ, area within which an adsorbed molecule translates around the adsorption site; m, mass of a molecule; k, Boltzmann constant; ν0 , constant characteristic of the adsorption system; J S, internal partition function of an adsorbed molecule, taking ²oS as energy zero for the adsorbed molecules; J G, internal partition function of a gaseous molecule, taking ²oG as energy zero for the gaseous molecules; zw, parameter of lateral interaction;1 L, Avogadro number; a, parameter of surface heterogeneity in adsorption). Following is an attempt to derive an expression for evaluating −(²oS − ²oG ) from the adsorption data, using the above cited statistical mechanical expression for the Temkin isotherm equation, actually that for the constant A, namely Eq. [2]. Taking logarithms of both sides of Eq. [2], assuming that J S = J G , and rearranging, we have £ ¤ £ ¡ ¢± log(AT 1/2 ) = log σ/(2π mk)1/2 ν0 + − ²oS − ²oG 2.303 ¤ [4] ×103 k (103 /T ). Due to the temperature independence of σ, ν0 , ²oS , and ²oG as implied by their characteristics, it is predicted from Eq. [4] that, for a given adsorption system obeying the Temkin isotherm equation at different temperatures, the plot of log(AT 1/2 ) against 103 /T should be a straight line with a slope of −(²oS − ²oG )/2.303 × 103 k. From the above prediction, we have the expression for evaluation of −(²oS − ²oG ), ¢ ¡ − ²oS − ²oG = [slope] × 2.303 × 103 k ¢ ¡ −L ²oS − ²oG = [slope] × 2.303 × 103 R,
[5] [6]
where [slope] is the slope of the log( AT 1/2 )–103 /T plot. From the adsorption data can be obtained the value of [slope] (the details of which will be given in the next section, Evaluation,) and hence that of −(²oS − ²oG ). The above method of evaluating −(²oS − ²oG ) is also, in principle, applicable to an adsorption system obeying other adsorption isotherm equations with its statistical mechanical expression being worked out. 3. EVALUATION
Adsorption data were recorded by Porter and Tompkins on chemisorption of hydrogen on evaporated iron films, covering a wide range of temperature (147 to 306 K) and pressure (10−5 to 10−1 mmHg) (11). The amounts adsorbed are given in millimoles per gram of iron, and denoted as sorption (mmol/g
1
Briefly, the parameter of lateral interaction (zw) characterizes the lateral interaction between the adsorbed molecules, the details of which have been given in Ref. (7), and hence is not repeated here. Intuitively, the larger zw is, the stronger the lateral interaction appears, and vice versa. As a limit case, zw = 0 means that there is no lateral interaction between the adsorbed molecules, which will be used in Section 4.2.
iron) in the figure and tables. The pressures P are expressed as logP(mmHg). When plotted as sorption(mmol/g iron)–logP (mmHg) isotherms, these adsorption data give straight lines, as shown in Fig. 1 in Ref. (11). The linearity of these isotherms indicates that this chemisorption obeys the Temkin isotherm equation (3). As stated in the previous section, Principle, an adsorption system obeying this isotherm equation is taken as an illustration of the principle of evaluating −(²oS − ²oG ). Therefore, an attempt is made to evaluate, from these adsorption data, the −(²oS − ²oG ) of this chemisorption as a sample evaluation. To evaluate the −(²oS − ²oG ), the above adsorption data are treated below. From the above sorption(mmol/g iron)– logP(mmHg) isotherms are taken sorptions at P = 1 mmHg and different temperatures. These sorptions are denoted by sorption(P = 1 mmHg), given in column 3 in Table 1, and, for the convenience of evaluation, are converted to θ(P = 1 mmHg), defined as the coverages at P = 1 mmHg and given in column 5, by the equation θ (P = 1 mmHg) = sorption(P = 1 mmHg)/saturation sorption.2 From the above isotherms are also obtained the slopes of isotherms at different temperatures, according to its definition, namely slope = 1 sorption/1 log P. These slopes are denoted by (slope)s , given in column 4, and, for the convenience of evaluation, are converted to (slope)θ , defined as the slopes of θ –log P(mmHg) isotherms and given in column 6, by the equation (slope)θ = (slope)s /saturation sorption. When applied to P = 1 mmHg, the Temkin isotherm equation, in the form of θ = (RT /α) ln(A P), reduces to θ (P = 1 mmHg) = (2.303RT /α) log A, where 2.303RT /α corresponds to the slope of the θ –log P(mmHg) isotherm (the graphical presentation of this isotherm equation), and hence the above (slope)θ . Substituting the above obtained θ(P = 1 mmHg) and (slope)θ into this reduced isotherm equation, we have the values of log A at different temperatures. These values are given in column 7. Now we evaluate −(²oS − ²oG ), using Eqs. [5] and [6], from the treated adsorption data given in Table 1. When log(AT 1/2 ) is plotted against 103 /T (Fig. 1), the points are shown to have a tendency to fall on a straight line with a slope of 11.6. Note that the slope of a log( AT 1/2 )–(103 /T ) plot is denoted by [slope], as stated in the previous section, Principle. Therefore, we have [slope] = 11.6. Substituting [slope] = 11.6 into Eqs. [5] and [6], we have the variation in energy of the lowest energy state of hydrogen molecules during its adsorption on iron films, ¢ ¡ − ²oS − ²oG = [slope] × 2.303 × 103 k = 26.7 × 103 k = 3.69 × 10−12 erg/molecule
¢ ¡ −L ²oS − ²oG = [slope] × 2.303 × 103 R = 26.7 × 103 R = 222.1 kJ/mol ∼ = 222 kJ/mol. 2 The saturation sorption is regarded as the Y -coordinate of the intercept of the extrapolated lines of sorption(mmol/g iron)–log P(mmHg) isotherms at different temperatures, which were plotted in Fig. 1 in Ref. (11), its value being 0.178 mmol/g iron.
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TABLE 1 Adsorption Data of Hydrogen Chemisorption on Iron Films Adsorption dataa (original) T (K)
103 /T
Sorption (P = 1 mmHg) (mmol/g iron)
147.5 177.8 195.3 217.4 250.3 273.2 306
6.78 5.62 5.12 4.60 4.00 3.66 3.27
0.167 0.165 0.162 0.159 0.155 0.150 0.148
a
Adsorption data (treated) (slope)s
× 103
θ (P = 1 mmHg)
(slope)θ × 103
log A
log (AT 1/2 )
0.938 0.927 0.910 0.893 0.871 0.843 0.832
14.4 18.0 20.1 22.6 26.0 28.5 38.2
65.1 51.5 45.3 39.5 33.5 29.6 21.8
66.2 52.6 46.5 40.7 34.7 30.8 23.0
2.57 3.20 3.57 4.03 4.63 5.07 6.80
From Ref. (11).
(It is necessary to emphasize the difference among [slope], (slope)s , and (slope)θ , so as to avoid confusion among their meanings. Their definitions are clearly presented in the above paragraphs.) 4. RESULTS AND DISCUSSION
4.1. Testing the Validity of the Evaluated Value of −(²oS − ²oG ) The validity is tested by comparing the heat of adsorption obtained from the evaluated value of −(²oS − ²oG ) with that confirmed by calorimetric measurement. Applying the Clapeyron– Clausius equation to isotherms at different temperatures (shown in Fig. 1 in Ref. (11)), Porter and Tompkins evaluated the heats of adsorption (−1H ) at different amounts adsorbed for chemisorption of hydrogen on iron films (11). The above results were presented in Fig. 3 in Ref. (11) as a −1H (kcal/mol)–amount adsorbed (mmol/g iron) plot. It was reported that the evaluated
values of −1H are confirmed by being in satisfactory agreement with those obtained by Beeck who used a calorimetric method (11). Of the values of −1H , only those in or near the middle range of coverage are, in this paper, taken as the standards for comparison, consistent with the Temkin isotherm equation being applicable in that range of coverage (9, 10). These values of −1H are given in Table 2. Besides, the heat of adsorption can be obtained from the value of −(²oS − ²oG ) as follows. Substituting Eq. [1] into the Clapeyron–Clausius equation, as q = −R[∂ ln P/∂(1/T )]θ , and assuming that J S = J G , we have the statistical mechanical expression for the heat of adsorption (q), ¡ ¢ q = (1/2)RT − L ²oS − ²oG − L[zw − T (dzw/dT ) (θ1 % θ % θ2 ),
+ 2kT 2 (da/dT )]θ
[7]
where dzw/dT is the derivative of zw with respect to T , and da/dT is the derivative of a with respect to T , both being constant for a given adsorption system at a given temperature. Physically, in Eq. [7], the terms Lzwθ and −L T (dzw/dT )θ can be regarded as contributions to the decrease in the heat of adsorption (q) by the lateral interaction between the adsorbed molecules, and the term 2RT 2 (da/dT )θ can be regarded as that by the surface heterogeneity in adsorption. It is shown in Eq. [7] TABLE 2 Comparison of Heats of Adsorption (q) with Heats of Adsorption (−∆H) Amount adsorbed
FIG. 1. log(AT 1/2 )–103 /T relationship for chemisorption of hydrogen on iron films.
Heats of adsorption, −1H a
Sorption (mmol/g iron)
Coverage θ
(Kcal/mole)
(kJ/mol)
Heats of adsorption, q (kJ/mol)
0.120 0.122 0.124 0.126 0.128 0.130
0.674 0.685 0.697 0.708 0.719 0.730
20.2 19.4 18.6 17.9 17.1 16.3
84.5 81.2 77.8 74.9 71.6 68.2
80.7 78.4 75.8 73.5 71.2 68.8
a
From Ref. (11).
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that, for a given adsorption system at constant temperature, q decreases linearly with increasing coverage, which is the form implied by the Temkin isotherm equation, thus leading to the verification of Eq. [7]. To evaluate the heat of adsorption from the value of −(²oS − G ²o ) using Eq. [7], some simplifying assumptions are made below. Assuming that dzw/dT and da/dT are small, and noting that the value of zw is 25.4 × 103 k (as evaluated in Ref. (7)), we would expect that, in the brackets in Eq. [7], dzw/dT and da/dT may numerically be neglected as compared with zw at not very high temperatures (as is usually the case for adsorption) without serious error. Based on the above assumptions, we have the simplified statistical mechanical expression for the heat of adsorption (q), ¡ ¢ q = (1/2)RT − L ²oS − ²oG − Lzwθ
(θ1 % θ % θ2 ). [8]
The verification of Eq. [8] is similar to that of Eq. [7], and hence is omitted here. Equation [8] is an approximate one, with the neglect of minute contributions to the decrease in the heat of adsorption (q) by the terms −L T (dzw/dT )θ and 2RT 2 (da/dT )θ. Due to its simplicity, yet retaining the essential feature of Eq. [7], Eq. [8] can give results of both theoretical and practical interest when employed to treat the adsorption data, even limited. Specifically, Eq. [8] will be employed not only to evaluate the heat of adsorption from the value of −(²oS − ²oG ) (as shown below), but also to examine the microscopic aspects of the heat of adsorption and to interpret the observed temperature independence of the heat of adsorption (as shown in Sections 4.2 and 4.3, respectively). Substituting T = 223.9 K,3 −(²oS − ²oG ) = 26.7 × 103 k (as evaluated in the section, Evaluation), and zw = 25.4 × 103 k (as evaluated in Ref. (7)), we have the values of the heat of adsorption (q) at different coverages (θ). (These coverages correspond to the amounts adsorbed, the heats of adsorption (−1H ) at which are taken as the standards for comparison.) The evaluated values of q at these coverages are given in Table 2. From Table 2, some results can be given as follows. In view of the probable error on the heat of adsorption obtained by Porter and Tompkins (−1H ) being 0.5–1.0 kcal/g iron (or 2.1–4.2 kJ/g iron) (11), the heats of adsorption obtained from the evaluated value of −L(²oS − ²oG ) (q) are close to the −1H confirmed by calorimetric measurement, thus confirming the q, and hence, to some extent, the evaluated value of −L(²oS − ²oG ).
3 The temperature is taken as 223.9 K, which is the average of 147.5, 177.8, 195.3, 217.4, 250.3, 273.2, and 306 K. From isotherms at these temperatures, which were plotted in Fig. 1 in Ref. (11), were obtained, by Porter and Tompkins, the heats of adsorption (−1H ), some of which are, in this paper, taken as the standards for comparison. The temperature is taken as such for the purpose of making a comparison between the heats of adsorption obtained by different methods (namely q and −1H ) under conditions of temperature as similar as possible.
4.2. Microscopic Understanding of Heat of Adsorption on a Quantitative Basis Following is an examination, with Eq. [8], of the microscopic aspects of the heat of adsorption. In Eq. [8], the first term (1/2)RT and the second term −L(²oS − ²oG ) can be regarded as contributions to the heat of adsorption (q) by the differences in translational energy and energy of the lowest energy state between the gaseous molecules and the adsorbed molecules, respectively. The third term −Lzwθ gives a linear decrease in q with increasing coverage (θ ) due to the lateral interaction actually existing in the practical adsorption system. It is shown from the above statement that, in a statistical mechanical sense, Eq. [8] leads, to some extent, to the microscopic understanding of the heat of adsorption. However, this microscopic understanding is of a qualitative nature. Therefore, an attempt is made to discuss the microscopic aspects of the heat of adsorption on a quantitative basis, using the evaluated value of −(²oS − ²oG ), the details of which are given below. On the basis of the significance of the parameter of lateral interaction zw, an adsorption systems with zw = 0 means that these adsorption systems are taken as those with no lateral interaction between the adsorbed molecules. (See footnote 1.) Substituting zw = 0 into Eq. [8], we have the statistical mechanical expression for the heat of adsorption for adsorption systems with no lateral interaction (qo ), ¡ ¢ qo = (1/2)RT − L ²oS − ²oG .
[9]
From Eq. [9], qo can theoretically be considered to be contributed by the variation in translational energy of molecules during adsorption, (1/2)RT , and that in energy of the lowest energy state of molecules during adsorption, −L(²oS − ²oG ). For chemisorption of hydrogen on iron films, the values of these contributions,4 together with that of qo , are given in Table 3. Substituting Eq. [9] into Eq. [8], we have another form of the simplified statistical mechanical expression for the heat of adsorption (q), q = qo − Lzwθ
(θ1 % θ % θ2 ).
[10]
From Eq. [10], the heat of adsorption for a practical adsorption system (obeying the Temkin isotherm equation) (q) can theoretically be considered to be the heat of adsorption for the adsorption system taken as the one with no lateral interaction (qo ) minus the decrease in the qo at a coverage of θ due to lateral interaction (Lzwθ ). For chemisorption of hydrogen on iron films, the values of Lzwθ , together with those of q, at the same coverages as those shown in Table 2, are also given in Table 3.
4 The value of temperature (T ) for evaluation of the contribution (1/2)RT is taken as 223.9 K, consistent with that for evaluation of the heats of adsorption (q) shown in Table 2.
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STATISTICAL MECHANICS OF VARIATION IN ENERGY
TABLE 3 Microscopic Understanding of Heat of Adsorption for H2 on Fe (Films) Contribution to qo by (1/2)RT
Contribution to qo by −L(²oS − ²oG )
Decrease in qo , Lzw θ
Heat of adsorption, qo (kJ/mol)
(kJ/mol)
(1/2)RT /qo
(kJ/mol)
−L(²oS − ²oG )/qo
Coverage, θ
(kJ/mol)
Lzwθ/qo
Heats of adsorption, q (kJ/mol)
223.0
0.9
0.4%
222.1
99.6%
0.674 0.685 0.697 0.708 0.719 0.730
142.3 144.7 147.2 149.5 151.8 154.2
63.8% 64.9% 66.0% 67.1% 68.1% 69.1%
80.7 78.4 75.8 73.5 71.2 68.8
From Table 3 and the above theoretical considerations, some quantitative results can be given for chemisorption of hydrogen on iron films. (1) The heat of adsorption (qo ) is invariably contributed by the variation in translational energy of molecules during adsorption (1/2)RT , and that in energy of the lowest energy state of molecules during adsorption −L(²oS − ²oG ). The relative importance of these contributions to the qo is as follows. The value of −L(²oS − ²oG ) is about 200 times larger than (1/2)RT at a temperature of 223.9 K, thus being the main contribution to the qo even at ordinary temperatures. (2) Due to lateral interaction between the adsorbed molecules, qo decreases by 63.8%–69.1% in the range of coverage θ = 0.674–0.730. The above results lead, to some extent, to the microscopic understanding of the heat of adsorption on a quantitative level. 4.3. Interpreting the Temperature Independence of the Heat of Adsorption Porter and Tompkins evaluated the heats of adsorption (−1H ) for hydrogen chemisorption on iron films, by applying the Clapeyron–Clausius equation to the isotherms at 147.5, 177.8, 195.3, 217.4, 250.3, 273.2, and 306 K, and then reported the fact that −1H is found to be independent of temperature (11). An attempt is made to interpret, mainly with the evaluated value of −L(²oS − ²oG ), this temperature independence of −1H as follows. Theoretically, for a given adsorption system (obeying the Temkin isotherm equation) at a given coverage, the increase in the heat of adsorption due to temperature increase can be evaluated, using the simplified statistical mechanical expression for the heat of adsorption (Eq. [8]), as follows, Increase in heat of adsorption = 1(1/2)RT = (1/2)R1T, [11] where 1T is the temperature increase. It is shown from Fig. 3 in Ref. (11) that the heat of adsorption for chemisorption of hydrogen on iron films obtained by Porter and Tompkins (−1H ) is dependent on the amount adsorbed. This implies that their above findings of temperature independence of −1H is made on the
condition of the given coverage. Therefore, the method of evaluating the increase in the heat of adsorption due to the temperature increase (namely Eq. [11]) can be applicable to this chemisorption under this condition. The range of temperature in which the values of −1H were obtained is 147.5 to 306 K, the temperature increase being 158.5 K. Substituting this temperature increase into Eq. [11], we have the increase in the heat of adsorption for this chemisorption due to this temperature increase, 1(1/2)RT = (1/2)R 1T = (1/2)R(158.5) = 0.659 kJ/mol, whereas the contribution to the heat of adsorption (expressed by Eq. [8]) by −L(²oS − ²oG ) is invariably 222 kJ/mol as evaluated in the section, Evaluation. The former (0.659 kJ/mol) is about 1/340 of the latter (222 kJ/mol), and hence can, in evaluation of the heat of adsorption with Eq. [8], be neglected as compared with the latter without serious error. The above comparison leads to the interpretation of the temperature independence of the heat of adsorption (−1H ), reported by Porter and Tompkins. (It is necessary to point out that the above interpretation is only valid for chemisorption without change in its mechanism, such as dissociation, in the range of temperature observed (3).) 5. SUMMARY AND CONCLUSION
The expression for evaluating the variation in energy of the lowest energy state of molecules during adsorption (−L(²oS − ²oG )) from the adsorption data was derived using the statistical mechanical expression for the constant A of the Temkin isotherm equation, namely Eq. [2]. The evaluated value of −L(²oS − ²oG ) is 26.7 × 103 R (or 222 kJ/mol) for chemisorption of hydrogen on iron films. The validity of this evaluated value was confirmed by the fact that the heats of adsorption obtained from this evaluated value are close to those confirmed by the calorimetric measurement. As one of its applications to the heat of adsorption, the evaluated value of −L(²oS − ²oG ) leads, to some extent, to the microscopic understanding of the heat of adsorption on a quantitative level, the main part of which is that, for chemisorption of hydrogen on iron films, the heat of adsorption decreases, due to the lateral interaction between the adsorbed molecules,
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by 63.8%–69.1% in the range of coverage θ = 0.674–0.730. Besides, it is shown from this evaluated value that, for this chemisorption, the contribution to the heat of adsorption (expressed by Eq. [8]) by −L(²oS − ²oG ) is numerically much larger than the theoretically evaluated increase in the heat of adsorption due to the temperature increase observed, thus suggesting the temperature independence of the heat of adsorption, as was found by Porter and Tompkins in this chemisorption. REFERENCES 1. Cerny, S., in “The Chemical Physics of Solid Surfaces and Heterogeneous Catalysis” (D. A. King and D. P. Woodroff, Eds.), Vol. 2, p. 1, Elsevier, Amsterdam, 1983.
2. Thomas, J. M., and Thomas, W. J., “Introduction to the Principles of Heterogeneous Catalysis,” Academic Press, London, 1967. 3. Hayward, D. O., and Trapnell, B. M. W., “Chemisorption,” 2nd ed., Butterworth, London, 1964. 4. Brunauer, S., “The Adsorption of Gases and Vapors,” Vol. 1, University Press, Princeton, 1945. 5. Jaroneic, M., and Madey, R., “Physical Adsorption on Heterogeneous Solids,” Elsevier, Amsterdam, 1988. 6. Kubo, R., “Statistical Mechanics,” North-Holland, Amsterdam, 1965. 7. Chung-hai Yang, J. Phys. Chem. 97, 7097 (1993). 8. Frumkin, A., and Slygin, A., Acta Physicochim. U.R.S.S. 3, 791 (1935). 9. Oscik, J., “Adsorption,” Ellis Horwood, Chichester, 1982. 10. Brunauer, S., Love, K. S., and Keenan, R. G., J. Am. Chem. Soc. 64, 751 (1942). 11. Porter, A. S., and Tompkins, F. C., Proc. R. Soc. A 217, 544 (1953).