The Jantti Approach In Case Of A Limited Adsorption With A Single Layer
The Jantti approach in case of a limited adsorption with a single layer
J.A. Poulis1, C.H. Massen1, E. Robens2
1Faculty of Technical Physics, Technical University Eindhoven,
Postbus 513, NL - 5600 MB Eindhoven, The Netherlands
2Institute for Inorganic Chemistry and Analytical Chemistry, Johannes Gutenberg University, Duisbergweg 10-14, D – 55099 Mainz, Germany
Abstract
Jantti introduced a method to calculate the adsorption equilibrium by measurement of actual adsorbed amount at three times after a change of the gas pressure. He applied that method for gas/solid systems in which simple adsorption processes occur and for an infinite number of adsorption sites. In the present paper we discuss the case that no adsorption is possible on a place which is already occupied (Langmuir isotherm) while the number of positions available for adsorption is considered to be limited. For the study of this limitation effect the advantage of the use of a gas pressure increasing linearly with time are evident. We discuss the use of Jantti\’s method for such measurements.
1. INTRODUCTION
In the history of the research on adsorption equation (1) plays an important role
(1)
where m (t) is the adsorbed amount observed as a function of time, mi is the asymptotic equilibrium value and x is a characteristic constant for the special gas/solid system.
Already in 1970 Jantti [1,2] suggested measuring of three points of the initial course of the kinetic adsorption curve and extrapolating the equilibrium value. When the specific molecular model of the adsorption of a gas on a solid surface is known and when it can be expected that only one kind of adsorption is at stake, this method delivers good results and allows a very fast stepwise
measurement of adsorption isotherms [3]. To get a quick estimate of the values of the parameters from measured values of m, Jantti introduced the function J: (2)
With Jantti\’s method the adsorbed mass is measured at times t1, t2 and t3 (where ) yielding the values m1, m2 and m3, respectively. Jantti used his method for adsorptions satisfying eq. (1). In the case of sorptions characterised by eq. (1), J (t) is independent of the values of both t and D t. In that case
(3)
It is useful to consider the limit J* of J for Dt approaching zero:
(4)
2. CASE OF A SINGLE LAYER
In former papers [4,5,6] we discussed different adsorption mechanisms. We introduced extra parameters and adaptations to eq. (1). We discussed for each case how the Jantti approach could be used .We restricted our calculations to a small coverage. In this paper we consider the effect of a limited number of the places available for adsorption. We restrict ourselves to the situations where no adsorption is possible on a place, which is already occupied.
The number of places available for adsorption we shall call m0. When working with a step in the gas pressure, the adsorption equation reads:
(5)
To apply the Jäntti procedure we use this equation in the form:
(6)
To study the influence of saturation many of these steps have to be applied and this is a time consuming procedure. The use of a gas pressure increasing linearly with time seems therefore obvious. In that case the adsorption equation reads:
(7)
To solve eq. (7) for small values of t we use:
(8)
Using eq. (8) into eq. (9) and considering the terms with equal powers of t we get:
(9)
(10)
(11)
So for small values of t we get
(12)
Using eq. (4) into eq. (12) we get again for small values of t:
(13)
For large values of t we use:
(14)
For convenience we introduce:
(15)
Using eq. (15) into eq. (7) and considering the terms with equal power of 1/t,we get:
(16)
(17)
(18)
For large values of t we get:
(19)
Using eq. (19) into eq. (7) we get, again for large values of t:
(20)
3. COMPUTER SIMULATION
In Fig. 1 a numerical example is shown as the result of a computer simulation. Using a chosen set of values of the parameters involved a numerical solution of eq. (7) has been calculated. The result is compared with the values calculated with the approximating eqs. (12), (13), (19), and (20). The striking feature of the J curve is the vertical asymptote which is characteristic for the saturation effect which reduces the value of the second derivative of the J curve. The use of the J curve is reduced by the fact that the inaccuracies of the measurement are magnified considerably by the use of Jantti\’s procedure.
Fig. 1. Computer simulation using m0 = 100, C = 1, t = 2. m1 calculated by solving eq. (7), m2 calculated with eq. (12), J1 calculated with m1 and eq. (4), J2 calculated with eq. (13), m3 calculated with eq. (19), J3 calculated with eq. (20).
References
1. O. Jantti, J. Junttila, E. Yrjanheikki: Mikropunnitusajan Lyhentamisesta Ekstrapolaatiomenetelmalla. (On curtailing the microweighing time by an extrapolation method). Suomen Kemistilehti A 43 (1970) 214-218.
2. O. Jantti, J. Junttila, E. Yrjanheikki: On curtailing the micro-weighing time by an extrapolation method. In: T. Gast, E. Robens (eds.): Progress in Vacuum Microbalance Techniques, Vol. 1. Heyden, London 1972, p. 345-353.
3. E. Robens, C.H. Massen, J.A. Poulis: On curtailing the time for gas adsorption measurements by extrapolation. IX. POROTEC-Workshop über die Charakterisierung von feinteiligen und porösen Festkörpern, 11.-12. 11 1998 und X Workshop, 15.-16.11.2000, Bad Soden (Taunus), Germany.
4. C.H. Massen, J.A. Poulis, E. Robens: Criticism on Jäntti\’s three point method on curtailing gas adsorption measurements. Adsorption 6 (2000) 229-232.
5. E. Robens, C.H. Massen, J.A. Poulis, P. Staszczuk: Fast measurements of adsorption on porous materials using Jantti\’s method. Adsorption Sci. and Tech. 17 (1999) 10, 801-804.
6. J.A. Poulis, C.H. Massen, E. Robens, K.K. Unger: A fast two-point method for gas adsorption measurements. In K.K. Unger, G. Kreysa, J.P. Baselt (eds.): Characterisation of Porous Solids V. Studies in Surface Science and Catalysis, Elsevier, Amsterdam 2000, p. 151-154.
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