Optimization of Parameters in Semiempirical Adsorption Isotherms P. GAJARDO AND J. CORTI~S Departamento de Qu$mica, Facultad de Cienclas F$sicas y Matemdticas, Universldad de Chile, Santiago, Chile Received June 10, 1974; accepted July 18, 1974 A rigorous approach to the selection of analytical adsorption isotherms that describe better the experimental results is proposed. By using a method for the optimization of parameters, the influence of parameters K and C in Anderson's adsorption isotherm are analyzed critically. INTRODUCTION
Very valuable experimental information on the study of multimolecular adsorption is found in the literature under the so called t curves or n curves, which result when the thickness of the adsorbed layer or the number of layers respectively is plotted against the relative equilibrium pressure, provided the solid is not porous and we are dealing with a specific temperature and adsorbate. The analytical interpretation of these curves has only been possible by means of empirical or semiempirical equations, despite the attempts to achieve it by theoretical interpretations using different approaches. However, as J. H. de Boer has pointed out, these equations provide many useful data regarding the texture of microporous substances such as adsorbents, catalysts or catalyst carriers (1). Perhaps the most theoretically sound of these equations is that of Frenkel-Halsey-Hill (FHH) (2). However, the most widely used ones have been those of the two-parameter Harkins-Jura (HJ) type (3), and Anderson's modification of the Brunauer-Emmett-Teller (BET) equation (4), which gives rise to a three-parameter equation that is applicable to nonporous solids. The latter equation has been recently analyzed theoretically by Brunauer (5).
In general a semiempirical equation of this kind, describing the adsorption of a gas on a solid, can be expressed as follows: ? = f(Xl X~,...,
X~, K1, K2,. . ., Ks), Eli
where the dependent variable !7 is a function of m independent variables X i and 1parameters Ki. In the particular case of curve n, the statistical number of layers appears usually as the dependent variable, and the relative pressure as the independent variable. The problem is then reduced to finding values for the parameters such that the equation accounts in the best way for the experimental results in a given interval of the variables. We can thus define a function N
= E (Yii=1
where N is the number of experimental points, 17i is the value of the analytic function (Eq. Eli) for point i, and Yi is the corresponding experimental value. We consider this function ¢ as a good criterion for determining which analytic isotherm describes better an experimental isotherm. This is a consequence of the principle of maximum similarity if it is admitted that the experimental points are distributed normally. In this paper we show, by using the modified 331
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Journal of Colloid and Interface Science, Vol. 50, No. 2, February 1975
GAJARDO AND CORTt~S
steepest-descent optimization method (6), the importance and the need of using rigorous methods, such as the one shown, for calculating the optimum parameters in some of the semiempirical equations for multimolecular adsorption. The modified steepest-descent method allows one to find the minimum value of function ¢. This is achieved by correcting the initial value given to Ki by means of the relation K~ (r+l) = Ki (r) + A K i (r),
where r is the iteration number. The correction of the Ki parameter is given by A K i (r) = a(r)Gi (~),
Gi(~) = El + (Ki(~))2~( O~K~)e).
The quantity c~(~) is the step size that determines the absolute magnitudes of the parameter corrections that must be adjusted from trial to trial. The value it will take in the rth iteration will depend on the magnitude and direction of the previous iteration's gradient. Following this we have applied the criterion (6) according to which a(r) = 0.25a (~-1) if
cos q~ < 0
and a (~")
Ol(r-1)(D1 + D2 cos4 ~)
cos ~ ) 0,
cos ~ = X
assuming that D1 = 0.5 and D2 = 1.0. OPTIMIZATIONOF THE K AND C PARAMETERS OF ANDERSON'S EQUATION Anderson's equation is expressed as (4)
CKX (1 - K x ) D
+ ( c - 1)KX.]
where V is the STP volume of gas adsorbed at the relative equilibrium pressure X = P/Po, Vm is the volume of a monolayer, C is the B E T constant, and K is a third parameter introduced by Anderson. We think that the application of this equation for determining the surface areas of nonporous solids, where there is no capillary condensation, can be useful in some systems because the range of relative pressures within which it describes satisfactorily the experimental results is greater than that of the B E T equation. This is particulary important for experimental reasons in solids with low surface areas, where adsorbates with low vapor pressures at the working temperature must be used. As can be seen, Anderson's equation has three parameters (Vm, C,K). However, these can be reduced to two if V,, is included in the dependent variable as n = VIVa,,. Brunauer applied Eq [-9.] in this way to Schull's experimental curve for nitrogen at 78°K (7, 8), assuming a value of 100 for C, and determining the value of K for the most reliable experimental point corresponding to the highest value of P/Po. According to Brunauer this is justified because of the negligible influence of C on the upper part of the curve when its value is of the order of 100, a usual figure for nitrogen as an absorbate, and because of his theoretical interpretation of the constant K, which would measure the strength of the attractive force field of the adsorbent. Parameter K determined in this way had a value of 0.79 Figures 1 and 2 show the results obtained when the modified steepest descent optimization method was applied to Schull's n curve. The experimental data are those reported by Pierce (8). We have designated as optimum parameters those corresponding to the minimum of the ¢ function (Eq. [-2.]) and are the result of the application of the method. In Fig. 1 function q~ is plotted against parameter C for a given value of K. The minimum observed for each curve is the optimum value of C for the corresponding value of K. Figure 2
Journal of Colloid and Interface Science, VoL 50, No. 2, F e b r u a r y 1975
SEMIEMPIRICAL ADSORPTION ISOTHERMS
334 171 K= O.787
FIG. 1. Plot of function ~ against C for different values of K.
FIG. 2. Plot of C optimum against K. J o u r n a l o f Colloid a n d I ~ t e r f a c e S c i e n c e , Vol. 50, No. 2, February 1975
GAJARDO AND CORTI~S TABLE i
VALUES CORRESPONDING TO OPTIMUM PARAMETERS K
0.780 0.786 0.787 0.788 0.789 0.790 0.80 0.81 0.82 0.83 0.84 0.85 0.87 0.89 0.91 0.93 0.95 0.97 0.99 1.00
1100 198 173 153 137 124 60 40 25 20 15 8.5 5.0 2.8 1.6 0.92 0.52 0.31 0.18 0.14
0.0330 0.0170 0.01673 0.01677 0.0174 0.0187 0.068 0.191 0.398 0.702 1.116 1.64 3.05 5.06 7.67 11.38 15.10 19.23 23.85 26.41
shows the curve that results when' these optimum values of C are plotted against K. Table I shows the values of q~ for each point of the curve in Fig. 2. I t is evident that the best parameters are those that give the lowest values for q~; in the case shown this occurs when C = 173 and K -- 0.787. I n the case of the H J equation, which is log P / P 0 = B -- A
decimal of K gives rise to large variations in C. For example, when K = 0.786, C = 198, and when K = 0.789, C = 137. However, in the literature only two decimals are reported for the value of K (1, 4, 5, 9). Thus it is not strange that both Anderson (4) and Brunauer (5) find that the values of C obtained from Eq [-91 "show no definite trend compared to B E T C values." This is a very definite demonstration of the need to determine the parameters rigorously, and any discussion of Eq. [-9], of the meaning of the parameters, or of a classification of adsorption systems based on parameter C, should consider an adequate method of determination. I n Table I it is seen that for K = 1.0, C optimum is 0.14 and ~ has its highest value. In order to show that the B E T equation is obeyed better, as is known, in an interval of P/Po smaller than about 0.35, Table I I was constructed, showing the values of C optimum calculated by the modified steepest descent optimization method, with K = 1.0. Each C optimum was obtained using the experimental points of Schull's n curve in the interval of P/Po between 0.2 and the relative pressure corresponding to each C optimum in Table II. The value of ¢ increases with increasing
the optimum values were 2.020 for parameter A and 0.061 for B. Function ~ then takes a value of 0.00949, showing that this equation describes Schull's experimental points better than Anderson's for the figures given in Table I. For Anderson's equation, one of the most interesting conclusions derived from the application of a rigorous method for determining the parameters, and one that is not mentioned in the literature, is the remarkable influence of the values of K on the corresponding optimum values of C. This can be observed in the very steep slope of the curve in Fig. 2 for a restricted interval of the abcissa K. Also, in Table I an increment in the third Journal of Colloid and Interface Science, Vol. 50, No. 2, February 1975
TABLE II C OPTIMUMVALUESOF THE BET EQUA'I'ION:FOR DIFFERENTVALUESOt"P/-Po. P/Po
0.30 O.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.825 0.85 0.875 0.90
34 29 24 20 16 13 10 7 5 3 2 0.75 0.40 0.24 0.14
0.002 O.0O9 0.02 0.05 0.10 0.20 0.50 0.90 1.8 3.5 7.0 10.71 14.79 19.56 26.40
SEMIEMPIRICAL ADSORPTION ISOTHERMS
P/Po, and therefore with increasing interval, but on the other hand C decreases. If results were available for the low pressure region, C of B E T would be obtained. The little influence of parameter C on the upper part of the n curve allowed Brunauer (5) to fit satisfactory Schull's experimental points for K = 0.79 and C = 100, even though these are not optimum values. The lack of precision with which K and C are commonly calculated can, however, lead to errors in the calculation of other constants of physicochemical interest that are derived from parameter C. To illustrate this statement we shall analyze two cases (a) The expression that determines the relative pressure at which the monolayer is completed (point B) is given by
(P/Po)m- K ( C -
With the values given by Brunauer (K = 0.79 and C = 100) we obtain (P/Po)m = 0.115. With the optimum parameters we have found (K = 0.787 and C = 173), (P/Po)~ = 0.0966 is obtained. (b) According to Brunauer (5), constant C of Eq. [-9] is
C = R'exp[-(Ex--E')/RT],
E 1 is the heat of adsorption of the first adsorbed layer, and E' that of each one of the upper layers, which is different from the heat of liquefaction, EL. Thus, with the questionable assumption that R' = 1, it is possible to calculate the net heat of adsorption (E1--E') from C. If the value C = 100 (which was used by Brunauer for the universal isotherm of nitrogen) is used, /~t - - E / = 707 cal/mole is obtained. With the value C = 173 that we determined from Schull's data, E 1 - E ~ = 791 cal/mole is obtained. For C optimum the value of E ~ - E' is larger than that obtained with Brunauer's value of C. I t is interesting to note that the net heat of adsorption obtained with C optimum is the same, within the margin of variation, as that given by Brunauer (10), which is 840 =t= 70 cal/mole
for the adsorption of nitrogen of twelve solids at 78°K. I t is well known that adsorption as a phenomenon can be described by an equation of the form f @ , ~ , T) = 0,
where ~ is the area per molecule, 7r is the spreading pressure, and T is the temperature. This is the equation of state of the adsorbed phase. Each function like (13) has an equation of state of the form F~ (P, V, T) = 0
corresponding to the gas phase. Expressions  and  can be obtained reciprocally using Gibbs' equation. At constant temperature function F2 represents an adsorption isotherm; among the best known are those of Brunauer, E m m e t t and Teller (11) and Harkins and Jura (3) because of their usefulness in determining the surface areas of solids. The H J isotherm assumes, at constant temperature, a linear function between and 7r for the adsorbed phase. According to Harkins' terminology, the adsorbed film is condensable in the interval where it occurs. On the other hand, Livingston (12) proved expressions  and V16], provided the B E T isotherm is followed:
K17r = log(Po--P q- C P ) / ( P o - P ) ,
K2~ = ( P 0 - P )
X (Po -- P + CP)/CPPo,
where constants K1 and K2 are independent of pressure. From Eqs. V15] and [16-1 one can obtain the equation of state of the absorbed phase corresponding to the B E T isotherm for each value of constant C. If P and P0 are given values, it is possible to plot K l v against K2~ for each P/Po, and this is shown in Fig. 3 for C = 100. I t is assumed, following Livingston, that ~r and ¢ are known with an accuracy of =L3%, so that each point is represented by a cross whose horizontal and vertical dimensions correspond to 6% of K2¢ and Kl~r, respectively. The relative pressures
Journal of Colloid and Interface Science, Vol. 50, No. 2, F e b r u a r y 1975
GAJARDO AND CORTI~S
0.724 0.671 1.9
0,342 O. 289
O.237 0. 197 0371 0.145
O. 237 0.197 0.171 0.145 1.I
PIYo Anderson K=0.79 i
K20FIe. 3. Pressure-area plot calculated from the Brunauer-Emmett-Teller isotherm and Anderson's isotherm (K = 0.79) with C = 100.
corresponding to each point are shown across them in the column at the right of the plot in Fig. 3. The interval of these pressures within which the equation represents a condensable film is that where the points lie on a straight line. As shown by Livingston, Journal of Colloid and Interface Science, Vol.
for C = 100 this interval lies between 0.04 and 0.72, and it represents the range in which the H J and the B E T equations can be used interchangeably to determine the surface areas of solids. In a similar way, if Anderson's equation is followed, the following expressions
50, No. 2. February 1975
SEMIEMPIRICAL ADSORPTION ISOTHERMS
surface area of solids is larger than that obtained with the B E T equation.
are readily obtained: Kl~r = l o g ( P o - - K P + C K P ) / + (Po -- K P ) ,
X (Po -- K P - 5 K C P ) / ( C K P P o ) .
K2o- = (Po -- K P )
In Fig. 3, the points denoted by circles, whose relative pressures are given on the left, were calculated for K = 0.79 and C = 100. It is seen that almost all the points calculated for the interval 0.04-1.0 of P / P o fall on a straight line, with a shortening of the interval of spreading pressures as compared with the case of the B E T equation. We had previously said that parameter K introduces a correction in the B E T equation which allows an improved fit of the experimental points in the analytic isotherm. I t is concluded from Fig. 3 that Anderson's equation assumes a condensable phase in an interval of relative pressures larger than that of the B E T equation. From this, which is specially valid for high pressures, it can be stated that the interval of relative pressures where the H J and Anderson's equations can be used interchangeably to determine the
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Journal of Colloid and Interface Science. Vol. 50, No. 2, February 1975