Better Refined Adsorption Isotherm than BET Equation

During studying the heat capacity of metals and brightening more than the original Lena’s image, we realize that the temperature increasing term obtained binomial expansion is used in the adsorption increasing term ( 1 − − n a n n N g N N ) and we could derive the total adsorption isotherm equation with it. In the first layer the quantization does not occur and from 2 nd to nth layer the quantization occurs. So as to get the total adsorption isotherm equation we add the quantized terms of the second to nth layers to the non-quantized term of the first layer. All terms are based on the unit surface sites. Hence the adsorption equations are derived much better than BET equation. The surface area is calculated through the integration of the adsorption isotherm equation from the inflection point to the wanted relative pressure.


Introduction
We derived the heat capacity equations of metals and then used consistent step multiplication of the appropriate binomial equations [1]. They are fitted to experimental data well [2]. The heat capacity equation (type V) and the adsorption equation (type II) draw sigmoid (S character) lines all together. And they are symmetry each other. The measurement gas of heat capacity is hydrogen or helium. The adsorption gases are water or nitrogen. The movements of their measurement gases are different. The former is expansion and the later contraction.
The most important term in the derivation of heat capacity equation was the temperature increasing term, [3], the more advanced adsorption equation than BET eq. comes out.

Statistical Modeling of Adsorption Isotherm
Suppose each level has one binomial equation for excitation and non-excitation. And (1 ) Let us multiply (1) and (2) side by side. Then for In the above In Eq. (3) the largest term dominates the equation. So the total differential of Eq. (3) becomes the zero which requires that the coefficients of all terms should be zero. Hence by using Stirling's approximation we solve the equation, at W . The first differential equation of Eq (3) becomes 1 1 2 In the above equations m is quantization constant. In the reference [4] m value should be corrected as those in the present figure [ Fig. 1]. The constants have three parts of As we see in Fig It is possible that a g is put as unit.
The next equations are In Eqs (4) and (5) In Eq. (6) add side by side and rearrange In Eq. (6) multiple side by side and rearrange We solve Eq. (8) with We solve Eq. (7) with Eq. (9) Therefore the total adsorption amount per unit surface ( m g ), that is, the adsorption isotherm for from first layer to the last (n) layer becomes by using Eqs (6) and (10) If m g is eliminated in Eq (12) N becomes as follows

Result and Discussion
The base of Eq. (10) and Eq. (12) is + a a β . So we may use the word, the rate without the dimensionless analysis. It affects the equation totally. Fig. 2 shows the total adsorption rate according to the values of a β . In accordance with the values of z approaching units, the total adsorption rates approach closely with one another. This seems to mean that the adsorption heat of the first layer is same as those of 2~n layers. We call m the quantization values. It seems to have same notion as the quantization appears in quantum mechanism. We are dealing statistical quantization which should exist in statistics. We can differentiate them, three cases.  Fig. 7 and Fig. 8. According to the above variations we optimized two kinds of the experimental adsorption data showed in Figs. 7 [6] and 8 [7] using trial and error method. The experimental data of Fig. 7 are obtained from the Figure 2-10 of the reference [6]. The experimental data are fitted to Eq. (12) well. As we see in Figs 7 and 8, BET isotherm can't imitate the experimental data except for beginning. What is the catalyst? As we see Fig. 9 in the above, 1, 2, 3, 4, 5, 6 and 7 molecules can function as the catalyst. That is, the molecules of the surface adsorption layer can't function like the catalyst. Because they use much energy in order to hold the surface. So they are not active. The molecules which lie on the surface layer adsorption molecules which hold the surface, can function as the real catalyst. Therefore in Eq. (12), Then the inflection points are obtained by Secant method [8] through the program showed in Appendix 1. Specific surface areas are changed according to the optional relative pressures. These are showed in Tables 1 and 2 precisely. The integrations with respect to z surface values give the total adsorption site numbers of the adsorbate. Before the inflection point the specific surface area of the adsorbent is not counted as a catalyst since it makes the strong surface film [9]. The adsorption rate increases consistently after the inflection point. The values of n of Fig. 7 and Fig. 8 match the range of the reference of BET [10]. The completion of 1 N to unit goes with the completion of N to the end as we see in . Its quality is poor. Our study have realized the saying that "After considerable work on the theory, Hill (1946) formed the opinion that any future improvement on it must be in the form of refinement rather than a modification on the basic theory" [11,12].