1. 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, . In case of adsorption it becomes of the same form. Here and are the total molecules at the layers of and . Let us put which is unit in data fitting as the constant. It is a testing constant. If is transformed into power of in every derivation of adsorption such as [3], the more advanced adsorption equation than BET eq. comes out.

2. Statistical Modeling of Adsorption Isotherm

Suppose each level has one binomial equation for excitation and non-excitation. And suppose molecules are excited on localized sites of the unit surface layer and molecules on to sites of the layers in sequence. Here m means quantization. The first adsorption layer has adsorption energy and. the second to layer . Hence the adsorption probability over the first layer is and the non-adsorption probability . In the above is the adsorption constant of the first adsorption layer. is similar to in heat capacity equation. The adsorption constant, is calculable quantity. It is the expression to combine the rotation and vibration energy of the adsorbed molecules in the first layer. [2] is similar to (Boltzmann’s constant). In the combination calculation of the first layer can take the values of from 0 to as variables in sequence. Then the binomial equation of the first layer becomes [1]

(1)

Next the adsorption probability of to layers is and then the non-adsorption probability . The constants of these are the same as those of layer in the above. The binomial expansion equations for to layers are

(2)

Let us multiply (1) and (2) side by side. Then for

(3)

In the above

(3.1)

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, . The first differential equation of Eq (3) becomes

(4)

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. 1, bonding constant mean quantization. If bonding occurs in many directions. If no quantization occur and if the adsorptions are interrupted in many directions. In Eq. (4)

(4.1)

(4.2)

It is possible that is put as unit.

The next equations are

(5)

In Eqs (4) and (5) let us put then which is same as Eq. (9) of the reference [5] solved by using the chemical potential

(6)

In Eq. (6) add side by side and rearrange

(7)

In Eq. (6) multiple side by side and rearrange

(8)

We solve Eq. (8) with in order to eliminate

(9)

We solve Eq. (7) with Eq. (9) in order to eliminate

(10)

Eq. (10) represents the adsorption amount of the first layer. It is Langmuir equation. The quantization values (m) and the numbers of the adsorption layer (n) are influential on the determination of . All values of the parameters directly participate in the determination of and following . The value of is much influential on the determination of the last term (). But it does not much increase or decrease of since it exists in the denominator and nominator. At it becomes

(11)

Therefore the total adsorption amount per unit surface that is, the adsorption isotherm for from first layer to the last (n) layer becomes by using Eqs (6) and (10) as follows

(12)

If is eliminated in Eq (12) becomes as follows

(12.1)

If is a linear function of four variables.

In the above

(12.2)

And the total adsorption rate is a linear function of z made of four constants The equation draws BET-like lines and fits BET type experimental data well. At it becomes

(13)

In the above

(13.1)

If the measurement gas is nitrogen, the general empirical formulae, is introduced into. to get the specific surface area of the adsorbent, [6]. Then is the molecular weight of nitrogen and density of nitrogen and Avogadro number, Hence from empirical formulae is used. The monolayer capacity, should also be substituted by the value of the integration of Eq. (12) subtracted by the surface adsorption isotherm

3. Result and Discussion

The base of Eq. (10) and Eq. (12) is 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 . 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 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. and The cases of explain bonding and explains existence of the interruptions for bonding. Fig 3 explains the increase of the adsorption rate according to m values. Less m than unit seem to interrupt the bonding. Fig. 4 represents the total adsorption rates according to n values. At less z than 0.5 the isotherms show the same adsorption rates. The constant showed in Fig 5 should be positive. When the constants draw the type II isotherm. At the same types of the isotherm appear. can include greater values than or equal unit. seems to be best. Fig. 6 shows with respect to the relative pressure like Langmuir’s lines which can’t become unit even if a is very large or much smaller. The parameter values of Fig. 2 come from the following 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), part must work in the catalytic reaction. Hence the real adsorption equation becomes

(14)

If this equation is integrated from the inflection point to each optional relative pressure we get the mono-area capacity with respect to the relative pressure (p/p₀). Tables 1 and 2 include those. Then the surface area for the catalyst obtained from the equation becomes with per number for nitrogen and per number for water (Table 2) [6] by substituting . in the equation of (2.60) of the reference [6] by in which =inflection point and =optional z value after the inflection point. In the above equation is the molecular weight (g) per g-mole and Avogadro number per g-mole.

Table 1. Sm (mono-area capacity) and S (surface area) for Fig. 7

Table 2. Sm (mono-area capacity) and S (surface area) for Fig. 8

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 to unit goes with the completion of N to the end as we see in Fig. 6 and its equation. But from the inflection point the pilling up of may begin characteristically.

Eq. (13) can be used in the data fitting with and . 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].

4. Conclusions

The total adsorption rate equations closely related with the past references are derived correctly and the figures according to four constants are also considered to describe BET-like figures (II) well. The quantization constants are useful in deriving the adsorption equations. In order to calculate the surface area of the catalysts the total adsorption equations excluding the first layer adsorption equation and their inflection points obtained are used appropriately. They fit the experimental data well.

ACKNOWLEDGEMENTS

The author thanks for the challenging encouragements of Yongduk Kim, an emeritus professor of Sogang University.

And the author thanks for helping the manuscript arrangement of Jones Cathy (http://www.sapub.org/), USA.

Appendix 1