The derivation of dynamic methods for Langmuir equation and B.E.T. equation

In this section, using kinetic methods, the adsorption equilibrium is regarded as the state when the adsorption rate and desorption rate are equal, and the Langmuir equation for molecular layer adsorption is derived（1）B.E.T. equation for adsorption with multiple molecular layers（3）.

(1)Langmuir equation

As in the second section, it is assumed that the solid surface is uniform and only adsorbs gas molecules in a single molecular layer.

The adsorption rate is obviously proportional to the pressure of the gas and also proportional to the empty surface area of the adsorbed gas molecules. Assuming the pressure of the gas is π, the percentage of empty surface area of unadsorbed gas molecules is θo, then the adsorption rate Rafor

Ra﹦aÞθo（3.62）

Where a is the proportionality coefficient.

On the other hand, the rate of desorption is inevitably proportional to the percentage θ of the surface area covered by the adsorbed gas molecules; The second is proportional to the fraction of molecules in the adsorbed gas molecules that have the energy to escape from the surface into space. Set one saThe lowest energy required for detachment from the surface, i.e. adsorption heat εaThe total number of molecules adsorbed on the surface is NaAmong them, the energy exceeds one εaThe number of molecules is Na（*）Then there is

Among them, f is the proportionality coefficient, and R is Bohr's constant. Therefore, the detachment speed Rdfor

Rd﹦a′θeεdRT （3.63）

Among them, a 'is the proportionality coefficient, θ is the percentage of specific surface area covered by various pollutants.

When reaching adsorption equilibrium, the adsorption rate should be equal to the adsorption rate, i.e. Ra=RdSo

aÞθo﹦a′θeεdRT

The percentage of empty surface area θoThe sum of the percentage of surface area covered by θ should be equal to 1, that is

θo＋ θ=1 （3.64）

Substituting the above equation yields the Langmuir monolayer adsorption equation

BH

θ﹦ （3.47）

1+ bathrooms

among which

a

b= e-6RT（3.65）

a′

From equation (3.65), it can be seen that the physical meanings of the factors in the expression of b are not as clear as in the derivation of statistical thermodynamics. Although the coefficients a and a 'can be further described after referencing molecular motion, a quantitative description of the adsorption heat ε a has not yet been made, and to achieve this, knowledge of statistical thermodynamics and quantum chemistry must be applied.

(2)）B. E.T. equation

The model used for derivation is the same as the previous third section, assuming that the solid surface is uniform and undergoes multi-layer adsorption. The adsorption starting from the second layer is regarded as condensation, so its adsorption heat is the condensation heat.

When reaching the overall adsorption equilibrium, a step-by-step equilibrium must be reached between each layer: that is, the rate at which the * layer is adsorbed on the zeroth layer (blank surface) is equal to the rate at which the * layer is adsorbed to form the zeroth layer; The adsorption rate of the i-th layer formed by adsorption on the (i-1) th layer is equal to the adsorption rate of the i-th layer formed by adsorption on the (i-1) th layer. If we assume θi（i=0，1，2，……）According to the principle of stepwise adsorption equilibrium and the method described in section (1), the percentage of the i-th adsorption layer occupying the total surface area can be determined

a1θoand a1（o）θ1eε1/RT

a2θ1and a2（o）θ2eε2/RT

……………………

aiθi-1and ai（o）θiei/RT

……………………

Among them, aiAnd ai（o）（i=1，2，……）The proportionality coefficient appearing in the equation representing the adsorption rate when the i-th layer is formed from the (i-1) layer and the adsorption rate when the i-th layer is formed from the (i-1) layer, ε1For the adsorption heat of the * layer, εi（i=2，3，……）The adsorption heat of the i-th layer. According to the assumptions of the model, there are

εi=εl（i=2，3，…） （3.67）

εlTo condense heat.

In the above equation, C, x, and y are some newly introduced symbols, and their physical meanings can be seen from the above equation. In the above equation, based on the assumption that adsorption above the second layer is condensation, it is reasonably assumed that

aia

ai′＝ a′ （i=2，3，…） （3.69）

From equation (3.68), it can be seen that

y a1a′

C= x ﹦ a1′a e(ɛi-ɛ1)/RT（3.70）

The total surface area occupied by each adsorption layer should be equal to the total surface area, so

n n

1=∑θi=θ0（1+C∑xi） （3.71）

I=0 i=1

Here n is the number of adsorption layers.

Now let's calculate the total adsorption capacity V. If VmIf the saturation and adsorption capacity of a single molecular layer are reached, the adsorption capacity of an i-layer adsorption layer is Vm（iθi）Therefore, the total adsorption capacity is

n n

V=Vm∑iθi=VmCθ0∑ixi

i=0 i=1