kb/data/en.wikipedia.org/wiki/BET_theory-1.md

---
title: "BET theory"
chunk: 2/4
source: "https://en.wikipedia.org/wiki/BET_theory"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T10:03:51.162056+00:00"
instance: "kb-cron"
---

Adsorptions occur only on well-defined sites of the sample surface (one per molecule)
The only molecular interaction considered is the following one: a molecule can act as a single adsorption site for a molecule of the upper layer.
The uppermost molecule layer is in equilibrium with the gas phase, i.e. similar molecule adsorption and desorption rates.
The desorption is a kinetically limited process, i.e. a heat of adsorption must be provided:
these phenomena are homogeneous, i.e. same heat of adsorption for a given molecule layer.
it is E1 for the first layer, i.e. the heat of adsorption at the solid sample surface
the other layers are assumed similar and can be represented as condensed species, i.e. liquid state. Hence, the heat of adsorption is EL is equal to the heat of liquefaction.
At the saturation pressure, the molecule layer number tends to infinity (i.e. equivalent to the sample being surrounded by a liquid phase)
Consider a given amount of solid sample in a controlled atmosphere. Let θi be the fractional coverage of the sample surface covered by a number i of successive molecule layers. Let us assume that the adsorption rate Rads,i-1 for molecules on a layer (i-1) (i.e. formation of a layer i) is proportional to both its fractional surface θi-1 and to the pressure P, and that the desorption rate Rdes,i on a layer i is also proportional to its fractional surface θi:

  
    
      
        
          R
          
            
              a
              d
              s
            
            ,
            i
            −
            1
          
        
        =
        
          k
          
            i
          
        
        P
        
          Θ
          
            i
            −
            1
          
        
      
    
    {\displaystyle R_{\mathrm {ads} ,i-1}=k_{i}P\Theta _{i-1}}
  

  
    
      
        
          R
          
            
              d
              e
              s
            
            ,
            i
          
        
        =
        
          k
          
            −
            i
          
        
        
          Θ
          
            i
          
        
        ,
      
    
    {\displaystyle R_{\mathrm {des} ,i}=k_{-i}\Theta _{i},}
  

where ki and k−i are the kinetic constants (depending on the temperature) for the adsorption on the layer (i−1) and desorption on layer i, respectively. For the adsorptions, these constants are assumed similar whatever the surface.
Assuming an Arrhenius law for desorption, the related constants can be expressed as

  
    
      
        
          k
          
            i
          
        
        =
        exp
        ⁡
        (
        −
        
          E
          
            i
          
        
        
          /
        
        R
        T
        )
        ,
      
    
    {\displaystyle k_{i}=\exp(-E_{i}/RT),}
  

where Ei is the heat of adsorption, equal to E1 at the sample surface and to EL otherwise.
Consider some substance A. The adsorption of A onto an available surface site 
  
    
      
        (
        ∗
        
          )
          
            0
          
        
      
    
    {\displaystyle (*)_{0}}
  
 produces a new site 
  
    
      
        (
        ∗
        
          )
          
            1
          
        
      
    
    {\displaystyle (*)_{1}}
  
 on the first layer. In summary,

  
    
      
        
          A
          
            (
            g
            )
          
          +
        
      
    
    {\displaystyle {\ce {A(g) +}}}
  
 
  
    
      
        (
        ∗
        
          )
          
            0
          
        
      
    
    {\displaystyle (*)_{0}}
  
 
  
    
      
        
          
            
              
                
                  
                    
                      ↽
                    
                    
                    
                    
                      −
                    
                  
                
              
              
                
                  
                    
                      −
                    
                    
                    
                    
                      ⇀
                    
                  
                
              
            
          
        
      
    
    {\displaystyle {\ce {<=>}}}
  
 
  
    
      
        (
        ∗
        
          )
          
            1
          
        
      
    
    {\displaystyle (*)_{1}}
  

Extending this to higher order layers one obtains

  
    
      
        
          A
          
            (
            g
            )
          
          +
        
      
    
    {\displaystyle {\ce {A(g) +}}}
  
 
  
    
      
        (
        ∗
        
          )
          
            1
          
        
      
    
    {\displaystyle (*)_{1}}
  
 
  
    
      
        
          
            
              
                
                  
                    
                      ↽
                    
                    
                    
                    
                      −
                    
                  
                
              
              
                
                  
                    
                      −
                    
                    
                    
                    
                      ⇀
                    
                  
                
              
            
          
        
      
    
    {\displaystyle {\ce {<=>}}}
  
 
  
    
      
        (
        ∗
        
          )
          
            2
          
        
      
    
    {\displaystyle (*)_{2}}
  

and similarly

  
    
      
        
          A
          
            (
            g
            )
          
          +
        
      
    
    {\displaystyle {\ce {A(g) +}}}
  
 
  
    
      
        (
        ∗
        )
      
    
    {\displaystyle (*)}
  

  
    
      
        n
        −
        1
      
    
    {\displaystyle n-1}
  
 
  
    
      
        
          
            
              
                
                  
                    
                      ↽
                    
                    
                    
                    
                      −
                    
                  
                
              
              
                
                  
                    
                      −
                    
                    
                    
                    
                      ⇀
                    
                  
                
              
            
          
        
      
    
    {\displaystyle {\ce {<=>}}}
  
 
  
    
      
        (
        ∗
        
          )
          
            n
          
        
      
    
    {\displaystyle (*)_{n}}
  

Denoting the activity of the number of available sites of the 
  
    
      
        n
      
    
    {\displaystyle n}
  
th layer with  
  
    
      
        
          θ
          
            n
          
        
      
    
    {\displaystyle \theta _{n}}
  
 and the partial pressure of A with 
  
    
      
        P
      
    
    {\displaystyle P}
  
, the last equilibrium can be written

  
    
      
        
          K
          
            n
          
        
        =
        
          
            
              θ
              
                n
              
            
            
              P
              
                θ
                
                  n
                  −
                  1
                
              
            
          
        
      
    
    {\displaystyle K_{n}={\frac {\theta _{n}}{P\theta _{n-1}}}}
  

It follows that the coverage of the first layer can be written

  
    
      
        
          θ
          
            1
          
        
        =
        
          K
          
            1
          
        
        P
        
          θ
          
            0
          
        
      
    
    {\displaystyle \theta _{1}=K_{1}P\theta _{0}}
  

and that the coverage of the second layer can be written

  
    
      
        
          θ
          
            2
          
        
        =
        
          K
          
            2
          
        
        P
        
          θ
          
            1
          
        
        =
        
          K
          
            2
          
        
        P
        
          K
          
            1
          
        
        P
        
          θ
          
            0
          
        
      
    
    {\displaystyle \theta _{2}=K_{2}P\theta _{1}=K_{2}PK_{1}P\theta _{0}}
  

Realising that the adsorption of A onto the second layer is equivalent to adsorption of A onto its own liquid phase, the rate constant for 
  
    
      
        n
        >
        1
      
    
    {\displaystyle n>1}
  
 should be the same, which results in the recursion

  
    
      
        
          θ
          
            n
          
        
        =
        (
        
          K
          
            ℓ
          
        
        P
        
          )
          
            n
            −
            1
          
        
        P
        
          K
          
            1
          
        
        
          θ
          
            0
          
        
      
    
    {\displaystyle \theta _{n}=(K_{\ell }P)^{n-1}PK_{1}\theta _{0}}
  

In order to simplify some infinite summations, let 
  
    
      
        x
        =
        
          K
          
            ℓ
          
        
        P
      
    
    {\displaystyle x=K_{\ell }P}
  
 and let 
  
    
      
        y
        =
        
          K
          
            1
          
        
        P
      
    
    {\displaystyle y=K_{1}P}
  
. Then the 
  
    
      
        n
      
    
    {\displaystyle n}
  
th layer coverage can written

  
    
      
        
          θ
          
            n
          
        
        =
        c
        
          θ
          
            0
          
        
        
          x
          
            n
          
        
        ,
        
        n
        >
        0
      
    
    {\displaystyle \theta _{n}=c\theta _{0}x^{n},\quad n>0}
  

if 
  
    
      
        c
        =
        y
        
          /
        
        x
      
    
    {\displaystyle c=y/x}
  
. The coverage of any layer is defined as the relative number of available sites. An alternative definition, which leads to a set of coverage's that are numerically to those resulting from the original way of defining surface coverage, is that 
  
    
      
        
          θ
          
            n
          
        
      
    
    {\displaystyle \theta _{n}}
  
 denotes the relative number of sites covered by only 
  
    
      
        n
      
    
    {\displaystyle n}
  
 adsorbents. Doing so it is easy to see that the total volume of adsorbed molecules can be written as the sum

  
    
      
        
          V
          
            ads
          
        
        =
        
          V
          
            m
          
        
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        n
        
          θ
          
            n
          
        
        =
        
          V
          
            m
          
        
        c
        
          θ
          
            0
          
        
        x
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        n
        
          x
          
            n
            −
            1
          
        
      
    
    {\displaystyle V_{\text{ads}}=V_{\text{m}}\sum _{n=1}^{\infty }n\theta _{n}=V_{\text{m}}c\theta _{0}x\sum _{n=1}^{\infty }nx^{n-1}}
  

where 
  
    
      
        
          V
          
            m
          
        
      
    
    {\displaystyle V_{\text{m}}}
  
 is the molecular volume. Employing the fact that this sum is the first derivative of a geometric sum, the volume becomes

  
    
      
        
          V
          
            ads
          
        
        =
        
          V
          
            m
          
        
        c
        
          θ
          
            0
          
        
        
          
            x
            
              (
              1
              −
              x
              
                )
                
                  2
                
              
            
          
        
        ,
        
        
          |
        
        x
        
          |
        
        <
        1
      
    
    {\displaystyle V_{\text{ads}}=V_{\text{m}}c\theta _{0}{\frac {x}{(1-x)^{2}}},\quad |x|<1}
  

Since the total coverage of a mono-layer must be unity, the full mono-layer coverage must be

  
    
      
        1
        =
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          θ
          
            n
          
        
      
    
    {\displaystyle 1=\sum _{n=1}^{\infty }\theta _{n}}
  

In order to properly make the substitution for 
  
    
      
        
          θ
          
            n
          
        
      
    
    {\displaystyle \theta _{n}}
  
, the restriction 
  
    
      
        n
        >
        1
      
    
    {\displaystyle n>1}
  
 forces us to take the zeroth contribution outside the summation, resulting in