BET Derivation

Consider a surface:

Definition:

q0, q1, ..., qn = Surface area (cm-2) covered by 0, 1, ..., n layers of adsorbed molecules.

At Equilibrium:

q0 must remain constant. 

        .      Rate of Evaporation        Rate of Condensation
       . .                           =
                 from First Layer          onto Bare Surface

Similarly, at equilibrium q1 must remain constant. 

  .       Rate of Condensation            Rate of Condensation 
 . .       on the Bare Surface             on the 1st Layer
                   +               =              +
          Rate of Evaporation             Rate of Evaporation 
           from the second layer           from the second layer

  .
 . .
k1Pq0 + k-2q2 = k2Pq1 + k-1q1

Substituting into (I) gives

k-2q2 = k2Pq1

Extending this argument to other layers,

Definitions:

Total surface area of the catalyst,

Total volume of gas adsorbed on surface


where v0 is the volume of gas adsorbed on one square centimeter of surface when it is covered with a complete layer. 

  .
 . .

where vm is the volume of gas adsorbed when the entire surface is covered with a complete monolayer.

From (I),

If we assume that the properties of the 1st, 2nd, ... layers are equivalent, then,

Similarly,

q3 =xq2 =x2q1

Generally,

qi =xqi-1 =xi-1q1 =xi-1yq0 =cxiq0 {c=x/y}

Substituting into (V),

Now,

Also, 

  .
 . .
At saturation pressure of gas P0, an infinite number of adsorbate layers must build up on the surface. From equation VII, for this to be possible,

must be infinite. This means that at P0, x must equal 1. 
  .
 . .
g = P0 (From definition of x) 

  .
 . .
x = P/P0

Substituting into VII, we arrive at the recognized form of the BET isotherm,

This can be rearranged to give,

Graphical form of the BET isotherm