Page 46 - Adsorption Technology & Design, Elsevier (1998)

Fundamentals of adsorption equilibria 43

fraction of sites) then, for dynamic equilibrium between the gas phase at
pressure p and the first layer of adsorbate,

alpOo = blZmOle -e~/R,r (3.14)
In equation (3.14) al is the number of molecules which would successfully
condense onto the bare surface per unit time per unit pressure and b~ is the
frequency with which molecules possessing sufficient energy E~ leave the
surface; the term e -e'/R,r in the equation is the probability that
molecules have an energy greater than E~ to escape from the first layer. For
layers of molecules subsequent to the first layer, the BET theory supposes
that the probability of molecules evaporating from those layers is equal and
given by e -EL/R,T where EL is identified as the heat of liquefaction. For
adsorbed molecules between the layers (i- 1) and i, therefore,
aipOi-1 = bizme -Edn~r i = 2, 3,...,n (3.15)

It is further assumed in the BET theory that ai/bi (= c) is constant
for a given temperature. The sum ~,~(i01) from i = 1 to i = n is
the fractional extent of adsorption (because Oi is the fraction of occupied
sites corresponding to the ith layer which have i molecules stacked one upon
the other). The actual number of molecules adsorbed is thus ZmY,,(iO~). Now
the ratio ZlZm is equivalent to the ratio qlqm of the quantity of adsorbate
adsorbed (expressed as either mass or volume at standard temperature and
pressure) to the total capacity of the adsorbent. Because the adsorbate
vapour totally condenses when the saturated vapour pressure p~ is reached,
then 01 = 02 when p = p~ and so

a2ps = b2zme -EdRRT (3.16)

Following lengthy algebraic manipulation, the BET equation is obtained
from equations (3.14), (3.15) and (3.16). In its most useful form the BET
equation is written
p 1 (c-1) p
= I . -- (3.17)
q(Ps -p) qmc qmc ps

At a fixed temperature a plot of the left-hand side of equation (3.17) against
p/ps would yield a slope (c- 1)/qmc and intercept 1/qmc thus enabling both qm
and c to be determined. The BET equation is extensively applied to the
determination of the surface area of porous adsorbents (q.v. Chapter 4).
The inherent assumptions in the BET theory which are important to note
are (i) no interaction between neighbouring adsorbed molecules and (ii) the
heat evolved during the filling of second and subsequent layers of molecules