Page 51 - Partition & Adsorption of Organic Contaminants in Environmental Systems

42 FUNDAMENTALS OF THE ADSORPTION THEORY

a (
1
k d q = k P - q) (4.1)
where q is the fraction of the total sites occupied by the vapor at an equilib-
rium partial pressure P, k d the desorption rate constant, and k a the adsorption
rate constant. Therefore,

kP (kk d )P
a
a
q= = (4.2)
k d + k P 1 +(kk d )P
a
a
Since the amount Q of vapor adsorbed by a unit mass of the solid is propor-
tional to q, one gets an adsorption isotherm as
QbP
m
Q = (4.3)
1 + bP
where Q m is the limiting (monolayer) adsorption capacity (i.e., when the
surface is covered with a complete monolayer of the adsorbed vapor) and b
= k a /k d is related to the heat of adsorption per unit mass (or per mole) of the
vapor, which is considered to be independent of the adsorbed amount.
As seen, at low P, where bP << 1, Q is proportional to P (i.e., Q = kP),
where k is a constant, and the relation between Q and P is therefore linear.
At high P, bP >> 1, Q approaches Q m asymptotically and the isotherm is
concave toward the P axis. The linear relation between Q and P at low P may
be referred to as the Henry region. The general shape of the Langmuir-type
isotherm falls under Brunauer’s classification of type I. Examples of systems
that closely meet Eq. (4.3) are the adsorption of relatively inert vapors of
nitrogen, argon, methane, and carbon dioxide on plane (open) surfaces of mica
and glass at liquid air or liquid nitrogen temperature (Langmuir, 1918).
Although Eq. (4.3) is intended originally only for vapor adsorption, a
similar form is frequently adapted to fit the adsorption data of a substance
(solute) from a solution, in which case the P term in Eq. (4.3) is replaced by
the equilibrium solute concentration. The constant Q m and b in the Langmuir
equation may be determined by rewriting the equation as

1 1 1
= + (4.4)
Q Q bP Q m
m
By Eq. (4.4), a plot of 1/Q versus 1/P gives a slope of 1/Q m b and an intercept
of 1/Q m . From the slope and intercept values, Q m and b can be calculated.
Although the adsorption data of many vapors or solutes on solids conform
to the general shape of the Langmuir equation, this is not necessarily a proof
that the system complies with the Langmuir model. For most solids, the
adsorption sites are energetically heterogeneous, and this energetic hetero-
geneity along with site limitations may give rise to a Langmuir-shape isotherm.