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156 Chapter 7
Uppermost layer is in equilibrium with vapor phase. iii) The heat of adsorption at all sites
is constant and equal to the heat of condensation from the second layer on; iv) There is no
lateral interactions between molecules. v) At saturation pressure, the number of layers
has infinite thickness. Hence, the BET equation could be derived:
p
n a (p 0 − p) = 1
n a m C
+
(C − 1)
n m a C
Here n a is the amount adsorbed, p is the pressure, p0 is the saturation pressure of the
adsorbed gas at a given temperature, n m
a
is the monolayer capacity and C an empirical
constant. The value of C can be associated to the magnitude of adsorbent-adsorbate
interactions. This equation is an adsorption isotherm and can be plotted as a BET plot
where 1/[n a (p0/p-1)] is the y-axis and p/p0 on the x-axis. The linear relationship of the
BET plot is kept only in a pressure region of p/p0= 0.05 - 0.30. For microporous materials
such as MOFs, a even smaller pressure range (p/p0 = 0.02 - 0.10) is usually applied.
Subsequently, the monolayer capacity n m
a
can be derived from the slope of the BET plot.
Consequently, the specific surface area SBET can be calculated from the following equation:
SBET = n m a N A a m /m
p
p 0
where NA is the Avogadro constant and am is molecular cross-section area occupied by a
single adsorbate molecule in the complete monolayer and m the mass of adsorbent. Still,
we should note that the BET theory is developed on basis of the oversimplified model. In
fact, SBET is greatly affected by the nature of the adsorbent as well as the strength of
adsorbate-adsorbent interactions. Microporous solids including MOFs, the pore surface of
which are rather heterogeneous, the above mentioned assumptions are not valid. The
obtained value of the BET surface area (SBET) can be widely used to compare the porosity
of different porous materials and should not be considered as a precise absolute value.
The adsorption isotherms of H2 adsorption at 77 K and 87 K were independently fit with
triple site Langmuir model
n = q sat,Ab A p
1 + b A p + q sat,Bb B p
1 + b B p + q sat,Cb C p
1 + b C p
where n is the molar loading of adsorbate (mmol/g), qsat is the saturation loading of site
A/B/C (mmol/g), b is the Langmuir parameter for A/B/C (bar -1 ), and p the pressure(bar).
After the triple-site Langmuir fits, the exact pressures correspond to the same amount