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