A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces

## t-plot and t/F-plot

(see an example in a surface area / pore volume analysis report)
(see a short summary of t-plot, t/F-plot and αs methods)
(see a comparison of t-plot and αs-plot)

## t-plot This method of isotherm data analysis was introduced by de Boer. It assumes, that in a certain isotherm region, the micropres are already filled-up, whereas the adsorption in larger pores occurs according to some simple equation, characteristic for a large class of solids. This equation should approximate adsorption in mesopores, macropores and on a flat surface in a narrow pressure range just above complete filling of micropores, but below vapour condensation in mesopores. Such assumption may be valid, if micropores are small and super-micropores are not present. Then the adsorption within this pressure region may be described by a simple linear dependence :
a(x) = amicro,max + k Sext tlayer(x)
where:
x = p/ps
amicro,max - adsorption in saturated micropores,
Sext - "external" surface area; here it is the surface area of pores larger than micropores,
tlayer(x) - estimated statistical thickness of adsorbed layer (in meso- and macropores) - usually by Harkins-Jura (HJ) and Halsey / Frenkel-Halsey-Hill (FHH) equations - often used are their experimental versions HJ and Halsey; in the t-plot calculated thickness values are within 0.354 nm (i.e. tmono) and 0.5 nm (it corresponds roughly to the range of relative pressures 0.1 - 0.2); for solids with small amounts of micropores, the Halsey isotherm may give negative value as an estimate of micropore volume/area, thus it sould be used with caution.
k - coefficient which depends on units used for the values of adsorption a(x), layer thickness t(x) and surface area Sext. E.g. for t [nm] , S [m2/g] and a [cm3/g STP] we obtain: k = 1 / (4.3532 tmono), where tmono = 0.354 nm is monolayer thickness for N2 adsorbed layers and 4.3532 is a coefficient for calculation of occupied area [m2/g] from adsorbed amount [cm3/g STP] (N2 adsorption at 78K).

## t/F-plot

Introduced by Kadlec in order to increase precision of t-plot analysis. In this method the Dubinin-Radushkevich isotherm is fitted in the micropore range (low pressures / adsorption range). This method may help to compensate for t-plots non-linearity in the case of solids containing some super-micropores. However much wider data range is required and t-plot method looses its great simplicity.

A preferred method (see comparison) should be αs-method. This method requires some standard isotherm data for comparisons, however it gives information largely free of adsorption model influences and additionally provides description of mesopore and - to some extent - macropore system.

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