© A.W.Marczewski 2002

A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces

Barret-Joyner-Halenda scheme (1955) for calculation of mesopore distribution from nitrogen adsorption data may be summarized in the formula:

In this formula, v_{ads}(x_{k}) is the volume of (liquid) adsorbate [cm^{3}/g] at relative pressure x_{k} (calculated from the value of adsorption expressed in [cm^{3}/g STP] by v_{ads}(x) = 0.0015468 a(x) ), pore volume V is given in [cm^{3}/g], S is surface area [m^{2}/g] and t is the thickness of adsorbed layer (in appropriate units).

This formula says, that the adsorbed amount at k-th point of adsorption isoterm may be divided into 2 distinct parts:

1^{st} is a volume in condensate in all pores smaller than some characteristic size depending on current relative pressure, r_{c}(x_{k}),

2

**NOTE!**

*One has to be always cautions as to identity of the radius used: in Kelvin equation x_{c} = x_{c}(r) and thickness equation t = t(x,r), radius is always determined by the menicus. However, in the calculation of ΔV and ΔS radius is a geometrical pore radius!*

Pore radiuses should be calculated by Kelvin equation for a given pore model. The diameter of solid pore is a sum of meniscus diameter + 2 * thickness of adsorbed film. In general the thickness of adsorbed film depends not only on relative pressure (like for flat surface - e.g simple HJ and FHH equations) but also is affected by meniscus radius.

For a given pore with radius r (or diameter d - better for slit-shaped pores), the pore area and pore volume are related by simple geometric relations, e.g.:

for spherical pores: **ΔV = d ΔS / 6**

for cylinders: **ΔV = d ΔS / 4**

for slits: **ΔV = d ΔS / 2**.

For adsorption (condensation) and desorption (evaporation) isotherm branches the calculation scheme looks the same - you always start from the point where all pores are filled-up (i.e. last point of adsorption branch, x_{ads,K} where the adsorbed amount becomes the same as on desorption branch x_{des,L}) and - preferably - the pore system is (apparently) oversaturated. The last requirement is important if there is a possibility of existence of bottle-shaped pores. In practice it is always true. Then you go along decreasing pressure, i.e. you iterate isotherm point counter downwards (x_{ads,i} i = K, K-1, K-2 ... ) whereas for desorption upwards (x_{des,i} i = L, L+1, L+2 ...).

In the 1st step, the change of adsorbate amount corresponds to proesses in a single (averaged!) pore size. This pore size is calculated with appropriate form of Kelvin equation (meniscus) and equation defining the adsorbed film thickness for two limiting radiuses (corresponding to maximum, x_{i}, and minimum, x_{i+1}, pressure). In the following steps the information on all opened pores (i.e. ΔS(r_{j} and ΔV(r_{j}, j < i) is utilized.

Send a message to *Adam.Marczewski AT@AT umcs.lublin.pl*