© A.W.Marczewski 2002

A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces

Reload Adsorption Guide

Barret-Joyner-Halenda (BJH) calculation scheme

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 [cm³/g] at relative pressure x_k (calculated from the value of adsorption expressed in [cm³/g STP] by v_ads(x) = 0.0015468 a(x) ), pore volume V is given in [cm³/g], S is surface area [m²/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),
2nd is a volume of adsorbed film on all larger pores, calculated a sum of terms: Σ (pore surface) (thickness of film in pore).

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.

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