© A.W.Marczewski 2002

A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces

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α_s method

Details of the method and an example of standard isotherm may be found e.g. in:

α_s method for N₂ adsorption (see other adsorbates)

α_s method was introduced by K.S.W. Sing. In some respects it is very similar to the de Boer's t-plot method, because it compares your adsorption data with a standard isotherm of adsorption on some non-porous solid. It is also assumed, that the adsorption in a certain region may be described by a straight line in which a y-intercept describes a saturated adsorption isotherm on micropores (i.e. maximum adsorption in micropores), whereas the slope is related to the adsorption on a non-microporous part. However, in contrast to the t-plot method, the standard isotherm in α_s-plot is usually some experimental isotherm on a non-porous adsorbent selected specifically for its chemical and structural similarity to the adsorbent in question.
The main equation of α_s-plot may be written as:
     a(x) = a_micro,max + k_std S_ext α_s(x)
or
     a(x) = a_micro,max + slope * α_s(x)
where:
x = p/p_s
a_micro,max - adsorption in saturated micropores,
S_ext - "external" surface area; here it is the surface area of pores larger than micropores,
α_s(x) = a_std(x) / a_std(x=0.4) (dimensionless value)
k_std = a_std(x=0.4) / S_std - where S_std is specific surface area of the standard used; its numerical value depends on the units used for the values of adsorption a(x) and surface area S.
Then the external surface area of the adsorbent may be calculated as:
     S_ext = slope [ S_std / a_std(x=0.4) ]

Another positive side of α_s method is its universality. It may be used in determination of mesopore volume, mesopore surface area, macropore volume and area etc. - it depends only on the data range available (in fact t-plot could be used in a similar manner as well). However, it also helps if pore size distribution has well defined peaks, i.e. there may be several types of quite distinct pores, without much of intermediates.
A series od lines approximating isotherm sections may be drawn:
a(x) = a_o,i + slope_i α_s ,    i = 1, 2 ...
Those lines may be interpreted as follows:

1. interpretation should be carried out starting from low adsorption values, with i=1 for micropores,
2. lines with the positive y-intercepts, a_o,i > 0, may be interpreted as the adsorption on "external" surface (surface of currently available pores), the y-intercept is the adsorbed amount in all completely filled-up pores, i.e. pores with condensation pressure x_c smaller than corresponding to a largest α_s(x) in the current linear section
3. if the "i"-th y-intercept a_o,i has value smaller than the previous estimated positive intercept ( a_o,i ≤ a_o,i-1 ) - or the slope_i is bigger than the previous one - it should be disregarded - this behaviour may be attributed to the adsorption on the surface that is increasing in the process (e.g. some pores did suddenly open) - in fact this happens if parallely to simple adsorption on available surface some pores are being filled-up by condensation (in the result available area is decreasing but as a total this effect is compensated by condensation)
4. lines with strongly negative y-intercept, a_o,i << 0 , reflect the process of rapid filling-up of some pores (their radius may be calculated from the corresponding relative pressure)
5. (positive) difference of y-intercepts ( a_o,i - a_o,i-1 > 0) corresponds to the volume of last filled pores; if adsorption is in [cm³/g STP]:
   ΔV_pore,i = 0.0015468 (a_o,i - a_o,i-1)    (for i > 1)
   V_micro = V_pore,1 = 0.0015468 a_o,1
6. (negative) difference of slopes (slope_i - slope_i-1 < 0) (i.e. current minus previous slope) corresponds to the surface area of previously filled pores:
   ΔS_pore,i = -(slope_o,i - a_o,i-1) [ S_std / a_std(x=0.4) ]    (for i > 1)

α_s method for C₆H₆ and other adsorbates

As opposed to the original α_s method as proposed originally and used for N₂ adsorption (some authors still use x=0.4 for e.g. benzene), the α_s should be defined differently:
   α_s(x) = a_std(x) / a_std(x=x_hist) (dimensionless value)
The point of opening/closing of hysteresis loop x_hist changes strongly with adsorbate (e.g. 0.4 for nitrogen, 0.175 for benzene). However, this changed value still corresponds to the point where the adsorbed layer is at least monoatomic and the micropores are filled-up. As of now I believe the problem is still open for discussion whether just x_hist is the best choice for characteristic point in α_s method. However it is still the simplest and the most logical one.

Some standard isotherms for nitrogen, benzene and n-hexane adsorption on carbonaceous adsorbents may be found e.g. in:

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