A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces

## αs method

(see a short summary of t-plot, t/F-plot and αs methods)
(see a comparison of t-plot and αs-plot)

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

"Standard Nitrogen Adsorption Data for Characterization of Nanoporous Silicas", M. Jaroniec, M. Kruk and J.P. Olivier, Langmuir (1999). αs 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 αs-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 αs-plot may be written as:
a(x) = amicro,max + kstd Sext αs(x)
or
a(x) = amicro,max + slope * αs(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,
αs(x) = astd(x) / astd(x=0.4) (dimensionless value)
kstd = astd(x=0.4) / Sstd - where Sstd 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:
Sext = slope [ Sstd / astd(x=0.4) ]

Another positive side of αs 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) = ao,i + slopei αs ,    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, ao,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 xc smaller than corresponding to a largest αs(x) in the current linear section
3. if the "i"-th y-intercept ao,i has value smaller than the previous estimated positive intercept ( ao,i ≤ ao,i-1 ) - or the slopei 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, ao,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 ( ao,i - ao,i-1 > 0) corresponds to the volume of last filled pores; if adsorption is in [cm3/g STP]:
ΔVpore,i = 0.0015468 (ao,i - ao,i-1)    (for i > 1)
Vmicro = Vpore,1 = 0.0015468 ao,1
6. (negative) difference of slopes (slopei - slopei-1 < 0) (i.e. current minus previous slope) corresponds to the surface area of previously filled pores:
ΔSpore,i = -(slopeo,i - ao,i-1) [ Sstd / astd(x=0.4) ]    (for i > 1)

### αs method for C6H6 and other adsorbates

As opposed to the original αs method as proposed originally and used for N2 adsorption (some authors still use x=0.4 for e.g. benzene), the αs should be defined differently:
αs(x) = astd(x) / astd(x=xhist) (dimensionless value)
The point of opening/closing of hysteresis loop xhist 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 xhist is the best choice for characteristic point in αs 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:

A.A. Isirikyan and A.V. Kiselev, J.Chem.Phys. 65(4) (1961) 601-607.

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