© A.W.Marczewski 2002

A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces

(see a comparison of t-plot and α

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).

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:

- interpretation should be carried out starting from low adsorption values, with i=1 for micropores,
- 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 - 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) - 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) - (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 [cm^{3}/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} - (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)

As opposed to the original α_{s} method as proposed originally and used for N_{2} 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:

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

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