Dear Seong Jun Yoon,

calculate the r from the p/p0 using the Kelvin equation, as you have it above. The volume you can calculate from the adsorbed volume. It seems that your adsorbed amounts are expressed in cc/g, and normalized to the maximum value. Best would be if you go back to the raw data of the analysis and figure out that maximum value. You'll also need to figure out which T is used for the adsorbed volume, sometimes that is 0°C, sometimes 25°C. Use the molar volume of the gas at that T to convert the cc/g to mol/g. Then divide by the molar volume of the liquid adsorptive at boiling point (the assumption here is that the adsorbate has liquid density, as in Gurvich's rule). Now you have the volume occupied by the adsorbate, which is assumed to be equal to the pore volume, and the associated pore radius from the Kelvin equation.

Plot these values in a p/p0 vs adsorbate volume graph, fit a function through the data points. The derivate of that function to ln(r) is the PSD you're after.

There are a couple of things you need to be aware of:

-This only works for gas physisorption, not for sorption of dissolved species in a solvent (at least not that I am aware of).

-Generally the Kelvin radius is not used directly, i.e. it is corrected by adding a t term for the thickness of the adsorbed layer at the considered p/p0 of capillary condensation/evaporation. That t comes from a statistical thickness curve, like the ones of Harkins and Jura or de Boer etc. The choice of t-curve depends on the adsorbent.

-for certain adsorptive-adsorbent systems the cos(theta) term in the Kelvin equation (i.e. the contact angle) is not 1.

-most of the software coming with gas adsorption devices will do all of these calculations for you

-PSD come in different forms, several of those are very unintuitive. dV/dlnr is the least intuitive if you ask me, and those distributions are very often misinterpreted. I personally prefer to stick to simple cumulative distributions. There is a very good paper by Meyer and Klobes about this:

Meyer, K., & Klobes, P. (1999). Comparison between different presentations of pore size distribution in porous materials. Fresenius' journal of analytical chemistry, 363(2), 174-178.

I hope that helps,

Pieter

Dear Yoon,

The technique of Barett-Joyner-Halenda method (BJH) is what you need to search about.

For further details about it see the next:

https://www.researchgate.net/post/What_is_the_difference_between_BET_and_BJH_methods_for_surface_analysis

and the attached file

Dear Pieter Bertier,

Thank you for your answer. Your kind explanation is quite helpful and now I can understand how's it going. One thing that I wonder is that

"Is it still possible to calculate PSD from adsorption isotherm, even though I don't have the absolute amount of adsorbate? "

Because the adsorption isotherm I got is not directly measrued. I can only get the relative value of the adsorbed volume, which also can be said as 'porosity'. 

I know the answer is YES obviously because a software give me a final PSD curve. I'm curious how it can be. And FYI, the adsorbate that I use is toluene @ RT.

if you do the calculation with the normalized adsorbed amounts, you'll also get a PSD, so you'll have an idea about the relative distribution of the volume as function of radius. But you won't be able to read any absolute volumes in specific radius ranges from that distribution curve. For dV/dlnr distributions that is not possible anyhow, because it is a derivate to lnr. For dV/dr distributions, the area under the curve is the volume for the considered radius range. Seems to me that for such a PSD (dV/dr), in your case with the normalized volumes, you'd get the fraction of the total volume from the area under the curve. That multiplied with the total adsorbate volume, the value you don't have, would give the absolute volume per radius range. 

Maybe that software of yours knows the total adsorbate volume? the PSD you calculate from the normalized data should have the same shape, but will be scaled by that total volume. If you can't get the total volume out of your software, you could calculate it from the relation between the software's and your calculated PSD curve.

Dear Pieter, 

There is no total adsorbate volume even in the software because the normalized adsorbed volume is calculated from ellipsometry data. My adsorption isotherm curve is derived from the change of refractive index as the adsorbate (toluene) adsorbed to the material. 

Actually, my main interest is in the calculation method of dV/dlnr PSD, so I don't have to know the absolute values of volumes in specific radius ranges, etc. To get dV/dlnr, can I assume the maximum value of adsorbed amount to any arbitrary values, e.g. V_total = 1, and follows the calculation procedure you mentioned at first? Or are there any other method when the maximum value is unknown?

For your understanding, I attached a figure showing what I exactly want.

It might be quite complicated to apply BJH model in your case as you will need the t-plot curve and this curve is specific of both adsorbant and adsorbate (see this ref for example https://www.researchgate.net/publication/278718088_Probing_the_microporosity_of_low-k_organosilica_films_MP_and_t-plot_methods_applied_to_ellipsometric_porosimetry_data). If you are not looking for a very accurate value of the pore size, Kelvin equation is already a good indication (at least if the pores are not very small). You must simply indicate that what you report is a kelvin pore radius. 

Regarding your concerns of not having the absolute values for the pore volume, it does not really matter as long as what you measure is proportional to the pore volume.

For the BJH PSD (if you still need to calculate it), although the principle is simple, the calculation itself is not that easy. I have attached an excel spreadsheet that describes the calculation (I downloaded it on this website -in french-: http://nte.mines-albi.fr/STP/co/C2_Contenu_05-03.html). You can also use it to determine the Kelvin PSD (with t=0).

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