A pity that the paper is removed. Please send me the pdf: h.dolhaine@gmx.de

For the time being I do not understand the reasoning of the authors.

I am roughly familiar with the PDE, but not statistical mechanics so I am reading about grand canonical partition function

\[Xi] == Sum[ (Binomial[L, N] q^N) (E^(\[Mu]/(k*T))^N) , {N,0,Inf} ]

q is not described further in the paper, wikipedia doesn't mention binomial coeff. times q,

The 2nd member seems to fit the form "The canonical ensemble assigns a probability P to each distinct microstate given by the following exponential" well, but it seems to be the old 'well known' Maxwell's speed distribution function from elementary physics / vel^2. I'm not far enough in chemistry to know (the many) equations involving T and the rate of reaction. I have plenty more reading about PDE to do as to "partitioning using hamiltonian". (Yet, wikipedia is often not a good way to judge because they can take an easily taught subject about a simple theorem and turn it into absolute oblivion - actually un-defining what was in books easy to define. A good example is they have explained Liouville by saying it is a element in lie algebra - which of course, is "just not so").

That being said... "p.134" (10 pages in pdf).

"N is an integral variable, but when an adsorbate molecule binds to the surface, ni sites are filled. Hence, variable N is substituted by s"

I'm unsure if you are arguing s cannot be substituted as prescribed. But do not use N==s n (that is a circular argument since we already are past that point and using s alone). Use s as the author suggests p. 134 (11).

That is my guess.

Effectively the binomial theorem will hold IFF (big IFF) I succeed in putting the expression in a form compatible with it.

The problems are:

1) The authors make a change of variable from N to s = 1/n N, where "1/n" is the same parameter as the "1/n" exponent in the Langmuir-Freundlich isotherm. So N = s n

2) Since the independent variable is now "s" and not "N" anymore, somehow I must sum "s" using some rule derived from N = s n, and I have no idea how to do a Summation with an iterator like N = s n, instead of a single-symbol one. I only get an "invalid iterator" error message.

3) Without the change of variable there is some result, instead of the error message above, but it is awfully complicated and includes bizarre hypergeometric functions.

Any idea how solve this problem? I have posted the paper, so maybe I missed something there, or maybe (I hope not) the paper is just wrong so I wasted my time.

Best regards,

Alberto Silva

Step 12? the last simplification looks similar to the binomial theorem (which I'm guessing Mathematica could perform?)

(x+y)^n == Sum_j->n C(n,j) x^n-j y^j

but no one asked.

As to 8, if the input does not follow this rule, then it is not "a single layer adsorption" but something else. It is probably a generic grid (nxm) which if populated, satisfies reactions == products == one layer / onto one layer - yet if not equal then it was not on the layer (it might be zero layer, many layer, or show another reaction captured some reactants so one layer wasn't populated, and etc). I shouldn't say more I'm bad at chemistry.

A few comments:

1) I will post the paper as an attachment if it is OK, I have not done that (yet) because I am not sure if there is Wolfram community rule forbidding me to do that.

It is permitted to attach papers here?

2) The problem is thermodynamical, related to partition functions in statistical mechanics, and yes, is a complicated one. That's why I am searching for help.

Nota bene: Gravity, quantum mechanics, etc, etc, have nothing to do with this chemical problem.

Best regards,

Alberto Silva

You should probably not take the formula too seriously. If adsorption is the description of molecules "sticking to the surface of a solid", then it is only a matter of spacing and counting (size of molecule, surface area if not flat and side effects could pose some advanced problems). The formula wouldn't include anything like gravity, miscibility, nothing. Just a simple counting routine. I would only take it as seriously as was the professor likely to put it on a test.

For me it is by no means clear how the authors arrive from eqs (4) and (8) at eq (12).

Look at

s1 = Sum[ Binomial[L, j] \!\(TraditionalForm\`
\*SuperscriptBox[\((\ \[Lambda]\ q\ )\), \(\(j\)\(\ \)\)]\), {j, 0, L}]
s2 = 1/np Sum[ Binomial[L, j] \!\(TraditionalForm\`
\*SuperscriptBox[\((\ \[Lambda]\ q\ )\), \(j\ /np\)]\), {j, 0, L}]
vals = {L -> 100, \[Lambda] -> .2, q -> 3.4, np -> 5}
s1 /. vals
s2 /. vals

And see remarks in the notebook attached

Attachments:

wcomstattherm.nb

Addendum: As discussed in the paper:

F. Kano et al. / Surface Science 467 (2000) 131–138; "Fractal model for adsorption on activated carbon surfaces: Langmuir and Freundlich adsorption"< sciencedirect link

A generalized Langmuir-Freundlich isotherm is derived from first principles, using the following formulas:

1) Adsorbing system "grand partition function":

2) Layer "partition function"

3) Definition of s and n:

4) Putting all together:

My problem to reproduce the result of the paper is that I don't know how to do a Summation with an arbitrary iterator (i.e. not just a symbol like "i", or "j"), in this case:

Sum[f[s/n], {s/n, 0, L}]

Nota bene: I try to do the summation with "Nt", I got a complicated answer that have nothing to do with the simple result obtained by the authors.

I hope to have made my question easier to understand. By the way, it is permitted to attach papers in this forum? I just quoted the paper to avoid copyright issues.

Best regards,

Alberto.

Erratum: re-reading my post, I found several typing mistakes. I wrote:

Z[N]=Binomial[L/m,Nt]*Product[Binomial[m,n],{n,1,Nt}]*q^Nt 
S[N, mu]=Sum[Z[Nt]*a^Nt, {Nt,0,L}]

I should have written this:

Z[Nt]=Binomial[L/m,Nt]*Product[Binomial[m,n],{n,1,Nt}]*q^Nt 
S[Nt,a]=Sum[Z[Nt]*a^Nt, {Nt,0,L}]

Note: The goal of this claculation is to derive a generalized Langmuir-Freundlich isotherm, as will be clear in a following addendum.