Editing Adsorption (section)

== Quantum mechanical – thermodynamic modelling for surface area and porosity ==
Since 1980 two theories were worked on to explain adsorption and obtain equations that work. These two are referred to as the chi hypothesis, the quantum mechanical derivation, and excess surface work (ESW).<ref>{{Cite book|last=Condon|first=James|title=Surface Area and Porosity Determinations by Physisorption, Measurement, Classical Theory and Quantum Theory, 2nd edition.|publisher=Elsevier|year=2020|isbn=978-0-12-818785-2|location=Amsterdam.NL|pages=Chapters 3, 4 and 5}}</ref> Both these theories yield the same equation for flat surfaces:

: <math>\theta=(\chi-\chi_c)U(\chi-\chi_c)</math>

where ''U'' is the unit step function. The definitions of the other symbols is as follows:

: <math>\theta:=n_\text{ads}/n_m ,\quad \chi := -\ln\bigl(-\ln\bigl(P/P_\text{vap}\bigr)\bigr)</math>

where "ads" stands for "adsorbed", "m" stands for "monolayer equivalence" and "vap" is reference to the vapor pressure of the liquid adsorptive at the same temperature as the solid sample. The unit function creates the definition of the molar energy of adsorption for the first adsorbed molecule by:

: <math>\chi_c =:-\ln\bigl(-E_a/RT\bigr)  </math>

The plot of <math>n_{ads}</math> adsorbed versus <math>\chi</math> is referred to as the chi plot. For flat surfaces, the slope of the chi plot yields the surface area. Empirically, this plot was noticed as being a very good fit to the isotherm by [[Michael Polanyi]]<ref>{{Cite journal|title= Über die Adsorption vom Standpunkt des dritten Wärmesatzes|last=Polanyi|first=M.|date=1914|journal=Verhandlungen der Deutschen Physikalischen Gesellschaft|language=de|volume=16|pages=1012}}</ref><ref>{{Cite journal|last=Polanyi|first=M.|date=1920|title=Neueres über Adsorption und Ursache der Adsorptionskräfte|journal=Zeitschrift für Elektrochemie|volume=26|pages=370–374}}</ref><ref>{{Cite journal|last=Polanyi|first=M.|date=1929|title=Grundlagen der Potentialtheorie der Adsorption|journal=Zeitschrift für Elektrochemie |volume=35|pages=431–432|language=de}}</ref> and also by [[Jan Hendrik de Boer]] and [[Cornelis Zwikker]]<ref>{{Cite journal|last1=deBoer|first1=J.H.|last2=Zwikker|first2=C.|date=1929|title=Adsorption als Folge von Polarisation|journal=Zeitschrift für Physikalische Chemie |volume=B3|pages=407–420|language=de}}</ref> but not pursued. This was due to criticism in the former case by [[Albert Einstein]] and in the latter case by Brunauer. This flat surface equation may be used as a "standard curve" in the normal tradition of comparison curves, with the exception that the porous sample's early portion of the plot of <math>n_{ads}</math> versus <math>\chi</math> acts as a self-standard. Ultramicroporous, microporous and mesoporous conditions may be analyzed using this technique. Typical standard deviations for full isotherm fits including porous samples are less than 2%.

Notice that in this description of physical adsorption, the entropy of adsorption is consistent with the Dubinin thermodynamic criterion, that is the entropy of adsorption from the liquid state to the adsorbed state is approximately zero.