Editing Physisorption (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 |edition=2nd |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_\text{c})U(\chi-\chi_\text{c})</math>

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

<math>\theta:=n_\text{ads}/n_\text{m}  \quad,\quad \chi := -\ln\bigl(-\ln\bigl(P/P_{\text{vap}}\bigr)\bigr)</math>
[[File:Ar alpha-s data.jpg|thumb|440x440px|Fig. 3. <math>\chi</math>-plot of the data by D. A. Payne, K. S. W. Sing, D. H. Turk, (J. Colloid Interface Sci. 43 (1973) 287.), which was used to create the <math>\alpha</math>-s plot.  <math>\chi</math>-plot is an excellent fit for the entire isotherm.]]
where "ads" stands for "adsorbed", "m" stands for "monolayer equivalence" and "vap" is reference to the vapor pressure ("ads" and "vap" are the latest IUPAC convention but  "m" has no IUAPC equivalent notation) 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_\text{c} =:-\ln\bigl(-E_\text{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 notice as being a very good fit to the isotherm by Polanyi<ref>{{Cite journal|last=Polanyi|first=M.|date=1914|journal=Verk. Deutsch. Physik, Gas|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 und Angewandte Physikalische Chemie|volume=26|issue=17–18 |pages=370–374 |doi=10.1002/bbpc.19200261706}}</ref><ref>{{Cite journal|last=Polanyi|first=M.|date=1929|title=Grundlagen der Potentialtheorie der Adsorption|journal= Zeitschrift für Elektrochemie und Angewandte Physikalische Chemie|volume=35|issue=7 |pages=431–432 |doi=10.1002/bbpc.1929035708}}</ref> and also by deBoer and Zwikker<ref>{{cite journal | doi=10.1515/zpch-1929-0332 | title=Adsorption als Folge von Polarisation | date=1929 | last1=Boer | first1=J. H. de | last2=Zwikker | first2=C. | journal=Zeitschrift für Physikalische Chemie | volume=3B | pages=407–418 }}</ref> but not pursued.  This was due to criticism in the former case by 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 typically less than 2%.

A typical fit to good data on a homogeneous non-porous surface is shown in figure 3.  The data is by Payne, Sing and Turk<ref>{{Cite journal |date=1973|title=Comparison of argon and nitrogen adsorption isotherms on porous and nonporous hydroxylated silica|journal= Journal of Colloid and Interface Science|volume=43|issue=2|pages=287–293|doi=10.1016/0021-9797(73)90376-7|bibcode=1973JCIS...43..287P |last1=Payne |first1=D.A |last2=Sing |first2=K.S.W |last3=Turk |first3=D.H }}</ref> and was used to create the <math>\alpha</math>-s standard curve. Unlike the BET, which can only be at best fit over the range of 0.05 to 0.35 of ''P''/''P''<small>vap</small>, the range of the fit is the full isotherm.