A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces ADSORPTION:
Data analysis - isotherm-specific methods
(some ideas)

General Integral Equation / GL (Generalized Langmuir) / All equations (preview)
Adsorption type ( Linear Langmuir plot / Graham plot / Consistency / Henry constant )
Popular isotherms ( Mono-, Multilayer, Experimental, Micro-, Mesoporous )
Data analysis: ( LSq data fitting / Heterogeneity: Global , σE / Linear plots / φ-function )
Heterogeneity and Molecular Size ( Theory and Prediction / Simple binary isotherm )
Linear plots | φ-function | Separation of Micro- and Mesopores

Data analysis - isotherm-specific methods:

• Linear plots: L / Everett / BET / LF / GL / Jov / F / DR / DA / HJ / FHH Try simple linear dependencies using only experimental adsorption and concentration (or their functions, e.g. logarithms) - methods are equation-specific and you must decide what type of equation should be checked (details of isotherm-specific linear plots):

NOTE 1. Some of the methods may require adsorption monolayer (adsorption capacity), am or e.g. characteristic micropore filling concentration, co, to be know in advance (estimated by independent method or LSQ-fitted - in this case, linear plot is only a verification).
NOTE 2. Replace concentration, c, by pressure, p, for gas adsorption. In dilute solute adsorption x=c/cs and in vapour adsorption x=p/ps, where index "s" indicates saturation concentration or pressure, respectively.

• Langmuir (see more and lines):
• Everett (see lines and basics):
• x1l / n1e vs. x1l / x2l
• x1l x2l / n1e vs. x1l (classic form, see Everett linear plots)
• x2l / n1e vs. 1/ x1l
• a(1-x) vs. a(1-x)/[x/(1-x)] (AWM: best, data is reduced to Langmuir monolayer)
• 1/[a(1-x)] vs. [(1-x)/x]
• x/[a(1-x)] vs. x (classic BET linear form)
Transform iso. equation in such a way that it would formally resemble Langmuir equation (same way as for BET above). Then use ideas of one of the linear plots for Langmuir. May be used for: Hüttig, LGD, LGDa eqns. For Sircar eq. an additional constant should be known in advance (fitted?), i.e. C .
• Langmuir-Freundlich and Langmuir (am must be estimated separately) (see also lines):
• log[θ/(1-θ)] vs. log(c)
• an/m vs. an/m/cm
• 1/an/m vs. 1/cm
• cm/an/m vs. cm
• Jovanovic and Jovanovic-Freundlich/Jov-m (am must be estimated separately) (see also lines):
• ln( -ln(1-θ)) vs. ln(c)
• log(a) vs. log(c)
• (-log θ)1/2 vs. log(c) (am must be estimated separately)
• log(a) vs. log2(co/c) (co must be estimated separately)
• (-log θ)1/n vs. log(c) (am must be estimated separately)
• log(a) vs. logn(co/c) (co must be estimated separately)
• log(c) vs. 1/a2
• log(c) vs. 1/a3

• Linear φ-plots: RP / Jos / exp / GL / LF / Jov / F / DR / DA Use a so-called: φ-function method (where φ = ∂ log(a)/ ∂ log(c)) (derivative of adsorption isotherm in logarithmic co-ordinates). φ-function is a dimensionless (independent of adsorption and concentration/pressure units as well as independent of type of logarithm you use). Various plots of φ-function versus adsorption, pressure etc. allow to determine isotherm type and isotherm parameters (or sometimes at least to reject some existing choices) (Here are model pictures)
NOTE. This method allows to determine equation type and parameters very precisely, however your data should be very good - evenly spaced, smooth and in a wide range of relative adsorption φ) - method is sensitive to data scatter.
• Radke-Prausnitz aka. Redlich-Peterson (RP) ( φ = 1 - n {(Kc)n / [1 + (Kc)n]} )
(3-step method):
• determine parameter m from linear dependence 1-φ vs. a/c
1 - φ = m - (m/amK)(a/c)
• determine parameter product amK from linear dependence (a/c) vs. (a/c)cm
(a/c) = amK - Km [(a/c)cm]
• determine parameter K and am from linear dependence (c/a) vs. (cm) (amK and m are already estimated).
(c/a) = 1/(amK) - Km/(amK) (cm)
• Jossens ( φ = 1 / [1 + b m am] = 1 / [1 + m ln(Kc/a)] )
(2-step to 3-step method):
• determine parameter m from linear dependence (1/φ - 1) vs. ln(a/c)
(1/φ - 1) = m ln(K) - m ln(a/c)
• determine parameters b and K from linear dependence ln(a/c) vs. am
ln(a/c) = ln(K) - b (am)
• improve K-value by using one of dependencies (use earlier obtained m and b):
• determine parameter K from linear dependence (a/c) vs. exp(-b am)
(a/c) = K exp(-b am)
• determine parameter K from linear dependence (c/a) vs. exp(b am)
(c/a) = K exp(b am)
• linear dependence (approximate!): φ vs. ln(a)
It corresponds to φ = n [ln(am) - ln(a)] or isotherm equation: a = am exp(-A/cn). This isotherm does not reduce to Henry isotherm for low concentrations and is not consistent with local Langmuir behaviour, however it may be treated as an approximation of GL equation for high adsorption values (close to monolayer) especially if strong lateral interactions are also involved (e.g. GL-FG). Then estimated parameters ( n = nGL, am = am,GL and A = mGL/(n Kn) ) may be used in GL equation.
• for GL isotherm ( φ = m / [1 + (Kc)n] = m [1 - θm/n]; in the case of LF equation φ = m [1 - θ] ) in order to obtain simple linear dependences, calculation of 2nd derivative is required. In such a case method becomes much more sensitive to experimental data scatter.
• for LF isotherm (special case of GL: m=n - see φ above) :
• determine K and am from: φ vs. a
φ = n [1 - a/am]
• for Jovanovic and Jovanovic-Freundlich/Jov-m isotherms ( φ = m (Kc)m / {exp[(Kc)m] - 1} = m [(1-θ)/θ] [ -ln (1-θ)] ) linear dependences are difficult to use (at least am is required in order to calculate θ):
• estimate m from: φ vs. [(1-θ)/θ] [ -ln(1-θ)]
φ = m [(1-θ)/θ] [ -ln(1-θ)]
much better idea is to use simple linear relationship not involving φ - both m and K are determined in one step (am is still required)
• Freundlich (F or DA with n=1): φ = const (and equal to B1RT = m ).
• Dubinin-Radushkevich (DR or DA with n=2), linear dependences:
• estimate ln(co) (and B2(RT)2) from: φ vs. ln(c)
φ = [2B2(RT)2] [ln(co) - ln(c)]
• estimate ln(am) (and B2(RT)2) from: φ2 vs. ln(a)
φ2 = [4B2(RT)2]2 [ln(am) - ln(a)]
• Dubinin-Astakhov (DA; n must be known in advance) linear dependences:
• estimate ln(co) (and Bn(RT)n) from: φ1/(n-1) vs. ln(c)
φ1/(n-1) = [n Bn(RT)n]1/(n-1) [ln(co) - ln(c)]
• estimate ln(am) (and Bn(RT)n) from: φn/(n-1) vs. ln(a)
φn/(n-1) = [nn Bn(RT)n]1/(n-1) [ln(am) - ln(a)]

• Isotherm analysis - separation of micropore and mesopore effects:
• t-plot (de Boer) compares experimental data with analytical "average" isotherm (usually HJ or Halsey/FHH expressed as a "statistical adsorbed layer thickness" and for N2 adsorption: tHJ and tHalsey, respectively) in a small pressure range just above monolayer (or micropore) filling (t = 3.54 - 5.0Å) in order to find micropore volume (by extrapolation) and "external" (or rather mesopore + macropore) surface area (from line slope)
• t/F-plot (Kadlec) as t-plot above, but additionally utilises DR isotherm to describe behaviour in the micropore filling range
• αs-method by Singh? - like t-plot but compares your experimental data on porous solids with adsorption on (arbitrarily) selected standard (should have the same chemical composition and structure, same surface properties - groups etc. - and no micro- or mesopores). Allows to determine not only the content of micropores but also mesopores as well as macropore/external area. Most often used for N2 data, but may be used for other adsorbates (the same for your sample and standard, of course) but every type of adsorbate requires determination of a "characteristic point" (x=0.40 for N2 and x=0.175 for benzene).

Linear plots | φ-function | Separation of Micro- and Mesopores
Adsorption type ( Linear Langmuir plot / Graham plot / Consistency / Henry constant )
Popular isotherms ( Mono-, Multilayer, Experimental, Micro-, Mesoporous )
Data analysis: ( LSq data fitting / Heterogeneity: Global , σE / Linear plots / φ-function )