© A.W.Marczewski 2002

A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces

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ADSORPTION:
Data analysis - isotherm-specific methods (some ideas)

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), a_m or e.g. characteristic micropore filling concentration, c_o, 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/c_s and in vapour adsorption x=p/p_s, where index "s" indicates saturation concentration or pressure, respectively.
  
  
  • Langmuir (see more and lines):
    • a vs. a/c (AWM: best, see Linear Langmuir plot)
    • 1/a vs. 1/c
    • c/a vs. c
  • Everett (see lines and basics):
    • x₁^l / n₁^e vs. x₁^l / x₂^l
    • x₁^l x₂^l / n₁^e vs. x₁^l (classic form, see Everett linear plots)
    • x₂^l / n₁^e vs. 1/ x₁^l
  • BET (see also lines):
    • 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)
  • Other multilayer isotherms (see also lines):
    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 (a_m must be estimated separately) (see also lines):
    • log[θ/(1-θ)] vs. log(c)
  • Generalized Langmuir (see also lines):
    • a^n/m vs. a^n/m/c^m
    • 1/a^n/m vs. 1/c^m
    • c^m/a^n/m vs. c^m
  • Jovanovic and Jovanovic-Freundlich/Jov-m (a_m must be estimated separately) (see also lines):
    • ln( -ln(1-θ)) vs. ln(c)
  • Freundlich (see also lines):
    • log(a) vs. log(c)
  • Dubinin-Radushkevich (see also lines):
    • (-log θ)^1/2 vs. log(c) (a_m must be estimated separately)
    • log(a) vs. log²(c_o/c) (c_o must be estimated separately)
  • Dubinin-Astakhov (see also lines):
    • (-log θ)^1/n vs. log(c) (a_m must be estimated separately)
    • log(a) vs. logⁿ(c_o/c) (c_o must be estimated separately)
  • Harkins-Jura (see also lines):
    • log(c) vs. 1/a²
  • Frenkel-Halsey-Hill (or Halsey) (see also lines):
    • 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/a_mK)(a/c)
    • determine parameter product a_mK from linear dependence (a/c) vs. (a/c)c^m
      (a/c) = a_mK - K^m [(a/c)c^m]
    • determine parameter K and a_m from linear dependence (c/a) vs. (c^m) (a_mK and m are already estimated).
      (c/a) = 1/(a_mK) - K^m/(a_mK) (c^m)
  • Jossens ( φ = 1 / [1 + b m a^m] = 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. a^m
      ln(a/c) = ln(K) - b (a^m)
    • 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 a^m)
        (a/c) = K exp(-b a^m)
      • determine parameter K from linear dependence (c/a) vs. exp(b a^m)
        (c/a) = K exp(b a^m)
  • linear dependence (approximate!): φ vs. ln(a)
    It corresponds to φ = n [ln(a_m) - ln(a)] or isotherm equation: a = a_m 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 = n_GL, a_m = a_m,GL and A = m_GL/(n Kⁿ) ) may be used in GL equation.
  • for GL isotherm ( φ = m / [1 + (Kc)ⁿ] = m [1 - θ^m/n]; in the case of LF equation φ = m [1 - θ] ) in order to obtain simple linear dependences, calculation of 2^nd 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 a_m from: φ vs. a
      φ = n [1 - a/a_m]
  • 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 a_m 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 (a_m 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(c_o) (and B₂(RT)²) from: φ vs. ln(c)
      φ = [2B₂(RT)²] [ln(c_o) - ln(c)]
    • estimate ln(a_m) (and B₂(RT)²) from: φ² vs. ln(a)
      φ² = [4B₂(RT)²]² [ln(a_m) - ln(a)]
  • Dubinin-Astakhov (DA; n must be known in advance) linear dependences:
    • estimate ln(c_o) (and B_n(RT)ⁿ) from: φ^1/(n-1) vs. ln(c)
      φ^1/(n-1) = [n B_n(RT)ⁿ]^1/(n-1) [ln(c_o) - ln(c)]
    • estimate ln(a_m) (and B_n(RT)ⁿ) from: φ^n/(n-1) vs. ln(a)
      φ^n/(n-1) = [nⁿ B_n(RT)ⁿ]^1/(n-1) [ln(a_m) - 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: t_HJ and t_Halsey, 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
  • α_s-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).

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