© A.W.Marczewski 2002

A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces

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Data analysis:
Linear Isotherm-specific plots
(Used mostly for raw isotherm data)

See other plot types: Graham | φ-plot | t-plot | α_s-plot

Try simple linear dependencies using only (if possible) experimental data of adsorption and concentration (or their functions, e.g. logarithms) - methods are equation-specific and you must decide what type of equation should be checked:

Warning. (LSq) Line fitting cannot be treated as a remedy for all troubles. It also brings in its own problems related to the scaling of measured data (i.e. adsorption and concentration or pressure) and data errors. In order to obtain reliable parameters, one has to take error scaling into account - by weighted LSq fitting, by orthogonal fitting (minimization of both x,y errors) - if x,y are independent - and/or by taking into account x,y-error correlation ( see discussion on LSq fitting problems).

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:
  (It is also a general, simple method of data assessment - may be used prior to any other isotherm-specific method, as well as Graham plot)
  • a vs. a/c   (linear Langmuir plot - an analog of Scatchard plot: a/c vs. a) (AWM: best; see Linear Langmuir plot and model lines)
    a = a_m [1 - (a/c)/(a_mK)]
  • 1/a vs. 1/c   (aka Lineweaver-Burke plot)
    (1/a) = (1/a_m) [1 + (1/c)(1/K)]
  • c/a vs. c   (aka classical linear Langmuir plot)
    (c/a) = [1/(a_mK)] [1 + Kc] = [1/(a_mK)] + (1/a_m) c
    The following is sometimes used in vapour adsorption with x = p/p_s:
    x/a vs. x   (an analog of classical BET plot but without multilayer)
    (x/a) = [1/(a_mK)] [1 + Kx] = [1/(a_mK)] + (1/a_m) x
• Everett (an analogue of Langmuir in binary liquid solutions, see also details for Everett linear plots) :
  (It is also a general, simple method of data assessment in binary liquid solutions - may be used prior to any other isotherm-specific method)
  • x₁^l / n₁^e vs. x₁^l / x₂^l
  • x₁^l x₂^l / n₁^e vs. x₁^l   (original Everett plot, see Everett linear plots)
  • x₂^l / n₁^e vs. 1/ x₁^l
• BET:
  for eq. in the form: a/a_m = [1/(1-x)] {K x/(1-x) / [1 + K x/(1-x)]}
  • a(1-x) vs. a(1-x)/[x/(1-x)]   (AWM: best; multilayer data is reduced to Langmuir monolayer plot)
    a(1-x) = a_m {1 - a(1-x)/[x/(1-x)] / (a_mK)}
  • 1/[a(1-x)] vs. [(1-x)/x]
    1/[a(1-x)] = (1/a_m) {1 + [(1-x)/x](1/K)}
  • x/[a(1-x)] vs. x   (classic BET linear form)
    x/[a(1-x)] = [1/(a_mK)] [1 + (K-1)x] = 1/(a_mK) + [(K-1)/(a_mK)] x
• Other multilayer isotherms:
  Transform iso. equation in such a way that it would formally resemble Langmuir equation (same way as for BET - see 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):
  • log[θ/(1-θ)] vs. log(c)   (aka Hill plot)
    log[θ/(1-θ)] = m log K + m log c
    any type of logarithm is OK here
• Langmuir-Freundlich analogues of Langmuir linear plot (heterogeneity parameter, m, must be determined separately):
  
  • a vs. a/c^m
    a = a_m [1 - (a/c^m)/(a_mK^m)]
  • 1/a vs. 1/c^m
    (1/a) = (1/a_m) [1 + (1/c^m)(1/K^m)]
  • c^m/a vs. c^m
    (c^m/a) = [1/(a_mK^m)] [1 + K^m c^m]
• Generalized Langmuir (aka. Marczewski-Jaroniec iso.) analogues of Langmuir linear plot (heterogeneity parameters, m and n, must be determined separately):
  
  • a^n/m vs. a^n/m/cⁿ
    a^n/m = a_m^n/m [1 - (a^n/m / cⁿ) / (a_m^n/m Kⁿ)]
  • 1/a^n/m vs. 1/cⁿ
    (1/a^n/m) = (1/a_m^n/m) [1 + (1/cⁿ) (1/Kⁿ)]
  • c^m/a^n/m vs. cⁿ
    (c^m/a^n/m) = [1/(a_m^n/mKⁿ)] [1 + Kⁿ cⁿ]
• Jovanovic and Jovanovic-Freundlich (JF/Jov-m) (a_m must be estimated separately):
  • ln( -ln(1-θ) ) vs. ln(c)
    ln( -ln(1-θ) ) = m ln K + m ln c
• Freundlich:
  • log(a) vs. log(c)
    log a = ( log a_m + m log K ) + m log c
  
  log a = ( log a_m - m log c_o ) + m log c
• Dubinin-Radushkevich:
  below: 2.303 = 1 / log₁₀e = ln 10
  • (-log θ)^1/2 vs. log(c) (a_m must be estimated separately)
    (-ln θ)^1/2 = (B₂)^1/2 (RT) (ln c_o - ln c)
    (-log₁₀θ)^1/2 = (2.303 B₂)^1/2 (RT) (log₁₀c_o - log₁₀c)
  • log(a) vs. log²(c_o/c) (c_o must be estimated separately - not the same as saturation concentration c_s)
    ln a = ln a_m - B₂(RT)² ln²(c_o/c)
    log₁₀a = log₁₀a_m - 2.303 B₂(RT)² [log₁₀(c_o/c)]²
• Dubinin-Astakhov:
  below: 2.303 = 1 / log₁₀e = ln 10
  • (-log θ)^1/n vs. log(c) (a_m must be estimated separately)
    (-ln θ)^1/n = (B_n)^1/n (RT) (ln c_o - ln c)
    (-log₁₀θ)^1/n = (2.303)^(n-1)/n (B_n)^1/n (RT) (log₁₀c_o - log₁₀c)
  • log(a) vs. logⁿ(c_o/c) (c_o must be estimated separately - not the same as saturation concentration c_s)
    ln a = ln a_m - B_n(RT)ⁿ lnⁿ(c_o/c)
    log₁₀a = log₁₀a_m - (2.303)^n-1 B_n(RT)ⁿ [log₁₀(c_o/c)]ⁿ
• Harkins-Jura:
  • log(c) vs. 1/a²
• Frenkel-Halsey-Hill (or Halsey):
  • log(c) vs. 1/a³

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