© A.W.Marczewski 2006-2013

A Practical Guide to Isotherms of ADSORPTION on Heterogeneous Surfaces

Reload Adsorption Guide: Kinetics

Kinetics of Adsorption
Some equations - see below, more theory - in the future.

General remarks:

• Do not use linear plots as a means to obtain fitted parameters - some plots are good for exposing deviations from the model used (e.g. alternative linear PSOE plot: a vs. a/t), whereas other plots are very good at masking such deviations (e.g. standard linear PSOE plot: t/a vs. t), thus suggesting that such data agrees with the model (though standard PSOE plot is a good tool to obtain estimate of equilibrium adsorbed amount.
• Do not compare fitting quality of various models while using different plots and/or fitting criteria (e.g. many authors compare fitting quality of PSOE and FOE by using corresponding R² ... for their specific linear plots!) - always use identical LSQ-fitting (or analogous) method with the same optimizing target.
• Linear plots are good for presentation purposes, rarely for fitting (notable exception is linear Freundlich plot, i.e. log a vs. log c, for constant relative deviation of adsorbed amount, i.e. local SD(a)/a ~ const).
  Sometimes linear plot may be used to extract/estimate a single parameter - e.g. standard calculation of S_BET (BET surface area, plot: x/[a(1-x)] vs. x) - estimate of constant C is often useless, standard PSOE plot for estimation of equilibrium adsorption (plot: t/a vs. t), etc.

In all equations:

  a - adsorbed amount,
  c- concentration,
  t - time, t_0.5 - half time
  subscripts: "o" - initial, "eq"- equilibrium,
  f_i - normalized to 1 term contribution factors,
  is mass balance, and
  F - adsorption/desorption progress (fractional attainment of equilibrium)
  adsorption progress for adsorption on pure surface
  u = (c_o-c)/c_o - solute uptake (relative, batch conditions),

My favourite empirical kinetic equations:

• Mixed 1,2-order (MOE) (see [5], Appl. Surf. Sci., 256, 5145-5152 (2010)). Empirical equation - generalization of the 1st (FOE/PFOE) and 2nd (SOE/PSOE) order equations - produces kinetic behaviours intermediate of the 1st (if f₁=1, f₂=0, integrated FOE/PFOE is obtained, MOE may be used) and 2nd order equations (if f₁=0, f₂=1, integrated SOE/PSOE is obtained, integrated form of MOE cannot be used). MOE for certain systems and parameter values is equivalent to the solution of the Langmuir rate theory (see IKL), however, it may be used also for non-complying systems, e.g. energetically homogenous systems.
  MOE may be also used to describe desorption (Ref. [12], Adsorption 19(2-4), 391-406 (2013)). Analysis of Langmuir kinetics (see gIKL, Ref. [12], Adsorption, 19(2-4), 391-406 (2013)) shows that for desorption f₂ may be negative (see below).
  See Refs. [5], Appl. Surf. Sci., 256, 5145-5152 (2010) and [12], Adsorption, 19(2-4), 391-406 (2013).
  
  MOE rate equation:
  and
  where the contributions terms are:
  MOE kinetic eq. (integrated form, only if f₂ < 1 ):
  and
  
  MOE kinetic eq. (linear form, analogous to the Lagergren eq.):
  
  
  NOTE: If f₂ = 1 and f₁ = 0 (i.e. k_1a = k_1c = k₁ = 0) integration yields SOE/PSOE eq.
  
  Adsorption half time is:
    t_0.5 = ln (2 - f₂) / k₁    if    f₂ < 1 (integral form of MOE)
    t_0.5 = 1 / k₂    if    f₂ = 1 (see SOE/PSOE)
  
  MOE generalization and extension to desorption:
  Analysis of Langmuir kinetics (see gIKL, Ref. [12], Adsorption19(2-4), 391-406 (2013)) shows that for MOE and desorption f₂ may be negative, too. However, while f₂ = f_L values corresponding to Langmuir kinetics (gIKL) must be in the range <-1; 1), for MOE there is no such lower limit constraint, and we have:
      f₂ <1 (mathematical constraint for MOE and gIKL)
  and adsorption/desorption rate maximum is observed for:
      f₂ <-1   at   F = 0.5(1+f₂)/f₂
  i.e. small maximum appears near F=0 for f₂ slightly below -1 and for further decreasing f₂ (i.e. increasing f₁) moves towards F=0.5. Relative height of the maximum (vs. initial rate) for f₂<-1 is: -0.25(1-f₂)²/f₂.

  f2	Fmax rate	max { (dF/dt)rel }
  -1	0	1
  -2	0.25	1.125
  -3	0.3(3)	1.3(3)
  -4	0.375	1.5625
  -5	0.4	1.8

  
  Important note
  Whereas values of f₂ between -1 and 1 may be easily justified by the way of comparison with generalization of Langmuir kinetics (both gIKL and LF-kinetics on heteregeneous surfaces), rate maxima are not so easily explained. In theory of kinetics of physical adsorption such rate maxima do appear, but after the lateral interactions are introduced (see mRSK, LF-mRSK kinetics). Thus, MOE (which is an empirical equation) for values of f₂ below -1 may well be used to describe in approximate way experimental systems where lateral interactions are important (energetic heterogeneity may mask lateral interaction effects - see e.g. Graham plot or global heterogeneity).
   
  

Kinetics based on Langmuir rate equation and Langmuir isotherm:

• Integrated Kinetic Langmuir equation (IKL) (see [9], Langmuir, 26(19), 15229–15238 (2010)). Corresponds to the equilibrium Langmuir isotherm and Langmuir rate equation. IKL may be also considered as the generalization of the 1st (FOE/PFOE) and 2nd (SOE/PSOE) order equations. It is controlled by Langmuir batch equilibrium factor, f_eq = θ_equ_eq, where θ_eq is given by equilibrium Langmuir isotherm θ_eq(c_eq) (homogeneous surface) and u_eq is equilibrium uptake. IKL produces kinetic behaviours intermediate of the 1st (if f_eq=0, integrated FOE/PFOE is obtained, IKL may be also used) and 2nd order equations (if f_eq=1, i.e. contribution of 2nd order kinetics f₂=1, integrated SOE/PSOE is obtained - general integrated kinetic Langmuir eq., IKL, cannot be used). Its mathematical form is identical to MOE, however IKL may be used only for systems where f_eq = θ_equ_eq and θ_eq is given by equilibrium Langmuir isotherm (homogeneous surface) and u_eq is equilibrium uptake, whereas MOE may be used also for other systems (e.g. for heterogeneous surfaces) where f₂ is not equal to θ_equ_eq.
  See also generalization of IKL (gIKL) (see Ref. [12], Adsorption, 19(2-4), 391-406 (2013)) to adsorption and desorption in systems with any possible combination of initial concentration and coverage.
  
  Langmuir rate equation was introduced by Irving Langmuir in several papers (1916-1918), but was not fully solved until 2010 (A.W. Marczewski, [9], Langmuir, 26(19), 15229–15238 (2010)) with more general solution in 2012 (A.W. Marczewski et al., Ref. [12]). Earlier attempts used approximations or lead to analytical but complicated equations (e.g. Azizian 2004), which could not allow to understand fully its properties.
  
  
  See Ref. [9], Langmuir 26(19), 15229–15238 (2010).
  
  General adsorption/desorption rate eq.:
  
  Langmuir rate equation, classic form:
  
  Langmuir isotherm: , where
  Relation of initial and equilibrium conditions in solution is given by solute uptake:
  and
  temporary concentration and adsorption coverage are expressed by equil. uptake and coverage and adsorption progress:
  and
  
  Langmuir isotherm by using initial concentration and equil uptake:
  
  Langmuir rate equation, new forms:
  
  
  Langmuir batch equilibrium factor:
  
  Langmuir rate equation, new forms, as adsorption progress rate dF/dt:
  
  or
  
  relation of rate coefficients of MOE (1st order coeff., k₁) and Langmuir rate equation (adsorption rate coeff., k_a):
  
  
  Integrated Kinetic Langmuir eq. (IKL) - various forms for adsorption progress:
  and
  
  Integrated Kinetic Langmuir eq. (IKL) - forms for (experimental) adsorption and concentration:
  and
  
  If f_eq = θ_equ_eq = 0, 1st order FOE/PFOE equation is obtained
  If f_eq = θ_equ_eq = 1, 2nd order SOE/PSOE equation is obtained
  If f_eq ≠ θ_equ_eq, mixed 1,2-order (MOE) equation is obtained (f₂ = f_eq)
  
  
  Adsorption half time is:
    t_0.5 = ln (2 - f_eq) / k₁    if    f_eq < 1
    t_0.5 = 1 / k₂    if    f_eq = 1 (see SOE/PSOE and MOE)    and    k₂ = k_a c_o.
  
  Lagmuir adsorption kinetics on pure surface - Langmuir rate and IKL plots:

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  Langmuir batch equilibrium coefficient f_eq - similarity to SOE/PSOE (red) and FOE/PFOE (blue) is indicated:
  
  

  ---

  Relative rate profile for Langmuir kinetics (IKL, adsorption on pure surface only):
  
  

  ---

  Langmuir kinetics (IKL) - in Lagergren plot and in alternative SOE/PSOE plot:
  
  

  ---

  Langmuir kinetics (IKL) - dependence on reduced time (linear and log scale):
  
  

  ---

  Langmuir kinetics (IKL) - compact plot:
  

  ---

  
  In the paper also:
  initial rate analysis; pseudo vs. non-pseudo equations (see also below);
  In the supporting info:
  on multiexponent (m-exp) equation; on n-th order (NOE) equations; on infinite order series expansion (ISE) eqns.
   
  
• generalized Integrated Kinetic Langmuir equation (gIKL) (see [12], Adsorption, 19(2-4), 391-406 (2013)). Corresponds to the equilibrium Langmuir isotherm and classic Langmuir rate equation. Whereas the IKL's simple form corresponds to adsorption on initially pure surfaces, gIKL describes Langmuir kinetics of adsorption and desorption with any possible combination of the initial "o" and equilibrium "eq" conditions.
  
  See Refs: [12], Adsorption, 19(2-4), 391-406 (2013) and (IKL only) [9], Langmuir 26(19), 15229–15238 (2010).
  
  General adsorption/desorption rate eq.:
  
  Langmuir rate equation, classic form:
  
  Langmuir isotherm: , where
  Both adsorption and desorption progress, F(t), may be defined by using initial ("o") and equilibrium ("eq") values of concentrations, c or adsorption, a (or relative adsorption coverage, θ = a / a_m) values (the relative adsorption uptake u_eq - see IKL - cannot be used for desorption process and is not sufficient to describe adsorption from pre-adsorbed state):
    F = (c - c_o) / (c_eq - c_o)
    F = (θ - θ_o) / (θ_eq - θ_o)
    F = (a - a_o) / (a_eq - a_o)
  hence, temporary values of concentration, c(t), or adsorbed amount, a(t) are expressed by using adsorption/desorption progress, F(t):
    c = c_o + F * (c_eq - c_o)
    θ = θ_o + F * (θ_eq - θ_o)
    a = a_o + F * (a_eq - a_o)
  
  Flying-start approach to of adsorption and desorption allows to solve general kinetic equations if a simple case is known or easily calculated. It is simple (even elementary) but powerful technique.
  By appropriate scaling of adsorption or desorption progress F (as well as θ, c, a) any kind of adsorption curve (with exception of diffusion models) may be shown as a part of the simple case of adsorption on pure surface (or any other easily solved case). It is useful both for solving analytical equations (like deriving gIKL from IKL) as well as for numerical purposes (e.g. calculation of LF-mRSK or LF-SRT kinetic curves), as it does not require often complicated expressions for a general case. Entire approach is explained in the supporting electronic material for the Ref. [12], Adsorption, 19(2-4), 391-406 (2013).
  
  gIKL has the same mathematical form as the IKL, however, its controlling parameter f_eq (values in the range <0;1)) is replaced by f_L with values in the range <0;1) for adsorption and <-1; 0> for desorption. Likewise, the rate constant k₁ (it is equal to the rate constant of the IKL's 1-st order term) is replaced by k_L. In the consequence for adsorption, the rate da/dt as a function of adsorption/desorption progress F decreases fast and then slower (f_L ≥ 0), whereas for desorption the rate initially decreases slowly and then faster ((f_L ≤ 0). It means that desorption curves a_des(t) will have longer initial linear part (almost constant rate, da/dt) than adsorption curves a_ads(t).
  
  In this paper (Ref. [12], Adsorption, 19(2-4), 391-406 (2013)) LF kinetics (i.e. IKL/gIKL with LF heterogeneity and no lateral interactions, i.e. LF-mRSK without lateral interactions) is also discussed.
  
  Generalized Langmuir kinetics - adsorption/desorption rate and gIKL plots

  ---

  Adsorption/desorption relative rate profiles (gIKL):
  
  

  ---

  Generalized Langmuir kinetics (gIKL) - dependences on reduced time (half-log and linear scale):
  
  

  ---

  Generalized Langmuir kinetics (gIKL) - compact plot:
  

  ---

  
  
  more soon ...
  
  In the paper also:
  - Analysis and discussion of Bangham plots for experimental data
  - Classic intraparticle diffusion model (IDM) (Crank) for non-constant solute concentration - initial rate analysis (analytical solution)
  - McKay's pore diffusion model (PDM)
  - Extension of MOE to describe wider range of behaviors
  - Extension of fractal-like MOE (Haerifar, Azizian) (fMOE) to adsorption/desorption
  - Enalysis of experimental data
  
  In the supporting info:
  - General method for extending kinetic equations to various initial and equilibrium conditions; wider analysis of equations and experimental data (pdf)
  - Experimental kinetic adsorption/desorption data (xls).
   
  
• mRSK and mixed LF-mRSK kinetic equations.
  
  See Refs: [11], J. Colloid Interface Sci., 361, 603-611 (2011)
  also: [5], Appl. Surf. Sci., 256, 5145-5152 (2010) and [9], Langmuir, 26(19), 15229–15238 (2010)
  
  
  Modifed regular solution kinetic model (mRSK) and mixed LF-mRSK model:
  It was shown that the FG lateral interactions model commonly used in desorption term of kinetic equations describing adsorption kinetics from dilute solutions within the framework of the Langmuir kinetic model (ART/TAAD) does not sufficiently describe interactions. By using regular solution model, RSK model was obtained with symmetrical interactions factors for adsorption and desorption (adsorption term grows with increasing interactions, desorption term decreases), which is in agreement with the equilibrium FG-model (mean-field approximation, MFA). By introducing asymmetric adsorption/desorption terms and specific (association) Kiselev factor in the desorption term, the mRSK model was obtained. After including Langmuir-Freundlich heterogeneity, the LF-mRSK model was derived and applied to analysis of experimental data. The properties of mRSK were fully investigated by model calculations for varying interaction terms - specific and non-specific, varying ratio of non-specific adsorption and desorption terms, varying relative solute uptake at equilibrium, u_eq, varying equilibrium coverage θ_eq (see also IKL and MOE).
  Many additional model figures, extended discussion and data plots are supplied as Supplementary Material (mmc1.pdf). The experimental data of adsorption kinetics of 4-nitrobenzoic acid (pH=2.2) on experimental mesoporous carbon W-120B with various plots are also supplied (mmc2.xls MS Excel file).
  
  LF kinetics (i.e. IKL/gIKL with LF heterogeneity and no lateral interactions) is discussed in Ref. [12], Adsorption, 19(2-4), 391-406 (2013).
  
  more soon ...
  
   
  

• Muli-exponential (m-exp) (see all refs above, especially [5], Appl. Surf. Sci., 256, 5145-5152 (2010)) - sum of Lagergren-like time-exponential terms. Describes well both concentration and adsorption changes for practically all adsorption systems (on porous and non-porous adsorbents). Very flexible empirical equation, may be considered as an approximate solution for description of many parallel or follow-up kinetic processes. In practice, 1-5 exponential terms may be fitted in a stable way - it depends on the data quality and the range of F and time. Analysis of its parameter spectrum, f_i(log k_i) allows to characterize an adsorption system and compare it with the other ones). Alternatively, you may use f_i(log t_0.5,i) parameter spectrum, where t_0.5,i = ln 2 / k_i is kinetic half time (corresponds to 50% adsorption progress for any given kinetic term).
  
  In the literature, this equation is most often used with only 2 terms. Typical names and abbreviations are: double exponent (DE), multiple exponent (ME). However, by generalizing FOE/PFOE and m-exp other equations are obtained using so called stretched-exponentials (SE), exp[-(kt)^m]: KEKAM (or stretched exponential, SE), double stretched exponential (DSE) or multiple stretched exponentials (MSE). Exponents m differing from 1 are supposed to take into account system's deviations from ideality or fractal nature of its structure. If the exponent m<1 then the initial rate will be infinite.
  See e.g. Ref. [5], Appl. Surf. Sci., 256, 5145-5152 (2010).
  
  Multi-exponential (m-exp) eq.:
  and
  where f_i is i-th process contribution in total adsorption and:
  Multi-exponential (m-exp) rate, dF/dt:
    where   (dF/dt)_ini = sum(f_i k_i)
  Multi-exponential (m-exp) rate, dc/dt:
    where   (dc/dt)_ini = c_ou_eq sum(f_ik_i)
  
  where partial uptakes of total solute in the system:
  and
  
  NOTE. If all coefficients f_i and k_i are positive then the relative rate profile will always be below the line for a single 1st order kinetics: (dF/dt)_FOE,rel = (1-F). Hence, (dF/dt)_m-exp,rel ≤ (1-F).
  
  Overall adsorption half time t_0.5 of m-exp equation must be calculated numerically. As long as the true equilibrium is not attained during the experiment, equilibrium adsorption a_eq will be an extrapolated value and in consequence time to a_0.5 = 0.5 a_eq (i.e. time to F=0.5) will inherit this property.
  
  Instead of using just one overall adsorption half time, one may calculate adsorption half time for each of the m-exp terms as:
    t_0.5,i = ln 2 / k_i.
  
  In order to compare various adsorption profiles it is advised to compare their respective parameter spectra as:
    f_i vs. k_i    or    a_eq,i vs. k_i
     where: a_eq,i = a_eqf_i is the amount adsorbed by i-th process.
  In many cases better representation is :
    f_i vs. ln k_i    or    a_eq,i vs. ln k_i
  However, the rate constants k_i are not very intuitive and it is much easier to analyze system behavior if we use adsorption half times t_0.5,i instead of rate constants k_i - it also allows to compare effect of each term on the overall adsorption half time t_0.5 as well as to the total time of the experiment t_max (estimated half times near or larger than this value may be considered as extrapolated values and as such are much less certain).
    f_i vs. ln t_0.5,i    or    a_eq,i vs. ln t_0.5,i
   
  
• Statistical rate theory (SRT)..
  SRT model was developed mainly by C.A. Ward and co-workers (1982-1997) and by W. Rudzinski and his group (1997-2012). Equilibrium of SRT kinetics corresponds to the Langmuir isotherm, however, its desorption rate term is a reciprocal of adsorption term, which is proportional to [c(1-θ)/θ] (for equilibrium Langmuir isotherm). Thus SRT in its standard form produces infinite initial rate and initially depends on square root of time, just like the IDM (a ~ t^1/2). If energetic heterogeneity is included (e.g. Langmuir-Freuindlich eq.: LF-SRT), the initial time dependence corresponds to an even smaller exponent (a ~ t^m/(1+m), where m is heterogeneity coefficient: 0 < m ≤1 ).
  This model is discussed e.g. in Refs. [9], Langmuir, 26(19), 15229–15238 (2010) and [11], J. Colloid Interface Sci., 361, 603-611 (2011).
  
  Classic SRT (homogeneous surface, no lateral interactions; at equilibrium Langmuir isotherm is obtained):
  
      (dθ/dt) = K_ls {(Kc)[(1-θ)/θ] - [θ/(1-θ)]/(Kc)}
      where K_ls defines liquid/solid exchange rate.
  
  If K_ls = const, then initially infinite rate is obtained (the same type of dependence as for intraparticle diffusion, i.e. IDM / Weber-Morris equations, as well as for empirical MPFO):
      dθ/dt) ~ 1/θ   and   θ ~ t^1/2
  
  Classic LF-SRT (heterogeneous surface - Langmuir-Freundlich energetic heterogeneity corresponding to the Langmuir-Freundlich isotherm/LF; non-specific lateral interactions - mean field approximation, corresponding to the Fowler-Guggenheim isotherm /FG; corresponds to the equilibrium LF-FG):
  
      (dθ/dt) = K_ls {(Kc)g_int(θ)[(1-θ)/θ]^1/m - [θ/(1-θ)]^1/m/[(Kc)g_int(θ)]}
      0 < m ≤ 1 - heterogeneity coefficient (see LF isotherm)
      g_int(θ) = exp(αθ) - lateral interaction factor (see FG isotherm) - random topography of adsorption sites is assumed.
  
  If K_ls = const, then initially infinite rate is obtained:
      dθ/dt) ~ 1/θ^1/m   and   θ ~ t^m/(1+m)
  
  NOTE. Quite often SRT proponents define K_ls as concentration and coverage dependent, i.e. K_ls = f(c,θ). While the above "classic SRT" (m=1) or "classic LF-SRT" (0<m≤1) correspond to infinite initial rates (initially (dθ/dt) ~ 1/θ^m/(1+m)), various modifications of K_ls may alter this behavior.
  
  NOTE. Big problems of this equation in its classic form is its infinite initial rate, however, some authors (e.g. Panczyk 2002 - although for gas adsorption only) modify classic form of SRT in a way that it allows them to get rid of the infinite initial rate.
  
  more soon ...
   
  
• KEKAM aka Avrami aka Erofeev aka stretched exponential (SE) eq..
  KEKAM equation was developed by Avrami, Erofeev and co-workers in 1930-s/1940-s. It is also knows as stretched exponential eq. (SE). It was explained by Sotalongo-Costa and Brouers (2006) as adsorption kinetics in disordered porous solid.
  
  KEKAM / Avrami / Erofeev / SE:
  
      F = 1 - exp[-(k₁t)^p]
  
  NOTE. Parameter p should be close to 1. If p=1 we obtain FOE/PFOE.
  NOTE. Data conforming to this equation has linear Bangham plot.
  
  Adsorption half time is:
      t_0.5 = (ln 2)^1/p / k₁
  
  This equation may produce inifinite initial rate (if p < 1).
  
  
  more soon ...
   
  
• Fractal-like MOE (fMOE)..
  Fractal-like MOE (Haerifar, Azizian 2012) may be considered as a modification/generalization of MOE by introducing factors corresponding to non-ideality according to KEKAM equation (aka. Avrami eq., aka. Erofeev eq., aka. stretched exponential eq. (SE)) and explained by Sotalongo-Costa and Brouers (2006) as fractal kinetics.
  This model is discussed e.g. in Ref. [12], Adsorption, 19(2-4), 391-406 (2013) (application for adsorption and desorption in porous solids).
  
  Fractal-like MOE (Haerifar, Azizian 2012), (deviation of p from 1 determines influence of system "fractality").
  
  If f₂ < 1, then analog of MOE is obtained:
      ln[(1-F)/(1-f₂F)] = -(k₁t)^p
      F = {1 - exp[-(k₁t)^p]} / {1 - f₂ exp[-(k₁t)^p]}
  
  If f₂ = 0 we obtain analog of FOE/PFOE i.e. KEKAM aka. Avrami aka. Erofeev aka. SE eq.:
      F = 1 - exp[-(k₁t)^p]
  
  If f₂ = 1 we obtain analog of SOE/PSOE :
      F = (k₂t)^p / [1 + (k₂t)^p]
  
  NOTE. Parameter p should be close to 1. If p=1 we obtain MOE and SOE/PSOE.
  
  Adsorption half time is:
      if f₂<1   then   t_0.5 = [ln (2-f₂)]^1/p / k₁
      if f₂=1   then   t_0.5 = 1/k₂
  
  f-MOE equation may produce inifinite initial rate (if p < 1).
  
  
  more soon ...
   
  

• First order (FOE/PFOE) (see [5], Appl. Surf. Sci., 256, 5145-5152 (2010)). Empirical equation - may be considered as a special/boundary solution of the Langmuir rate theory and special case of the MOE equation. FOE is concentration dependent, adsorption dependent is PFOE - both forms are equivalent if mass balance is used and equation presented as adsorption progress F(t).
  
  See Refs. [5], Appl. Surf. Sci., 256, 5145-5152 (2010) and [9], Langmuir, 26(19), 15229–15238 (2010)
  Eq. was introduced by Lagergren in 1898.
  
  FOE/PFOE rate equations:
  PFOE: and FOE: or
  equivalently for FOE/PFOE: where: k₁=k_1a=k_1c.
  
  FOE/PFOE time-integrated equations:
  
  
  FOE/PFOE time-integrated equations as linear Lagergren plots:
  PFOE: and FOE: or
  equivalently for FOE/PFOE:
  
  NOTE. Serious problem with using this so called Lagergren plot or linear Lagergren equation lies in the necessity to obtain (possibly independently) equilibrium data (c_eq, a_eq) prior to plotting this data. Hence when we draw line through the initial point (t=0, ln(a_eq) we may see whether this data is 1st order or not. However, if we try to fit straight line to such a plot, the plot-estimated a_eq and value a_eqused to calculate data will be different. Moreover, quite rarely the estimated equilibrium will be the real one.
  
  Adsorption half time is:
    t_0.5 = ln 2 / k₁ ≈ 0.693 / k₁
  
  
   
  
• Second order (SOE/PSOE) (see [5], Appl. Surf. Sci., 256, 5145-5152 (2010)). Empirical equation - may be considered as a special/boundary solution of the Langmuir rate theory and a limiting solution of the MOE rate equation. SOE is concentration dependent, adsorption dependent is PSOE - both forms are equivalent if mass balance is used and equation presented as adsorption progress F(t).
  
  See Refs. [5], Appl. Surf. Sci., 256, 5145-5152 (2010) and [9], Langmuir, 26(19), 15229–15238 (2010)
  Eq. introduced and disscussed by: Ritchie 1977, Blanchard 1984, Ho & McKay 1998.
  
  SOE/PSOE rate equations:
  PSOE: and SOE or
  equivalently for SOE/PSOE: where
  
  SOE/PSOE time-integrated equations:
  PSOE: and SOE/PSOE:
  
  SOE/PSOE time-integrated equations as linear plots:
  PSOE standard linear plot: - good for estimation of adsorption equilibrium
  PSOE alternative linear plot: - good as a tool exposing all discrepancies from SOE/PSOE conformance.
  
  Adsorption half time is:
    t_0.5 = 1 / k₂    where    k₂ = k_2aa_eq
  
  
   
  
• N-th order (NOE/PNOE) (introduced by Ritchie (J.Chem.Soc.Faraday Trans.I, 1977), discussed by C.W. Cheung, J.F. Porter, G. McKay (Water Res. 2001); see also Refs. [5], Appl. Surf. Sci., 256, 5145-5152 (2010) and [9], Langmuir, 26(19), 15229–15238 (2010)). Empirical equation - 1st and 2nd order equations are its special cases. NOE is concentration dependent, adsorption dependent is PNOE - both forms are equivalent if mass balance is used and equation presented as adsorption progress F(t).
  
  See Refs. [5], Appl. Surf. Sci., 256, 5145-5152 (2010) and [9], Langmuir, 26(19), 15229–15238 (2010)
  Eq. introduced by Ritchie (1977) and disscussed by e.g. C.W. Cheung, J.F. Porter, G. McKay, (2001):
  ref: A.G. Ritchie, J. Chem. Soc. Faraday Trans. I, 1977, 73, 1650-1653.
  ref: C.W. Cheung, J.F. Porter, G. McKay, Water Res. 2001, 35(3), 605-612.
  
  NOTE. There are problems in definitions of θ in Cheung, Porter, McKay (2001):
  

  1. In Nomenclature: θ = q_t/q_e is called "surface coverage" (but seems to be relative adsorption progress, instead).
  2. In eq. (2), Langmuir rate equation: θ is probably just surface coverage (here:: θ = a/a_m).
  3. In eq. (3), Langmuir isotherm: q_e is clearly adsorbed amount for a given concentration (there is no indication that it might be maximum adsorbed amount, as required by the definition present in the Nomenclature) (here: a_eq = q_e). Moreover, adsorption capactity (max. adsorption) in this equation is q_max = K_LFa_LF (here:: adsorption monolayer capacity, a_m).
  4. In eqns. 7-15 equations θ is used in two ways - like a true coverage, but also - when we take into account C.P.M. definition (θ = q_t/q_e) - like relative adsorption progress F (here:: F= a/a_eq=θ/θ_eq).
  5. It does not cause too much of a problem when adsorption occurs on pure surface (i.e. θ_o=0), however, such conflicting definitions become problematic for adsorbent with some pre-adsorbed adsorbate. Both definitions do not clash only if q_e = q_max (in C.P.M. paper, from eq.(3): q_max = K_LFa_LF). It would correspond to θ_eq=1, which is clearly never mentioned (nor implied) in the C.P.M. paper.

  
  In the following equations corrected approach (AWM: see Refs. [5] and [9]) is used, where θ is relative coverage and F is relative progress.
  
  PNOE rate equation (pseudo n-th order):
      (dF/dt) = k_n (1-F)ⁿ
      (dθ/dt) = k_nθ (θ_eq - θ)ⁿ   where   k_nθ = k_n/(θ_eq)^n-1
      (da/dt) = k_na (a_eq - a)ⁿ   where   k_na = k_n/(a_eq)^n-1
      (dc/dt) = -k_nc (c - c_eq)ⁿ   where   k_nc = k_n/(c_o - c_eq)^n-1
  
  As usual, for batch adsorption (volume V, adsorbent mass m, adsorption capacity a_m):
      (da/dt) = -(m/V)(dc/dt)
      (da/dt) = a_m(dθ/dt)
      (dθ/dt) = θ_eq(dF/dt)
  
  
  If initially F_o = 0, i.e. θ_o = 0 (adsorption on pure surface), and n≥2 (if n=1 PFOE is obtained) then:
      F = {1 - 1/[1 + k_n(n-1)t]^1/(n-1)}
  hence (because θ = θ_eqF):
      θ = θ_eq {1 - 1/[1 + k_n(n-1)t]^1/(n-1)}
  
  Adsorption half time is then (n≥2):
      t_0.5 = (2^n-1 - 1)/ [k_n(n-1)]
  
  If initially F_o > 0, i.e. θ_o=θ_eqF_o > 0 (pre-adsorption) , and n≥2 then:
      (1-F) = (1-F_o) / [1 + k_n(n-1)Δt(1-F_o)^n-1 ]^1/(n-1)
      where Δt = t-t_o is "new" experimental time (no knowledge of t_o is required, however).
  Again, θ(t) = θ_eqF(t).
  
  NOTE. A different simple yet general approach to pre-adsorption or desorption is considered in Ref. [12], Adsorption, 19(2-4), 391-406 (2013). For a given experiment only current observation is considered with its adsorption/desorption progress always scaled from 0 to 1 (i.e. from a_o, c_o for F=0 to a_eq, c_eq for F=1). However, this simple flying start approach (see description) is not valid for equations involving diffusion in porous media.
  
  more soon ...
   
  
• Infinite series expansion (ISE/PISE) (see Refs. [5], Appl. Surf. Sci., 256, 5145-5152 (2010) and [9], Langmuir, 26(19), 15229–15238 (2010)). Empirical equation - 1st, 2nd or n-th order equations are its special cases, as well as MOE (mixed 1,2-order eq.). ISE is concentration dependent, adsorption dependent is PISE - both forms are equivalent if mass balance is used and equation presented as adsorption progress F(t).
  If the 1st order term of ISE/PISE has non-zero contribution then its near-equilibrium behavior will mimic 1st order (exponential decay: FOE/PFOE, MOE, IKL/gIKL, IDM, MPFO - but not SOE/PSOE). In general the lowest-exponent with non-zero contribution determines ISE/PISE near-equilibrium behavior.
  See Refs. [5], Appl. Surf. Sci., 256, 5145-5152 (2010) and [9], Langmuir, 26(19), 15229–15238 (2010)
  
  General expression for ISE/PISE rate:
      dF/dt = (dF/dt)_ini sum[ f_i (1-F)ⁱ , i=1, ∞]
  Where initial rate (dF/dt)_ini is finite and term contributions are normalized to 1:
      sum[ f_i , i=1, ∞] = 1
  
  As we can see from the general Langmuir (IKL, gIKL) and MOE, term contributions may be positive (IKL for adsorption) or negative (gIKL, MOE for desorption) and not necessarily fractional (e.g. f₁=3, f₂=-2 and f₁+f₂=1), however, in such a general case (see MOE) rate maximum may appear.
  
  more soon ...
   
  
• Pseudo- vs. non-pseudo-n-th order kinetic equations.
  Pseudo n-th order rate equations - rate depends on adsorbed amount: (a-a_eq)
  Non-pseudo n-th order rate equations - rate depends on concentration (or pressure): (c-c_eq)
  
  See Refs: [5], Appl. Surf. Sci., 256, 5145-5152 (2010) and [9], Langmuir, 26(19), 15229–15238 (2010)
  
  Pseudo- vs. non-pseudo-n-th order kinetic equations:
  It was shown by easy analysis (ref. [5] and [9]) that in batch experiments n-th order rate equations (FOE, SOE, general n-th order NOE, ISE etc.) are mathematically equivalent to the pseudo n-th order rate equations (PFOE, PSOE, general nt-th order PNOE, PISE etc.). This distiction was strongly promoted e.g. by Y.S. Ho. However, the problem with using solely pseudo-n-th order equations is not only the problem of naming convention. As it was shown in [9] the description of kinetic experiments by using adsorption vs. time data only (a(t) or q(t)) and not concentration vs. time data (c(t)) moves focus away from the important factor which is the magnitude of relative uptake at equilibrium, u_eq. In fact Langmuir kinetics may be considered as a perfectly symmetric with respect to the pseudo- and non-pseudo rate equation problem, which is nicely exposed in the key role of "Langmuir batch equilibrium factor" f_eq = θ_equ_eq.
  
  PFOE: and FOE: or for FOE/PFOE: where: k₁=k_1a=k_1c.
  
  PSOE: and SOE or for SOE/PSOE: where
  
  
  For the n-th order (NOE/PNOE), mixed 1,2-order MOE and infinite expansion series (ISE/PISE) the same applies, i.e.:
      k_ia = k_ic (m/V)^i-1
  
   
  

• Elovich equation.
  Elovich equation describes kinetics of chemisorption, adsorbed amount has no max. value. Some authors (e.g. Aharoni and co-workers: Tompkins, Suzin, Ungarish) used this equation to describe middle part of diffusion kinetics.
  References (some):
  1. Aharoni, Tompkins, Trans. Faraday. Soc. 1970, 66, 434-444
  2. Aharoni, Suzin, J. Chem. Soc. Faraday Trans. 1, 1982, 78, 2321-2327
  
  Elovich equation:
  adsorption rate (chemisorption):
  
  
  integrated forms (initially: a=0, t=0):
  
  
  
  simplified form used in experimental plots:
  where
  
  If we integrate not from the initial a=0, t=0, but we assume, that at t=t₀ adsorbed amount is a_o, then:
  
  and we may plot data as:
  
  
  Aharoni with co-workers use Elovich equation to approximate intraparticle diffusion equations. Then in the initial part the Weber-Morris equation may be used:
  
  Elovich equation is then used above t = t_p:
  
  where t_p is the inflection point of the plot:
  
   
  
• Modified pseudo-first order eq. (MPFO).
  MPFO eq. (X.Y. Yang, B. Al-Duri, J. Colloid Interface Sci. 2005, 287, 25-34) is an empirical modification of PFOE. It's behavior quite well approximates systems, where the initial very fast adsorption (infinite initial rate, analogous to the IDM) is followed by a typical 1st order middle range and near-equilibrium behavior.
  
  MPFO eq.:
      F + ln(1-F) = -k₁t
  
  MPFO rate:
      dF/dt = k₁(1-F)/F
      Hence for t → 0 (F→0): dF/dt ≈ k₁/F
         and for t → ∞ (F→1): dF/dt ≈ k₁(1-F)
  
  Initial kinetics/rate:
  Initial rate may be calculated directly from the rate expression (above) or by using MPFO eq. itself and its Taylor expansion series'.
  For F → 0 we obtain (Taylor series expansion, -1≤F<1):
      F + ln(1-F) = -sum[Fⁱ/i, i=2, ∞] ≈ -F²/2
  and then for the initial part of MPFO:
      F ≈ (2 k₁t)^1/2 or equivalently, dF/dt ≈ k₁/F
  i.e. relation identical to Weber-Morris approximation of the IDM model or for the low-time part of SRT.
  
  Near-equilibrium rate may be approximated by FOE/PFOE (or equivalence of Boyd equation), i.e. exponentially nearing equilibrium (with a time scale slightly shifted to the left, if compared to FOE/PFOE with the same rate constant):
      F ≈ 1 - exp(-k₁t - 1) = 1 - exp[-k₁(t - Δt_o)]   where   Δt_o = -1/k₁
  
  MPFO half time (approx. 28% of the FOE/PFOE halftime for the same k₁):
      t_0.5 = (ln 2 - 0.5) / k₁ ≈ 0.193 / k₁
  
  more soon ...
   
  

Diffusion-based sorption kinetics

• Intraparticle diffusion model (IDM), Crank et al.
  Standard IDM model describes occlusion (this term is used by e.g. Aharoni and Tompkins) by diffusion into porous particle. The driving force of occlusion is diffusion only and adsorption phenomena are never directly included. Sorption/adsorption is included in the effective diffusion coefficient (Henry isotherm only - it does not change the math. solution) and indirectly by allowing particle to occlude more adsorbate than results from particle volume (concentration of adsorbate is greater than in solution).
  This model is discussed e.g. in Ref. [12], Adsorption, 19(2-4), 391-406 (2013) and some in [11], J. Colloid Interface Sci., 361, 603-611 (2011).
  
  Introduced and discussed by many authors, e.g. Boyd, Myers, Adamson 1947, Reichenberg 1953, Crank 1953, 1975 etc. Simplified versions are due to (Boyd, Myers and Adamson) and Reichenberg. So called Boyd eq. was developed by Boyd, Myers and Adamson. So-called Weber-Morris eq. (1963) was earlier proposed by Boyd, Myers and Adamson.
  In all equations effective diffusion coefficient D_a is used:
      D_a = D / [τ ( 1 + ρK_Hε_p ]
  where: D is molecular diffusion coeff., τ is (dimensionless) pore tortuosity factor, ρ is particle density, K_H is (adsorption) Henry constant, ε_p is particle porosity.
  
  Near-equilibrium behavior is similar to the FOE/PFOE, MOE or IKL, MPFO, m-exp or any any equation with non-zero first order term in its expension series:
  
  Boyd eq. (Boyd, Myers, Adamson 1947) - simplified, spherical particles (radius r), near equilibrium:
      F ≈ 1 - (6/π²) exp (-π² D_a t / r²)
  
  Initial behavior: For all various solutions (spheres or slabs, constant or varying concentration) the infinite initial rate (da/dt)_ini is obtained.
  
  Simplified IDM (Boyd, Myers, Adamson 1947), spherical particles, low times:
      F ≈ (6/r)(D_at/π)^1/2
  
  So-called Weber-Morris eq. (1963) - simplified IDM, low times (the same as obtained by Boyd, Myers and Adamson in 1947):
      a ~ t^1/2
      (One should note that in such a case: da/dt ~ 1/a, see also MPFO and SRT)
  
  Marczewski et al. (see Ref. [12]) for spherical particles and varying concentration, low times:
      F ≈ (1-u_eq)(6/r)(D_at/π)^1/2
  hence, rate dF/dt ~ 1/F:
      dF/dt ≈ 0.5(1-u_eq)²(6/r)²(D_a/π) / F
  For concentrations this solution gives:
      c_o - c ≈ c_ou_eq (1-u_eq)(6/r)(D_at/π)^1/2
  
  Good approximation (relative error < 10^-9) for low times (F < 0.5), sperical particles (radius r) and constant concentration (uptake u_eq=0) (Reichenberg 1947):
      F ≈ (6/π^3/2)(Bt)^1/2 - (3/π²)(Bt)
      Bt ≈ 2π - π²F/3 - 2π(1 - πF/3)^1/2
      where B ≈ π²D_a / r²
  
  Hence IDM half time (spherical particles, const. concentration) is approximately:
      Bt_0.5 ≈ 0.3015
  
  Big problems of all these equations is the infinite initial rate. This problem may be solved by introducing some additional constraints, e.g. resistance of the external film transfer (see e.g. PDM/McKay)
  
  
  IDM equations are always given as series formulas. However, those series diverge and thus usually such formulas are suitable only for short and medium times, especially in the case of variable concentration (i.e. u_eq > 0).
  Good approximations and their variants are suggested by Boyd, Myers and Adamson (1947), Carslaw and Jaeger (1947), Reichenberg (1953), Carman and Haul (1954), Crank (1956-1975), Ruthven (1984) and recently by Haynes and Lucas (2007).
  
  Influence of model parameters is shown in model figures (A.W. Marczewski (c) 2013).
  

  ---

  Effect of relative uptake u_eq = 0 ... 1 (fixed D_a/t^1/2 = 0.02) shown as scaled Weber-Morris plots F ~ t^1/2 (log-log scales and linear scales):
  
  
  more soon ...
   
  
• Pore diffusion model (PDM), McKay et al..
  Simple shrinking core approach. It avoids the initial infinite rate present in the IDM by introducing resistance of the external film transfer. However, by simplifying description of diffusion through replacing diffuse adsorbate front with a sharp concentration drop, a finite time to equilibrium is obtainded. This particular model was introduced by McKay at al. 1996.
  
  This model is discussed e.g. in Ref. [12], Adsorption, 19(2-4), 391-406 (2013). It is also compared to the IDM and MOE.
  
  Pore Diffusion Model (Ref: McKay, El Geundi, Nassar, Process Saf. Environ. Prot. 1996, 74, 277-288).
  
  Position of concentration front, r_c within granule of radius r:
  
  Sorption progress rate, where τ_s is the non-dimensional experiment time:
  
  After integration we obtain reduced (dimensionless) time τ_s needed to reach a particular sorption progress F:
  
  where for simplification some variables are defined:
  ,
  So called capacity factor Ch is equivalent to the relative equilibrium uptake:
  
  other variables are defined by using Bi (Biot number), K_f (external film transfer coefficient) and D_p (pore diffusion coefficient):
  
  
  Due to the simplifications used (sharp adsorbate front: shrinking core approach), a finite equilibrium time, t(F=1), is obtained (See Ref. [12], Adsorption, 19(2-4), 391-406 (2013)). Equilibrium dimensionless time, τ_s,eq is:
  
  In order to obtain real experimental time, t, instead of the "dimensionless time" τ_s simple relations are used:
  
  where experimental half time is:
  
  Influence of model parameters is shown in model figures (A.W. Marczewski (c) 2013).
  

  ---

  Effect of relative uptake u_eq (for fixed Biot number, Bi=1, 2, 10, 100):
  
  

  ---

  Effect of Biot number Bi (for fixed relative uptake, u_eq = 0.001, 0.5, 0.9, 0.999):
  
  Note that for high Biot numbers (Bi > 200), rate profiles feature a very sharp initial drop and McKay's PDM may describe behaviors similar to the infinite initial rate (e.g. IDM). However, the magnitude of this initial drop is also important, e.g. if such a drop is observed in a narrow range, say below F=0.05 (and followed by a slow rate change), than the fast stage will cover only the narrow initial range from 0 to 5% of equilibrium adsorption (very often difficult to observe or measure with sufficient precision).
  
   
  

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