© 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.

Kinetics: ( basics | symbols | half time | plots )
Equations: MOE | Langmuir kinetics ( IKL | gIKL | mRSK/LF-mRSK )
Various: m-exp, ( SRT/LF-SRT ), f-MOE
Other: ( FOE/PFOE | SOE/PSOE | NOE/PNOE | ISE/PISE ), Pseudo/non-pseudo, ( Elovich | MPFO)
Diffusion: ( IDM/Crank | W-M | Boyd ), ( PDM/McKay )
References (see all my papers and ResearcherID):
  1. "Kinetics and equilibrium of adsorption of organic solutes on mesoporous carbons", A.W. Marczewski, Appl. Surf. Sci., 253, 5818-5826 (2007) (pdf available upon e-mail request; doi: 10.1016/j.apsusc.2006.12.037)
  2. ”Kinetics and equilibrium of adsorption of dissociating solutes from aqueous solutions on mesoporous carbons”, A.W. Marczewski, Polish J. Chem., 82, 271–281 (2008) (pdf available upon e-mail request; see vol. 82 abstracts)
  3. “Adsorption of dyes on mesoporous carbons”, A. Deryło-Marczewska, A. W. Marczewski and Sz. Winter, Annales UMCS Sec. AA, vol. LXIII, 287-299 (2008) (pdf available at journal site: doi: 10.2478/v10063-009-0016-0 : Open Access)
  4. "Study of structure properties of organized silica sorbents synthesized on polymeric templates", A.W. Marczewski, A. Deryło-Marczewska, I. Skrzypek, S. Pikus and M. Kozak, Adsorption, 15, 300-305 (2009) (pdf available upon e-mail request; doi: 10.1007/s10450-009-9183-8)
  5. In this paper, MOE (see below) is introduced, m-exp details are also analysed:
    "Application of mixed order rate equations to adsorption of methylene blue on mesoporous carbons", A.W. Marczewski, Appl. Surf. Sci., 256, 5145-5152 (2010) (pdf available upon e-mail request; doi: 10.1016/j.apsusc.2009.12.078)
  6. "Studies of Adsorption Equilibria and Kinetics in the Systems: Aqueous Solution of Dyes – Mesoporous Carbons", A. Deryło-Marczewska, A.W. Marczewski, Sz. Winter, and D. Sternik, Appl. Surf. Sci., 256, 5164-5170 (2010) (pdf available upon e-mail request; doi: 10.1016/j.apsusc.2009.12.085)
  7. "Adsorption of selected herbicides from aqueous solutions on activated carbon", A. Deryło-Marczewska, M. Błachnio, A.W Marczewski, A. Swiatkowski, B. Tarasiuk, J. Therm. Anal. Calorimetry, 110, 785-794 (2010) (pdf by OpenAccess from SpringerLink; doi: 10.1007/s10973-010-0840-7)
  8. "Studies of adsorption equilibria and kinetics of o-, m-, p- nitro- and chlorophenols on microporous carbons from aqueous solutions", A. Deryło-Marczewska, K. Mirosław, A.W. Marczewski, D. Sternik, Adsorption, 16(4-5), 359-375 (2010) (pdf by OpenAccess from Adsorption on SpringerLink; doi: 10.1007/s10450-010-9247-9)
  9. In this paper, Integrated Kinetic Langmuir Equation (IKL) (see below) is introduced:
    "Analysis of Kinetic Langmuir Model. Part I: Integrated Kinetic Langmuir Equation (IKL) - A New Complete Analytical Solution of the Langmuir Rate Equation", A.W. Marczewski, Langmuir, 26(19), 15229–15238 (2010), doi: 10.1021/la1010049 (abstract, pdf and supporting info from Langmuir (ACS))
  10. "Kinetyka desorpcji kwasu salicylowego z mezoporowatych materiałów krzemionkowych" (in Polish; "Desorption kinetics of salicylic acid from mesoporous siliceous materials") , A.W. Marczewski, A. Deryło-Marczewska, P. Adamek, A. Słota, In: "Nauka i przemysł - metody spektroskopowe w praktyce, nowe wyzwania i mozliwosci" (Ed: Z. Hubicki), UMCS Press, Lublin, Poland, 2010 (ISBN 978-83-227-3050-8), pp. 617-620 (pdf in Polish available upon e-mail request)
  11. In this paper, mRSK and LF-mRSK (see below) models are introduced:
    "Extension of Langmuir Kinetics in Dilute Solutions to Include Lateral Interactions According to Regular Solution Theory and The Kiselev Association Model", A.W. Marczewski, J. Colloid Interface Sci., 361, 603-611 (2011), doi:10.1016/j.jcis.2011.06.013 with supplementary material (.pdf and .xls) (experimental data is included).
  12. In this paper, gIKL is introduced (see below) with corresponding MOE generalization:
    "Adsorption and desorption kinetics of benzene derivatives on mesoporous carbons", A.W. Marczewski, A. Deryło-Marczewska, A. Słota, Adsorption, 19(2-4), 391-406 (2013) (is available as pdf (+ 2* supporting materials) by OpenAccess from Adsorption on SpringerLink; doi: 10.1007/s10450-012-9462-7) (experimental data is included).
  13. Adsorption halftimes halftimes are used to study temperature dependence of adsorption kinetic (uses m-exp, MOE, f-MOE, IDM and PDM):
    "Adsorption equilibrium and kinetics of selected phenoxyacid pesticides on activated carbon - effect of temperature", A.W. Marczewski, M. Sęczkowska, A. Deryło-Marczewska, M. Błachnio, Adsorption, 22(4), 777-790 (2016) (is available as article and pdf (+ supporting material, pdf) by OpenAccess from Adsorption on SpringerLink; doi: 10.1007/s10450-016-9774-0).
  14. Adsorption of HSA and BSA proteins on templated carbons (m-exp, MOE, f-MOE, IDM, PDM equations):
    "Kinetics of protein adsorption by nanoporous carbons with different pore size", A.M. Puziy, O.I. Poddubnaya, A. Deryło-Marczewska, A.W. Marczewski, M. Blachnio, M.M. Tsyba, V.I. Sapsay, D.O. Klymchuk, Adsorption, 22(4), 541-552 (2016) (is available as article and pdf (+ supporting materials) from Adsorption on SpringerLink; doi: 10.1007/s10450-015-9723-3).
  15. Adsorption of chlorophenoxy pesticides on granular carbons (m-exp, MOE, f-MOE, IDM, PDM equations):
    "Adsorption of chlorophenoxy pesticides on activated carbon with gradually removed external particle layers", A. Derylo-Marczewska, M. Blachnio, A.W. Marczewski, A. Swiatkowski, B. Buczek, Chemical Enginneering Journal, 308, 408-418 (2017) (is available as Article and pdf (+ supplementary data mmc1.pdf) from ScienceDirect (Elsevier); doi: 10.1016/j.cej.2016.09.082).
    Important note: all kinetic equations discussed in the paper are defined in supplementary data file mmc1.pdf.

Other References (see e.g. this ResearcherID):
  1. "Novel zeolite composites and consequences for rapid sorption processes", A. Brandt, M. Bülow, A. Deryło-Marczewska, J. Goworek, J. Schmeißer, W. Schöps, B. Unger, Adsorption, 13, 267-279 (2007) doi: 10.1007/s10450-007-9019-3)

General remarks:

In all equations:

  a - adsorbed amount,
  c- concentration,
  t - time, t0.5 - half time
  subscripts: "o" - initial, "eq"- equilibrium,
  fi - 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 = (co-c)/co - solute uptake (relative, batch conditions),

Adsorption half time:
In physics: "half time" (sometimes "half-time" or "halftime") is defined as a time necessary for the system to attain 50% of some change. In the case of adsorption kinetics it is time needed to attain half of the equilibrium adsorption value (a(t0.5) = 0.5aeq), half of the concentration change (c(t0.5) = 0.5(co + ceq), half of the adsorption progress (F=0.5).

Overall adsorption half time t0.5 is a general and simple model-independent parameter allowing to assess kinetic properties of a system - it may be also used to compare various systems without any specific assumptions. If we use any kind of real data (including data scatter, e.g. missing near equilibrium part), then half time must be determined numerically.

As long as the true equilibrium is not attained during the kinetic experiment, equilibrium adsorption aeq will be an extrapolated value and in consequence time to a0.5 = 0.5 aeq (i.e. time to F=0.5) will inherit this property. Of course, when the maximum relative experimental uptake umax is near 1, then umax ≤ ueq ≤ 1 and the error in estimation of equilibrium adsorption becomes negligible. However, the error in determination of equilibrium concentration ceq expressed in fractional (ceq,est / ceq,real) or in log scale, log (ceq,est / ceq,real), may be very significant. (On the other hand, if maximum experimental uptake umax and extrapolated/fitted value ueq are not close to 1, then the relative error of ceq is smaller at the cost of larger error of aeq).

Due to the general properties of m-exp (below) it may be used as a general kinetic fitting/smoothing equation (polynomials are unstable), as well it allows to estimate in a reliable way the equilibrium adsorption and concentration (extrapolation!). Depending on the equation used adsorption half time may be calculated analytically (FOE/PFOE, SOE/PSOE, IKL, gIKL, MOE, MPFO etc.) or must be calculated numerically (m-exp, mRSK, LF-mRSK, SRT, IDM, PDM). For equations that do not show equilibrium (e.g. Elovich - adsorption goes to infinity) it cannot be calculated.
Kinetic plots and fitting:
It is strongly advised to use non-linear fitting (general LSQ optimization) methods instead of various popular linear plots, or at least to use such plots with extrene caution. The main reason is the transformation of data deviations caused by using such plots. Of course, by using appropriate weighted fitting such problem may be overcome, however, simple linear LSQ becomes inadequate. Another problem is that such linear plots may be used only for a few simplest equations (e.g. 1st order Lagergren plot is not a true linear plot - one parameter must be known in advance).
However, various plot are great tools allowing to notice and understand character of deviations from the fitted/assumed theoretical model (e.g. systematic or random). Pure optimizations are not good (no more than linear plots) as the only criterion for assessment of model suitability for description of some system data.

Selected standard plots:
- 2nd order SOE and PSOE linear plots:
    standard SOE/PSOE plot:
    t/a vs. t
    NOTE. Good to estimate equilibrium adsorbed amount, aeq.

    alternative SOE/PSOE plot:
    a vs. a/t
    NOTE. Good to expose deviations from PSOE.

- 1st order Lagergren plot:
    ln(a - aeq) vs. t
    NOTE. Sensitive to even minor deviations near equilibrium. Due to its mathematical properties this plot should not be used near the equilibrium (even for true 1st order data).

- Bangham plot:
    standard form (log() or ln() may be used alternatively):
    log{log[co / (co - ma/V)]} vs. log t
    it is equivalent to the much simpler:
    ln[ln(co / c)] vs. ln t
    NOTE. Helpful in data analysis/comparisons, but overrated as a tool for detecting effects of diffusion - fully linear plot actually means that the data conforms to: a/aeq = 1-exp(-ktn), i.e. the KEKAM/Avrami/Erofeev eq. (recently obtained also by using fractal approach to kinetics). If the slope of Bangham plot is 0.5 it may mean diffusion, however, similar behaviors are also obtained e.g. for SRT (slope 0.5), LF-SRT (slope ≤ 0.5), fractal kinetics etc.
Following Aharoni (1984) and other authors it is usually assumed, that experimental Bangham plots with slope <0.5 correspond to energetic heterogeneity or chemisorption, whereas those in the range 0.6-0.7 are rather diffusive in nature.

- so-called Weber-Morris plot:
    a or c vs. t1/2     NOTE. Linearity of the initial part of kinetics strongly suggests high impact of diffusion (see IDM) but may also correspond to SRT kinetics. This plot has also a compacting nature.

New plots:
These plots require determination of adsorption halftime, t0.5 (once it is determined, aeq and F(t) are also known). Then instead of time the relative (reduced) time (tau) τ = t / t0.5 is used. No model is assumed - their role is to compact experimental kinetic data is such a way, that the features of both initial and near-equilibrium data ranges are well visible.

- compact plots (linear for SOE/PSOE; all data within limited range):
    a vs. τ/(1+τ) or c vs. τ/(1+τ) or F vs. τ/(1+τ)
    NOTE. Time scale t ∈[0;∞) is transformed into τ ∈[0,1].

- half-log plots:
    a vs. ln(1+τ) or c vs. ln(1+τ) or F vs. ln(1+τ)
    NOTE. Time scale ln(1+τ) is initially linear (≈τ for t << t0.5) and logarithmic near equilibrium (≈ ln(τ) for t >> t0.5)

Rate profile plots:
- rate profile:
    da/dt vs. F, dc/dt vs. F, dθ/dt vs. F, dF/dt vs. F,
    NOTE. x-scale is F, however, for experimental data it may be replaced by a, c or θ.

- relative rate profile ("ini" denotes value at F=0):
    (dF/dt)rel vs. F,   where:
    (dF/dt)rel = (dF/dt) / (dF/dt)ini = (da/dt) / (da/dt)ini = (dθ/dt) / (dθ/dt)ini = (dc/dt) / (dc/dt)ini
    NOTE: if the initial rate is finite then the relative rate is always = 1 at (t=0,F=0). Equations/systems with the infinite initial rate (SRT, IDM, MPFO) or infinite adsorption, i.e. no equilibrium (Elovich) cannot be analysed by using this concept.


My favourite empirical kinetic equations:

Kinetics based on Langmuir rate equation and Langmuir isotherm:

Various kinetic equations:
Other kinetic equations:

Diffusion-based sorption kinetics

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