Skip to main content
SpringerLink
Log in

Discrimination between adsorption isotherm models based on nonlinear frequency response results

  • Published:
Adsorption Aims and scope Submit manuscript

Abstract

A number of criteria are established for distinguishing between different adsorption isotherm types. These criteria are defined based on the adsorption isotherm derivatives up to the third order, which, on the other hand, can be estimated from nonlinear frequency response data. The criteria for five favourable (Langmuir, Freundlich, Sips, Toth and Unilan) isotherms and two complex isotherms (BET and quadratic) are presented. These criteria enable unique identification of the underlying adsorption isotherm relation if the values of the local first, second and third order isotherm derivatives at several points are known. The method is applied to experimental data from our previous publications, for one case of a favourable and one case of a complex isotherm.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Abbreviations

\(\tilde{a}\) :

Dimensionless first order derivative of the adsorption isotherm

\(\tilde{b}\) :

Dimensionless second order derivative of the adsorption isotherm

b :

Model parameter for Langmuir, Sips, Toth and Unilan isotherms

b L , b S :

BET isotherm model parameters

b 1 , b 2 :

Quadratic isotherm model parameters

C (g/dm3):

Concentration in the fluid phase

c :

Dimensionless concentration in the fluid phase

\(\tilde{c}\) :

Dimensionless second order derivative of the adsorption isotherm

CR :

Criterion

\(F_{n} (\omega_{1} ,\omega_{2} , \ldots \omega_{n} )\) :

n-th order frequency response function on the particle level

\(G_{n} (\omega_{1} ,\omega_{2} , \ldots \omega_{n} )\) :

n-th order frequency response function on the adsorber level

K :

Freundlich isotherm model parameter

L (cm):

Column length

n :

Model parameter for Freundlich and Sips isotherm

Q (g/dm3):

Concentration in the solid phase

Q 0 (g/dm3):

Model parameter for Langmuir, Sips, Toth, Unilan, BET and quadratic isotherms

q :

Dimensionless concentration in the solid phase

t :

Model parameter for Toth isotherm

u :

Interstitial column velocity

V (dm3):

Volume

\(\dot{V}\)(dm3/min):

Volumetric flow-rate

ε:

Porosity

ω:

Frequency, general and dimensionless

ω* (rad/min):

Frequency, dimensional

BET :

BET

F :

Freundlich

L :

Langmuir

QUAD :

Quadratic

S :

Sips

s :

Steady-state

T :

Toth

U :

Unilan

d :

Dimensional

DC:

Non-periodic (DC) term

FR:

Frequency response

FRF:

Frequency response function

NFR:

Nonlinear frequency response

References

  • Brunauer, S., Emmett, P.H., Teller, E.: Adsorption of gases in multimolecular layers. J. Am. Chem. Soc. 60, 309–319 (1938)

    Article  CAS  Google Scholar 

  • Do, D.D.: Adsorption Analysis: Equilibria and Kinetics. Imperial College Press, London (1998)

    Google Scholar 

  • Freundlich, H.M.F.: Over the adsorption in solution. J. Phys. Chem. 57, 385–470 (1906)

    CAS  Google Scholar 

  • Foo, K.Y., Hameed, B.H.: Insights into the modeling of adsorption isotherm systems. Chem. Eng. J. 156, 2–10 (2010)

    Article  CAS  Google Scholar 

  • Gritti, F., Guiochon, G.: Band splitting in overloaded isocratic elution chromatography II. New competitive adsorption isotherms. J. Chromatogr. A 1008, 23–41 (2003)

    Article  CAS  Google Scholar 

  • Guiochon, G., Golshan-Shirazi, S., Katti, A.: Fundamentals of Preparative and Nonlinear Chromatography. Academic Press, New York (1994)

    Google Scholar 

  • Ho, Y.S., Porter, J.F., Mckay, G.: Equilibrium isotherm studies for the sorption of divalent metal ions onto peat: copper, nickel and lead single component systems. Water Air Soil Pollut. 141, 1–33 (2002)

    Article  CAS  Google Scholar 

  • Ilić, M., Petkovska, M., Seidel-Morgenstern, A.: Nonlinear frequency response method for estimation of single solute adsorption isotherms. Part II. Experimental study. Chem. Eng. Sci. 62, 4394–4408 (2007)

    Article  Google Scholar 

  • Langmuir, I.: The adsorption of gases on plane surfaces of glass, mica and platinum. J. Am. Chem. Soc. 40, 1361–1403 (1918)

    Article  CAS  Google Scholar 

  • Petkovska, M.: Nonlinear frequency response method for investigation of equilibria and kinetics in adsorption systems. In: Spasic, A.M., Hsu, J.P. (eds.) Finely Dispersed Particles: Micro Nano and Atto-Engineering, p. 283. CRC Taylor & Francis, Boca Raton (2006)

    Google Scholar 

  • Petkovska, M.: Nonlinear FR-ZLC method for investigation of adsorption equilibrium and kinetics. Adsorption 14, 223–239 (2008)

    Article  CAS  Google Scholar 

  • Petkovska, M., Do, D.D.: Nonlinear frequency response of adsorption systems: isothermal batch and continuous flow adsorbers. Chem. Eng. Sci. 53, 3081–3097 (1998)

    Article  CAS  Google Scholar 

  • Petkovska, M., Seidel-Morgenstern, A.: Nonlinear frequency response of a chromatographic column. Part I: application to estimation of adsorption isotherms with inflection points. Chem. Eng. Commun. 192, 1300–1333 (2005)

    Article  CAS  Google Scholar 

  • Seidel-Morgenstern, A.: Experimental determination of single solute and competitive adsorption isotherms. J. Chromatogr. A 1037, 255–272 (2004)

    Article  CAS  Google Scholar 

  • Sips, R.: Combined form of Langmuir and Freundlich equations. J. Chem. Phys. 16, 490–495 (1948)

    Article  CAS  Google Scholar 

  • Toth, J.: Adsorption: Theory, Modelling, and Analysis. Dekker, New York (2002)

    Google Scholar 

Download references

Acknowledgments

This work was supported by the Ministry of Education and Science of the Republic of Serbia in the frame of Project No. 172022.

Author information

Authors and Affiliations

  1. Department of Chemical Engineering, Faculty of Technology and Metallurgy, University of Belgrade, Karnegijeva 4, 11000, Belgrade, Serbia

    Menka Petkovska

Authors
  1. Menka Petkovska
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Menka Petkovska.

Appendix: Functions f 1(C) to f 14(C) in Table 1

Appendix: Functions f 1(C) to f 14(C) in Table 1

$$f_{1} (C) = \frac{{(Cb)^{4/n} (2n^{2} + 3n - 1) + (Cb)^{3/n} (8n^{4} + 12n^{3} - 4n^{2} - 4) + (Cb)^{2/n} (2n^{2} - 3n + 1)}}{{\left( {(Cb)^{1/n} (n - 1) + (Cb)^{2/n} (n + 1)} \right)^{2} }}$$
(52)
$$f_{2} (C) = \frac{2}{{n((Cb)^{1/n} + 1)}} - \frac{n - 1}{n}$$
(53)
$$f_{3} (C) = \frac{2}{{n((Cb)^{1/n} + 1)}} - \frac{n - 1}{n}$$
(54)
$$f_{4} (C) = - \frac{{(Cb)^{1/n} (3n((Cb)^{2/n} - 1) + (2n^{2} + 1)((Cb)^{2/n} + 1) + (Cb)^{2/n} (4n^{2} - 4)}}{{n((Cb)^{1/n} (n - 1) + (Cb)^{2/n} (n + 1))((Cb)^{1/n} + 1)}}$$
(55)
$$f_{5} (C) = \frac{{(n^{2} - 1)((Cb)^{1/n} + 1)^{4} }}{{Q_{0}^{2} (Cb)^{2/n} }}$$
(56)
$$f_{6} (C) = \frac{1}{t - 1}\left( {\frac{2}{{(Cb)^{t} }} + 1} \right) - \frac{1}{{(Cb)^{t} }} + 1$$
(57)
$$f_{7} (C) = - \frac{{(Cb)^{t} (t + 1)}}{{(Cb)^{t} + 1}}$$
(58)
$$f_{8} (C) = \frac{(t - 1)(t + 1)}{{\left( {(Cb)^{t} + 1} \right)^{2} }} - (t - 1)(t + 1)$$
(59)
$$f_{9} (C) = - \frac{{((Cb)^{t} + 1)^{2t} ((t^{2} + 2t - 1)(Cb)^{2t} - (2t^{2} + 2t + 2)(Cb)^{t} )}}{{C^{2} b^{2} Q_{0}^{2} }}$$
(60)
$$f_{10} (C) = \frac{{\cosh (2s) - (3\cosh (s)^{2} /2 + 1/2)}}{{(\cosh (s) + Cb)^{2} }} + 3/2$$
(61)
$$f_{11} (C) = - \frac{{2e^{s} C^{2} b^{2} + (e^{2s} + 1)Cb}}{{(e^{s} + Cb)(Cbe^{s} + 1)}}$$
(62)
$$f_{12} (C) = \frac{{C^{2} b^{2} (e^{2s} - 1)^{2} }}{{(e^{s} + Cb + e^{2s} Cb + C^{2} b^{2} e^{s} )^{2} }}$$
(63)
$$f_{13} (C) = - \frac{{2e^{s} C^{2} b^{2} + (e^{2s} + 1)Cb}}{{(e^{s} + Cb)(Cbe^{s} + 1)}}$$
(64)
$$f_{14} (C) = \frac{{2Cb(e^{4s} + 3Cbe^{3s} + 3Cbe^{s} + 1) + 2Cbe^{2s} (3C^{2} b^{2} + 1)}}{{(e^{2s} + 2e^{s} Cb + 1)(e^{s} + Cb + e^{2s} Cb + C^{2} b^{2} e^{s} )^{2} }}$$
(65)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Petkovska, M. Discrimination between adsorption isotherm models based on nonlinear frequency response results. Adsorption 20, 385–395 (2014). https://doi.org/10.1007/s10450-013-9571-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10450-013-9571-y

Keywords

Search

Navigation