Skip to main content
SpringerLink
Log in

Kinetic and Equilibrium Models of Adsorption

  • Chapter
  • First Online:
Carbon Nanomaterials as Adsorbents for Environmental and Biological Applications

Part of the book series: Carbon Nanostructures ((CARBON))

Abstract

In adsorption study, isotherms and kinetics of adsorption process provide pieces of information underlying the mechanisms and dynamics of the process. Several equilibrium and kinetic models are usually employed for performing the experimental design of an adsorption system. In this chapter, the Langmuir, Freundlich, Sips, Liu, Redlich–Peterson nonlinear equations, as well as other unusual isotherm models (Hill, Khan, Radke–Prausnitz, Toth) are discussed. For the kinetics of adsorption, the pseudo-first-order, pseudo-second-order, general-order, Avrami fractionary order, and Elovich chemisorption models are explained. The importance of statistical parameters such as coefficient of determination (\(R^{2}\)), adjusted coefficient of determination (\(R_{\text{adj}}^{2}\)), and standard deviation (root of mean square error) are highlighted. The usage of linearized and nonlinearized equations are illustrated and explained. Some common mistakes commonly committed in the literature using linearized equilibrium and kinetic adsorption models as well as other polemic points in adsorption research are pointed out. Analytical techniques together with thermodynamical data of enthalpy and entropy changes are needed to ascertain if an adsorption process is a chemical or a physical process.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Abbreviations

DB-53:

Direct Blue 53

DOF:

Degree of freedom

FTIR:

Fourier transform infrared

MWCNT:

Multi-walled carbon nanotube

SWCNT:

Single-walled carbon nanotube

RB-4:

Reactive Blue 4

RO-16:

Reactive Orange-16

R :

Correlation coefficient

\(R^{2}\) :

Coefficient of determination

\(R_{\text{adj}}^{2}\) :

Adjusted coefficient of determination

SD:

Standard deviation (root of mean square error)

TGA:

Thermogravimetric analysis

q e :

Adsorption capacity at the equilibrium (mg g−1) or amount adsorbed by the adsorbent at the equilibrium

q t :

Amount adsorbed by the adsorbent at any time (mg g−1)

C e :

Concentration of the adsorbate at the equilibrium (mg L−1)

C o :

Initial adsorbate concentration (mg L−1)

Q max :

Maximum amount adsorbed (mg g−1)

K L :

Langmuir equilibrium constant (L mg−1)

K F :

The Freundlich equilibrium constant (mg g−1(mg L−1)−1/n)

n F :

The Freundlich exponent (dimensionless)

K S :

The Sips equilibrium constant (mg L−1)−1/ns

n S :

The Sips exponent (dimensionless)

K g :

The Liu equilibrium constant (L mg−1)

n L :

Dimensionless exponent of the Liu equation

K RP :

Redlich–Peterson constants (L g−1)

a RP :

Redlich–Peterson constant (mg L−1)−g

g :

The Redlich–Peterson exponent (dimensionless)

K H :

The Hill equilibrium isotherm constant [(mg L−1)nH]

n H :

The Hill exponent (dimensionless)

K K :

The Khan equilibrium isotherm constant (L mg−1)

n K :

The Khan exponent (dimensionless)

K RAP :

The Radke–Prausnitz equilibrium isotherm constant (L mg−1)

n RP :

The Radke–Prausnitz exponent (dimensionless)

K T :

The Toth equilibrium isotherm constant (L mg−1)

n T :

The Toth exponent (dimensionless)

dq :

Differential of q (adsorption capacity)

dt :

Differential of time

k f :

The pseudo-first-order rate constant (min−1)

t :

Contact time (min)

k s :

The pseudo-second-order rate constant (g mg−1 min−1)

h o :

Initial sorption rate (mg g−1 min−1)

k N :

The rate constant for the order N

n :

The order of adsorption with respect to the effective concentration of the adsorption active sites present on the surface of adsorbent

θ t :

Number of the active sites available on the surface of adsorbent for adsorption

α :

The initial adsorption rate (mg g−1 min−1)

β :

Related to the extent of surface coverage and the activation energy involved in chemisorption (g mg−1)

k AV :

The Avrami kinetic constant (min−1)

n AV :

Is a fractional adsorption order

k id :

The intraparticle diffusion rate constant (mg g−1 min−0.5)

C :

A constant related to the thickness of boundary layer (mg g−1)

A :

The Arrhenius constant

E a :

Is the activation energy (kJ mol−1)

R :

The universal gas constant (8.314 J K−1 mol−1)

T :

The absolute temperature (K)

K D :

The distribution equilibrium constant (dimensionless)

References

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

    Article  Google Scholar 

  2. Freundlich H (1906) Adsorption in solution. Phys Chem Soc 40:1361–1368

    Google Scholar 

  3. Sips R (1948) On the structure of a catalyst surface. J Chem Phys 16:490–495

    Article  Google Scholar 

  4. Liu Y, Xu H, Yang SF, Tay JH (2003) A general model for biosorption of Cd2+, Cu2+ and Zn2+ by aerobic granules. J Biotechnol 102:233–239

    Article  Google Scholar 

  5. Redlich O, Peterson DL (1959) A useful adsorption isotherm. J Chem Phys 63:1024–1027

    Article  Google Scholar 

  6. Largegren S (1898) About the theory of so-called adsorption of soluble substances. Kungliga Suensk Vetenskapsakademiens Handlingar 24:1–39

    Google Scholar 

  7. Blanachard G, Maunaye M, Martin G (1984) Removal of heavy metals from waters by means of natural zeolites. Water Res 18:1501–1507

    Article  Google Scholar 

  8. Alencar WS, Lima EC, Royer B et al (2012) Application of aqai stalks as biosorbents for the removal of the dye Procion Blue MX-R from aqueous solution. Sep Sci Technol 47:513–526

    Article  Google Scholar 

  9. Vaghetti JCP, Lima EC, Royer B et al (2009) Pecan nutshell as biosorbent to remove Cu(II), Mn(II) and Pb(II) from aqueous solutions. J Hazard Mater 162:270–280

    Article  Google Scholar 

  10. Prola LDT, Machado FM, Bergmann CP et al (2013) Adsorption of Direct Blue 53 dye from aqueous solutions by multi-walled carbon nanotubes and activated carbon. J Environ Manage 130:166–175

    Article  Google Scholar 

  11. Foo KY, Hameed BH (2010) Review—insights into the modelling of adsorption isotherm systems. Chem Eng J 156:2–10

    Article  Google Scholar 

  12. Liu Y, Liu YJ (2008) Review—biosorption isotherms, kinetics and thermodynamics. Sep Purif Technol 61:229–242

    Article  Google Scholar 

  13. Ho YS (2006) Review of second-order models for adsorption systems. J Hazard Mater 136:681–689

    Article  Google Scholar 

  14. Liu Y, Shen L (2008) A general rate law equation for biosorption. Biochem Eng J 38:390–394

    Article  Google Scholar 

  15. Vaghetti JCP, Lima EC, Royer B et al (2008) Application of Brazilian-pine fruit coat as a biosorbent to removal of Cr(VI) from aqueous solution. Kinetics and equilibrium study. Biochem Eng J 42:67–76

    Article  Google Scholar 

  16. Lopes ECN, dos Anjos FSC, Vieira EFS, Cestari AR (2003) An alternative Avrami equation to evaluate kinetic parameters of the interaction of Hg(II) with thin chitosan membranes. J Colloid Interface Sci 263:542–547

    Article  Google Scholar 

  17. Machado FM, Bergmann CP, Lima EC et al (2012) Adsorption of Reactive Blue 4 dye from water solutions by carbon nanotubes: experiment and theory. Phys Chem Chem Phys 14:11139–11153

    Article  Google Scholar 

  18. Adebayo MA, Prola LDT, Lima EC et al (2014) Adsorption of Procion Blue MX-R dye from aqueous solutions by lignin chemically modified with aluminium and manganese. J Hazard Mater 268:43–50

    Article  Google Scholar 

  19. Krowiak AW (2012) Analysis of temperature-dependent biosorption of Cu2+ ions on sunflower hulls: kinetics, equilibrium and mechanism of the process. Chem Eng J 192:13–20

    Article  Google Scholar 

  20. Weber-Jr WJ, Morris JC (1963) Kinetics of adsorption on carbon from solution. J Sanit Eng Div Am Soc Civil Eng 89:31–59

    Google Scholar 

  21. El-Khaiary MI, Malash GF (2011) Common data analysis errors in batch adsorption studies. Hydrometallurgy 105:314–320

    Article  Google Scholar 

  22. Kumar KV (2007) Optimum sorption isotherm by linear and non-linear methods for malachite green onto lemon peel. Dyes Pigm 74:595–597

    Article  Google Scholar 

  23. Machado FM, Bergmann CP, Fernandes THM (2011) Adsorption of Reactive Red M-2BE dye from water solutions by multi-walled carbon nanotubes and activated carbon. J Hazard Mater 192:1122–1131

    Article  Google Scholar 

  24. Prola LDT, Acayanka E, Lima EC et al (2013) Comparison of Jatropha curcas shells in natural form and treated by non-thermal plasma as biosorbents for removal of Reactive Red 120 textile dye from aqueous solution. Ind Crop Prod 46:328–340

    Article  Google Scholar 

  25. Cardoso NF, Lima EC, Royer B (2012) Comparison of Spirulina platensis microalgae and commercial activated carbon as adsorbents for the removal of Reactive Red 120 dye from aqueous effluents. J Hazard Mater 241–242:146–153

    Article  Google Scholar 

  26. Calvete C, Lima EC, Cardoso NF, Vaghetti JCP, Dias SLP, Pavan FA (2010) Application of carbon adsorbents prepared from Brazilian-pine fruit shell for the removal of reactive orange 16 from aqueous solution: kinetic, equilibrium, and thermodynamic studies. J Environ Manage 91:1695–1706

    Article  Google Scholar 

  27. El-Khaiary MI, Malash GF, Ho YS (2010) On the use of linearized pseudo-second-order kinetic equations for modelling adsorption systems. Desalination 257:93–101

    Article  Google Scholar 

  28. Ho YS, McKay G (2010) The kinetics of sorption of divalent metal ions onto sphagnum moss peat. Water Res 34:735–742

    Article  Google Scholar 

Download references

Acknowledgments

The authors acknowledge funding from Brazilian agencies CNPq, CAPES, and TWAS.

Author information

Authors and Affiliations

  1. Instituto de Química, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, 91501970, Brazil

    Éder Cláudio Lima

  2. Department of Chemistry, Federal University of Agriculture, Abeokuta, 110008, Ogun State, Nigeria

    Matthew Ayorinde Adebayo

  3. Centro de Desenvolvimento Tecnológico, Universidade Federal de Pelotas, Pelotas, RS, 96010610, Brazil

    Fernando Machado Machado

Authors
  1. Éder Cláudio Lima
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matthew Ayorinde Adebayo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Fernando Machado Machado
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Éder Cláudio Lima .

Editor information

Editors and Affiliations

  1. Escola de Engenharia Departamento de Materials - LACER, The Federal University of Rio Grande do Sul, Porto Alegre, Brazil

    Carlos P. Bergmann

  2. Center of Technological Development, Federal University of Pelotas, Pelotas-RS, Brazil

    Fernando Machado Machado

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Lima, É.C., Adebayo, M.A., Machado, F.M. (2015). Kinetic and Equilibrium Models of Adsorption. In: Bergmann, C., Machado, F. (eds) Carbon Nanomaterials as Adsorbents for Environmental and Biological Applications. Carbon Nanostructures. Springer, Cham. https://doi.org/10.1007/978-3-319-18875-1_3

Download citation

Publish with us

Policies and ethics

Search

Navigation