Journal of Mathematical Chemistry

, Volume 57, Issue 2, pp 616–631 | Cite as

Full analytical solution of a nucleation-growth type kinetic model of nanoparticle formation

  • Rebeka Szabó
  • Gábor LenteEmail author
Original Paper


The analytical solution of a specific kinetic model describing nanoparticle formation is presented. The model starts from a monomer unit, two of which combine in a slow second-order seed formation reaction. The other process is second-order particle growth between a particle and a monomer unit, the rate constant of which is proportional to the mass of the growing nanoparticle. Exact analytical solutions are derived for the time dependence of the concentrations of all different kinds of nanoparticles. The average number of monomer units, the average size and polydispersity is also given by exact formulas. It is shown that the final size distribution of nanoparticles is described by a monotonically decreasing function under all conditions. Possibilities for the comparison of these modeling results with actual experimental data are also considered.


Kinetics Moments Nanoparticle growth Analytical solution 



The research was financed by the Higher Education Institutional Excellence Programme of the Ministry of Human Capacities in Hungary, within the framework of the 1st thematic programme of the University of Pécs.

Supplementary material

10910_2018_975_MOESM1_ESM.docx (205 kb)
Supplementary material 1 (DOCX 204 kb)


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Inorganic and Analytical ChemistryUniversity of DebrecenDebrecenHungary
  2. 2.Department of General and Physical ChemistryUniversity of PécsPecsHungary

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