Rate Laws Based on Nucleation

Many chemical reactions in the solid state follow rate laws that are based on the process of nucleation. The active sites have been observed microscopically in some cases, and the phenomenon of nucleation is well established. Although they will not be described in detail, several other processes have nucleation as an integral part of at least the early stages. For example, crystal growth has been successfully modeled by this type of rate law. Condensation of droplets is also a process that involves nucleation. Consequently, kinetics of a wide variety of transformations obey rate laws that have some dependence on a nucleation process. 

The general form of the rate law that is used to describe nucleation processes is known as the Avrami (or Avrami—Erofeev) rate law. 

In this rate law, n is called the index of reaction, and it usually has values of 1.5, 2, 3, or 4. These rate laws are generally abbreviated as A1.5, A2, etc. In particular, the A1.5 rate law (when = 1.5, l/ is 2/3, so this rate law is sometimes identified in that way) has been used to describe crystaUization processes in some soHds. The rate laws having n = 2 and n = 3 are associated with two- and three-dimensional growth of nuclei, respectively. 

Nuclei may be present initially or they may grow in at certain locations by a process that is usually considered to be first-order. The derivation of the Avrami rate law can be accomplished by several procedures (see Young, 1966), all of them rather complicated. Therefore, a detailed derivation will not be presented nor is it needed to see the appHcations of this type of rate law. In general, assumptions are made regarding the rate of change in the 

Taking the cube root of this equation and letting W = C k Y yields an equation of the form 

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