single particle grows in a solution. This equation describes where the boundary is between small, shrinking particles and large, growing particles. They finally conclude that the average radius of the particles , grows as follows:
: where

Note that the quantity is different from , and that the statement that goes as relies on being zero; but because nucleation is a separate process from growth, this places outside the bounds of validity of the equation. In contexts where the actual value of is irrelevant, an approach that respects the meanings of all terms is to take the time derivative of the equation to eliminate and . Another such approach is to change the to with the initial time having a positive value.