# Vat population density

## Constant nucleation without growth

The number of particles increases in proportion to time.

## Growth without nucleation

Function n_{o} does not change by L that goes from L_{o}to L_{o} + Gt
The number of crystals does not vary:

## Nucleation with growth

The partial derivative equation, with n as the solution, is:

A possible solution is:

At the beginning of this operation, t#0 and we obtain:

Experience confirms this result that represents the *population density of detectable seeds at the beginning of the operation.*

# Choice of crystallization

## Cooling or vaporization

The process by cooling is required if the relationship between temperature and concentration to saturation is such that:

Vaporization is required *a priori* if:

Thus, the solubility of sodium chloride is practically independent of temperature, as that of K_{2}CO_{3} and (NH_{4})_{2}CO_{3}. However, many crystallizations occur by vaporization for another reason. In fact, vaporization allows us to reach yields approaching 1 (if the purge is low), which is not the case for cooling.

Cooling a solution leads to a mediocre depletion of the mother liquor. If we cool too much, we risk crystallizing a more hydrated crystal species than desired and, in extreme cases, crystallizing the water itself.

If we want to obtain a crystal species that is greatly hydrated, the evaporation technique requires low temperature, that is, a high vacuum, and occasionally, we are obliged to proceed by cooling anyway.

In order to crystallize by cooling, we must ensure that solubility varies in a significant way with temperature, although this is not a sufficient condition and, very often, we crystallize such products by evaporation for the reasons of yield addressed above.