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Vat population densityConstant nucleation without growthThe 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 growthThe 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 crystallizationCooling or vaporizationThe 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. 
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