## Ideal Solubility of Polymorphs

### Quick Start

However good your solvent is, for a crystalline solute you cannot get a concentration higher than the "ideal solubility". This is because the solvent first has to virtually melt the solid before the molecules can dissolve. A high MPt and high enthalpy of fusion (and a low virtual heat capacity) automatically means a low solubility. If your formulation needs a solubility higher than the ideal then give up now - it can't be done.

When dealing with polymorphs, their ideal solubility curves may be more-or-less parallel (monotropes) or will cross (enantiotropes). Whether, and where, they cross depends on all 3 parameters, and your ability to crystallize one or the other depends on the outcome; see the Polymorphs app

### Credits

This is part of my series of apps trying to wrestle with the complexities of crystallization.

### Ideal Polymorphs

MPt1 °C
ΔHf1 kJ/mole.K
ΔCp1 J/mole.K
Tmin
Tmax
MPt2 °C
ΔHf2 kJ/mole.K
ΔCp2 J/mole.K

Any solubility approach such as HSP or COSMO-RS can do a reasonable job of explaining why a given molecule is more compatible with a given solvent than with another. However, two molecules with identical solvent compatibility will show very different solubilities if one is low MPt and the other is high MPt. The reason is obvious - to dissolve the molecules, energy has to be provided to "melt" the solid. Only then does that virtual liquid have a chance to interact more or less well with the solvent.

Predicting the effects of the crystallinity on solubility is somewhere between hard and impossible. Hard if you have data on the molecules melting behaviour, impossible with no such data. One popular approach which uses melting data is to work out the "ideal solubility" defined as the solubility of the molecule in an ideal solvent, i.e. where there is no enthalpic loss when solvent and solute mix. This is different from the best solvent which might have some positive reasons to interact via, say, acid-base interactions. In these calculations the ideal solubility is expressed in mole fraction, x, of the solute in the ideal solvent at a given temperature. By definition, x=1 at the MPt of the solid.

The key ideas are that the solubility depends on the enthalpy of fusion, ΔHF, the melting point Tm and on the change of heat capacity ΔCp of the molecule between the solid and the "virtual liquid" state. As it is very hard to know what ΔCp is the term is somewhat unhelpful. The equation for the solubility x at temperature T and with gas constant R becomes:

ln(x)={ΔH_F(1/T_m-1/T)+ΔC_p[T_m/T – ln(T_m/T)-1]}/R

### Applying this to polymorphs

Polymorphs are two crystal forms of the same molecules. Usually you need to crystallize one form rather than the other. As you can see in the Polymorphs app, whether you get one or the other depends on the solubility curves, where they cross and from where you start your crystallization process. In this app we see the standard explanation for why the polymorph cooling curves are what they are - a subtle dependence on all three parameters. Because MPt and ΔHf can be measured, we are well on the way to fully understanding these curves. Unfortunately, the ΔCp term is to do with virtual, supercooled liquids and you have to be very lucky to be able to measure the values. So why do we bother with the app? Good question. I've written it for my own benefit to better understand what's going on, so I can write a better book chapter. If I'm the only one who ever uses it, that's fine. If you find it helpful, that's even better.