Calculating the theoretical percent purity of a recrystallization

Question

The sample contains some percent A and some percent B with A being the majority. The solubility for both is given at two different temperatures. One where both are very soluble and one where both are much less soluble. From these five values I'm supposed to determine the theoretical purity of the recrystallized compound.

I'm surprised that this is possible and I have no idea how to even begin approaching it. I always thought that crystallization was structurally depended as well. Shouldn't the lattice favor the dominant compound? What if the impurity is less soluble at the low temperature than the majority compound? How can we mathematically represent such a (seemingly) complex phenomena with such little information?

Note: I don't mean percent yield, the question specifically stated purity.

Answer 1

I can give you a hint, imagine you have a mixture containing one consitutant $A$ only in your solution with a solvant.

Doing the balanced equations about the mass of $A$ in each phases you have :

$$\begin{cases} m_A^\mathrm{t}=m_A^\mathrm{c}+m_A^\mathrm{liq} \\ m_A^\mathrm{i}=x_A^\mathrm{c}\cdot m^\mathrm{c}+x_A^\mathrm{liq} \cdot m^\mathrm{liq} \end{cases}$$

Where $m^\mathrm{t}$, $m^\mathrm{c}$, $m^\mathrm{liq}$ and $m^\mathrm{i}$ denote respectively the mass of $A$ total, in the cristal form, dissolute in the solvant (liquid), and at the begining (initial) and $x$ the corespondant mass-fractions.

Then you must obtain, $$m^\mathrm{c}=m^\mathrm{t}\left(\frac{\frac{m_A^\mathrm{i}}{m^\mathrm{t}}-x_A^\mathrm{c}}{x_A^\mathrm{liq}-x_A^\mathrm{c}}\right)$$

Notice that $\frac{m_A^\mathrm{i}}{m^\mathrm{t}}=x_{A,\mathrm{i}}^\mathrm{liq}$ which is the mass-fraction at the begining in the liquid and $x_A^\mathrm{liq}$ is the mass-fraction in the liquid at the end.

If you know how what you put at the begining, you can determine the yield. Now if you want to know the purity, you'll need to do the same with both of your component to see how much of them have cristalised at the end and then determine the purity of your marjoritory compound.

I hope it can help, have a good day ! :)

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Be careful when you do the balanced equations if there is also a gas phase which can happend sometime.

Answer 2

For such seemingly complex problem, you are usually expected to use an ideal case as an approximation. Here, I suppose you should use "ideal crystallization" so that A and B will form pure crystals. So the contamination of A by impurity B will only depend on their relative solubility in the medium.

HTH