# Resolution Distribution Calculator

If you are carrying out crystallisation resolution of two chemical components, here is a simple and useful calculator. It can be used for example to calculate the percentage yield of crystals, given the initial composition and those of the crystals and liquors obtained. Instructions are given afterwards.

## Instructions

The calculator form has three Sections, labelled **1**, **2** and **3**. Which is used depends on the data you have and
what you need to find out. Replace the initial default figures in the yellow cells with your own values in any of the
Sections and press the 'Calculate' button. The blue cells will then contain some corresponding computed figures. Mostly,
composition values are input as percentage excesses; for example a diastereoisomeric excess (de) or enantiomeric excess (ee) if it
is a chiral resolution. So, for 50%ee, enter '50' in the appropriate cell, not '0.5'. Be careful with the sign of these
composition values; if the crystals are enriched in one component and the liquors in the other (a common scenario) then the crystal
composition should be entered as positive and the liquor composition as negative. Composition percentages must be in the range
-100 to +100%. In the top Section there is the option to enter mole fractions instead of percentage excesses. To do this change the radio
button on the left. Mole fractions must be in the range 0 to 1 and it is best to enter the crystal composition as the mole fraction
of the major component, in the range 0.5 to 1. Mole fractions may often be more convenient to use when purifying a compound from
a mixture

The top Section of the calculator will work out the excess percentages from the mole fractions, or vice-versa
depending on which radio button is checked. They can alternatively be readily interconverted according
to the equations:

*Excess*= [(2 x

*Mole-Fraction*) - 1] x 100%

*Mole-Fraction*= [(

*Excess*/100%) + 1]/2.

The calculator provides some measures of the distribution that might be useful for various comparisons. The value
*D* gives the extent of solid-solution formation if the phases are in equilibrium and if this is the cause of the distribution
seen (0% = no solid-solution; 100% = complete solid solution). This parameter is discussed in more detail in a dedicated page
on solid-solutions. while the surrounding theory is published in:
*Tetrahedron Letters*, **2007**, *48*, 869-872
doi:10.1016/j.tetlet.2006.11.131. The value *D* may be considered more generally as a simple measure of
distribution for comparing different resolution experiments regardless of the reason. Section **3** of the calculator gives the opportunity
to predict the outcome of experiments with a different starting composition or yield once the value of *D* is established.
Here, the value should be entered as a percentage, e.g. as '10' for 10% and not 0.1.

Section **1** of the calcuator additionally gives values of *D′* and *S*. The
*D′* value is a transposition of *D* according to the equation:
*D′* = (1 + *D* - 2.√*D*)/(1 - *D*). It represents the maximum yield
of a resolution that is possible for a given value of *D* (at 50% yield), and may be taken as a measure of
resolution efficiency. The value *S* is known as the Fogassy parameter (*Tetrahedron Lett*.,
**1980**, *21*, 647-650) and is extensively used in classical resolution studies. When starting from
racemate, its value is twice the product of the crystal diastereoisomeric excess and the yield. See the 'Advanced
Considerations' section below for the treatment of the *S*-value when starting with a non-racemic composition.

## Advanced Considerations

Because the value of *D* is formally the ratio of two equilibrium constants (or rather partition
coefficients of the two components between the crystal and liquor phases), it may attain a value either below or
above 1 (100%). A value of *D* greater than 1 will result if the liquors become more enriched in the major
component than the crystals. With figures given in percentages this would appear to give a spoiled result.
Instead, if the computed value of *D* in Sections **1** or **2** of the calculator is more than 1, the
value displayed will be the negative reciprocal and thereby be in the range -100% to 0%. Correspondingly in
Section **3** of the calculator, a *D*-value can be input in the range -100% to +100%. By treating the
value of *D* in this way, it retains the same property as that of *D′* which ranges
from -100% to +100%. For *D′* a positive value means that the crystallisation causes the
compositions to diverge, a negative value means that the composition converges (mixing rather than resolution).

It may be noticed that if certain figures are entered, a negative yield may result; however it will correspond to
an impossible scenario; e.g. if the crystals and liquors are *both* more enriched in a particular component
than the initial composition.

With regard to the Fogassy parameter *S*, this classically applies for a racemic starting composition. More
generally for any starting composition an appropriate equation is shown below, and is used in the calculator above.
It only returns a positive value for *S* when the excess by mass of required isomer in the crystals is
greater than it was in the starting mixture.

*S*= 2 x [(

*crystal de*x

*yield*) -

*starting de*] / [1 -

*starting de*]

A common practise in classical resolutions is to obtain a crystal crop and then recrystallise it to improve
the purity. The recrystallisation will start with a non-racemic composition. In that event, a relevant value
of *S* for the recrystallisation step is given by 2 x [(crystal de x yield) - starting de]. This may give
a positive or negative value depending upon whether the recrystallisation increases or decreases the excess
(by mass) of the required component. Also the figure needs to be adjusted down in proportion to the yield of the
first (main) crystallisation so that the overall *S*-value is the sum of the values for the main
crystallisation and recrystallisation. The above calculator does not take into account such adjustment, but it
can easily be computed by hand.

It will be found that for a crystallisation resolution at optimal yield, the values of *S* and *D′* will
often be similar; and they represent complementary measures of resolution efficiency. The former is particularly sensitive to the
yield obtained, whilst the latter is sensitive to the crystal purity (composition).