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Partition coefficient in solvent extraction

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Given that the partition coefficient for a solute is K in a water-ether solvent extraction system, is it more effective to do one extraction with V mL of ether or three extractions with V/3 mL of ether. Assume that the volume of water in the system to be V0 mL

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Nearly all liquid- liquid extractions rely on the process of partition, where the analyte or solute is distributed between two liquid phases in contact. This partition of a solute between two immiscible phases is an equilibrium phenomenon governed by Nernst law.
The equilibrium can be written for a solute species "A" as :
Aaq--> Aorg
Where the subscripts refer to the aqueous and organic phases
According to the distribution law, at equilibrium where the solute phase is allowed to distribute between water and organic phase, the ratio of activities of the solute A in each phase is a constant independent of the total quantity of A. That is, at ...

Solution Summary

A 50 ml of water containing 0.1 g of analyte undergoes extraction with 25 ml of ether, Distribution ratio is given as 0.01176. Now let us compare one extraction with all 25mL of ether and also with three extractions each using 8.33 mL ether.
First case: After single extraction using 25 mL
the weight of solute remaining in the aqueous layer is
An = Aaq [ KV/(KV+v)] exp n
Weight remaining =(0.01176 x 50 x 0.1)/ (0.01176 x 50) +25= 0.0023g
Weight extracted = 0.1-0.0023=0.0977g
Second case : three extractions each using 8.33 mL ether and combining the extract ;
Weight extracted = 0.1- { 0.1 [ (0.01176x50)/ (0.01176x50+ 8.33)]3}=0.099 999 g
That is ; virtually complete extraction

It is always better to use several small portions of solvent (for eg; three times) to extract a sample than to extract with one large portion (say for eg; single step)