An aqueous dilute fermentation broth contains \(0.5 \mathrm{wt} \%\) of penicillin \(\mathrm{F}\). It is to be extracted with amyl acetate. At room temperature and \(\mathrm{pH}=3.2\) water and amyl acetate are essentially insoluble, and the distribution coefficient for the penicillin is \(K_{D}=80\).

(a) If \(100 \mathrm{~kg}\) of the fermentation broth is extracted with \(6 \mathrm{~kg}\) of the pure solvent in a single ideal stage, calculate the fractional recovery of penicillin and the penicillin concentration in the final raffinate and extract.

(b) What would be the recovery with two-stage crosscurrent extraction if \(6 \mathrm{~kg}\) of fresh solvent is used in each stage?

(c) How many ideal stages would be needed to give the same recovery as in part

(b) if a countercurrent cascade were used with \(6 \mathrm{~kg}\) of solvent/100 kg feed?