A $5$ mol$/$L solution of uranyl nitrate $($UN$)$ in water is treated with tributyl phosphate $($TBP$)$ to

remove uranyl nitrate in batchwise equilibrium contacts. Assume that water and TBP are

mutually insoluble. The distribution coefficient for UN is K$$DB $=($cB$)$C $/($cB$)$A $=3\mathrm{.}5,$ where $($cB$)$C

$=$ concentration of UN in TBP and $($cB$)$A $=$ concentration of UN in water, both in mol$/$L$.$

a$)$ If $10$L of UN in water is fed into a single batch extraction, what is the percent of UN

obtained if $10$L of TBP is used. $($Hint: start with a mass balance!$)$

b$)$ If instead you used three batch extractions with one$-$third of the solvent $($TBP$)$ used in

each batch, how much total solvent is needed to achieve the same percent of UN

extracted in part a$,$ keeping the amount of feed the same.

Now, assume that we operate with N stages under crosscurrent flow, but with an uneven

distribution of solvent over the various stages $($i$.$e$.$ solvent fed into stage i is fiS, where S is the

total amount of solvent used, and fi is the fraction of solvent fed to stage i$,$ such that $\backslash$Sigma $1$N fi $=1).$

c$)$ Derive an expression to determine the percent extraction for a given solvent to feed ratio

S$/$F and distribution coefficient K$$DB$.$

d$)$ Calculate the percent extraction of UN if three batch extractions are done with $3$L solvent

in the first, $4$L solvent in the second, and $3$L solvent in the third batch. Feed is still $10$L$.$

e$)$ Calculate the percent extraction for a continuous countercurrent process with three stages,

assuming the same flow rate of solvent calculated in part b was used.

f$)$ Calculate the amount of solvent used in a continuous countercurrent process with infinite

stages, using the same percent extraction found in part e$.$