Problem $4.$

Radioactive isotopes play an important role in Medicine for radiation therapy and medical imaging.

Solutions with radioactive isotopes lose quickly use their "potency" due to the radioactive decay.

It is therefore often necessary to isolate radioactive isotopes $($produced in a synchrotron or nuclear

reactor$)$ from their decay products immediately before its medical use. The efficiency of this

purification process may be increased through a recycling process. In a good approximation the

decay of a radioactive isotope can be treated in analogy to chemical reactions.

A diagnostic medical procedure is based on the detection of the decay of a radioactive isotope $A.$

Isotope A decays to another isotope B via a radioactive decay:

$AB$

This process occurs with a time constant $k=0\mathrm{.}05mi{n}^{-1}$ according to

$n_A=n_{A,0}$exp$(-kt)$

A mixture of A and B is isolated from a nuclear production process. The medical procedure

requires the pure isotope $A.$ Therefore, A and $B$ are undergoing a continuous separation process

at the site of the medical procedure. We are analyzing this separation step:

Consider a constant feed of an aqueous solution of $100m\frac{l}{?}$ hour containing $10mg$ of isotope A

and $50mg$ of isotope B$.$ This isotope mixture enters a chromatographic separation process. It

takes $20$ minutes for the feed to pass through this separator. Two streams exit the column. The

desired stream contains an aqueous solution of pure A leaving the reactor with a rate of

$1m\frac{g}{?}$ hour. The remaining stream contains components A and B$.$ It is split in two equal parts

such that $50\%$ of this stream is recycled, the other $50\%$ are discarded as nuclear waste. All flows

associated with the recycling process are sufficiently fast such that you may ignore nuclear

decays. However, you may not ignore any decay processes during the separation process.

Calculate the concentration of component A and B in all streams.

Hints: As stated above the nuclear decay can be treated in very good approximation as chemical

reaction, use equation $($i$)$ to calculate the conversion occurring in the separator. $(2)$ In a very

good approximation the molecular weights of A and $B$ can be considered identical.