= Solution
The useful mechanism is <radioactive decay> during retention, not chemical destruction of <carbon dioxide>. With the time in seconds,
$$
\lambda=\frac{\ln 2}{20.3(60)}=5.69086\times10^{-4} \mathrm{s^{-1}},\qquad
a(t)=10^{14}e^{-\lambda t} \mathrm{Bq\,m^{-3}}.
$$
Here $a$ is a <radioactive activity concentration>; the supplied units are per volume rather than per mass. The original plot follows this <concentration> over four hours.
\Image[/past-exam-of-the-mphil-in-nuclear-energy-course-of-the-university-of-cambridge/2021/paper-ne4-carbon11-delay.png]
{title=Carbon-11 activity concentration during ideal radioactive gas holdup}
A practical illustrative choice is \b[a three-hour guaranteed <residence time>], at which $a/a_0=0.00214175$, giving about a factor $467$ reduction. On a plug-flow basis, the carbon-dioxide-equivalent hold-up volume is
$$
V_{\mathrm{CO_2}}=Q\tau=(3.7\times10^{-7})(3\times3600)=\boxed{3.996\times10^{-3} \mathrm{m^3}=4.00 \mathrm L}.
$$
Use a contained delay line or a validated retention arrangement that prevents early breakthrough, followed by monitored discharge. A retention bed could instead hold the radioactive <carbon dioxide>, but its capacity and breakthrough performance would have to be established. Gas retention and monitored discharge are waste-treatment functions, independent of the eventual use of the <isotope>.<Predisposal Management of Radioactive Waste from Nuclear Power Plants and Research Reactors by the International Atomic Energy Agency>[SSG-40, section 6, gaseous-waste treatment and selection of decay devices]{ref}
\b[The $4.00$ L result is a flow-delay volume, not a fill-and-immediately-dump tank specification.] It credits <radioactive decay> during the guaranteed transit time and no subsequent storage after discharge from the delay system. Gas entering a filling tank near the end of filling has not aged for the entire filling time. For a literal tank filled for $T$ and immediately emptied, the <radioactive activity> at emptying is instead
$$
A_{\rm fill}(T)=\int_0^T a_0Qe^{-\lambda(T-s)}\,ds=\frac{a_0Q}{\lambda}(1-e^{-\lambda T}).
$$
That alternative must not be analysed by multiplying the whole collected inventory by $e^{-\lambda T}$. If “no post filling <radioactive decay>” is intended to prescribe this literal fill-and-dump arrangement, the three-hour calculation does not establish compliance for it.
The <nitrogen> fraction is also unspecified. If $y$ is the <carbon dioxide> volume fraction at the stated volume reference conditions, a gas delay line requires $V_{\rm mixture}=V_{\mathrm{CO_2}}/y$, not necessarily $4.00$ L of physical vessel volume. The graph alone does not define a unique optimum: a cost objective, operating calendar and residence-time model are needed. Parts (c) and (d) quantify the discharge constraint for the stated guaranteed-delay design.
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