A volatile organic compound $\text{(VOC)}$ is to be adsorbed from air onto a bed of activated carbon. The equilibrium capacity of activated carbon at the feed conditions is $0.4$ grams $\text{VOC}$ per gram of activated carbon. The column contains $4$ grams of activated carbon per $\mathrm{cm}^{2}$ of cross-section. The feed rate into the adsorber column is $0.2$ grams $\text{VOC} \text{ cm}^{-2} \mathrm{~h}^{-1}$. Breakthrough time is defined as the time at which the concentration at the exit of the bed (c) reaches a value of $0.05 c_{0}$, where $c_{0}$ is the concentration of $\text{VOC}$ in the feed. The breakthrough time for the bed is $2.1 \text{ h}.$ The area under $c / c_{0}$ curve between the initial and breakthrough times is $0.1 \text{ h}.$ Which one of the following is the fraction of unused bed at breakthrough?
- $0$
- $0.25$
- $0.50$
- $0.75$
