Regarding MAP technology, for perforated films, the permeability to $C{O}_2$ is only $0\mathrm{.}77$ times that of $O_2.$ As a result, in a package relying on perforations for $C{O}_2$ exchange, $C{O}_2$ levels climb to roughly the same extent as $O_2$ levels decline $($i$.$e$.$ the gradients are nearly equal$)$ such that the sum of $O_2$ and $C{O}_2$ partial pressures is usually in the range of $18$ to $20\%.$ For any given $O_2$ levels, therefore, the perforated package will have a considerably higher level of $C{O}_2$ compared to the continuous film package.

$($a$)$ Show $($with calculations that if a package were designed to have a $10\%O_2$ steady stateatmosphere using perforations as the route of gas exchange, the $C{O}_2$ level would be approximately $74\%,$ which is about $6-$fold greater than in the continuous film package. Assume $C{O}_2$ inside $=0\%.$

Graduate students: State one additional assumption.

$($b$)$ Graduate students: Determine the $C{O}_2$ level inside the package if using a continuous LDPE film with a permeability ratio of $4\mathrm{.}44$ and compare your answer with that from $($a$),2$ pts$.$ bonus for undergraduates.