You are working for a company that is trying to use membrane GP to remove carbon dioxide from natural gas (methane). Carbon dioxide has a higher permeability than methane in polymer membranes. Unfortunately, none of the commercially available membranes have a high enough selectivity \(\alpha_{\mathrm{CO} 2-\mathrm{CH} 4}\) at an acceptable value of the \(\mathrm{CO}_{2}\) permeance, \(\left(\mathrm{P}_{\mathrm{CO} 2} / \mathrm{t}_{\mathrm{ms}}\right)\). A small membrane research company claims they can design a polymer membrane that is selective for \(\mathrm{CO}_{2}\) and will have a high enough \(\mathrm{CO}_{2}\) permeance. Your company's design group wants to separate feed that is \(40 \mathrm{~mol} \%\) carbon dioxide and \(60 \mathrm{~mol} \%\) methane. The goal is to produce \(84 \mathrm{~mol} \%\) methane retentate and recover \(70 \%\) of the methane in the retentate using a single perfectly mixed membrane module with a retentate pressure of \(6.2 \mathrm{~atm}\) and a permeate pressure of \(1.04 \mathrm{~atm}\). The membrane area required for \(1.0 \mathrm{kmol} / \mathrm{s}\) of feed should be \(70,000 \mathrm{~m}^{2}\) or less. Find the cut \(\theta\), selectivity \(\alpha_{\mathrm{CO} 2}\)