Methanol is synthesized in a catalytic reactor using $CO$ and $H_2.$ However, a side reaction is observed to occur which leads to the formation of formaldehyde, HCHO, The fresh feed to the process contains $30\%$molCO,$60\%mol{H}_2,$ and $10\%$ mol $N_2.$ The combined feed stream to the reactor, containing $22\%$ mol $N_2,$ consists of $4$mol recycle per mol of fresh feed. In the reactor, $9$ moles of $C{H}_{3}OH$ is produced per mole of HCHO. The reactor effluent gas goes to the condenser where two streams emerge: a gas stream containing all $CO,H_2,$ and $N_2$ and a liquid stream $C{H}_{3}OH$ and HCHO. The gas stream is split into two fractions: one stream is purged and the other is combined with the fresh feed to the reactor.

a$.$ What is the fraction of $CO$ in the purge stream?

b$.$ What is the single pass conversion with respect to $CO?$

c$.$ What is the overall conversion with respect to $C?$