The following reaction is carried out in a batch reactor in which there is pure A initially.

The following values are available: $k_1=0\mathrm{.}05mi{n}^{-1},k_2=0\mathrm{.}03mi{n}^{-1},k_3=0\mathrm{.}02mi{n}^{-1},k_4=0\mathrm{.}04mi{n}^{-1},$ and $C_{A0}=0\mathrm{.}25mo\frac{l}{d}{m}^3.$

a$)(40$ pts$)$Derive an expression for the ratio $\frac{C_A}{C_{A0}}$ as a function of time. What is this ratio after $30$ minutes?

b$)(50$ pts$)$ Derive equations for the concentration of $B,C,$ and D as a function of time. Leave these equations in differential form $($and they may contain $C_B$ and $C_D).$

c$)(60$ pts$)$ Derive an analytical expression for $C_B.$ Calculate the time when the concentration of B is a maximum.The following reaction is carried out in a batch reactor in which there is pure A initially.

The following values are available: $k_1=0\mathrm{.}05mi{n}^{-1},k_2=0\mathrm{.}03mi{n}^{-1},k_3=0\mathrm{.}02mi{n}^{-1},k_4=0\mathrm{.}04mi{n}^{-1},$ and $C_{A0}=0\mathrm{.}25mo\frac{l}{d}{m}^3.$

a$)(40$ pts$)$Derive an expression for the ratio $\frac{C_A}{C_{A0}}$ as a function of time. What is this ratio after $30$ minutes?

b$)(50$ pts$)$ Derive equations for the concentration of $B,C,$ and D as a function of time. Leave these equations in differential form $($and they may contain $C_B$ and $C_D).$

c$)(60$ pts$)$ Derive an analytical expression for $C_B.$ Calculate the time when the concentration of B is a maximum.