Consider the self$-$condensation of an $A-R_1-B$ monomer $(R_1$ is an alkane$)$ with $N_R$ monomers charged in a reactor.

a$.$ In a perfect world with no contamination in the reactor, what would be the numberaverage molecular weight of the final polymer?

b$.$ Now consider contamination with another reactant $R_2-B$ which can undergo a condensation reaction with the A groups $(R_2$ is just another alkane$).$ If there are $N_Q$ contaminants, derive an expression for $\frac{?}{\text{b}\text{a}\text{r}}(x_n)$ in terms of the conversion of $A(p)$ and $r^{\text{'}}=\frac{N_Q}{N_R}.$

c$.$ Consider the step$-$growth polymerization of $10g$ of $N{H}_2-(C{H}_2{)}_5-\frac{C}{O}OH$ to give nylon. The reactor is cleaned carefully with acetic acid $(C{H}_3-\frac{C}{O}OH)$ and dried. However, $0\mathrm{.}01g$ of acetic acid is left in the reactor. What are $\frac{?}{\text{b}\text{a}\text{r}}(x_n)$ and $\frac{?}{\text{b}\text{a}\text{r}}(x_w)$ for the polymer made $($compute the latter using the Flory most probable distribution and assume $p=0\mathrm{.}99.$

d$.$ If you wish to make ultra$-$high molecular$-$weight nylon $)>({10}^6,$ what amount of residual acetic acid is permitted after cleaning?