Polymer Engineering $4.$ Show that azeotropic copolymerization occurs when the feed composition is given as: $f_1=\frac{1-r1}{2-r1-r2}$

Derive this equation based on the condition for the azeotropic copolymerization, i$.$e$.,F1=f1,$ using

the Mayo & Lewis equation.

Instantaneous copolymerization $=$ Mayo & Lewis equation:

Equation list from textbook:

$k_{ij}=P_i{Q}_j$exp$(-e_i{e}_j)$

$r_1\frac{k_{11}}{k_{12}}=(\frac{Q_1}{Q_2})exp[-e_1(e_1-e_2)]$

$r_2=\frac{k_{22}}{k_{21}}=(\frac{Q_2}{Q_1})exp[-e_2(e_2-e_1)].$

$\frac{d[M_1]}{d[M_2]}=\frac{[M_1]}{[M_2]}(\frac{r_1[M_1]+[M_2]}{[M_1]+r_2[M_2]}).$

$f_1=\frac{[M_1]}{[M_1]+[M_2]}$

where $f_1+f_2=1.$ In a similar fashion, the mole fraction of monomer $1$ in the copoly$-$

mer, $F_1,$ is given by the relative differential change in monomer concentration as

$F_1=\frac{d[M_1]}{d[M_1]+d[M_2]}$

where $F_1+F_2=1.$ If the numerator and denominator of eq$.(2\mathrm{.}47)$ are each divided

by $d[M_2],$ use of eqs. $(2\mathrm{.}45)$ and $(2\mathrm{.}46)$ gives the following useful relationship:

$F_1=\frac{r_1{f}_1^2+f_1{f}_2}{r_1{f}_1^2+2f_1{f}_2+r_2{f}_2^2}.$