For the homogeneous free radical polymerization of pure acrylic acid with $H_2{O}_2($initiator$)$ at $70C$ in an isothermal, batch reactor, a synthetic condition is provided as follows:

$[M]=1mo\frac{l}{L}$;$[l]=1\mathrm{.}0{10}^{-3}mo\frac{l}{L}$;$f(initiator$ efficiency$)=0\mathrm{.}37$

$k_p=8\mathrm{.}40{10}^{-2}\frac{L}{\text{m}\text{o}\text{l}s}$;$k_d=4\mathrm{.}51{10}^{-5}{s}^{-1}$;$k_t=6{10}^{-7}\frac{L}{\text{m}\text{o}\text{l}s}$

Estimate kinetic chain length.

Calculate the $($instantaneous$)$ number average chain length $({\stackrel{}{x}}_n)$ and number average molecular weight $(\frac{?}{\text{b}\text{a}\text{r}}(M{)}_n),$ respectively, for the two cases:

a$)$ When the disproportionation termination mode is dominant.

b$)$ When the combination termination mode is dominant.

If necessary, use the following information: atomic weight of $C$:$12,H$:$1.$ When calculating the molecular weight, do not consider the molecular weight of the initiator.

On the basis of $\frac{?}{\text{b}\text{a}\text{r}}(M{)}_n$ obtained under the condition $2-$a$)$ and $2-$b$),$ respectively, calculate the $\frac{?}{\text{b}\text{a}\text{r}}(M{)}_n$ when there are chain transfer reactions to the initiator. The value of chain transfer constant to the initiator $(C_l)$ is $0\mathrm{.}0028.$ Assumption: except the chain transfers to the initiator, there is no other chain transfer reaction during the polymerization.

Assume that $E_p=6$kca$\frac{l}{m}ol,E_t=2$kca$\frac{l}{m}ol,$ and $E_d=30$kca$\frac{l}{m}ol.$ As such, determine $\frac{?}{\text{b}\text{a}\text{r}}(M{)}_n$ when the reaction temperature was decreased to $60C.$ Assume that there is no chain transfer reaction $($therefore$,\frac{?}{\text{b}\text{a}\text{r}}(M{)}_n$ under the condition $2-$a$)$ and $2-$b$)$ should be used at $70C).$ For the calculation, use the data: gas constant $R=1\mathrm{.}987ca\frac{l}{K\text{m}\text{o}\text{l}}.$