Determine the freely$-$jointed and freely$-$rotating root$-$mean$-$square, end$-$toend distance for three samples of poly$($vinyl alcohol$)$ of $\frac{?}{\text{b}\text{a}\text{r}}(M{)}_n=88,000\frac{,}{b}ar(M{)}_n=176,000,$ and $\frac{?}{\text{b}\text{a}\text{r}}(M{)}_n=264,000.$

If $C_n=6,$ determine also the unperturbed root$-$mean$-$square end$-$to$-$end distance.

Then plot these three parameters as functions of $\frac{?}{\text{b}\text{a}\text{r}}(M{)}_n.$ Comment whether or not doubling of the molecular weight doubles the molecular size.