Although we know that flexible polymer coils have, on average, a Gaussian

distribution of segments from the center of mass to the extremities, it is often convenient

to represent the chains as a sphere with radius $R_g$ and a uniform density of segments. This

is known as a "mean$-$field" approximation.

a$.)$ Calculate how many macromolecules are contained in one coil volume for $M={10}^6$

$\frac{g}{m}ol$ poly$($ethylene$)($PE$)$ and ${10}^6\frac{g}{m}ol$ poly$($vinyl cyclohexane$)(PCHE)$ at $140C.$

Note that $M$ implies monodisperse polymer. Table $6\mathrm{.}1$ in the textbook has relevant

information. Note that the data is for a segment volume of $100{}^3$; be sure to take the

density into account. $($Also note the Table legend erroneously labels this as $100).$

Hint: $R_g$ is invariant. Therefore, $b^{\text{'}}(\frac{N^{\text{'}}}{6}{)}^{\frac{1}{2}}=b(\frac{N}{6}{)}^{\frac{1}{2}}$ where in Table $6\mathrm{.}1)$

indicates the reference condition $($e$.$g$.v=100{}^3)$ and $b$ and $N$ refer to the statistical

segment length and degree of polymerization based on the repeat unit volume.