Polystyrene $($PS$)$ has a statistical segment length of $0\mathrm{.}67nm$ in the melt and a density of

$1\mathrm{.}05gc{m}^{-3}.$

a$.$ Find the characteristic ratio $C_{}$ for polystyrene using the known bond distances

and angles of the PS backbone.

b$.$ Find the contour length of a PS chain of molar mass $M=70000gmo{l}^{-1}.$

c$.$ What is the ratio of the length of a fully extended PS chain of this molecular mass

to the RMS end$-$to$-$end distance of this chain in the melt.

d$.$ It is possible to determine the radius of gyration of the PS chain

$70000gmo{l}^{-1})$ by small$-$angle neutron scattering if it is diluted by PS chains whose

hydrogens have been replaced by deuteriums. What value of the radius of gyration

will be measured if the chain obeys random coil statistics?