Given the hypothetical distribution of chain lengths:

$f_n(x)=C(1-{10}^{-3}x),0x{10}^3$

where $f_n(x)$ is the number fraction of polymer with degree of polymerization of $x,$

a$.$ Determine the value of constant $C.$

b$.$ Calculate the number$-$average $({\stackrel{}{x}}_n)$ and weight$-$average $({\stackrel{}{x}}_w)$ degree of polymerization and the polydispersity index for this distribution.

c$.$ Find the expression for the weight fraction of the chains as a function of molecular length $($degree of polymerization$)x.$ Plot the number and weight fraction distributions vs$.$ degree of polymerization.

d$.$ Obtain the analytical expressions for the mole$-(F_n)$ and weight$-$fraction $(F_w)$ cumulative distributions; plot the $F_n(x)$ and $F_w(x)$ cumulative distributions.