QUESTION $2$: $(30$ pts$)$: Molecular weight averages and distribution, characterizations

The most important feature of polymers compared to small molecules is their sizes. The sizes

of the macromolecules primarily determine their physical properties. When we say size, we

may refer to either the degree of polymerization $($molecular weights$)$ or the spatial arrangement

of the polymers which gives its actual shape in $3-$D$.$ The synthetic polymers come with a

distribution of molecular weights, and it is important to know the average and the dispersity.

$($a$)(7$ pts$)$ Show that $\frac{1}{M_n}={\displaystyle \underset{i}{\overset{?}{}}}\frac{w_i}{M_i}$ where $w_i$ is the weight fraction of chains with molar

mass $M_i$ and $M_n$ is the number average molecular weight.

$($b$)(8$ pts$)$ Four samples of a polymer is mixed without reaction. Each individual sample

has narrow molecular weight distribution. Calculate the number and weight average

molecular weights of the mixture and the resulting dispersity.

$($c$)(7$ pts$)$ The ${?}^{1}H$ NMR spectrum shown below corresponds to a sample of polyisoprene,

$(C_5{H}_8{)}_n,$ containing a sec$-$butyl initiating group and a hydroxyl terminating group. What

is $M_n$ for this polymer $($notice that $m$ and $n$ are for the same monomers but different

isomers, so both contribute to the degree of polymerization$)?$ What is the ratio of $\frac{m}{n}?$

What is the main limitations of the NMR technique for polymer characterization?

$($d$)(8$ pts$)$ The figure below shows the osmotic pressure data obtained for PS in

cyclohexane at three different temperatures. The straight lines are linear regression fits.

Estimate the molecular weight of PS and the second virial coefficients at each

temperature. Is this a number or a weight average molecular weight? At what

temperature, the chains are expanded due to excluded volume effects?