Molar mass measurements play an important role in characterizing polymer solutions. Number-average molar mass $\left(\bar{M}_{n}\right)$ is defined as the total molar mass (given by $\Sigma N_{i} M_{i}$ ) divided by the total number of molecules:
$$
\bar{M}_{n}=\frac{\Sigma N_{i} M_{i}}{\Sigma N_{i}}
$$
where $N_{i}$ is the number of molecules with molar mass $M_{i}$. Another important definition is the weightaverage molar mass $\left(\bar{M}_{w}\right)$ where
$$\bar{M}_{w}=\frac{\Sigma N_{i} M_{i}^{2}}{\Sigma N_{i} M_{i}}$$
The difference between these two definitions is that $\bar{M}_{w}$ is based on experimental measurements that are affected by the size of molecules.
(a) Consider a
solution containing five molecules of molar masses $1.0,3.0,4.0,4.0,$ and $6.0 \mathrm{~kg} / \mathrm{mol} .$ Calculate
both $\bar{M}_{n}$ and $\bar{M}_{w}$. (b) $\bar{M}_{w}$ is always greater than $\bar{M}_{n}$ because of the square term in the definition. However, if all the molecules have identical molar mass, then we have $\bar{M}_{n}=\bar{M}_{w} .$ Show that this is the case if we have four molecules having the same molar mass of $5 \mathrm{~kg} / \mathrm{mol} .$ (c) Explain how a comparison of these two average molar masses gives us information about the distribution of the size of synthetic polymers like polyethylene and poly(vinyl chloride). (d) Proteins like myoglobin and cytochrome $c$ have the same $\bar{M}_{n}$ and $\bar{M}_{w},$ while this is not the case for hemoglobin. Explain.