Express molecular mass of a polymer on the basis of weight average molecular mass.

Explanation

Let `N_1, N_2, N_3`,------ be the number of molecules having molecular masses `M_1, M_2, M_3`,------- respectively. Hence, total mass of `N_1 = N_1 xx M_1 =N_1 M_1`
Similarly, total mass of `N_2 = N_2 xxM_2 = N_2M_2` and so on.
Now, by multiplying these total masses with their molecular masses, we get
`N_1 M_1 xx M_1, N_2 M_2 xx M_2, N_2M_3 xxM_3`--------
By adding these products, we get ` = N_1 M_1M_1 + N_2M_2M_2 + N_3 M_3 M_3+---------- = N_1M_1^2 + N_2M_2^2 + N_3M_3^2 +----------underset(i=1)overset(n)Sigma N_1M_i^2`
Also, total mass of all molecules of polymer `= N_1 M_1 + N_2M_2 + --------- underset(i = 1)overset(n) Sigma N_1 M_1`
`:.` weight average molecular mass `= bar M_w = (underset(i = 1)overset(n)Sigma N_1M_1^2)/(underset(i = 1)overset(n) Sigma N_i M_i)`
Thus, weight average molecular mass is the ratio of product of total masses of particular group of molecules with their molecular masses to the total inass of all molecules.