Weight average molecular weight

In linear polymers the individual polymer chains rarely have exactly the same degree of polymerization and molar mass, and there is always a distribution around an average value. The molar mass distribution (or molecular weight distribution) describes the relationship between the number of moles of each polymer species (N_i) and the molar mass (M_i) of that species. The molar mass distribution of a polymer may be modified by polymer fractionation.

Different average values can be defined, depending on the statistical method applied. In practice four averages are used, representing the weighted mean taken with the mole fraction, the weight fraction, and two other functions which can be related to measured quantities:

${\displaystyle M_{n}={\frac {\sum M_{i}N_{i}}{\sum N_{i}}},\quad M_{w}={\frac {\sum M_{i}^{2}N_{i}}{\sum M_{i}N_{i}}},\quad M_{z}={\frac {\sum M_{i}^{3}N_{i}}{\sum M_{i}^{2}N_{i}}},\quad M_{v}=\left[{\frac {\sum M_{i}^{1+a}N_{i}}{\sum M_{i}N_{i}}}\right]^{\frac {1}{a}}}$

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Wikipedia