In polymer chemistry, the molar mass distribution (or molecular weight distribution) describes the relationship between the number of moles of each polymer species (Ni) and the molar mass (Mi) of that species.[1] In linear polymers, the individual polymer chains rarely he exactly the same degree of polymerization and molar mass, and there is always a distribution around an erage value. The molar mass distribution of a polymer may be modified by polymer fractionation.
IUPAC definition for erage degree of polymerization in polymer chemistry.
Definitions of molar mass erage[edit]
Different erage values can be defined, depending on the statistical method applied. In practice, four erages 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:
Number erage molar mass (Mn), also loosely referred to as number erage molecular weight (NAMW). Mass erage molar mass (Mw), where w stands for weight; also commonly referred to as weight erage or weight erage molecular weight (WAMW). Z-erage molar mass (Mz), where z stands for centrifugation (from German Zentrifuge). Viscosity erage molar mass (Mv).M n = ∑ M i N i ∑ N i M w = ∑ M i 2 N i ∑ M i N i M z = ∑ M i 3 N i ∑ M i 2 N i M v = [ ∑ M i 1 + a N i ∑ M i N i ] 1 a {\displaystyle {\begin{aligned}M_{\mathrm {n} }&={\frac {\sum M_{i}N_{i}}{\sum N_{i}}}&&M_{\mathrm {w} }={\frac {\sum M_{i}^{2}N_{i}}{\sum M_{i}N_{i}}}\\M_{\mathrm {z} }&={\frac {\sum M_{i}^{3}N_{i}}{\sum M_{i}^{2}N_{i}}}&&M_{\mathrm {v} }=\left[{\frac {\sum M_{i}^{1+a}N_{i}}{\sum M_{i}N_{i}}}\right]^{\frac {1}{a}}\end{aligned}}}
![{\displaystyle {\begin{aligned}M_{\mathrm {n} }&={\frac {\sum M_{i}N_{i}}{\sum N_{i}}}&&M_{\mathrm {w} }={\frac {\sum M_{i}^{2}N_{i}}{\sum M_{i}N_{i}}}\\M_{\mathrm {z} }&={\frac {\sum M_{i}^{3}N_{i}}{\sum M_{i}^{2}N_{i}}}&&M_{\mathrm {v} }=\left[{\frac {\sum M_{i}^{1+a}N_{i}}{\sum M_{i}N_{i}}}\right]^{\frac {1}{a}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e9d870c42232d77308da6f617720a6a0609ee4e)
Here, a is the exponent in the Mark–Houwink equation that relates the intrinsic viscosity to molar mass.[2]
Measurement[edit]These different definitions he true physical meaning because different techniques in physical polymer chemistry often measure just one of them. For instance, osmometry measures number erage molar mass and small-angle laser light scattering measures mass erage molar mass. Mv is obtained from viscosimetry and Mz by sedimentation in an analytical ultra-centrifuge. The quantity a in the expression for the viscosity erage molar mass varies from 0.5 to 0.8 and depends on the interaction between solvent and polymer in a dilute solution. In a typical distribution curve, the erage values are related to each other as follows: M n