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) in a polymer describes the relationship between the number of moles of each polymer species (N_{i}) and the molar mass (M_{i}) of that species.^{[1]} The molar mass distribution of a polymer may be modified by polymer fractionation.
Contents

Definition of molar mass averages 1

Measurement 2

Number average molar mass 2.1

Mass average molar mass 2.2

Zaverage molar mass 2.3

See also 3

References 4
Definition of molar mass averages
Different average values can be defined depending on the statistical method that is 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:

Number average molar mass or M_{n} (also loosely referred to as Number Average Molecular Weight (NAMW))

Mass average molar mass or M_{w} (w is for weight; also commonly referred to as weight average (Weight Average Molecular Weight (WAMW))

Z average molar mass or M_{z} (z is for centrifugation; from the German Zentrifuge)

Viscosity average molar mass or M_{v}
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} ^{[2]}
Here a is the exponent in the Mark–Houwink equation that relates the intrinsic viscosity to molar mass.
Measurement
These different definitions have true physical meaning because different techniques in physical polymer chemistry often measure just one of them. For instance, osmometry measures number average molar mass and smallangle laser light scattering measures mass average molar mass. M_{v} is obtained from viscosimetry and M_{z} by sedimentation in an analytical ultracentrifuge. The quantity a in the expression for the viscosity average 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 average values are related to each other as follows: M_{n} < M_{v} < M_{w} < M_{z}. The dispersity (also known as the polydispersity index) of a sample is defined as M_{w} divided by M_{n} and gives an indication just how narrow a distribution is.^{[2]}^{[3]}
The most common technique for measuring molecular mass used in modern times is a variant of highpressure liquid chromatography (HPLC) known by the interchangeable terms of size exclusion chromatography (SEC) and gel permeation chromatography (GPC). These techniques involve forcing a polymer solution through a matrix of crosslinked polymer particles at a pressure of up to several hundred bar. The limited accessibility of stationary phase pore volume for the polymer molecules results in shorter elution times for highmolecularmass species. The use of low dispersity standards allows the user to correlate retention time with molecular mass, although the actual correlation is with the Hydrodynamic volume. If the relationship between molar mass and the hydrodynamic volume changes (i.e., the polymer is not exactly the same shape as the standard) then the calibration for mass is in error.
The most common detectors used for size exclusion chromatography include online methods similar to the bench methods used above. By far the most common is the differential refractive index detector that measures the change in refractive index of the solvent. This detector is concentrationsensitive and very molecularmassinsensitive, so it is ideal for a singledetector GPC system, as it allows the generation of mass v's molecular mass curves. Less common but more accurate and reliable is a molecularmasssensitive detector using multiangle laserlight scattering  see Static Light Scattering. These detectors directly measure the molecular mass of the polymer and are most often used in conjunction with differential refractive index detectors. A further alternative is either lowangle light scattering, which uses a single low angle to determine the molar mass, or RightAngleLight Laser scattering in combination with a viscometer, although this latter technique does not give an absolute measure of molar mass but one relative to the structural model used.
The molar mass distribution of a polymer sample depends on factors such as chemical kinetics and workup procedure. Ideal stepgrowth polymerization gives a polymer with dispersity of 2. Ideal living polymerization results in a dispersity of 1. By dissolving a polymer an insoluble high molar mass fraction may be filtered off resulting in a large reduction in M_{m} and a small reduction in M_{n} thus reducing dispersity.
Number average molar mass
The number average molar mass is a way of determining the molecular mass of a polymer. Polymer molecules, even ones of the same type, come in different sizes (chain lengths, for linear polymers), so the average molecular mass will depend on the method of averaging. The number average molecular mass is the ordinary arithmetic mean or average of the molecular masses of the individual macromolecules. It is determined by measuring the molecular mass of n polymer molecules, summing the masses, and dividing by n.
\bar{M}_n=\frac{\sum_i N_iM_i}{\sum_i N_i}
The number average molecular mass of a polymer can be determined by gel permeation chromatography, viscometry via the (Mark–Houwink equation), colligative methods such as vapor pressure osmometry, endgroup determination or proton NMR.^{[4]}
High NumberAverage Molecular Mass Polymers may be obtained only with a high fractional monomer conversion in the case of stepgrowth polymerization, as per the Carothers' equation.
Mass average molar mass
The mass average molar mass (often loosely termed weight average molar mass) is another way of describing the molar mass of a polymer. Some properties are dependent on molecular size, so a larger molecule will have a larger contribution than a smaller molecule. The mass average molar mass is calculated by
\bar{M}_w=\frac{\sum_i N_iM_i^2}{\sum_i N_iM_i}
where N_i is the number of molecules of molecular mass M_i.
The mass average molecular mass can be determined by static light scattering, small angle neutron scattering, Xray scattering, and sedimentation velocity.
The ratio of the mass average to the number average is called the dispersity or the polydispersity index.^{[3]}
The massaverage molecular mass, M_{w}, is also related to the fractional monomer conversion, p, in stepgrowth polymerization as per Carothers' equation:

\bar{X}_w=\frac{1+p}{1p} \quad \bar{M}_w=\frac{M_o\left(1+p\right)}{1p}, where M_{o} is the molecular mass of the repeating unit.
Zaverage molar mass
The zaverage molar mass is the third moment or third power average molar mass, which is calculated by
\bar{M}_z=\frac{\sum M_i^3 N_i} {\sum M_i^2 N_i}\quad
The zaverage molar mass can be determined with ultracentrifugation. The melt elasticity of a polymer is dependent on M_{z}.^{[5]}
See also
References

^ I. Katime "Química Física Macromolecular". Servicio Editorial de la Universidad del País Vasco. Bilbao

^ ^{a} ^{b} R.J. Young and P.A. Lovell, Introduction to Polymers, 1991

^ ^{a} ^{b} Stepto, R. F. T.; Gilbert, R. G.; Hess, M.; Jenkins, A. D.; Jones, R. G.; Kratochvíl P. (2009). "Dispersity in Polymer Science" Pure Appl. Chem. 81 (2): 351–353. DOI:10.1351/PACREC080502.

^ Polymer Molecular Weight Analysis by 1H NMR Spectroscopy Josephat U. Izunobi and Clement L. Higginbotham J. Chem. Educ., 2011, 88 (8), pp 1098–1104 doi:10.1021/ed100461v

^ Seymore, R.B and Caraher, C.E. Polymer Chemistry: An Introduction, 1992.
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