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| 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<sub>i</sub>) and the molar mass (M<sub>i</sub>) of that species.<ref>I. Katime "Química Física Macromolecular". Servicio Editorial de la Universidad del País Vasco. Bilbao</ref> The molar mass distribution of a polymer may be modified by [[polymer fractionation]].
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| == Definition of molar mass averages ==
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| Different average values can be defined depending on the statistical method that is applied. The [[weighted mean]] can be taken with the weight fraction, the mole fraction or the volume fraction:
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| *Number average molar mass or M<sub>n</sub>
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| *Mass average molar mass or M<sub>w</sub>
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| *Viscosity average molar mass or M<sub>v</sub>
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| *Z average molar mass or M<sub>z</sub>
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| <math>
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| M_n=\frac{\sum M_i N_i} {\sum N_i},\quad
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| M_w=\frac{\sum M_i^2 N_i} {\sum M_i N_i},\quad
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| M_z=\frac{\sum M_i^3 N_i} {\sum M_i^2 N_i},\quad
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| M_v=\left[\frac{\sum M_i^{1+a} N_i} {\sum M_i N_i}\right]^\frac{1} {a}
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| </math>
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| <ref name="ReferenceA">R.J. Young and P.A. Lovell, Introduction to Polymers, 1991</ref>
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| Here a is the exponent in the [[Mark-Houwink equation]] that relates the [[intrinsic viscosity]] to molar mass.
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| ==Measurement==
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| 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 small-angle [[laser]] [[scattering#Electromagnetic scattering|light scattering]] measures mass average molar mass. M<sub>v</sub> is obtained from [[viscometer|viscosimetry]] and M<sub>z</sub> 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<sub>n</sub> < M<sub>v</sub> < M<sub>w</sub> < M<sub>z</sub>. [[Polydispersity]] of a sample is defined as M<sub>w</sub> divided by M<sub>n</sub> and gives an indication just how narrow a distribution is.<ref name="ReferenceA"/>
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| The most common technique for measuring molecular mass used in modern times is a variant of high-pressure 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 [[cross-linked]] polymer particles at a pressure of up to several hundred [[Bar (unit)|Bar]]. The limited accessibility of stationary phase pore volume for the polymer molecules results in shorter elution times for high-molecular-mass species. The use of low polydispersity 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.
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| 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 concentration-sensitive and very molecular-mass-insensitive, so it is ideal for a single-detector GPC system, as it allows the generation of mass v's molecular mass curves. Less common but more accurate and reliable is a molecular-mass-sensitive detector using multi-angle laser-light scattering - see [[Static Light Scattering]]. These detectors directly measure the molecular mass of the polymer and are most often used in conjunction with differental refractive index detectors. A further alternative is either low-angle light scattering, which uses a single low angle to determine the [[molar mass]], or Right-Angle-Light 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.
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| The molar mass distribution of a polymer sample depends on factors such as [[chemical kinetics]] and work-up procedure. Ideal [[step-growth polymerization]] gives a polymer with polydispersity of 2. Ideal [[living polymerization]] results in a polydispersity of 1. By dissolving a polymer an insoluble high molar mass fraction may be filtered off resulting in a large reduction in M<sub>m</sub> and a small reduction in M<sub>n</sub> thus reducing polydispersity.
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| ===Number average molecular mass===
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| The '''number average molecular 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''.
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| <math>\bar{M}_n=\frac{\sum_i N_iM_i}{\sum_i N_i}</math>
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| The number average molecular mass of a polymer can be determined by [[gel permeation chromatography]], [[viscometry]] via the ([[Mark-Houwink equation]]), [[colligative properties|colligative methods]] such as [[vapor pressure osmometry]], [[end-group]] determination or [[proton NMR]].<ref>''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}}</ref>
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| An alternative measure of the molecular mass of a polymer is the mass average molecular mass. The ratio of the ''mass average'' to the ''number average'' is called the [[polydispersity index]].
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| ''High Number-Average Molecular Mass Polymers'' may be obtained only with a high '''fractional monomer conversion''' in the case of [[step-growth polymerization]], as per the [[Carothers' equation]].
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| ===Mass average molecular mass===
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| The '''mass average molecular mass''' is a way of describing the [[molecular mass]] of a [[polymer]]. Polymer [[molecule]]s, even if of the same type, come in different sizes (chain lengths, for linear polymers), so we have to take an average of some kind. For the mass average molecular mass, this is calculated by
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| <math>\bar{M}_w=\frac{\sum_i N_iM_i^2}{\sum_i N_iM_i}</math>
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| where <math>N_i</math> is the number of molecules of molecular mass <math>M_i</math>.
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| If the mass average molecular mass is ''m'', and one chooses a random monomer, then the polymer it belongs to will have a mass of ''m'' on average (for a homopolymer).
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| The mass average molecular mass can be determined by [[static light scattering]], [[small angle neutron scattering]], [[X-ray scattering techniques|X-ray scattering]], and [[Analytical_ultracentrifugation#Analytical_ultracentrifuge|sedimentation velocity]].
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| The ratio of the ''mass average'' to the ''number average'' is called the [[polydispersity]] index.
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| The ''mass-average molecular mass'', ''M''<sub>w</sub>, is also related to the ''fractional monomer conversion'', ''p'', in [[step-growth polymerization]] as per [[Carothers' equation]]:
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| :<math>\bar{X}_w=\frac{1+p}{1-p} \quad \bar{M}_w=\frac{M_o\left(1+p\right)}{1-p}</math>, where ''M''<sub>o</sub> is the molecular mass of the repeating unit.
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| ==See also==
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| *[[Distribution function]]
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| *[[Chemical equilibrium]]
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| *[[Mass distribution]]
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| *[[Sedimentation]]
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| ==References==
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| {{Reflist}}
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| [[Category:Polymer chemistry]]
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Hello! My name is Shantell.
It is a little about myself: I live in Brazil, my city of Mogi Das Cruzes.
It's called often Eastern or cultural capital of SP. I've married 3 years ago.
I have two children - a son (Nidia) and the daughter (Jaimie). We all like Gymnastics.
Feel free to visit my website; gourmet food PR Crier PR