weight-average molecular weight
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The molar mass distribution (or molecular weight distribution) describes the relationship between the number of
moles Moles can refer to: * Moles de Xert, a mountain range in the Baix Maestrat comarca, Valencian Community, Spain * The Moles (Australian band) *The Moles, alter ego of Scottish band Simon Dupree and the Big Sound People *Abraham Moles, French engin ...
of each
polymer A polymer (; Greek '' poly-'', "many" + ''-mer'', "part") is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic a ...
species (Ni) and the
molar mass In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance which is the number of moles in that sample, measured in moles. The molar mass is a bulk, not molecular, ...
(Mi) of that species. In linear polymers, the individual polymer chains rarely have exactly the same
degree of polymerization The degree of polymerization, or DP, is the number of monomeric units in a macromolecule or polymer or oligomer molecule. For a homopolymer, there is only one type of monomeric unit and the ''number-average'' degree of polymerization is given b ...
and molar mass, and there is always a distribution around an average value. The molar mass distribution of a polymer may be modified by polymer fractionation.


Definitions of molar mass average

Different average values can be defined, depending on the statistical method applied. In practice, four averages are used, representing the
weighted mean The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The ...
taken with the
mole fraction In chemistry, the mole fraction or molar fraction (''xi'' or ) is defined as unit of the amount of a constituent (expressed in moles), ''ni'', divided by the total amount of all constituents in a mixture (also expressed in moles), ''n''tot. This ex ...
, the weight fraction, and two other functions which can be related to measured quantities: *''Number average molar mass'' (Mn), also loosely referred to as ''number average molecular weight'' (NAMW). *''Mass average molar mass'' (Mw), where ''w'' stands for weight; also commonly referred to as ''weight average'' or ''weight average molecular weight'' (WAMW). *''Z-average molar mass'' (Mz), where ''z'' stands for centrifugation (from German "''Zentrifuge''"). *''Viscosity average molar mass'' (Mv). M_n=\frac ,\quad M_w=\frac ,\quad M_z=\frac ,\quad M_v=\left frac \right\frac Here, a is the exponent in the
Mark–Houwink equation The Mark–Houwink equation, also known as the Mark–Houwink–Sakurada equation or the Kuhn–Mark–Houwink–Sakurada equation or the Landau–Kuhn–Mark–Houwink–Sakurada equation or the Mark-Chrystian equation gives a relation between intr ...
that relates the
intrinsic viscosity Intrinsic viscosity \left \eta \right/math> is a measure of a solute's contribution to the viscosity \eta of a solution. It should not be confused with inherent viscosity, which is the ratio of the natural logarithm of the relative viscosity to th ...
to molar mass.R.J. Young and P.A. Lovell, Introduction to Polymers, 1991


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 small-angle
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fi ...
light scattering Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...
measures mass average molar mass. Mv is obtained from viscosimetry and Mz by
sedimentation Sedimentation is the deposition of sediments. It takes place when particles in suspension settle out of the fluid in which they are entrained and come to rest against a barrier. This is due to their motion through the fluid in response to the ...
in an analytical ultra-centrifuge. 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: Mn < Mv < Mw < Mz. The
dispersity In chemistry, the dispersity is a measure of the heterogeneity of sizes of molecules or particles in a mixture. A collection of objects is called uniform if the objects have the same size, shape, or mass. A sample of objects that have an inconsi ...
(also known as the ''polydispersity index'') of a sample is defined as Mw divided by Mn and gives an indication just how narrow a distribution is.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/PAC-REC-08-05-02.
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 Size-exclusion chromatography (SEC), also known as molecular sieve chromatography, is a chromatographic method in which molecules in solution are separated by their size, and in some cases molecular weight. It is usually applied to large molecules ...
(SEC) and
gel permeation chromatography Gel permeation chromatography (GPC) is a type of size-exclusion chromatography (SEC), that separates analytes on the basis of size, typically in organic solvents. The technique is often used for the analysis of polymers. As a technique, SEC was fi ...
(GPC). These techniques involve forcing a polymer solution through a matrix of
cross-linked In chemistry and biology a cross-link is a bond or a short sequence of bonds that links one polymer chain to another. These links may take the form of covalent bonds or ionic bonds and the polymers can be either synthetic polymers or natural ...
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 high-molecular-mass 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 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 Static light scattering is a technique in physical chemistry that measures the intensity of the scattered light to obtain the average molecular weight ''Mw'' of a macromolecule like a polymer or a protein in solution. Measurement of the 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 low-angle light scattering, which uses a single low angle to determine the
molar mass In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance which is the number of moles in that sample, measured in moles. The molar mass is a bulk, not molecular, ...
, 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. The molar mass distribution of a polymer sample depends on factors such as
chemical kinetics Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with chemical thermodynamics, which deals with the direction in ...
and work-up procedure. Ideal step-growth polymerization gives a polymer with dispersity of 2. Ideal
living polymerization In polymer chemistry, living polymerization is a form of chain growth polymerization where the ability of a growing polymer chain to terminate has been removed. This can be accomplished in a variety of ways. Chain termination and chain transfer ...
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 Mw and a small reduction in Mn thus reducing dispersity.


Number average molar mass

The number average molar mass is a way of determining the
molecular mass The molecular mass (''m'') is the mass of a given molecule: it is measured in daltons (Da or u). Different molecules of the same compound may have different molecular masses because they contain different isotopes of an element. The related quanti ...
of a
polymer A polymer (; Greek '' poly-'', "many" + ''-mer'', "part") is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic a ...
. 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_n=\frac The number average molecular mass of a polymer can be determined by
gel permeation chromatography Gel permeation chromatography (GPC) is a type of size-exclusion chromatography (SEC), that separates analytes on the basis of size, typically in organic solvents. The technique is often used for the analysis of polymers. As a technique, SEC was fi ...
,
viscometry A viscometer (also called viscosimeter) is an instrument used to measure the viscosity of a fluid. For liquids with viscosities which vary with flow conditions, an instrument called a rheometer is used. Thus, a rheometer can be considered as a spe ...
via the (
Mark–Houwink equation The Mark–Houwink equation, also known as the Mark–Houwink–Sakurada equation or the Kuhn–Mark–Houwink–Sakurada equation or the Landau–Kuhn–Mark–Houwink–Sakurada equation or the Mark-Chrystian equation gives a relation between intr ...
), colligative methods such as
vapor pressure osmometry Vapor phase osmometry (VPO), also known as vapor-pressure osmometry, is an experimental technique for the determination of a polymer's number average molecular weight The molar mass distribution (or molecular weight distribution) describes the relat ...
,
end-group End groups are an important aspect of polymer synthesis and characterization. In polymer chemistry, they are functional groups that are at the very ends of a macromolecule or oligomer (IUPAC). In polymer synthesis, like condensation polymerizati ...
determination or
proton NMR Proton nuclear magnetic resonance (proton NMR, hydrogen-1 NMR, or 1H NMR) is the application of nuclear magnetic resonance in NMR spectroscopy with respect to hydrogen-1 nuclei within the molecules of a substance, in order to determine the struct ...
. ''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 In step-growth polymerization, the Carothers equation (or Carothers' equation) gives the degree of polymerization, , for a given fractional monomer conversion, . There are several versions of this equation, proposed by Wallace Carothers, who inve ...
.


Mass average molar mass

The mass average molar mass (often loosely termed ''weight average molar mass'') is another way of describing the
molar mass In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance which is the number of moles in that sample, measured in moles. The molar mass is a bulk, not molecular, ...
of a
polymer A polymer (; Greek '' poly-'', "many" + ''-mer'', "part") is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic a ...
. 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_w=\frac 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 Static light scattering is a technique in physical chemistry that measures the intensity of the scattered light to obtain the average molecular weight ''Mw'' of a macromolecule like a polymer or a protein in solution. Measurement of the scattering ...
,
small angle neutron scattering Small-angle neutron scattering (SANS) is an analytical technique, experimental technique that uses elastic neutron scattering at small scattering angles to investigate the structure of various substances at a mesoscopic scale of about 1–100&nbs ...
,
X-ray scattering X-ray scattering techniques are a family of non-destructive analytical techniques which reveal information about the crystal structure, chemical composition, and physical properties of materials and thin films. These techniques are based on observ ...
, and sedimentation velocity. The ratio of the ''mass average'' to the ''number average'' is called the
dispersity In chemistry, the dispersity is a measure of the heterogeneity of sizes of molecules or particles in a mixture. A collection of objects is called uniform if the objects have the same size, shape, or mass. A sample of objects that have an inconsi ...
or the ''polydispersity'' index. The ''mass-average molecular mass'', ''M''w, is also related to the ''fractional monomer conversion'', ''p'', in step-growth polymerization (for the simplest case of linear polymers formed from two monomers in equimolar quantities) as per
Carothers' equation In step-growth polymerization, the Carothers equation (or Carothers' equation) gives the degree of polymerization, , for a given fractional monomer conversion, . There are several versions of this equation, proposed by Wallace Carothers, who inve ...
: :\bar_w=\frac \quad \bar_w=\frac, where ''M''o is the molecular mass of the repeating unit.


Z-average molar mass

The z-average molar mass is the third moment or third power average molar mass, which is calculated by \bar_z=\frac \quad The z-average molar mass can be determined with ultracentrifugation. The melt elasticity of a polymer is dependent on Mz.Seymore, R.B and Caraher, C.E. Polymer Chemistry: An Introduction, 1992.


See also

* Distribution function * Flory–Schulz distribution * Mass distribution *
Sedimentation Sedimentation is the deposition of sediments. It takes place when particles in suspension settle out of the fluid in which they are entrained and come to rest against a barrier. This is due to their motion through the fluid in response to the ...


References

{{Authority control Polymer chemistry