Write the formula to determine ‘weight average molecular weight’ of polymers.
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Polymer molecular weight is defined as a distribution rather than a specific number because polymerization occurs in such a way to produce different chain lengths. Weight average molecular weight (MW) and number average molecular weight (MN) are two ways we can characterize the polymer molecular weight. Polymer MW is defined as follows.
M
W
=
∑
i
=
1
N
w
i
MW
i
.
Here, wi is the weight fraction of polymer chains having a molecular weight of MWi. The MW is typically measured by light scattering experiments. When light passes through a liquid polymer-solvent solution, it is scattered by the individual molecules suspended in the solution. The degree of scattering arises from the molecule size and, thus, a molecular weight distribution can be mathematically resolved from the total scattering created by the sample.
MN is an average based on the number of polymer chains in a sample and is defined as follows.
M
N
=
∑
i
=
1
N
X
i
MW
i
.
In this equation, MWi refers to the molecular weight of each individual polymer chain and Xi refers to the mole fraction of each chain length included in the summation. This number average means that each chain, regardless of its molecular weight, contributes equally to the reported MN.
Gel permeation chromatography (GPC), also known as size exclusion chromatography, is used to determine this property. In GPC, a dissolved polymer passes over a material held in a cylindrical column. The material, the substrate, contains holes that temporarily trap low molecular weight molecules. The lower the individual molecule’s molecular weight, the more time they spend trapped and, thus, the longer they spend on the column. The polymer molecule concentration in the solvent escaping the column is measured, and residence time on the column is related to the inverse of the molecular weight.
The ratio of (Mw/Mn) allows us to define the polymer polydispersity. Fig. 3.3 illustrates Mw and Mn for a high- and low-polydispersity polymer.
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Figure 3.3. Molecular weight distributions for a high- and low-polydispersity index polypropylene. In both materials, Mw>Mn. The polydispersity for the left figure is higher because of the broader molecular weight distribution which means that the high molecular weight materials influence MW more than they influence Mn.
For all real polymers, MnView chapterPurchase book
M
W
=
∑
i
=
1
N
w
i
MW
i
.
Here, wi is the weight fraction of polymer chains having a molecular weight of MWi. The MW is typically measured by light scattering experiments. When light passes through a liquid polymer-solvent solution, it is scattered by the individual molecules suspended in the solution. The degree of scattering arises from the molecule size and, thus, a molecular weight distribution can be mathematically resolved from the total scattering created by the sample.
MN is an average based on the number of polymer chains in a sample and is defined as follows.
M
N
=
∑
i
=
1
N
X
i
MW
i
.
In this equation, MWi refers to the molecular weight of each individual polymer chain and Xi refers to the mole fraction of each chain length included in the summation. This number average means that each chain, regardless of its molecular weight, contributes equally to the reported MN.
Gel permeation chromatography (GPC), also known as size exclusion chromatography, is used to determine this property. In GPC, a dissolved polymer passes over a material held in a cylindrical column. The material, the substrate, contains holes that temporarily trap low molecular weight molecules. The lower the individual molecule’s molecular weight, the more time they spend trapped and, thus, the longer they spend on the column. The polymer molecule concentration in the solvent escaping the column is measured, and residence time on the column is related to the inverse of the molecular weight.
The ratio of (Mw/Mn) allows us to define the polymer polydispersity. Fig. 3.3 illustrates Mw and Mn for a high- and low-polydispersity polymer.
Sign in to download full-size image
Figure 3.3. Molecular weight distributions for a high- and low-polydispersity index polypropylene. In both materials, Mw>Mn. The polydispersity for the left figure is higher because of the broader molecular weight distribution which means that the high molecular weight materials influence MW more than they influence Mn.
For all real polymers, MnView chapterPurchase book
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Answer:
Explanation:
The weight average molecular weight depends not only on the number of molecules present, but also on the weight of each molecule. To calculate this, Ni is replaced with NiMi.
MW¯¯¯¯¯¯¯¯¯=∑i=1∞NiM2i∑i=1∞NiMi
This can also be written as:
MW¯¯¯¯¯¯¯¯¯=∑i=1∞wiMi
where wi is the weight fraction of polymer with molecular weight Mi.
The weight average molecular weight is therefore weighted according to weight fractions.
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