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Question: If \[{M_w}\] is the weight average molecular weight and \({M_n}\) is the number average molecular we...

If Mw{M_w} is the weight average molecular weight and Mn{M_n} is the number average molecular weight of a polymer, the polydispersity index (PDI) of the polymer is given by:
A. MˉnMw\dfrac{{{{\bar M}_n}}}{{{M_w}}}
B. MˉnMˉw\dfrac{{{{\bar M}_n}}}{{{{\bar M}_w}}}
C. Mˉw×Mˉn{\bar M_w} \times {\bar M_n}
D. 1Mˉw×Mˉn\dfrac{1}{{{{\bar M}_w} \times {{\bar M}_n}}}

Explanation

Solution

A polymer material is denoted by the term disperse, or non-uniform, if its chain lengths vary over a wide range of molecular masses. This is characteristic of man-made polymers. Natural organic matter produced by the decomposition of plants and wood debris in soils (humic substances) also has a pronounced polydisperse character.

Complete answer:
In chemistry, the dispersion 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 inconsistent size, shape and mass distribution is called non-uniform. The objects can be in any form of chemical dispersion, such as particles in a colloid, droplets in a cloud, crystals in a rock, or polymer macromolecules in a solution or a solid polymer mass. Polymers can be described by molecular mass distribution; a population of particles can be described by size, surface area, and/or mass distribution; and thin films can be described by film thickness distribution.
The polydispersity index (PDI) or heterogeneity index, is a measure of the distribution of molecular mass in a given polymer sample. The PDI of a polymer is the ratio of weight average molecular weight and the number average molecular weight. Mathematically, this can be denoted as:
PDI=PDI = MˉnMˉw\dfrac{{{{\bar M}_n}}}{{{{\bar M}_w}}}

Thus, the correct option is B.

Note:
The dispersity indicates the distribution of individual molecular masses in a batch of polymers. Polydispersity index has a value equal to or greater than 1, but as the polymer chains approach uniform chain length, PDI approaches unity (1). For some natural polymers PDI is almost taken as unity.