Name: Polymers: Defining Molecular Weight
Uploaded: 2023-04-30T11:23:34
Description: Learn by watching this video about Polymers: Defining Molecular Weight at JoVE.com

Unlike a small molecule with a definite molecular weight, a polymer can be considered as a mixture of molecules with various molecular weights.
The overall molecular weight of a polymer is expressed in two types of averages: number average molecular weight and weight average molecular weight.
The number average is the summation of the number fraction of each unique polymer chain times its molecular weight. For each unique chain, the number fraction is the ratio of the number of just those chains to the total number of polymer chains in the sample.
The weight average is the summation of the weight fraction of each unique polymer chain times its molecular weight. For each unique chain, the weight fraction is the ratio of the total weight of just those chains to the total weight of the sample.
The weight average is higher than the number average unless all the polymer chains have the same molecular weight.

<ol>
	<li>Oullette, R. J., &#38; Rawn, J. D. (2014). Organic Chemistry: Structure, Mechanism, and Synthesis. Elsevier, 993.</li>
	<li>Smith, J. G. (2008). Organic Chemistry. New York: McGraw-Hill Higher Education, 1145.</li>
	<li>DoITPoMS. University of Cambridge. Polymer Basics: Molecular weight.&#160; <a href="https://www.doitpoms.ac.uk/tlplib/polymerbasics/mw.php">https://www.doitpoms.ac.uk/tlplib/polymerbasics/mw.php</a></li>
	<li>IUPAC. (2007) Quantities, Units, and Symbols in Physical Chemistry. 3rd edition. RSC Publishing. <a href="https://iupac.org/what-we-do/books/greenbook/">https://iupac.org/what-we-do/books/greenbook/</a> Prepared for publication by Cohen, E. R., Cvitas, T., Frey, J. G., Holmstrom, B., Kuchitsu, K., Marquardt, R., Mills, I., Pavese, F., Quack, M., Stohner, J., Strauss, H., Takami, M., &#38; Thor, A.J.</li>
</ol>

polymers: defining molecular weight

Unlike small molecules with definite molecular weights, polymers are a mixture of individual polymer chains of varying lengths, each with a unique molecular weight.&#160; So, the molecular weight of a polymer is expressed as an average value based on the average size of the polymer chains. The two most common forms of averages used for polymers are the number average molecular weight and weight average molecular weight.
The number average molecular weight (Mn) is the summation of the number fractions (x) of the unique polymer chains multiplied by their molecular weights (M).
<img alt="Figure1" src="/files/ftp_upload/12914/12914_PT_Figure_1.svg" style="text-align: center; height: 25px;" />
Here, each number fraction (xj) is the ratio of the number of polymer chains with a particular molecular weight (nj) to the total number of polymer chains.
<img alt="Figure2" src="/files/ftp_upload/12914/12914_PT_Figure_2.svg" style="text-align: center; height: 45px;" />
The number average molecular weight is a good measure of the chain length of the polymer.
Table 1: Calculation of the number fraction and number average molecular weight (NAMW) of a polymer sample with polymer chains of different weights
<table style="margin-left: auto; margin-right: auto;">
	<tbody style="border: 1px solid black;">
		<tr>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">Chain</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">No. of chains</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">MW of chain</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">Number fraction</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">NAMW</td>
		</tr>
		<tr>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">1</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">3</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">10</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">3/21</td>
			<td rowspan="5" style="border-style: solid; border-color: black; color: black; text-align: center;">30.71</td>
		</tr>
		<tr>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">2</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">5</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">20</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">5/21</td>
		</tr>
		<tr>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">3</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">9</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">35</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">9/21</td>
		</tr>
		<tr>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">4</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">4</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">50</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">4/21</td>
		</tr>
		<tr>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">Total</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">21</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;"></td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;"></td>
		</tr>
	</tbody>
</table>
Alternatively, the weight average molecular weight (Mw) is the sum of the weight fractions (w) of the polymer chains multiplied by their molecular weights.
<img alt="Figure3" src="/files/ftp_upload/12914/12914_PT_Figure_3.svg" style="text-align: center; height: 25px;" />
Here, each weight fraction (wj) is the ratio of the total weight of all polymer chains with a specific molecular weight (njMj) to the total weight of the polymer sample.
<img alt="Figure4" src="/files/ftp_upload/12914/12914_PT_Figure_4.svg" style="text-align: center; height: 45px;" />
Weight average molecular weight gives an idea about the statistical size of the polymer.
Table 2: Calculation of the weight fraction and weight average molecular weight (WAMW) of a polymer sample with polymer chains of different weights.
<table style="margin-left: auto; margin-right: auto;">
	<tbody style="border: 1px solid black;">
		<tr>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">Chain</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">No. of chains</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">MW of chain</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">Total weight</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">Weight fraction</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">WAMW</td>
		</tr>
		<tr>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">1</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">3</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">10</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">30</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">30/645</td>
			<td rowspan="5" style="border-style: solid; border-color: black; color: black; text-align: center;">82.66</td>
		</tr>
		<tr>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">2</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">5</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">20</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">100</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">100/645</td>
		</tr>
		<tr>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">3</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">9</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">35</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">315</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">315/645</td>
		</tr>
		<tr>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">4</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">4</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">50</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">200</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">200/645</td>
		</tr>
		<tr>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">Total</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">21</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;"></td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;">645</td>
			<td style="border-style: solid; border-color: black; color: black; text-align: center;"></td>
		</tr>
	</tbody>
</table>
Typically, the weight average molecular weight is higher than the number average molecular weight because the larger chains in a sample weigh more than the smaller chains.

Learn by watching this video about Polymers: Defining Molecular Weight at JoVE.com