Name: polymer molecular weight averages
Uploaded: 2023-04-30T11:23:34.000Z
Duration: 1 min 1 s
Channel: JoVE

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.

polymers: defining molecular weight

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. 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="Polymer molecular weight equation; summation formula; static equilibrium; chemical analysis." src="/files/ftp_upload/12914/12914_PT_Figure_1.svg" style="text-align: center; height: 25px;" data-altupdatedat="1761830439000" data-alt="Figure1" />
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="Mole fraction formula, mathematical equation showing X_j = n_j/&#931;n_i from 1 to &#8734;." src="/files/ftp_upload/12914/12914_PT_Figure_2.svg" style="text-align: center; height: 45px;" data-altupdatedat="1761830439000" data-alt="Figure2" />
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="Molecular weight equation \(M_w = \sum_{i=1}^{\infty} w_i M_i\), formula representation." src="/files/ftp_upload/12914/12914_PT_Figure_3.svg" style="text-align: center; height: 25px;" data-altupdatedat="1761830439000" data-alt="Figure3" />
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="Weighted mean formula, \( w_j = \frac{n_j M_j}{\sum_{i=1}^{\infty} n_i M_i} \), statistical analysis equation." src="/files/ftp_upload/12914/12914_PT_Figure_4.svg" style="text-align: center; height: 45px;" data-altupdatedat="1761830439000" data-alt="Figure4" />
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.

Unlike small molecules with definite molecular weights, polymers are a mixture of individual polymer chains of varying lengths, each with a unique ...

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KEY TERMS AND CONCEPTS

polymer molecular weight averages

Polymer molecular weight is often described with averages because a polymer sample contains chains of many different lengths. Unlike a small molecule, which has one definite molecular weight, each polymer chain can have a unique molecular weight. The result is a sample made up of chains with varying sizes.

The number average molecular weight, or Mn, uses the number fraction of each unique chain. A number fraction is the number of chains with a given molecular weight divided by the total number of chains. Mn is a useful measure of the chain length of the polymer.

The weight average molecular weight, or Mw, uses the weight fraction of each chain type. A weight fraction is the total weight of chains with one molecular weight divided by the total weight of the whole polymer sample. Mw gives an idea of the statistical size of the polymer.

For a sample with chains of different weights, the two averages can be different. In the example shown, the number average molecular weight is 30.71, while the weight average molecular weight is 82.66. The weight average molecular weight is typically higher because the larger chains contribute more mass than the smaller chains.

Polymer Molecular Weight Averages

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key terms and concepts

Learn how polymer molecular weight is calculated with number average and weight average values, and why polymers are described by averages rather than one exact number.

Polymer Molecular Weight Averages | JoVE K12

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Polymer Molecular Weight Averages

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