The number average and weight average molecular weights are two distinct ways to describe the average molecular weight of a polymer sample or any mixture containing molecules of varying sizes. They provide different perspectives on the distribution of molecular weights within the sample.
Number Average Molecular Weight (Mn)
The number average molecular weight (M̄n) considers the number of molecules present. It is calculated by summing the molecular weights of all individual molecules and then dividing by the total number of molecules. The reference provided defines this as:
M ¯ n = ∑ N i M i / ∑ N i = ∑ w i / ∑ ( w i / M i )
Where:
- Mi is the molecular weight of the ith species.
- Ni is the number of molecules of the ith species.
- wi is the weight fraction of the ith species.
In simpler terms, each molecule contributes equally to the average, regardless of its size. This is very useful for understanding colligative properties, which depend on the number of particles and not necessarily the size of the particles (like boiling point elevation, freezing point depression, etc.)
Example: If you have a mixture with 10 molecules of 10,000 g/mol and 20 molecules of 20,000 g/mol:
- M̄n = (10 10,000 + 20 20,000) / (10 + 20)
- M̄n = 500,000 / 30
- M̄n = 16,666.67 g/mol
Weight Average Molecular Weight (Mw)
The weight average molecular weight (M̄w) takes into account the weight fraction of each species. Heavier molecules contribute more to the average because they make up a larger portion of the total mass. The reference provides the following definition for this:
M ¯ w = ∑ w i M i / ∑ w i
Where:
- Mi is the molecular weight of the ith species.
- wi is the weight fraction of the ith species.
Essentially, it's a weighted average where each molecule's contribution is scaled by its molecular weight. This type of average is more influenced by the presence of higher molecular weight molecules and is more relevant to the mechanical properties of a polymer.
Example: Using the same mixture as above (10 molecules of 10,000 g/mol and 20 molecules of 20,000 g/mol):
- Total weight of 10,000 g/mol molecules = 10 * 10,000 = 100,000 g
- Total weight of 20,000 g/mol molecules = 20 * 20,000 = 400,000 g
- Total Weight of sample = 500,000 g
- Weight fraction of 10,000 g/mol molecules = 100,000 / 500,000 = 0.2
- Weight fraction of 20,000 g/mol molecules = 400,000 / 500,000 = 0.8
- M̄w = (0.2 10,000 + 0.8 20,000)/ 1 = 18,000 g/mol
Key Differences and Implications
Here’s a table summarizing the key differences:
| Feature | Number Average Molecular Weight (Mn) | Weight Average Molecular Weight (Mw) |
|---|---|---|
| Focus | Number of molecules | Weight fraction of molecules |
| Sensitivity | Equally sensitive to all molecules | More sensitive to larger molecules |
| Calculation | Sum of all molecular weights divided by the number of molecules | Weighted sum of molecular weights by their weight fractions |
| Relevance | Colligative properties | Mechanical properties |
| Typical Relationship | M̄n ≤ M̄w | M̄w ≥ M̄n |
In a sample with a range of molecular weights, M̄w is always equal to or greater than M̄n because it is more influenced by the presence of larger molecules. The ratio of M̄w to M̄n is known as the polydispersity index (PDI), which is used to measure the breadth of the molecular weight distribution. A PDI of 1 indicates a perfectly monodisperse sample, where all molecules have the same molecular weight.
Understanding the difference between these two averages is essential for characterizing polymer materials and predicting their behaviors.
