Calculating the molecular weight of a polymer is not as straightforward as with simple molecules because polymers are made up of chains of varying lengths. Instead of a single molecular weight, polymers are characterized by average molecular weights. The most common average is the number-average molecular weight.
Understanding Average Molecular Weights
Polymers are typically a mixture of chains with different lengths. This is why we use average molecular weights instead of a single, defined value. The two most common ways to express molecular weight are:
-
Number-Average Molecular Weight (Mn): This is the total weight of all polymer molecules divided by the total number of polymer molecules. It is often expressed by the formula:
Mn = (Total Weight of Polymer) / (Total Number of Molecules)
This value is particularly sensitive to low molecular weight chains.
-
Weight-Average Molecular Weight (Mw): This is based on the weight fraction of each chain length. It is always higher than the number average.
How to Calculate Number-Average Molecular Weight (Mn)
Since the question focuses on general ways to calculate average molecular weights, we will concentrate on the number-average molecular weight. Here's how to determine it:
-
Determine the Molecular Weight of Each Chain Length: Analyze your polymer sample to determine the molecular weight of each chain length. This can be done using techniques like size-exclusion chromatography (SEC) or gel permeation chromatography (GPC). These techniques separate polymer chains based on their size, making it possible to quantify the amount of polymer with specific molecular weights.
-
Determine the Number of Chains for Each Length: For each molecular weight value obtained, you'll need to determine the total number of molecules.
-
Multiply Molecular Weight by Number of Chains: Multiply the molecular weight of each chain length by the number of chains having that specific molecular weight.
-
Sum the Products: Add up all the results obtained in step 3 to get the total weight of the polymer sample.
-
Sum the Total Number of Chains: Add up the number of polymer chains you found in step 2 to determine the total number of molecules.
-
Calculate Mn: Divide the total weight obtained in step 4 by the total number of molecules obtained in step 5.
- Mn = (Σ NiMi) / (Σ Ni)
where:- Ni = Number of molecules of chain length i
- Mi = Molecular weight of chain length i
- Mn = (Σ NiMi) / (Σ Ni)
Practical Example
Imagine a polymer sample with two chain lengths:
| Chain Length | Molecular Weight (Mi) | Number of Chains (Ni) | Ni * Mi |
|---|---|---|---|
| Short | 10,000 g/mol | 100 | 1,000,000 |
| Long | 20,000 g/mol | 50 | 1,000,000 |
| Total | 150 | 2,000,000 |
Then,
- Total weight of polymer = 1,000,000 + 1,000,000 = 2,000,000 g
- Total number of molecules = 100 + 50 = 150
- Number-average molecular weight, Mn = 2,000,000 / 150 = 13,333 g/mol.
Conclusion
Calculating the molecular weight of a polymer involves determining the average molecular weight due to the presence of chains of varying lengths. The number-average molecular weight is calculated by dividing the total weight of the polymer by the total number of molecules. Analytical techniques like chromatography are essential to determine these chain lengths and quantities before any molecular weight calculation.
