The number average molecular mass (Mn) is a measure of the average molecular mass in a sample, considering the number of molecules of each mass present. It's calculated by dividing the total weight of the sample by the total number of molecules.
Calculating Mn for Monodisperse and Polydisperse Systems
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Monodisperse System: A monodisperse system contains only molecules of one size. In this simpler case, the calculation is straightforward:
Mn = W / N
Where:
- W = total weight of the sample (grams)
- N = total number of molecules
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Polydisperse System: A polydisperse system, the more common scenario, contains molecules of various sizes. The calculation involves summing the product of each molecule's mass and its number, then dividing by the total number of molecules.
Mn = Σ (Ni * Mi) / Σ Ni
Where:
- Ni = number of molecules with molecular mass Mi
- Mi = molecular mass of the ith type of molecule
- Σ Ni = total number of molecules
Example:
Let's say we have a polydisperse polymer sample with:
- 100 molecules with a molecular mass of 1000 g/mol
- 50 molecules with a molecular mass of 2000 g/mol
- 25 molecules with a molecular mass of 3000 g/mol
Mn = [(100 1000) + (50 2000) + (25 * 3000)] / (100 + 50 + 25) = 1437.5 g/mol
Understanding the References
The provided references consistently define the number average molecular weight as the total weight of the polymer divided by the total number of molecules. This is the basis of the formulas presented above, applying to both simple (monodisperse) and complex (polydisperse) systems. The reference at https://pslc.ws/macrog/average.htm clearly states this definition. Similarly, https://www.doitpoms.ac.uk/tlplib/polymerbasics/mw.php reinforces the concept. Other references expand upon this, discussing molar mass distribution and the implications for various polymer properties.
