The number-average molecular mass is an average molecular weight determined by considering the number of molecules of each size in a polymer sample. It represents the total weight of the sample divided by the total number of molecules.
Understanding Number-Average Molecular Mass (Mn)
The number-average molecular mass (Mn) is a crucial parameter in polymer science, providing insight into the average size of polymer chains within a sample. It's calculated by:
Mn = Σ(Ni * Mi) / ΣNi
Where:
- Ni is the number of molecules with molecular mass Mi
- Mi is the molecular mass of those molecules
Essentially, each molecule contributes equally to the average, regardless of its size.
Significance of Number-Average Molecular Mass
- Colligative Properties: Mn is particularly important when considering colligative properties (e.g., osmotic pressure, boiling point elevation, freezing point depression) which depend on the number of solute particles present, not their mass.
- Polymer Synthesis: Understanding Mn allows scientists to control and optimize polymerization processes.
- Material Properties: Mn influences various material properties such as tensile strength, viscosity, and elasticity.
Example Calculation
Imagine a polymer sample containing:
- 10 molecules with a molecular mass of 10,000 g/mol
- 5 molecules with a molecular mass of 20,000 g/mol
Then the number-average molecular mass (Mn) would be:
Mn = (10 10,000 + 5 20,000) / (10 + 5) = (100,000 + 100,000) / 15 = 200,000 / 15 ≈ 13,333.33 g/mol
Comparison with Weight-Average Molecular Mass (Mw)
It's important to distinguish Mn from weight-average molecular mass (Mw). Mw is more sensitive to the presence of high molecular weight molecules. Mw is calculated as:
Mw = Σ(Wi Mi) / ΣWi = Σ(Ni Mi^2) / Σ(Ni * Mi)
Where:
- Wi is the weight fraction of molecules with molecular mass Mi
Typically, Mw > Mn for a polydisperse polymer sample (i.e., a sample with chains of varying lengths). The ratio Mw/Mn is known as the polydispersity index (PDI), which provides a measure of the distribution of molecular weights in the sample. A PDI close to 1 indicates a narrow distribution (nearly monodisperse).
Measurement Techniques
Several experimental techniques are used to determine Mn, including:
- Osmometry: Measures osmotic pressure, which is directly related to the number of molecules.
- End-group analysis: Chemically titrates the number of end groups in a polymer sample.
- Vapor pressure osmometry: Measures the vapor pressure lowering of a solution.
In conclusion, the number-average molecular mass represents an average molecular weight that is highly sensitive to the number of molecules present in a sample. It is essential for understanding and controlling the properties of polymers.
