The average molar mass, often referred to as the number average molar mass, is calculated by taking into account the number of molecules (or polymer chains) of each size present in a sample. Here's how you can find it:
Understanding Number Average Molar Mass
The Number Average Molar Mass (Mn) is a measure of the average molecular weight of a polymer sample. This is particularly important in polymer chemistry where samples often contain a distribution of chain lengths. The reference defines it as:
Number Average Molar Mass refers to the average mass of polymer chains in a sample, calculated by summing the product of each chain's mass and number of chains, and dividing by the total number of chains.
Calculating Number Average Molar Mass
Here's a step-by-step guide to calculate the average molar mass using the method described:
Step-by-Step Calculation
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Identify the molecules/chains present: Determine the various molecular species (or polymer chains) in your sample and their respective molar masses (Mi).
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Determine the number of molecules/chains: Count the number of molecules (Ni) or polymer chains of each type present.
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Calculate the product of mass and number for each type: For each type of molecule, multiply the molar mass (Mi) by the number of molecules (Ni). That is, Mi Ni.
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Sum the products: Add all the products obtained in Step 3. Mathematically, this is ∑(Mi Ni).
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Sum the number of molecules/chains: Add all the numbers of molecules or polymer chains to find the total number of molecules (∑Ni).
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Divide to find the average molar mass: Divide the sum of the products (from Step 4) by the total number of molecules (from Step 5). The number average molar mass (Mn) is:
Mn = ∑(Mi Ni) / ∑Ni
Example Calculation
Let's say you have a sample with the following molecular species:
| Molecule Type | Molar Mass (g/mol) (Mi) | Number of Molecules (Ni) |
|---|---|---|
| Type A | 1000 | 5 |
| Type B | 2000 | 3 |
| Type C | 3000 | 2 |
Here's the calculation:
- Mi Ni for Type A: 1000 g/mol * 5 = 5000
- Mi Ni for Type B: 2000 g/mol * 3 = 6000
- Mi Ni for Type C: 3000 g/mol * 2 = 6000
- ∑(Mi Ni)= 5000 + 6000 + 6000 = 17000
- ∑Ni = 5 + 3 + 2 = 10
- Mn = 17000 / 10 = 1700 g/mol
Therefore, the number average molar mass of this sample is 1700 g/mol.
Key Points to Remember
- Number average: The method emphasizes the number of chains (or molecules) over their individual mass. A greater number of small molecules will significantly influence this average.
- Polymers: The average molar mass is particularly useful in polymer chemistry, as these materials rarely consist of chains with uniform size.
- Applications: This average is essential for determining properties of polymeric materials that depend on the number of molecules present, such as colligative properties.
