In a polymer sample, `30%` of molecules have a molecualr mass of `20,000, 40%` have 30,000 and the rest 60,000. What is the weight average molecular mass of the polymer?

Options

A. 40300
B. 30600
C. 43333
D. 50400

Answer

To find the weight average molecular mass of the polymer, we can follow these steps:

### Step 1: Define the molecular masses and their percentages
- 30% of molecules have a molecular mass of 20,000 g/mol.
- 40% of molecules have a molecular mass of 30,000 g/mol.
- The remaining 30% of molecules have a molecular mass of 60,000 g/mol.

### Step 2: Assume a total number of molecules
For simplicity, we can assume there are 100 molecules in total. This makes it easier to calculate the number of molecules for each mass category:
- Number of molecules with 20,000 g/mol = 30% of 100 = 30 molecules
- Number of molecules with 30,000 g/mol = 40% of 100 = 40 molecules
- Number of molecules with 60,000 g/mol = 100 - (30 + 40) = 30 molecules

### Step 3: Calculate the contribution to the weight average molecular mass
Using the formula for weight average molecular mass (Mw):

\[ M_w = \frac{\sum (n_i \cdot M_i)}{\sum n_i} \]

where \( n_i \) is the number of molecules and \( M_i \) is the molecular mass.

#### Calculate the numerator:
- Contribution from 20,000 g/mol:
\[ 30 \cdot 20,000 = 600,000 \]
- Contribution from 30,000 g/mol:
\[ 40 \cdot 30,000 = 1,200,000 \]
- Contribution from 60,000 g/mol:
\[ 30 \cdot 60,000 = 1,800,000 \]

Adding these contributions together:
\[ 600,000 + 1,200,000 + 1,800,000 = 3,600,000 \]

#### Calculate the denominator:
The total number of molecules is:
\[ 30 + 40 + 30 = 100 \]

### Step 4: Calculate the weight average molecular mass
Now, substituting the values into the formula:
\[ M_w = \frac{3,600,000}{100} = 36,000 \text{ g/mol} \]

### Final Answer:
The weight average molecular mass of the polymer is **36,000 g/mol**.

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