The correct formula to calculate the weight average molecular weight of polymers is

Options

A. `(sum N_(i)M_(i))/(sum N_(i))`
B. `(sum N_(i)M_(i)^(2))/(sum N_(i))`
C. `(sum N_(i)M_(i))/(sum N_(i)M_(i)^(2))`
D. `(sum N_(i)M_(i)^(2))/(sum N_(i)M_(i))`

Answer

To find the correct formula for calculating the weight average molecular weight of polymers, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Terms**:
- Let \( n_i \) be the number of chains of molecular weight \( m_i \).
- The weight fraction \( w_i \) can be defined as the product of the number of chains and the molecular weight of that chain.

2. **Defining Weight Fraction**:
- The weight fraction \( w_i \) is given by:
\[ w_i = n_i \cdot m_i \]
- This means that the weight fraction is directly proportional to the number of chains and their respective molecular weights.

3. **Calculating Average Molecular Weight**:
- The formula for the weight average molecular weight \( M_w \) can be expressed as:
\[ M_w = \frac{\sum (w_i \cdot m_i)}{\sum w_i} \]
- Here, \( w_i \) is the weight fraction and \( m_i \) is the molecular weight of the chains.

4. **Substituting Weight Fraction**:
- Substitute \( w_i \) in the formula:
\[ M_w = \frac{\sum (n_i \cdot m_i \cdot m_i)}{\sum (n_i \cdot m_i)} \]
- This can be simplified to:
\[ M_w = \frac{\sum (n_i \cdot m_i^2)}{\sum (n_i \cdot m_i)} \]

5. **Final Formula**:
- The final formula for the weight average molecular weight of polymers is:
\[ M_w = \frac{\sum (n_i \cdot m_i^2)}{\sum (n_i \cdot m_i)} \]
- This matches the fourth option provided in the question.

### Conclusion:
Thus, the correct formula to calculate the weight average molecular weight of polymers is:
\[ M_w = \frac{\sum (n_i \cdot m_i^2)}{\sum (n_i \cdot m_i)} \]