For most of the synthetic polymers

Options

A. `bar(M)_(w) gt bar(M)_(n)`
B. `bar(M)_(n) gt bar(M)_(w)`
C. `bar(M)_(n) = bar(M)_(w)`
D. `bar(M)_(n) = 2bar(M)_(w)`

Answer

To solve the question regarding the relationship between the weight average molecular weight (Mw) and the number average molecular weight (Mn) for most synthetic polymers, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Terms**:
- **Mw (Weight Average Molecular Weight)**: This is the average molecular weight of a polymer sample, weighted by the mass of each species.
- **Mn (Number Average Molecular Weight)**: This is the average molecular weight of a polymer sample, weighted by the number of molecules.

2. **Introduce Polydispersity Index (PDI)**:
- The **Polydispersity Index (PDI)** is defined as the ratio of Mw to Mn:
\[ \text{PDI} = \frac{M_w}{M_n} \]
- PDI provides insight into the distribution of molecular weights in a given polymer sample.

3. **Analyze PDI Values**:
- For **natural polymers**, the PDI is typically equal to 1, indicating a monodisperse system where all molecules have the same molecular weight.
- For **synthetic polymers**, the PDI is generally greater than 1, indicating a polydisperse system where there is a distribution of molecular weights.

4. **Determine the Relationship**:
- Since for synthetic polymers, the PDI is greater than 1, we can infer:
\[ \frac{M_w}{M_n} > 1 \]
- This implies:
\[ M_w > M_n \]

5. **Select the Correct Option**:
- Based on the analysis, the correct answer to the question "For most of the synthetic polymers" is that the weight average molecular weight (Mw) is greater than the number average molecular weight (Mn). Therefore, the correct option is:
- **Mw is greater than Mn**.

### Final Answer:
The correct answer is **Mw > Mn**.

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