If `bar(M)_(w)` is the weight -average molecular weight and `bar(M)_(n)` is the number -average molecular weight of polymer, the poly dispersity index (PDI) of the polymer is given by

Options

A. `(bar(M)_(n))/(bar(M)_(w))`
B. `(bar(M)_(w))/(bar(M)_(n))`
C. `bar(M)_(w) xx bar(M)_(n)`
D. `(1)/(bar(M)_(w) xx bar(M)_(n))`

Answer

To solve the question regarding the polydispersity index (PDI) of a polymer, we can follow these steps:

### Step 1: Understand the Definitions
- **Weight-average molecular weight (\( \bar{M}_w \))**: This is the average molecular weight of a polymer sample, weighted by the mass fraction of each component.
- **Number-average molecular weight (\( \bar{M}_n \))**: This is the average molecular weight calculated by considering the number of molecules of each component.

### Step 2: Define the Polydispersity Index (PDI)
The polydispersity index (PDI) is a measure of the distribution of molecular mass in a given polymer sample. It is defined as the ratio of the weight-average molecular weight to the number-average molecular weight.

### Step 3: Write the Formula for PDI
The formula for the polydispersity index can be expressed as:
\[ \text{PDI} = \frac{\bar{M}_w}{\bar{M}_n} \]

### Step 4: Interpret the PDI Value
- A PDI value of 1 indicates a monodisperse polymer, meaning all the polymer chains have the same molecular weight.
- A PDI value greater than 1 indicates a polydisperse polymer, meaning there is a distribution of molecular weights among the polymer chains.

### Conclusion
Thus, the polydispersity index (PDI) of the polymer is given by the formula:
\[ \text{PDI} = \frac{\bar{M}_w}{\bar{M}_n} \]