Which of the following represence the units of van der Waal constant `a` ?

Options

A. `L mol^(-1)`
B. `atm L^(2) mol^(-2)`
C. `L atm. Mol^(-1)`
D. `L atm. Mol^(-2)`

Explanation

To find the units of the van der Waals constant \( a \), we can start from the van der Waals equation for real gases, which is given by:

\[ \left( P + \frac{a n^2}{V^2} \right)(V - nb) = nRT \]

Where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles
- \( R \) = universal gas constant
- \( T \) = temperature
- \( a \) and \( b \) = van der Waals constants

We can isolate the term involving \( a \):

\[ P = \frac{a n^2}{V^2} - \frac{nRT}{(V - nb)} \]

To find the units of \( a \), we can focus on the term \( \frac{a n^2}{V^2} \) and rearrange it to solve for \( a \):

\[ a = \frac{P V^2}{n^2} \]

Now, we need to determine the units of each component:

1. **Pressure \( P \)**: The unit of pressure is typically in atmospheres (atm).
2. **Volume \( V \)**: The unit of volume is in liters (L).
3. **Number of moles \( n \)**: The unit of the number of moles is in moles (mol).

Now substituting the units into the equation:

- The unit of \( P \) is \( \text{atm} \).
- The unit of \( V^2 \) is \( \text{L}^2 \).
- The unit of \( n^2 \) is \( \text{mol}^2 \).

Now substituting these units into the equation for \( a \):

\[ a = \frac{\text{atm} \cdot \text{L}^2}{\text{mol}^2} \]

Thus, the units of the van der Waals constant \( a \) are:

\[ \text{atm} \cdot \text{L}^2 \cdot \text{mol}^{-2} \]

### Final Answer:
The units of the van der Waals constant \( a \) are \( \text{atm} \cdot \text{L}^2 \cdot \text{mol}^{-2} \).