Which of the following expressions represents the value and unit of van der Waals constant a?

Options

A. `a = (V)/(n) ," L mol"^(-1)`
B. `a = (PV)/(n) , "atm L"^(2) mol^(-1)`
C. `a = (PV^(2))/(n^(2)) , "atm L"^(2) mol^(-1)`
D. `a = (P)/(n) ," atm mol"^(-1)`

Answer

To solve the question regarding the van der Waals constant \( a \), we will follow these steps:

### Step 1: Understand the van der Waals equation
The van der Waals equation for real gases is given by:
\[ \left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT \]
where:
- \( P \) = pressure of the gas
- \( V_m \) = molar volume of the gas
- \( T \) = temperature
- \( R \) = universal gas constant
- \( a \) = van der Waals constant for the attractive forces between particles
- \( b \) = volume occupied by the gas particles

### Step 2: Rearranging the equation
From the van der Waals equation, we can isolate the term involving \( a \):
\[ P = \frac{RT}{V_m - b} - \frac{a}{V_m^2} \]
This shows that \( a \) is related to the pressure and volume of the gas.

### Step 3: Identify the units of \( a \)
To find the units of \( a \), we can look at the term \( \frac{a}{V_m^2} \). The units of pressure \( P \) are typically in atmospheres (atm), and the units of molar volume \( V_m \) are in liters per mole (L/mol).

Thus, we can express the units of \( a \) as follows:
- Rearranging gives us \( a = P \cdot V_m^2 \).
- The units of pressure \( P \) are atm, and the units of molar volume \( V_m \) are L/mol.

### Step 4: Calculate the units of \( a \)
Substituting the units into the equation:
\[ \text{Units of } a = \text{atm} \cdot \left(\frac{\text{L}}{\text{mol}}\right)^2 = \text{atm} \cdot \frac{\text{L}^2}{\text{mol}^2} \]
Thus, the units of \( a \) are:
\[ \text{Units of } a = \text{atm} \cdot \text{L}^2 \cdot \text{mol}^{-2} \]

### Step 5: Conclusion
The expression representing the value and unit of the van der Waals constant \( a \) is:
\[ \frac{P \cdot V^2}{n^2} \quad \text{with units } \text{atm} \cdot \text{L}^2 \cdot \text{mol}^{-2} \]

### Final Answer
The correct option is:
\[ \text{Option C: } \frac{P}{V^2} \cdot n^{-2} \text{ with units } \text{atm} \cdot \text{L}^2 \cdot \text{mol}^{-2} \]