The dimensions of Van der Waals constants a and b are respectively,

Options

A. bar `L^(2) mol^(-2) and L mol^(-1)`
B. bar `L^(2) mol^(-1) and L^(-1) mol^(-1)`
C. bar `L^(2) mol^(2) and L^(-1) mol^(-1)`
D. `"bar"^(-1) L^(2) mol^(-2) and L^(-2) mol^(-1)`

Answer

To determine the dimensions of the Van der Waals constants \( a \) and \( b \), we start from the Van der Waals equation:

\[ \left( P + \frac{aN^2}{V^2} \right) (V - Nb) = NRT \]

### Step 1: Analyze the term involving constant \( a \)

The term \( \frac{aN^2}{V^2} \) represents a pressure correction. We can rearrange this to isolate \( a \):

\[ P = \frac{aN^2}{V^2} \]

### Step 2: Identify the dimensions of pressure \( P \)

Pressure \( P \) has the dimensions of force per unit area. In SI units, this is expressed as:

\[ [P] = \text{Force} / \text{Area} = \frac{MLT^{-2}}{L^2} = ML^{-1}T^{-2} \]

### Step 3: Identify the dimensions of \( N \) and \( V \)

- \( N \) (number of moles) has the dimension of \( [N] = \text{mol} \).
- \( V \) (volume) has the dimension of \( [V] = L^3 \).

### Step 4: Substitute the dimensions into the equation

Substituting the dimensions into the equation \( P = \frac{aN^2}{V^2} \):

\[ ML^{-1}T^{-2} = \frac{a (\text{mol})^2}{(L^3)^2} \]

This simplifies to:

\[ ML^{-1}T^{-2} = \frac{a \cdot \text{mol}^2}{L^6} \]

### Step 5: Solve for the dimensions of \( a \)

Rearranging gives:

\[ a = \frac{ML^{-1}T^{-2} \cdot L^6}{\text{mol}^2} = \frac{ML^{5}T^{-2}}{\text{mol}^2} \]

Thus, the dimensions of \( a \) are:

\[ [a] = ML^{5}T^{-2} \cdot \text{mol}^{-2} \]

### Step 6: Analyze the term involving constant \( b \)

The term \( b \) represents the volume correction. In the Van der Waals equation, we have:

\[ V = V - Nb \implies Nb = V - V_{real} \]

From this, we can express \( b \):

\[ b = \frac{V}{N} \]

### Step 7: Solve for the dimensions of \( b \)

Substituting the dimensions:

- \( V \) has dimensions of \( L^3 \).
- \( N \) has dimensions of \( \text{mol} \).

Thus:

\[ [b] = \frac{L^3}{\text{mol}} = L^3 \cdot \text{mol}^{-1} \]

### Final Result

The dimensions of the Van der Waals constants are:

- \( a: \, [a] = ML^{5}T^{-2} \cdot \text{mol}^{-2} \)
- \( b: \, [b] = L^{3} \cdot \text{mol}^{-1} \)