For a moles of gas ,Van der Weals equation is `(p = (a)/(V^(-2))) (V - b) = nRT`ltbr. Find the dimensions of a `a and b `, where `p = pressure` of gas` ,V = volume` of gas and` T = temperature of gas` .

Explanation

To find the dimensions of \( a \) and \( b \) in the Van der Waals equation, we will analyze the equation step by step.

### Step 1: Understand the Van der Waals Equation
The Van der Waals equation can be expressed as:
\[ p = \frac{a}{V^2}(V - b) = nRT \]
Where:
- \( p \) = pressure
- \( V \) = volume
- \( T \) = temperature
- \( n \) = number of moles
- \( a \) and \( b \) are constants.

### Step 2: Identify the Dimensions of Each Variable
1. **Pressure \( p \)**:
- Pressure is defined as force per unit area.
- Dimensions of force = mass × acceleration = \( [M][L][T^{-2}] \)
- Area = \( [L^2] \)
- Therefore, dimensions of pressure \( p \) are:
\[ [p] = \frac{[M][L][T^{-2}]}{[L^2]} = [M][L^{-1}][T^{-2}] \]

2. **Volume \( V \)**:
- Volume is defined as length cubed.
- Therefore, dimensions of volume \( V \) are:
\[ [V] = [L^3] \]

3. **Temperature \( T \)**:
- The dimension of temperature is denoted as:
\[ [T] = [\Theta] \quad (\text{where } \Theta \text{ is the dimension of temperature}) \]

### Step 3: Find the Dimensions of \( b \)
- From the equation \( V - b \), since both \( V \) and \( b \) are in the same term, they must have the same dimensions.
- Therefore, the dimensions of \( b \) are:
\[ [b] = [V] = [L^3] \]

### Step 4: Find the Dimensions of \( a \)
- Rearranging the Van der Waals equation gives us:
\[ p = \frac{a}{V^2} \implies a = p \cdot V^2 \]
- Now substituting the dimensions:
\[ [a] = [p] \cdot [V^2] = [M][L^{-1}][T^{-2}] \cdot [L^6] = [M][L^{5}][T^{-2}] \]

### Final Dimensions
- The dimensions of \( a \) are:
\[ [a] = [M][L^5][T^{-2}] \]
- The dimensions of \( b \) are:
\[ [b] = [L^3] \]

### Summary
- Dimensions of \( a \): \( [M][L^5][T^{-2}] \)
- Dimensions of \( b \): \( [L^3] \)