The van der Waal's equation of state for some gases can be expressed as :
`(P + (a)/( V^(2))) ( V - b) = RT`
Where `P` is the pressure , `V` is the molar volume , and `T` is the absolute temperature of the given sample of gas and `a, b , and R` are constants.
In the above problem , the dimensional formula for `RT` is same as that of

Options

A. Energy
B. Force
C. Specific heat
D. Latent heat

Explanation

To find the dimensional formula for \( RT \) in the van der Waals equation, we can start by recalling the ideal gas equation, which is given by:

\[ PV = RT \]

### Step 1: Understand the variables
- \( P \) is the pressure.
- \( V \) is the volume (specifically, molar volume).
- \( T \) is the absolute temperature.
- \( R \) is the universal gas constant.

### Step 2: Find the dimensional formula for pressure \( P \)
Pressure \( P \) is defined as force per unit area. The dimensional formula for force is given by:

\[ \text{Force} = \text{mass} \times \text{acceleration} = M \cdot L \cdot T^{-2} \]

where:
- \( M \) is the dimension of mass,
- \( L \) is the dimension of length,
- \( T \) is the dimension of time.

The area has the dimension:

\[ \text{Area} = L^2 \]

Thus, the dimensional formula for pressure \( P \) is:

\[ P = \frac{\text{Force}}{\text{Area}} = \frac{M \cdot L \cdot T^{-2}}{L^2} = M \cdot L^{-1} \cdot T^{-2} \]

### Step 3: Find the dimensional formula for volume \( V \)
Volume \( V \) has the dimensional formula:

\[ V = L^3 \]

### Step 4: Substitute into the ideal gas equation
From the ideal gas equation \( PV = RT \), we can express \( R \) in terms of \( P \), \( V \), and \( T \):

\[ R = \frac{PV}{T} \]

### Step 5: Find the dimensional formula for \( RT \)
Now, substituting the dimensional formulas we found:

1. The dimensional formula for \( P \) is \( M \cdot L^{-1} \cdot T^{-2} \).
2. The dimensional formula for \( V \) is \( L^3 \).
3. The dimensional formula for \( T \) is \( \Theta \) (temperature).

Substituting these into the formula for \( R \):

\[ R = \frac{M \cdot L^{-1} \cdot T^{-2} \cdot L^3}{\Theta} = \frac{M \cdot L^{2} \cdot T^{-2}}{\Theta} \]

Now, we can find the dimensional formula for \( RT \):

\[ RT = R \cdot T = \left(\frac{M \cdot L^{2} \cdot T^{-2}}{\Theta}\right) \cdot \Theta = M \cdot L^{2} \cdot T^{-2} \]

### Conclusion
Thus, the dimensional formula for \( RT \) is:

\[ M \cdot L^{2} \cdot T^{-2} \]