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.
Which of the following does not have the same dimensional formula as that for `RT?`

Options

A. `PV`
B. `Pb`
C. `(a)/(V^(2))`
D. `(ab)/(V^(2))`

Explanation

To solve the problem, we need to analyze the dimensions of the terms involved in the van der Waals equation and determine which option does not have the same dimensional formula as that for \( RT \).

### Step 1: Understand the van der Waals equation
The van der Waals equation is given by:

\[ (P + \frac{a}{V^2})(V - b) = RT \]

Where:
- \( P \) = Pressure
- \( V \) = Molar volume
- \( T \) = Absolute temperature
- \( a, b, R \) = Constants

### Step 2: Identify the dimensions of \( RT \)
The term \( RT \) can be expressed in terms of its dimensions:
- The universal gas constant \( R \) has dimensions of energy per temperature, which can be expressed as:

\[ [R] = \frac{[M][L^2]}{[T^2][\Theta]} \]

Where:
- \( [M] \) = Mass
- \( [L] \) = Length
- \( [T] \) = Time
- \( [\Theta] \) = Temperature

- The absolute temperature \( T \) has dimensions of:

\[ [T] = [\Theta] \]

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

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

### Step 3: Analyze the dimensions of \( P \) and \( V \)
- The dimensions of pressure \( P \) are:

\[ [P] = \frac{[F]}{[A]} = \frac{[M][L]}{[L^2]} = \frac{[M]}{[L][T^2]} \]

- The dimensions of molar volume \( V \) are:

\[ [V] = [L^3] \]

### Step 4: Determine the dimensions of \( a \) and \( b \)
From the van der Waals equation, we know that:

1. The term \( \frac{a}{V^2} \) must have the same dimensions as pressure \( P \).
- Therefore, we can write:

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

This implies:

\[ [a] = [P][V^2] = \left(\frac{[M]}{[L][T^2]}\right)[L^6] = [M][L^5][T^{-2}] \]

2. The term \( b \) must have the same dimensions as volume \( V \):

\[ [b] = [L^3] \]

### Step 5: Compare dimensions
Now we can summarize:
- Dimensions of \( RT \): \(\frac{[M][L^2]}{[T^2]}\)
- Dimensions of \( P \): \(\frac{[M]}{[L][T^2]}\)
- Dimensions of \( a \): \([M][L^5][T^{-2}]\)
- Dimensions of \( b \): \([L^3]\)

### Step 6: Identify the option that does not match \( RT \)
Among the options provided (not listed in the question), we need to find the one that does not have the same dimensional formula as \( RT \).

Given our analysis:
- \( P \) has dimensions of \(\frac{[M]}{[L][T^2]}\)
- \( a \) has dimensions of \([M][L^5][T^{-2}]\)
- \( b \) has dimensions of \([L^3]\)

The only term that does not match the dimensions of \( RT \) is \( P \), as it has a different dimensional formula.

### Final Answer
Thus, the option that does not have the same dimensional formula as that for \( RT \) is:

**Option 3: Pressure \( P \)**