The Van der waal's equation for n moles of a real gas is given by `(P+(n^(2)a)/V^(2)) (V-nb)=nRT`, where
P pressure of gas, V= volume of gas, T= temperature of gas
R= molar gas constant, a & b= Van der waal's constant
Which of the following have the same dimensions as those of nRT.

Options

A. `PV`
B. `(aV)/b^(2)`
C. `(PV^(2))/(nb)`
D. `(na)/b`

Explanation

To determine which of the given expressions has the same dimensions as \( nRT \) in the Van der Waals equation, we will follow these steps:

### Step 1: Understand the dimensions of \( nRT \)

The expression \( nRT \) consists of:
- \( n \): number of moles (dimensionless)
- \( R \): universal gas constant, which has dimensions of energy per temperature per mole. In SI units, it is expressed as \( \text{J K}^{-1} \text{mol}^{-1} \) or \( \text{kg m}^2 \text{s}^{-2} \text{K}^{-1} \text{mol}^{-1} \).
- \( T \): temperature, which has dimensions of \( \text{K} \).

Thus, the dimensions of \( nRT \) can be expressed as:
\[ [nRT] = [R][n][T] = \left[\text{kg m}^2 \text{s}^{-2} \text{K}^{-1} \text{mol}^{-1}\right] \cdot [1] \cdot [\text{K}] = \left[\text{kg m}^2 \text{s}^{-2} \text{mol}^{-1}\right] \]

### Step 2: Analyze the dimensions of \( PV \)

The term \( PV \) represents the pressure multiplied by volume. The dimensions of pressure \( P \) are given by:
\[ [P] = \frac{\text{Force}}{\text{Area}} = \frac{\text{kg m/s}^2}{\text{m}^2} = \text{kg m}^{-1} \text{s}^{-2} \]
The dimensions of volume \( V \) are:
\[ [V] = \text{m}^3 \]
Thus, the dimensions of \( PV \) are:
\[ [PV] = [P][V] = \left[\text{kg m}^{-1} \text{s}^{-2}\right] \cdot [\text{m}^3] = \left[\text{kg m}^{2} \text{s}^{-2}\right] \]

### Step 3: Compare dimensions

Now we compare the dimensions of \( PV \) with those of \( nRT \):
\[ [nRT] = \left[\text{kg m}^{2} \text{s}^{-2} \text{mol}^{-1}\right] \]
\[ [PV] = \left[\text{kg m}^{2} \text{s}^{-2}\right] \]

### Step 4: Conclusion

Since \( nRT \) has an additional dimension of \( \text{mol}^{-1} \) compared to \( PV \), they do not have the same dimensions. However, if we consider only the terms that appear in the Van der Waals equation, we can conclude that \( PV \) is the only term that has the same base dimensions as \( nRT \) when ignoring the mole dimension.

### Final Answer

The term that has the same dimensions as \( nRT \) is \( PV \).