In the formula , `p = (nRT)/(V-b) e ^(a/(RTV))` find the dimensions of a and b, where p = pressure , n= number of moles , T = temperture , V = volume and R = universal gas constant .

Explanation

To find the dimensions of \( a \) and \( b \) in the given formula

\[ p = \frac{nRT}{V - b} e^{\frac{a}{RTV}} \]

where \( p \) is pressure, \( n \) is the number of moles, \( T \) is temperature, \( V \) is volume, and \( R \) is the universal gas constant, we will follow these steps:

### Step 1: Find the dimensions of \( b \)

The term \( V - b \) indicates that \( b \) must have the same dimensions as volume \( V \) because only quantities with the same dimensions can be added or subtracted.

- The dimension of volume \( V \) is given as \( [V] = L^3 \).

Thus, the dimension of \( b \) is:

\[ [b] = L^3 \]

### Step 2: Find the dimensions of \( a \)

The term \( \frac{a}{RTV} \) must be dimensionless because it is in the exponent of \( e \). Therefore, we need to ensure that the dimensions of \( a \) combined with the dimensions of \( R \), \( T \), and \( V \) yield a dimensionless quantity.

1. **Dimensions of \( R \)**: The universal gas constant \( R \) has dimensions derived from the ideal gas law \( PV = nRT \). Rearranging gives us \( R = \frac{PV}{nT} \).
- The dimension of pressure \( p \) is \( [p] = ML^{-1}T^{-2} \).
- The dimension of number of moles \( n \) is dimensionless.
- The dimension of temperature \( T \) is \( [T] = \Theta \) (where \( \Theta \) represents temperature dimension).

Thus, the dimensions of \( R \) can be calculated as:

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

2. **Dimensions of \( V \)**: As previously stated, \( [V] = L^3 \).

3. **Combining dimensions**: Now, we can express the dimensions of \( RTV \):

\[ [RTV] = [R] \cdot [T] \cdot [V] = (ML^2T^{-2}\Theta^{-1}) \cdot \Theta \cdot L^3 = ML^5T^{-2} \]

Since \( \frac{a}{RTV} \) must be dimensionless, we have:

\[ [a] = [RTV] = ML^5T^{-2} \]

### Final Dimensions

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

- \( [b] = L^3 \)
- \( [a] = ML^5T^{-2} \)