Question

What is the dimensional formula of \( ab^{-1} \) in the equation \[ \left( P + \frac{a}{V^2} \right) (V - b) = RT, \] where letters have their usual meaning.

• A. \( [M^0 L^3 T^{-2}] \)
• B. \( [M L^2 T^{-2}] \)
• C. \( [M^{-1} L^5 T^3] \)
• D. \( [M^6 L^7 T^4] \)

Answer 1

The Correct Option is B

Given:

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

where:
- \( P \) denotes pressure,
- \( V \) denotes volume,
- \( R \) is the universal gas constant,
- \( T \) denotes temperature.

Step 1: Determine the dimensions of the given quantities
- From \( [V] = [b] \), the dimension of \( b \) is:

\[ [b] = [L^3] \quad (\text{volume}) \]

- The dimensional formula for pressure \( P \) is:

\[ [P] = \left[\frac{F}{A}\right] = \left[\frac{MLT^{-2}}{L^2}\right] = [ML^{-1}T^{-2}]. \]

Step 2: Determine the dimension of \( a \)
The term \( \frac{a}{V^2} \) has the same dimension as pressure \( P \):

\[ \left[\frac{a}{V^2}\right] = [P] = [ML^{-1}T^{-2}]. \]

Consequently, the dimensional formula of \( a \) is:

\[ [a] = [P] \times [V^2] = [ML^{-1}T^{-2}] \times [L^6] = [ML^5T^{-2}]. \]

Step 3: Calculate the dimensional formula of \( ab^{-1} \)
The dimensional formula of \( b \) is \( [L^3] \). Therefore, the dimensional formula of \( ab^{-1} \) is:

\[ ab^{-1} = \frac{[a]}{[b]} = \frac{[ML^5T^{-2}]}{[L^3]} = [ML^2T^{-2}]. \]

The correct dimensional formula for \( ab^{-1} \) is \( [ML^2T^{-2}] \).