Question

In the equation \( \left( P + \frac{a}{V^2} \right) (V - b) = RT \), where \( P \) is pressure, \( V \) is volume, \( T \) is temperature, \( R \) is the universal gas constant, and \( a, b \) are constants. The dimensions of \( a \) are:

• A. \( ML^{-1} T^{-2} \)
• B. \( ML^5 T^{-2} \)
• C. \( M^0 L^3 T^0 \)
• D. \( ML^3 T^{-2} \)

Hint:

To determine the dimensions of a physical quantity, express it in terms of fundamental quantities \( M, L, T \) and equate dimensions accordingly.

Answer 1

The Correct Option is B

Step 1: Identifying the Dimensions of Given Quantities
- The equation given is: \[ \left( P + \frac{a}{V^2} \right) (V - b) = RT. \] - The dimensions of pressure \( P \) are: \[ [P] = ML^{-1} T^{-2}. \] - The dimensions of volume \( V \) are: \[ [V] = L^3. \] Step 2: Finding Dimensions of \( a \)
Since \( P + \frac{a}{V^2} \) must have the same dimensions as \( P \), we equate: \[ \frac{a}{V^2} = P. \] Rearranging for \( a \): \[ a = P \cdot V^2. \] Substituting dimensions: \[ [a] = (ML^{-1} T^{-2}) \cdot (L^6). \] \[ [a] = ML^5 T^{-2}. \] Step 3: Conclusion
Thus, the correct dimensional formula for \( a \) is: \[ \mathbf{ML^5 T^{-2}}. \]