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

Answer: (B)

$[M L^2 T^{-2}]$

Answer 2

Given:

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

where:
- $P$ is pressure,
- $V$ is volume,
- $R$ is the universal gas constant,
- $T$ is temperature.

Step 1: Dimensions of the Given Quantities
- $[V] = [b]$, so 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: Dimension of $a$
From the term $\frac{a}{V^2}$ having the same dimension as pressure $P$:

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

Thus, the dimensional formula of $a$ is:

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

Step 3: Calculating the Dimensional Formula of $ab^{-1}$
The dimensional formula of $b$ is $[L^3]$. Therefore, the dimensional formula of $ab^{-1}$ is:

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

Therefore, the correct dimensional formula of $ab^{-1}$ is $[ML^2T^{-2}]$.