JEE PYQ: Units & Measurements - Question ID 466a53d0a4c8 (JEE Main 2024)

Core Formula & Concept:

The given equation is the van der Waals equation of state for real gases: $\left(P + \frac{a}{V^2}\right)(V - b) = RT.$ Here:

• $P$ is pressure,
• $V$ is molar volume,
• $T$ is absolute temperature,
• $R$ is the universal gas constant,
• $a$ and $b$ are van der Waals constants that correct for intermolecular forces and finite molecular size, respectively.

Dimensional homogeneity principle: Every additive term in a physically meaningful equation must have the same dimensions. We exploit this to find the dimensions of $a$ and $b$.

Step-by-Step Derivation:

Step 1: Dimensions of known quantities

• Pressure $P$: $[P] = \mathrm{ML}^{-1}\mathrm{T}^{-2}$
• Volume $V$: $[V] = \mathrm{L}^{3}$
• Temperature $T$: $[T] = \mathrm{\Theta}$
• Gas constant $R$: $[R] = \mathrm{ML}^{2}\mathrm{T}^{-2}\mathrm{\Theta}^{-1}$

Step 2: Isolate the term containing $a$
The term $\displaystyle \frac{a}{V^2}$ must have the same dimensions as $P$ (since they are added): $\left[\frac{a}{V^2}\right] = [P].$ Therefore $[a] = [P] \cdot [V^2] = \mathrm{ML}^{-1}\mathrm{T}^{-2} \cdot (\mathrm{L}^{3})^{2} = \mathrm{ML}^{5}\mathrm{T}^{-2}.$

Step 3: Dimensions of $b$
The term $b$ is subtracted from $V$, so $[b] = [V] = \mathrm{L}^{3}.$

Step 4: Dimensions of $a b^{-1}$
We now compute $\left[a b^{-1}\right] = \frac{[a]}{[b]} = \frac{\mathrm{ML}^{5}\mathrm{T}^{-2}}{\mathrm{L}^{3}} = \mathrm{ML}^{2}\mathrm{T}^{-2}.$

Step 5: Match with the given options
The dimensional formula $\mathrm{ML}^{2}\mathrm{T}^{-2}$ corresponds exactly to option D.

Common Traps & Exam Tip:

1. Confusing $a$ and $b$ roles: Students sometimes mix up which constant corrects for volume and which for pressure. Remember $a$ is associated with the pressure correction $\displaystyle \frac{a}{V^2}$, while $b$ is the excluded volume.
2. Incorrect exponent on $V$: A frequent error is writing $[V^2] = \mathrm{L}^{6}$ instead of $\mathrm{L}^{6}$ (correct), leading to wrong dimensions for $a$.
3. Forgetting to invert $b$: The question asks for $a b^{-1}$, not $a$ or $b$ alone. Always read the question carefully.