What is the dimensional formula of $a b^{-1}$ in the equation…

What is the dimensional formula of $a b^{-1}$ in the equation $\left(\mathrm{P}+\frac{\mathrm{a}}{\mathrm{V}^2}\right)(\mathrm{V}-\mathrm{b})=\mathrm{RT}$, where letters have their usual meaning.
  1. $\left[\mathrm{M}^{-1} \mathrm{~L}^5 \mathrm{~T}^3\right]$
  2. $\left[\mathrm{M}^6 \mathrm{~L}^7 \mathrm{~T}^4\right]$
  3. $\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]$
  4. $\left[\mathrm{M}^0 \mathrm{~L}^3 \mathrm{~T}^{-2}\right]$

Solution

$\because[\mathrm{V}]=[\mathrm{b}]$ $\therefore$ Dimension of $b=\left[\mathrm{L}^3\right]$ $\begin{aligned} & \&[\mathrm{P}]=\left[\frac{\mathrm{a}}{\mathrm{V}^2}\right] \\ & {[\mathrm{a}]=\left[\mathrm{PV}^2\right]=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]\left[\mathrm{L}^6\right]} \end{aligned}$ Dimension of $\mathrm{a}=\left[\mathrm{ML}^5 \mathrm{~T}^{-2}\right]$ $\therefore \mathrm{ab}^{-1}=\frac{\left[\mathrm{ML}^5 \mathrm{~T}^{-2}\right]}{\left[\mathrm{L}^3\right]}=\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]$

Asked in: JEE Main 2024 (05 Apr Shift 2)

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