Q 11-01-176JEE MainJEE Main 2021 (24 Feb, Shift 1)Medium
The work done by a gas molecule in an isolated system is given by $W = \alpha\beta^2 e^{-\frac{x^2}{\alpha kT}}$, where $x$ is the displacement, $k$ is the Boltzmann constant and $T$ is the temperature. $\alpha$ and $\beta$ are constants. Then the dimensions of $\beta$ will be:
Answer: (C) $[\text{MLT}^{-2}]$
The exponent must be dimensionless, so $[\alpha] = \dfrac{[x^2]}{[kT]} = \dfrac{\text{L}^2}{\text{ML}^2\text{T}^{-2}} = [\text{M}^{-1}\text{T}^2]$ (since $kT$ is an energy).
Then $[\beta^2] = \dfrac{[W]}{[\alpha]} = \dfrac{\text{ML}^2\text{T}^{-2}}{\text{M}^{-1}\text{T}^2} = [\text{M}^2\text{L}^2\text{T}^{-4}]$, so $[\beta] = [\text{MLT}^{-2}]$.
Solution by Sreeraj P, M.Sc Physics