Q 11-01-184JEE MainJEE Main 2021 (26 Feb, Shift 1)Medium
In a typical combustion engine the work done by a gas molecule is given by $W = \alpha^2 \beta e^{-\frac{\beta x^2}{kT}}$, where $x$ is the displacement, $k$ is the Boltzmann constant and $T$ is the temperature. If $\alpha$ and $\beta$ are constants, dimensions of $\alpha$ will be:
Answer: (D) $[\text{M}^0\text{LT}^0]$
The exponent must be dimensionless, so $[\beta] = \dfrac{[kT]}{[x^2]} = \dfrac{[\text{ML}^2\text{T}^{-2}]}{[\text{L}^2]} = [\text{MT}^{-2}]$.
Since the exponential is dimensionless, $[W] = [\alpha^2][\beta]$:
$$[\alpha^2] = \frac{[\text{ML}^2\text{T}^{-2}]}{[\text{MT}^{-2}]} = [\text{L}^2] \Rightarrow [\alpha] = [\text{M}^0\text{LT}^0]$$
Solution by Sreeraj P, M.Sc Physics