The workdone 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 dimension of $\beta$ will be

Q: The workdone 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 dimension of $\beta$ will be

b $\frac{ x ^{2}}{\alpha kT } \rightarrow$ dimensionless $\Rightarrow[\alpha]=\frac{\left[ x ^{2}\right]}{[ kT ]}=\frac{ L ^{2}}{ ML ^{2} T ^{-2}}= M ^{-1} T ^{2}$ Now $[ W ]=[\alpha][\beta]^{2}$ $[\beta]=\sqrt{\frac{ ML ^{2} T ^{-2}}{ M ^{-1} T ^{2}}}= M ^{1} L ^{1} T ^{-2}$

Question

A$\left[ M L ^{2} T ^{-2}\right]$
B$\left[ M L T ^{-2}\right]$
C$\left[ M ^{2} L T ^{2}\right]$
D$\left[ M ^{0} L T ^{0}\right]$

JEE MAIN 2021, Diffcult

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