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Question:
The workdone by a gas molecule in an isolated
system is given by, $\mathrm{W}=\alpha \beta^{2} \mathrm{e}^{-\frac{x^{2}}{\alpha k T}}$, where $\mathrm{x}$ is the displacement, $\mathrm{k}$ is the Boltzmann constant and $\mathrm{T}$ is the temperature, $\alpha$ and $\beta$ are constants. Then the dimension of $\beta$ will be :
Correct Option: , 2
Solution:
$\frac{x^{2}}{\alpha \mathrm{kT}} \rightarrow$ dimensionless
$\Rightarrow[\alpha]=\frac{\left[\mathrm{x}^{2}\right]}{[\mathrm{kT}]}=\frac{\mathrm{L}^{2}}{\mathrm{ML}^{2} \mathrm{~T}^{-2}}=\mathrm{M}^{-1} \mathrm{~T}^{2}$
Now $[\mathrm{W}]=[\alpha][\beta]^{2}$
$[\beta]=\sqrt{\frac{\mathrm{ML}^{2} \mathrm{~T}^{-2}}{\mathrm{M}^{-1} \mathrm{~T}^{2}}}=\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-2}$
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