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Question:
In a typical combustion engine the work don
by a gas molecule is given $W=\alpha^{2} \beta e^{\frac{-\beta x^{2}}{k T}}$
where $x$ is the displacement, $k$ is the Boltzmann constant and $\mathrm{T}$ is the temperature. If $\alpha$ and $\beta$ are constants, dimensions of $\alpha$ will be :
Correct Option: , 2
Solution:
kT has dimension of energy
$\frac{\beta x^{2}}{k T}$ is dimensionless
${[\beta]\left[\mathrm{L}^{2}\right]=\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right] }$
${[\beta]=\left[\mathrm{MT}^{-2}\right] }$
$\alpha^{2} \beta$ has dimensions of work
${\left[\alpha^{2}\right]\left[\mathrm{MT}^{-2}\right]=\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right] }$
${[\alpha]=\left[\mathrm{M}^{0} \mathrm{LT}^{0}\right] }$
Ans. 2
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