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# In a typical combustion engine the work don

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 :

1. $\left[\mathrm{MLT}^{-2}\right]$

2. $\left[\mathrm{M}^{0} \mathrm{LT}^{0}\right]$

3. $\left[\mathrm{M}^{2} \mathrm{LT}^{-2}\right]$

4. $\left[\mathrm{MLT}^{-1}\right]$

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