Units & Dimensions

JEE Main level, single correct, medium

108 of 149

Question 108 of 149

In a typical combustion engine the workdone by a gas molecule is given by W=α2βe−βx2kTW={\alpha }^{2}\beta {e}^{\dfrac{-\beta {x}^{2}}{kT}}, where xx is the displacement, kk is the Boltzmann constant and TT is the temperature. If α\alpha and β\beta are constants, dimensions of α\alpha will be:

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Answer: (D) [M0LT0]\left[{M}^{0}{LT}^{0}\right]

kTkT has dimension of energy
βx2kT\dfrac{\beta {x}^{2}}{kT} is dimensionless
[β][L2]=[ML2T−2]\left[\beta \right]\left[{L}^{2}\right]=\left[{ML}^{2}{T}^{-2}\right]
[β]=[MT−2]\left[\beta \right]=\left[{MT}^{-2}\right]
α2β{\alpha }^{2}\beta has dimensions of work
[α2][MT−2]=[ML2T−2]\left[{\alpha }^{2}\right]\left[{MT}^{-2}\right]=\left[{ML}^{2}{T}^{-2}\right]
[α]=[M0LT0]\left[\alpha \right]=\left[{M}^{0}{LT}^{0}\right]

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