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In a typical combustion engine the work done by a gas molecule is given by $W = α^{2} β e^{\frac{- β x^{2}}{K T}}$ where X is the displacement, K is the Boltzmann constant and “T’’ is the temperature. If $α, β$ are constants, dimensions of " $α$ " will be

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$[M L T^{- 2}]$

$[M^{2} L T^{- 2}]$

$[M^{0} L T^{0}]$

$[M L T^{- 1}]$

answer is B.

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Detailed Solution

$W = α^{2} β e^{\frac{- β x^{2}}{k T}}$

As exponents are dimensionless so, $\frac{β x^{2}}{K t}$ should be dimensionless.

$∴ [β] = \frac{K T}{x^{2}} = \frac{M L^{2} T^{- 2}}{L^{2}} = M T^{- 2}$

From the dimensional homogeneity $α^{2} β$ should have dimensions of work

$∴ [α^{2} β] = [M L^{2} T^{- 2}]$

$\Rightarrow α^{2} = \frac{M L^{2} T^{- 2}}{M T^{- 2}}$

$\Rightarrow α = [M^{0} L T^{0}]$

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