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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 $α$ and $β$ are constants, dimensions of $α$ will be:

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

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

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

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

answer is B.

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Detailed Solution

$Power of exponent is dimensionless.$

$S o, [β x^{2}] = [k T] = [M^{1} L^{2} T^{- 2}]$

$\Rightarrow [β] = [M^{1} L^{0} T^{- 2}]$

$And exponential term is also dimensionless.$

$So, dimension of [α^{2} β] i s s a m e a s d i m e n s i o n o f w o r k .$

$\begin{array}{l} \Rightarrow [α^{2} β] = [M^{1} L^{2} T^{- 2}] \\ \Rightarrow [α^{2}] = \frac{[M^{1} L^{2} T^{- 2}]}{[M^{1} L^{0} T^{- 2}]} = [M^{0} L^{2} T^{0}] \\ \Rightarrow [α] = [M^{0} L^{1} T^{0}] \end{array}$

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