The work done by a gas molecule in an isolated system is given by, $W = α β^{2} e^{- \frac{x^{2}}{α k T}},$ where $x$ is the displacement, k is the Boltzmann constant and T is the temperature. $α$ And $β$ are constants. Then the dimensions of $β$ will be

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

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

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

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

answer is B.

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Detailed Solution

$\begin{array}{l} \frac{x^{2}}{α K T} i s \dim e n s i o n l e s s, [α] = \frac{[x^{2}]}{k T} = π^{- 1} T^{2} \\ [w] = [α] {[β]}^{2} \\ [β] = \sqrt{\frac{M L^{2} T^{- 2}}{M^{- 1} T^{2}}} = M L T^{- 2} \end{array}$

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