The work done by a gas molecule in an isolated system is given by,...

$\dfrac{{x}^{2}}{\alphakT} \to$ dimensionless

$⇒ \left[\alpha\right] = \dfrac{\left[{x}^{2}\right]}{\left[kT\right]} = \dfrac{{L}^{2}}{{ML}^{2}{T}^{- 2}} = {M}^{- 1}{T}^{2}$

Now \left[W\right] = \left[\alpha\right]{\left[\beta\right]}^{2}$</p><p>\left[\beta\right] = \sqrt{\dfrac{{ML}^{2}{T}^{- 2}}{{M}^{- 1}{T}^{2}}} = {M}^{1}{L}^{1}{T}^{- 2}$