An expression for a dimensionless quantity P is given by P =...

Given $p = \dfrac{\alpha}{\beta}\ln \left(\dfrac{kT}{\betax}\right)$

We know that expression inside log is dimensionless.

Therefore,\left[\dfrac{kT}{\betax}\right]$is dimensionless</p><p>$⇒ \left[\beta\right] = \left[\dfrac{kT}{x}\right] = {ML}^{2}{T}^{- 2}{K}^{- 1} \times K \times {L}^{- 1} = {MLT}^{- 2}$</p><p>Since$p$is dimensionless$∴ \left[p\right] = $\left[\dfrac{\alpha}{\beta}\right] = {M}^{0}{L}^{0}{T}^{0}$ $⇒ \left[\alpha\right] = \left[\beta\right]$ $⇒ \left[\alpha\right] = {MLT}^{- 2}$