where, k is Boltzmann constant, T is temperature and x is displacement. We know that, ‌
x2
αkT
is a dimensionless quantity. ∴‌‌[‌
x2
αkT
]=[M0L0T0]⇒[α]=‌
[x2]
[k][T]
⇒‌‌[α]=‌
[L2]
[k][T]
Since, dimensions of k are [k]=[M1L2T−2K−1] ... (i) Dimensions of temperature are [T]=[K] Substituting Eqs. (ii) and (iii) in Eq. (i), we get [α]‌‌=‌
[L2]
[M1L2T−2K−1][K]
[α]‌‌=[M−1T2] According to dimensional analysis, [W]‌=[αβ2] ⇒‌‌[β2]‌=‌