Pauling scale is based on an emperical relation between the energy of a bond and the electronegativities of bonded atoms.
Let us consider A-B between two dissimilar atoms A and B of a molecule AB.
Let EA-B, EB-B and EA-B be the bond energies of A-A, B-B and A-B respectively
It can be seen that the bond dissociation energy of A-B is almost higher than the geometric mean of the bond dissociation energies of A-A and B-B bonds i.e,
EA-B is greater than √EA-A x EB-B
According to the following equation, their difference (∆) is related to the difference in the electronegativities of A and B
∆ = EA-B – √(EA-A x EB-B) = (XA -XB)2
(XA – XB)2 = 0.208 √(EA-B – √EA-A x EB-B)
Here, XA and XB are the electronegativities of A and B respectively