Question #44694

The classical Bohr model fails for heavy elements due to relativistic effects. For a hydrogen-like ion, the electron velocity in the $n^{th}$ orbit is given by $v_n = \frac{Z c}{137 n}$ (assuming the fine-structure constant $\alpha \approx 1/137$). As $v_n$ approaches the speed of light $c$, the relativistic mass of the electron increases, leading to a contraction of the Bohr radius ($r \propto 1/m$).

Consider the hydrogen-like Lead ion, \ce{Pb^{81+}} ($Z=82$). Let $X$ be the percentage contraction (rounded to the nearest integer) of its 1s orbital radius relative to the classical non-relativistic Bohr radius. This relativistic contraction cascades to the valence shell, fundamentally causing the 'Inert Pair Effect' in heavy p-block elements. Let $Y$ be the sum of the thermodynamically most stable oxidation states of Thallium (\ce{Tl}) and Lead (\ce{Pb}) in aqueous solution at 298 K. Let $W$ be the number of radial nodes in the valence s-orbital of neutral Lead.

Calculate the value of $(X + Y + W)$.