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Mercury shows variable valency while zinc does not. Its electronic configuration is $\ce{[Xe]\:4f^14 5d^10 6s^2}$. So it can donate the $\ce{6s^2}$ electrons and should only be able to form $\ce{Hg^2+}$, right?

  1. How can it show $+1$ oxidation state?

  2. Why does not $\ce{Zn}$ show $+1$ oxidation state?

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On the contrary, zinc(I) compounds do exist, though they are rare, and relatively unstable. Most zinc(I) compounds contain a $\ce{[Zn2]^{2+}}$ core, which is analogous to the $\ce{[Hg2]^{2+}}$ cation. The $\ce{[Zn2]^{2+}}$ ion does, however, rapidly disproportionate into zinc metal and zinc(II), and has only ever been obtained by cooling a solution of metallic zinc in molten $\ce{ZnCl2}$.

This should show you that the chemistry of zinc and mercury really aren't all that different. The $\ce{[Hg2]^{2+}}$ ion's much greater stability relative to $\ce{[Zn2]^{2+}}$ is due to the atypically large ionization enthalpy of the $\ce{Hg}$ atom ($100.7\:\mathrm{kJ\:mol^{-1}}$ greater than $\ce{Zn}$), which in turn due to relativistic stabilization of the $\mathrm{6s}$ orbital. Relativistic effects account for a myriad of the properties that are characteristic of heavy metals, including why mercury is the only metal that is a liquid at room temperature, and even gold's color.

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