How to find enthalpy of solvation for gas phase cation using Hess's law?
One approach to compute the enthalpy change for the reaction
$$\ce{H2(g)-&gt;2H+(aq) + 2e-}$$
is to combine the bond dissociation enthalpy and hydrogen ionization energy (both refer to gas phase species) and the hydration enthalpy of the hydrogen ion:
$$\ce{H2(g)-&gt;2H(g)}~~~~~~~~{\Delta H = \pu{439 kJ/mol}  }\\
\ce{2H(g)-&gt;2H+(g) +2e-(g)}~~~~~~~~{\Delta H = \pu{262 kJ/mol}  }\\
\ce{2H+(g)-&gt;2H+(aq)}~~~~~~~~{\Delta H = \pu{-2260 kJ/mol}  }$$
If you add up the enthalpies you obtain as the solvation enthalpy of $\ce{Ca^{2+}}$ the value $\Delta_{\text{solv}} H = -1546 \pu{kJ/mol}$ which is quite reasonable (see eg a table here)
The result is clearly very different from my earlier answer which I am still pondering:
The catch is that $\Delta H^\circ=0$ for this reaction. The reason is because $\Delta G^\circ=0$ for the reaction (by definition, see standard reduction potential) at all temperatures, and then, since
$$\Delta H = -T^2\left( \frac{\partial (\Delta G/T)}{\partial T}\right)_p$$
it follows that  $\Delta H^\circ=0$. I suspect the problem with this approach is due to a missing term involving enthalpy of solvation of the electrons (or a similar term involving the electrons) but I haven't put my finger on it.