Q.Define the first and second ionization enthalpies of an element, writing the equation for each in the case of magnesium. Why is the second ionization enthalpy of an element always greater than its first?

Enthalpy, $H$, is a state function defined as $H = U + PV$, introduced because the overwhelming majority of real chemical reactions take place in open vessels at essentially constant atmospheric pressure, rather than at constant volume. At constant pressure, the change in enthalpy equals the heat exchanged with the surroundings, $\Delta H = q_p$ — precisely the quantity a calorimeter measures for a reaction carried out in an open flask, container, or solution. Enthalpy and internal energy are related, for gas-phase reactions at constant temperature, through $\Delta H = \Delta U + \Delta n_g RT$, where $\Delta n_g$ is the change in moles of gas across the reaction.

Chemists have defined a family of precisely standardized enthalpy changes, each describing one specific, carefully defined process, denoted by the standard-state superscript $^\circ$ (usually $298\ \text{K}$, $1\ \text{bar}$):

• Standard enthalpy of formation ($\Delta_fH^\circ$) — one mole of a compound formed from its elements in their standard states; by convention, an element in its standard state has $\Delta_fH^\circ = 0$.
• Standard enthalpy of combustion ($\Delta_cH^\circ$) — one mole of a substance completely burnt in excess oxygen; always exothermic (negative), and readily measured by bomb calorimetry.
• Standard enthalpy of atomization ($\Delta_aH^\circ$) — one mole of a substance completely converted into gaseous atoms; always endothermic (positive).
• Bond dissociation enthalpy — the energy to break one mole of a specific covalent bond in a gaseous molecule; dividing an atomization enthalpy by the number of equivalent bonds gives an average bond enthalpy.
• Enthalpy of ionization (first, second, …) — the energy to remove one mole of electrons from one mole of gaseous atoms (first) or from the resulting gaseous cations (second, always larger than the first, since removing an electron from an already-positive species is harder). …