Q.Using the idea of exchange energy, explain why the half-filled configuration $2p^3$ of nitrogen is more stable than a hypothetical arrangement in which the three electrons are not all unpaired with parallel spin.

Exchange energy arises because two electrons that have the same spin and occupy different orbitals of the same subshell are, quantum mechanically, indistinguishable from one another in a way that allows them to 'exchange' places without any physically measurable change in the atom. Every such possible exchange between a pair of parallel-spin electrons lowers the total energy of the atom by a certain fixed amount, so an atom's stability increases with the number of possible exchanges available to it.

In nitrogen's actual ground state, $2p^3$ has all three electrons unpaired, one in each of $2p_x$, $2p_y$, $2p_z$, all with parallel spin (by Hund's rule). With three electrons all of the same spin in three separate orbitals, there are $\binom{3}{2} = 3$ distinct pairs of electrons, and each pair can exchange -- giving three stabilising exchange interactions.

Now compare this with a hypothetical alternative arrangement in which the three electrons are not all parallel and unpaired -- for example, two electrons paired (opposite spin) in one $2p$ orbital and the third electron alone, with either spin, in a second orbital, leaving the third orbital empty. In such an arrangement, the two paired electrons (opposite spins) cannot exchange with each other, and there are no other same-spin pairs available either, so the number of possible exchanges drops sharply -- to zero or, at best, a much smaller number depending on the exact hypothetical arrangement chosen. …

Count the number of same-spin electron pairs available for exchange in the real (Hund's-rule) arrangement versus a hypothetical alternative, and relate a high …

Explaining stability only in terms of 'symmetry' without connecting it to the countable number of exchangeable same-spin pair …