The elements P , S , K , and Ca form stable ions to achieve the...

The elements $P$, $S$, $K$, and $Ca$ form the stable ions $P^{3-}$, $S^{2-}$, $K^{+}$, and $Ca^{2+}$ respectively to achieve the nearest noble gas configuration of Argon ($18$ electrons).

These ions are isoelectronic species, meaning they have the same number of electrons but different nuclear charges.

For isoelectronic species, the ionic radius is inversely proportional to the effective nuclear charge (or atomic number $Z$). As the nuclear charge increases, the electrons are pulled more strongly towards the nucleus, resulting in a smaller ionic radius.

The atomic numbers of the given elements are $P (15)$, $S (16)$, $K (19)$, and $Ca (20)$.

Since $Ca$ has the highest nuclear charge ($Z = 20$), $Ca^{2+}$ is the smallest ion. Since $P$ has the lowest nuclear charge ($Z = 15$), $P^{3-}$ is the largest ion.

The order of ionic radii is $Ca^{2+} < K^{+} < S^{2-} < P^{3-}$.

Thus, the smallest and largest ions are $Ca^{2+}$ and $P^{3-}$, respectively.

Answer: $Ca^{2+}$ and $P^{3-}$