$\ce{S^{2-}}$, $\ce{Cl-}$, $\ce{K+}$, and $\ce{Ca^{2+}}$ each have the same number of electrons. Their effective ionic radii are 182, 181, 138, and 100 picometers, respectively, and the number of protons for each are 16, 17, 19, and 20. My question is, why does the effective ionic radius drastically go from 181 to 138 picometers between chlorine and potassium? Is it really necessarily because they are in different periods or is there something that I am missing here?
Answer
The whole concept of ionic radius is not sharply defined one. Note, that the same is true for the notion of atomic radius, as well as for any other kind of radius or any other notion of size. In general, the notion of size of an object looses its usual (i.e. classical) meaning in the microscopic realm since objects there do not have well-defined physical boundaries. So from the very beginning one should not expect a perfect trend in a not so well defined property.
Anyway, values of the ionic radii are based on crystallographic data, but what is actually determined in X-ray crystallography is the distance between two ions, not their radii. Then it is basically up to you how to divide this distance into the radii of ions in question. But whatever way the division is done, the ionic radius of an ions is not a fixed, perfectly defined property, since for the same ion, the radius can differ in different crystal lattices due certain variables such as coordination number and electron spin.
For instance, you mentioned the ionic radius of 138 pm for $\ce{K+}$. But this value is for 6-coordinate compounds only while for 8- and 12-coordinate compounds of $\ce{K+}$ the ionic radius is 151 pm and 160 pm respectively. Same story for $\ce{Ca^2+}$: 100 pm, 112 pm, and 135 pm for 6-, 8-, and 12-coordinate compounds respectively.1
Interestingly, for anions there is a little variation in the ionic radius depending on coordination number, so that for many of them only one value independent of coordination number is specified. To be honest I could not come up with a quick explanation why, so it feels like I have to dig into the matter deeper. But I'm quite reluctant to do so, since, as I said in the beginning, the whole concept of ionic radius looks suspicious, so I expect to find just some obscure rule arriving out of the blue which will explain this little variation in the anionic radius relatively to cationic ones.
Slater's rules to the rescue!
Now, I guess, I know which rule can be used to explain this drastic change in ionic radii. Let us examine the application of the good-ol' Slater's rules to the matter. These rules allows us to calculate the effective nuclear charge $Z_{\mathrm{eff}}$, which is the net positive charge experienced by an electron in a multi-electron atom. $$ Z_{\mathrm{eff}} = Z - s \, , $$ where $s$ is the screening constant. Slater's rules provide a way to calculate the values for the screening constants, and consequently, to obtain the values of effective nuclear charges.
All the ions $\ce{S^2-}$, $\ce{Cl-}$, $\ce{K+}$, $\ce{Ca^2+}$ are isoelectronic: the electron configuration of them is that of $\ce{Ar}$, i.e. $\mathrm{[Ne] 3s^2 3p^6}$. Thus, we have to compare effective nuclear charges for $\mathrm{3p}$ electrons in these ions. And as a first approximation we could compare effective nuclear charges for $\mathrm{3p}$ electrons in the corresponding atoms. Now without going into the details (listing the actual Slater's rules & applying them) I will just leave here the final numbers, effective nuclear charges for $\mathrm{3p}$ electrons for the corresponding atoms:
$$ \begin{array}{c|cccc} & \ce{S} & \ce{Cl} & \ce{K} & \ce{Ca} \\ \hline Z & 16 & 17 & 19 & 20 \\ Z_{\mathrm{eff}}(\mathrm{3p}) & 5.482 & 6.116 & 7.726 & 8.658 \\ r_{\mathrm{ion}} & 182 & 181 & 138 & 100 \\ \end{array} $$
Now you can see that despite a relatively small change in the nuclear charge going from $\ce{Cl}$ to $\ce{K}$ $$ (19-17)/17 * 100 \% \approx 11.8 \% \, , $$ there is a rather drastic change in effective nuclear charge $$ (7.726-6.116)/6.116 * 100 \% \approx 26.3 \% \, . $$ And the drastic change in ionic radius is exactly in line with this drastic change in effective nuclear charge $$ (138-181)/181 * 100 \% \approx -23.8 \% \, . $$ To summarize: the effective nuclear charge for $\mathrm{3p}$ electrons increases by about 26.3% when going from $\ce{Cl}$ to $\ce{K}$ and the ionic radius accordingly decreases by 23.8%.
Note that we used effective nuclear charges of the atoms just as the first approximation for that of the ions. For anions an additional electron will occupy an outer orbital and there will be an increased electron-electron repulsion. This will push the electrons further apart and increase the shielding, so that $Z_{\mathrm{eff}}$ for anions will be smaller than for the corresponding atoms. For cations the effect of the removal of an electron is the opposite: when an electron is lost, electron-electron repulsion and therefore, shielding decreases, so that $Z_{\mathrm{eff}}$ for cations will be bigger than for the corresponding atoms. It would not change the above mentioned trend though; in fact, it will make it even more prominent.
1 Ionic radii values are taken from here.
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