Coulomb's Law in Atoms needs one more piece. A valence electron doesn't feel the nucleus's full charge, because the inner electrons, from your Electron Configurations, sit between them and cancel part of it. The charge left over is the effective nuclear charge, $Z_\text{eff}$.
Key idea
$$Z_\text{eff} \approx Z - (\text{number of core electrons})$$ Core (inner-shell) electrons shield the nucleus almost completely; electrons in the same shell shield each other only a little. So across a period $Z_\text{eff}$ rises by about one per element, and down a group it stays about the same while the distance grows.
Why it works
A core electron spends nearly all its time much closer to the nucleus than a valence electron does, so from outside it looks like it sits on the nucleus and cancels one proton. An electron in the same shell is as often farther out as closer in, so it cancels very little. Sodium ($1s^2 2s^2 2p^6 3s^1$) has 10 core electrons: $Z_\text{eff} \approx 11 - 10 = 1$. Chlorine ($[\text{Ne}]\,3s^2 3p^5$) has the same 10 core electrons and 17 protons: $Z_\text{eff} \approx 7$. Its valence electrons feel about seven times the pull, at a similar shell, so chlorine's atom is smaller and much harder to ionize.
| Element | $Z$ | Core electrons | $Z_\text{eff}$ (simple) | Covalent radius |
|---|---|---|---|---|
| Li | 3 | 2 | 1 | 128 pm |
| C | 6 | 2 | 4 | 76 pm |
| F | 9 | 2 | 7 | 57 pm |
| Na | 11 | 10 | 1 | 166 pm |
| Cl | 17 | 10 | 7 | 102 pm |
| K | 19 | 18 | 1 | 203 pm |
Worked example: Finding Z_eff
Estimate $Z_\text{eff}$ for a valence electron of phosphorus ($Z = 15$).
- Configuration: $[\text{Ne}]\,3s^2\,3p^3$. The core is the neon part: 10 electrons.
- $Z_\text{eff} \approx 15 - 10 = 5$.
Answer: About $+5$.
Worked example: Using Z_eff to compare sizes
Which atom is smaller, magnesium or sulfur? Explain with $Z_\text{eff}$.
- Both have valence electrons in shell 3 and a neon core of 10 electrons.
- Mg: $Z_\text{eff} \approx 12 - 10 = 2$. S: $Z_\text{eff} \approx 16 - 10 = 6$.
- Sulfur's valence electrons feel three times the pull and are drawn in closer.
Answer: Sulfur (105 pm against 141 pm).
Common mistake
Thinking electrons in the same shell shield each other as well as core electrons do. If they did, every element in a period would have $Z_\text{eff} = 1$, and atoms would not shrink from sodium to argon, though they clearly do. Count only the inner-shell electrons as shielding. Check: within one period, the simple $Z_\text{eff}$ should equal the number of valence electrons for main-group elements.
The simple formula overstates the effect a little, because same-shell electrons do shield somewhat. Slater's rules, a better estimate, give $Z_\text{eff} = 2.2$ for sodium's 3s electron and $6.1$ for chlorine's 3p electrons. The trend is the same.