Effective Nuclear Charge Calculator
Sodium's outer electron feels 2.20 of its eleven protons, which is why it comes off so readily.
The formula
Zeff = Z - S, with S from Slater's rules
What the outer electron actually feels
Sodium has eleven protons, but its single 3s electron does not experience eleven. The ten inner electrons stand between, and Slater's rules put a number on how effectively: 0.85 for each electron in the shell just inside, 1.00 for anything deeper, 0.35 for others sharing the same shell. Sodium's 3s electron feels about 2.20, which is why it comes off so easily.
Why atoms shrink across a period
Going from lithium to fluorine, each step adds a proton and an electron to the same shell. The new electron screens its neighbours at only 0.35, so the effective charge climbs steadily and pulls the whole shell inwards. Atomic radius therefore falls across a period even though electrons are being added — a result that looks backwards until the screening is counted.
The d electrons screen badly
Slater treats every electron to the left of a d electron as screening fully, which is why iron's 3d electron feels 6.25 while its 4s electron feels 3.75. The rules are a fit to spectroscopic data from 1930 and they are crude, but they capture the trends that matter.
Models, and where they stop
Slater's rules, the Kapustinskii equation and Pauling's ionic-character expression are all empirical fits, not derivations. They were built to reproduce measured numbers with arithmetic simple enough to do by hand, and they succeed at that within perhaps five or ten per cent. Where a page here quotes one, it says which — a figure from a fitted rule and a figure from a Born-Haber cycle are not the same kind of thing, even when they agree.