Potassium screening model

Consider $=19$ potassium. As a rough approximation, assume that each of its $=1$ electrons

orbits $19$ protons and half an electron, i$.$e$.$ on average, half its fellow $=1$ electron. Assume that

each of its $=2$ electrons orbits $19$ protons, two $1$ electrons, and half of the seven other $=$

$2$ electrons. Continue the process, assuming that electrons at each $$ orbit a correspondingly

reduced positive charge.

$($a$)$ Calculate in terms of $0$ the orbit radii of hydrogenlike atoms of these effective $.$

$($b$)$ The radius of potassium is often quoted at around $0\mathrm{.}22$ nm$.$ In view of this, are your $=$

$1$ through $=3$ radii reasonable?

$($c$)$ About how many more protons would have to be $$unscreened$$ to the $=4$ electron to

agree with the quoted radius of potassium?

$($d$)$ Considering the shape of its orbit, should potassium$$s $=4$ electron orbit entirely

outside all the lower$-$ electrons? Explain.