Question

from the standpoint of atomic structure, what determines which element will be the first and which is the last is the last in a period of the periodic table

Answer 1

From the standard point of atomic structure, the periodic table is based on Atomic number of an element.
atomic number means we consider here the number of electrons and the way their are arranged in its shell and sub shells.
For example ARGON
Atomic number is 18.
electronic configuration is 1s2 2s2 2p6 3s2 3p6
valence electrons are 8 so group number is 10+8 [since it is p block]
period number is maximum principal quantum number - 3
hence electronic configuration plays an important role in deciding its period and group

Answer 2

The number of shells in an element determines the elements which should be placed first and last in a period.