The number of valence electrons in an atom's valence electron shell is related to the position of the element on the periodic table. The periodic table is organized in a way that elements with similar chemical properties are grouped together.
The number of valence electrons corresponds to the group number (also called the family number) of the element. Elements in the same group have the same number of valence electrons. For example, all elements in Group 1 (such as hydrogen, lithium, sodium) have one valence electron, while elements in Group 17 (such as fluorine, chlorine, bromine) have seven valence electrons.
The period (or row) of an element on the periodic table indicates the number of electron shells (energy levels) that element's atoms have. As you move across a period from left to right, the number of valence electrons increases by one, starting from one and ending at eight. However, there are exceptions for elements in the transition metals and inner transition metals, which have varying numbers of valence electrons due to the complex electronic configurations of those elements.
In summary, the number of valence electrons in an atom's valence electron shell generally corresponds to the group number of the element, while the period indicates the number of electron shells.