Element reactivity is all about electron configurations, i.e. the stability of an atom's electron configuration.
Outermost shell electrons are called valence electrons, while electrons located in the inner shells of the atom are called core electrons.
With the exception of Helium, a noble gas that has 2 electrons in its stable configuration, all other noble gases have complete octets - 8 electrons in their outermost shells.
This is essentially what all atoms, with the exception of Hydrogen (it will try and reach Helium's stable configuration), will try to obtain, so their reactivities will depend on how close they are to completing their octet.
An element's reactivity is determined by the number of valence electrons it has. Elements closer to having a full outermost shell will be the most reactive.
Elements in group 1 - alkali metals - are highly reactive because they have 1 valence electron; these elements are willing to give up that electron in order to have a stable electron configuration.
LIkewise, group 17 elements - halogens - are also highly reactive because they have 7 electrons in their outermost shell; these elements will try and gain 1 electron in order to complete their octet.
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