Groups- vertical
In the periodic table, the group number is the same as the number of electrons in the outer shell. (e.g. Group 1 atoms have 1 electron in their outer shells.)
Because the chemical properties of an element depends on the number of electrons in its outer shells, this means that all the elements in a particular group react in a similar way.
Atoms like to gain a stable structure (i.e. a full outer shell). If they are in group 1,2,3 they lose electrons. If they are in group 5,6,7 they gain electrons. Group 4 shares electrons. Atoms of elements in the same group behave in similar ways to either lose/gain/share electrons and thus get a full outer shell.
Notice that group 8 electrons have a full outer shell. As a result they are unreative.
Periods- horizontal
The number of shells an element needs to accomodate all its electrons is the same as its period number.
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