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Carbon atoms do not form $ {C^{4 - }} $ anion and $ {C^{4 + }} $ cation. Why?

Answer
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Hint: The valence electrons in carbon are $ 4 $ , and this element is short of $ 4 $ elements to attain the nearest inert gas configuration like neon. But this atom does not form $ {C^{4 - }} $ anion and $ {C^{4 + }} $ cation due to their unstable nature and some conditions which were discussed as below.

Complete answer:
A modern periodic table was discovered by Mendeleev in which the chemical elements were arranged in the increasing number of atomic numbers in vertical columns and horizontal rows. Vertical columns were known as groups and the horizontal rows were known as periods.
The elements that belong to group $ 14 $ are known as the carbon family. Carbon is an element with atomic number $ 6 $ that has the electronic configuration of $ 1{s^2}2{s^2}2{p^2} $ . By this electronic configuration, it was clear that there were four valence electrons. Thus, carbon can gain or lose four electrons to form $ {C^{4 - }} $ anion and $ {C^{4 + }} $ cation.
But to form an anion $ {C^{4 - }} $ , carbon should gain four electrons which need more energy. Due to this condition, it does not form $ {C^{4 - }} $ anion.
To form a cation $ {C^{4 + }} $ , carbon should lose electrons. but an atom with two electrons and six protons is a highly unstable atom.
Thus, the carbon atom does not form $ {C^{4 - }} $ anion and $ {C^{4 + }} $ cation.

Note:
In every atom, based on the valence electrons only, the atoms can turn into ions. But carbon atoms are a different case. Though it exhibits tetravalency it does not form ions like $ {C^{4 - }} $ and $ {C^{4 + }} $ . But it shares its four electrons with atom bonds and participates in a covalent bond by sharing its four electrons.