Give reasons why copper exhibits variable valency?

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Hint: The Combining Capacity of an element is called Valency. The number of bonds that an atom can form as part of a compound is expressed by the valency of the element.
We all know how electrons in an atom are arranged in shells/orbitals in a specific order. Those electrons which are present in the outermost orbit of the atom are considered to be valence electrons. Valency of copper based on the number of valence electrons.

Complete answer:
Copper belongs to the transition elements of the periodic table. The electronic configuration of transition elements in the ground state is\[[Ar]n{{s}^{x}}n{{d}^{x}}\]. For the transition elements, the energy level is\[\left( n-1 \right)\]. ‘x’ here in\[n{{s}^{x}}\] and \[n{{d}^{x}}\] denotes the number of electrons in the s and d orbitals respectively. We know that s orbital can accommodate a maximum of 2 electrons and d orbital can accommodate a maximum of 10 electrons. The atomic number of copper is 29.
Copper belongs to \[{{4}^{th}}\] now but as transitions elements have \[\left( n-1 \right)\] level, the electrons configuration of copper in ground state is\[\left[ Ar \right]3{{d}^{10}}4{{s}^{1}}\].
Here we write \[4{{s}^{1}}\] and \[3{{d}^{10}}\]because half filled and fully filled subshells have extra Stability.
For \[C{{u}^{+}}\] the ions, the one electron from\[4s\] subshell gets removed and the configuration becomes\[\left[ Ar \right]3{{d}^{10}}\]
For \[C{{u}^{+}}\] the cut of the one electron from \[4s\] and \[3d\] one also gets removed. Hence the variation in oxidation state of copper is seen due to the involvement of d orbitals along with s orbital.

Note: Transition metals like nickel, tin iron also show variable valency. The d-block elements are all metals and most have one or more chemically active d-orbital electrons as they are the valence electrons. Because there is a relatively small difference in the energy of the different d-orbital electrons, the number of electrons participating in chemical bonding can vary and so they show multiple oxidation states and valances.