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what does it mean:

"Transition metals either have incompletely filled d subshells or readily give rise to cations that have incompletely filled d subshells.

I don't understand the portion starting from " or readily give rise to..."

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Take the electronic configuration of copper (atomic number $29$) $\ce{[Ar] 3d^10 4s^1}$. According to the first portion of the definition, copper doesn't have incompletely filled subshells. But, copper readily gives rise to cation $\ce{Cu^{2+}}$ that has the electronic configuration $\ce{[Ar] 3d^9 4s^0}$ with incompletely filled d subshells. So, copper is considered as a transition metal.

Now, you can guess why zinc (atomic number $30$) is not considered as a transition metal.

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