Question
Why $PCl _5$ exist and $NCl _5$ do not?

Answer

Nitrogen is an element of second period which have 5 electrons in valence shell but have only 3 unpaired electrons.
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Nitrogen do not have vacant d-orbital. Hence in this electron do not get in excited state. Hence in this octet do not get expanded, hence its valency can't be 5. Due to this $NCl _5$ do not exist. While P of $PCl _5$ is the element of third period. Hence in this electron get in excited state and valency get 5 and $PCl _5$ get easily formed.
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