Question
Why does $NH_3$ form hydrogen bond but $PH_3$ does not?

Answer

Formation of hydrogen bonding depend on the size of molecule. smaller the size greater the hydrogen bonding. Nitrogen has smaller size than phosphrous thus having a more abilty to form hydrogen bond.
N—H bond is reasonably polar and this leads to hydrogen bonding. As the bond polarity of the P—H bond is almost negligible, $PH_3$ is not involved in hydrogen bonding.

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