safely; without incident; successfully
Used when a journey, task, or event ends without trouble or harm. Often appears with verbs of returning, arriving, or completing.
He returned home safely.
peacefully; quietly; in peace
Describes a state or action free from disturbance or conflict. Less common than the 'safely' sense; often used in wishes or descriptions of a calm life.
Please spend your time in peace.
He arrived safely at the airport.
The operation was completed successfully.
The village lived quietly.
The surgery ended successfully.
The soldiers came back safely.
The family is living peacefully.
平穏 is a na-adjective/noun describing a calm, undisturbed state, while 無事に is the adverb used to say something happened or was done peacefully.
無事に (safely, without incident) shares the kanji 事 with 大事 (important, serious) and learners sometimes mix them up despite the very different meanings.
From 無事 (safety, peace) + adverbial particle に. 無事 is a Sino-Japanese compound of 無 (without) and 事 (matter, incident).