Is expiation (kaffara) obligatory for breaking an oath for someone who swore not to tell anyone a secret, but then told a social worker for the purpose of seeking help?

Disclosing a secret is forbidden and contrary to trustworthiness. If you swore an oath not to tell anyone what your relative confided in you, then the oath is binding according to its intention. If you intended not to disclose the matter itself, then you have broken your oath and atonement for breaking an oath is required of you. If you intended not to disclose her person, and at that time you mentioned her name or she was identified by description or circumstantial evidence, then you have broken your oath. However, if your words were merely a general consultation unrelated to your relative, then you have not broken your oath.