Does a divorce occur if the husband says to his wife, "You are divorced," while in a state of extreme anger and threatening, after an argument in which the wife verbally abused him and insulted him, and then the husband hit her? And is there any expiation for that?

What was uttered by the husband is a clear and binding divorce, and there is no expiation for it. If it was the first or second divorce, he may reconcile with his wife before her waiting period (iddah) ends by saying: "I take you back," or "I restore you to my marital bond," or by performing an action that indicates this, such as touching or intercourse with the intention of reconciliation. Intercourse constitutes reconciliation even without intention, according to the sounder opinion. Both spouses should avoid cursing, reviling, and invoking maledictions. It is forbidden for the wife to disrespect her husband with insults or curses, as this constitutes unlawful defiance (nushuz). Similarly, it is forbidden for the husband to curse his wife, as this contradicts the principle of treating one another kindly.