What is one power unique to the Senate and why did the framers give this power to the Senate?

The Senate shall have the sole power to try all impeachments. Basically, the House serves as a grand jury that may indict an office holder. The Senate conducts the actual trial. The framers intentionally split these duties between the houses because the House is less exclusive. The Senate members serve longer terms (less elections) and are for that reason more resistant to the frequent whims of the people and thus are less likely to convict office holders and remove them from office based on fickle passions and witch-hunts. The thought is that Senators are more reasoned. An indictment can be conducted by passionate congressmen because their decisions are checked by the cool deliberation of the Senate.