Why does he confess to the murder when the police arrive?
The speaker lets the police into his apartment in order to reassure them that he has nothing to hide from them. However, the longer they stay, the more he begins to grow agitated. He feels that he begins to hear a sound, and he thinks that "the noise was not within [his own] ears" but coming from outside of himself. He says that it is a "low, dull, quick sound—much such a sound as a watch makes when enveloped in cotton," a sound he also claimed to have heard just prior to murdering the old man. When he heard it the first time, he interpreted it as "the beating of the old man's heart," pounding loudly because of the old man's fear. However, it is unlikely that he, then, heard the old man's heart beating from across the room, and it is impossible that he should hear the dead old man's heart beating from beneath the floorboards now. It seems most likely that the sound the speaker hears is actually his own heart beating harder and faster as a result of his own adrenaline pumping. He feels certain that the police must hear it, though they do not (further evidence that it is the narrator's own heart). He fears detection, which makes his heart race; he mistakes the sound for the old man's heartbeat, and he assumes that the police will hear it. He can no longer stand the suspense of waiting for them to acknowledge the sound and his guilt, and so he confesses rather than having to endure the tension any longer.
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