If my question is incorrect then please also mention the correct thing which happens to them.
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No. An electron getting absorbed by a nucleus is called Electron Capture, or sometimes K Capture, but this is a nuclear reaction mediated by the Weak force, i.e. rare, and usually with electrons from the innermost K shell, not quasi-free electrons that form a current. Typical energies are 10s to 100s of keV.
Instead, electrons that form a current move in the conduction band. These energy levels only have energies of eV. So the electrons simply bounce around the conductor. This effect is what makes the Ohmic resistance of a conductor increase with temperature: hotter means more atom movement means more bouncing means more electrical resistance.
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