Final answer:
The de Broglie wavelength of an electron released from a large distance from a proton is longer at 1 m due to lower kinetic energy and shorter at 0.1 nm due to higher kinetic energy, as the wavelength decreases with increasing momentum.
Step-by-step explanation:
The question relates to the de Broglie wavelength of an electron in the presence of a proton at varying distances. According to de Broglie's hypothesis, the wavelength of a particle is inversely proportional to its momentum. An electron released at a large distance from a proton has relatively low potential energy, which increases as it moves closer to the proton due to the electrostatic attraction.
As the electron falls into the potential well of the proton, its kinetic energy increases, which in turn increases its momentum. Consequently, as the momentum of the electron increases, its wavelength decreases.
(i) At a distance of 1 m, the electron is still far from the proton, so it has lower kinetic energy and, hence, longer wavelength. (ii) At a distance of 0.1 nm, the electron is very close to the proton, with significantly higher potential and kinetic energy, resulting in a much shorter wavelength. Therefore, the answer is (a) (i) Longer, (ii) Shorter, because as the electron approaches the proton, it gains kinetic energy and its wavelength becomes shorter.