Why is the following situation impossible? In the Bohr model...

Why is the following situation impossible? In the Bohr model of the hydrogen atom, an electron moves in a circular orbit about a proton. The model states that the electron can exist only in certain allowed orbits around the proton: those whose radius $r$ satisfies $r=n^{2}(0.0529 \mathrm{nm}),$ where $n=1,2$ $3, \ldots$ For one of the possible allowed states of the atom, the electric potential energy of the system is $-13.6 \mathrm{eV}$.

Key Concepts

Energy Quantization

Within the Bohr model, the energy of the electron is quantized, with levels determined by a principal quantum number (n). The total energy for the hydrogen atom's ground state is -13.6 eV, and each higher energy level corresponds to a less negative energy value. This quantization explains the observed discrete spectral lines from hydrogen.

Coulomb Potential Energy

The electron-proton interaction is governed by Coulomb's law, which yields a potential energy that is negative due to the attractive force between the oppositely charged particles. In the Bohr model, the potential energy (derived from the Coulomb interaction) is not equal to the total energy; in fact, it is typically twice the magnitude (but negative) of the total energy in bound Coulomb systems.

Bohr Model of the Hydrogen Atom

This model describes the electron as confined to specific circular orbits around the nucleus, where each orbit corresponds to a certain energy level. The allowed orbit radii and energies are quantized, meaning that electrons can only reside in these discrete states rather than any arbitrary orbit, which was a key step in developing early quantum theory.

Virial Theorem for Coulombic Systems

The virial theorem provides a relation between the kinetic and potential energies in systems bound by inverse-square forces like the Coulomb force. For the hydrogen atom, it establishes that the total energy is half the potential energy. Specifically, if the ground state total energy is -13.6 eV, the corresponding potential energy should be -27.2 eV, not -13.6 eV, showing the inconsistency and impossibility of the situation described.

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