In classical physics, the potential energy of an electron in a hydrogen atom when it is far away from the nucleus can be approximated by considering the electrostatic interaction between the electron and the proton.
The potential energy of a system of two charged particles can be given by the equation:
U = k * (q1 * q2) / r
where U is the potential energy, k is the electrostatic constant (approximately 8.99 x 10^9 N m^2/C^2), q1 and q2 are the charges of the two particles (in this case, the charge of the electron and the proton, respectively), and r is the distance between the particles.
When the electron is far away from the nucleus, the distance between the electron and the proton becomes very large, and the potential energy approaches zero. This is because the electrostatic force between the electron and the proton decreases as the distance between them increases.
However, it's important to note that the classical physics description breaks down at the atomic scale, and the behavior of electrons in atoms is better described by quantum mechanics. In quantum mechanics, the concept of potential energy is not as straightforward, and the electron's energy is described by quantized energy levels. The energy levels in a hydrogen atom are given by the Rydberg formula:
E = -13.6 eV / n^2
where E is the energy of the electron, n is the principal quantum number, and -13.6 eV is the ionization energy of hydrogen. As n approaches infinity, the energy of the electron approaches zero, indicating that it becomes unbound or free from the influence of the nucleus.