The energy of an electron orbital, particularly for a one-electron atom or ion (like Hydrogen), can be calculated using a specific formula derived from the Bohr model and quantum mechanics.
Calculating Electron Orbital Energy
The formula to calculate the energy (E) of an electron orbital in a hydrogen-like atom is:
E = -2.18 x 10-18 J * (Z2 / n2)
Where:
- E is the energy of the electron orbital (in Joules).
- -2.18 x 10-18 J is the Rydberg constant (RH), representing the ionization energy of hydrogen.
- Z is the atomic number (number of protons in the nucleus). For Hydrogen, Z = 1. For Helium ion (He+), Z=2.
- n is the principal quantum number, representing the energy level of the electron (n = 1, 2, 3, ...). n=1 is the ground state, n=2 is the first excited state, and so on.
Explanation:
- The negative sign indicates that the energy is relative to the ionization energy (the energy required to remove the electron completely from the atom), which is defined as zero. The electron is bound to the nucleus, so its energy is lower (more negative) than zero.
- The atomic number (Z) is squared because the nuclear charge has a stronger effect on the electron's energy. A larger nuclear charge results in a more negative (lower) energy.
- The principal quantum number (n) is squared in the denominator, indicating that as the electron moves to higher energy levels (larger n), its energy becomes less negative (higher). The energy levels get closer together as n increases.
Example: Calculating the Energy of the Ground State (n=1) of Hydrogen (Z=1)
E = -2.18 x 10-18 J * (12 / 12)
E = -2.18 x 10-18 J
This shows that the energy of an electron in the ground state (n=1) of hydrogen is -2.18 x 10-18 J.
Important Considerations:
- This formula is most accurate for one-electron species (hydrogen-like atoms/ions).
- For multi-electron atoms, electron-electron interactions complicate the energy level calculations, and more sophisticated methods are needed. These methods often involve approximations and computational techniques.
- The formula provides the potential energy of the electron within the electric field of the nucleus. The total energy would require consideration of kinetic energy components if dealing with more advanced scenarios.