What is the formula for the energy of an electron?

The energy of an electron in a hydrogen-like atom (an atom with only one electron) can be calculated using a specific formula derived from the Bohr model and quantum mechanics.

Energy of an Electron Formula

The formula for the energy (E) of an electron in the nth energy level of a hydrogen-like atom is:

E = -2.178 x 10^-18 (Z²/n²) J

Where:

• E = Energy of the electron (in Joules)
• Z = Atomic number (number of protons in the nucleus)
• n = Principal quantum number (energy level, n = 1, 2, 3, ...)
• -2.178 x 10^-18 J is a constant representing the Rydberg constant (R_H) for hydrogen.

Explanation of the Formula

• Negative Sign: The negative sign indicates that the electron's energy is relative to a zero-energy state when the electron is infinitely far from the nucleus. The electron is bound to the nucleus, hence the negative energy.
• Z² Term: The square of the atomic number (Z) signifies that the energy is directly proportional to the square of the nuclear charge. A higher nuclear charge results in a stronger attraction to the electron, leading to lower (more negative) energy.
• 1/n² Term: The inverse square of the principal quantum number (n) means that as the electron moves to higher energy levels (larger n), its energy becomes less negative (higher), approaching zero as n approaches infinity. This reflects that electrons in higher energy levels are less tightly bound to the nucleus.

Example

Let's calculate the energy of an electron in the ground state (n=1) of a hydrogen atom (Z=1):

E = -2.178 x 10^-18 (1²/1²) J
E = -2.178 x 10^-18 J

Important Considerations

• This formula is most accurate for hydrogen-like atoms (ions with only one electron).
• For multi-electron atoms, electron-electron interactions complicate the energy level calculations, and this simple formula is no longer sufficient. More complex quantum mechanical methods are required.
• The formula gives the potential energy of the electron. The total energy will also include kinetic energy considerations, although these are implicitly included through the virial theorem within the context of the Bohr model derivation.