Calculate Energy of Bohr Electron N

The Bohr model describes the hydrogen atom as a small, positively charged nucleus orbited by electrons in specific energy levels. This calculator computes the energy of an electron in a hydrogen atom using the quantum number n.

What is the Bohr Model?

Developed by Niels Bohr in 1913, the Bohr model is a simplified representation of the hydrogen atom. It proposes that electrons orbit the nucleus in fixed circular paths called energy levels or shells, each associated with a specific energy.

The model explains why atoms have discrete emission and absorption spectra. When an electron moves from a higher to a lower energy level, it emits a photon of light with energy equal to the difference between the levels.

Energy Formula

The energy of an electron in the Bohr model is given by:

E_n = -R_∞hc / n²

Where:

E_n = Energy of the electron in the nth level (in electron volts, eV)
R_∞ = Rydberg constant (109,737.31 cm^-1)
h = Planck's constant (6.626 × 10^-34 J·s)
c = Speed of light (2.998 × 10⁸ m/s)
n = Principal quantum number (positive integer)

The negative sign indicates that the electron is bound to the nucleus. The energy becomes less negative (more positive) as n increases, meaning the electron is less tightly bound.

How to Calculate

Determine the principal quantum number n (n = 1, 2, 3, ...)
Square the quantum number (n²)
Divide the Rydberg constant by the squared quantum number (R_∞ / n²)
Multiply by the product of Planck's constant and the speed of light (hc)
Take the negative of the result to get the energy in electron volts

Note: The Rydberg constant is often used in its simplified form (13.605693 eV) when working with hydrogen-like atoms.

Example Calculation

Let's calculate the energy of an electron in the n=2 level:

n = 2
n² = 4
R_∞ / n² = 13.605693 / 4 = 3.401423 eV
E₂ = -3.401423 eV

This means the electron in the n=2 level has -3.401423 electron volts of energy, which is higher than the ground state (n=1) but still bound to the nucleus.

Limitations

The Bohr model has several limitations:

It only applies to hydrogen-like atoms (one electron)
It doesn't explain the fine structure of spectral lines
It doesn't account for the wave-like nature of electrons
It fails to predict the relative intensities of spectral lines

Modern quantum mechanics uses wave functions and the Schrödinger equation to describe atomic structure more accurately.

FAQ

What is the ground state energy of hydrogen?: The ground state energy (n=1) is approximately -13.605693 eV, which is the ionization energy of hydrogen.
Can the Bohr model be used for multi-electron atoms?: No, the Bohr model is only valid for hydrogen-like atoms with a single electron. For atoms with multiple electrons, quantum mechanics must be used.
What happens when n approaches infinity?: As n approaches infinity, the energy approaches zero, meaning the electron is no longer bound to the nucleus and is free.
How does the energy change with n?: The energy becomes less negative (more positive) as n increases, meaning the electron is less tightly bound to the nucleus.
What is the difference between energy levels and shells?: In the Bohr model, energy levels and shells are often used interchangeably, but in modern quantum mechanics, shells refer to groups of subshells with similar energies.