Energy Level Formula

A quantum mechanical system or particle that is bound, that is, spatially constrained, may only take on discrete energy values known as energy levels. On the other hand, classical particles can absorb any amount of energy. The term most commonly refers to the energy levels of electrons bound by the electric field of the nucleus in atoms, ions, or molecules, but it may also refer to nuclei's energy levels or vibrational or rotational energy levels in molecules. A system with such different energy levels is considered to have a quantized energy spectrum.

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Energy Level Formula

When it comes to the rotational energy levels formula, the Bohr model of the hydrogen atom must be considered. An assumption was made about atom quantization in the Bohr model of the hydrogen atom. According to this hypothesis, electrons orbit around the nucleus in predefined orbits or shells with a definite radius. Only shells whose radius was given by the equation below were allowed. Furthermore, there could not be any electrons present between the shells.

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Mathematically, the expression of the energy level equation the allowed value of the atomic radius is given by,

r(n) = n² + r(1)

Hence the above formula is also called Bohr’s formula for energy levels for radius.

Electron Energy Level Formula

Bohr used electrons in circular and quantized orbits to calculate the energy of an electron at the nth level of the hydrogen atom. This can be seen below in the form of the electron energy level formula,

\[E(n)=-\frac{1}{n^{2}}\times13.6eV\]

Where 13.6 eV is the smallest possible energy constant value of a hydrogen electron.

The energy in the form of photons is absorbed by an electron, which allows it to be exciting to a higher energy level. The excited electron is less stable after moving to a higher energy level, also known as the excited state, and would soon emit a photon to return to a lower and more stable energy level. For a given transition, the released energy is equal to the difference in energy between the two energy levels. The following energy level equation may be used to compute the energy by using 

\[hv=\Delta E=(\frac{1}{n^{2}_{Low}}-\frac{1}{n^{2}_{High}})13.6eV\]

The hydrogen atom energy levels equation is given by

\[E=\frac{E_{0}}{n^{2}}\]

where E₀ is 13.6 eV and n is 1,2,……and so on

Solved Examples

Ex.1. The hydrogen spectrum in terms of electrons absorbing and emitting photons to change energy levels with the longest wavelength (n_high) is 4 and the smallest wavelength  (n_low) is 2. Calculate the energy by using the Bohr energy level formula.

Solution:

Given,

n_high = 4

n_low = 2

By using the Bohr formula energy levels we get,

\[\Delta E=(\frac{1}{n^{2}_{low}}-\frac{1}{n^{2}_{high}})13.6eV\]

\[\Delta E=(\frac{1}{2^{2}}-\frac{1}{4^{2}})13.6eV\]

ΔE = (0.25 - 0.0625) x 13.6eV

ΔE = 0.1875 x 13.6eV

ΔE =2.55eV