Energy Level FormulaElectron Energy level Formula (original) (raw)

Last Updated : 23 Jul, 2025

When a quantum mechanical system or particle is bound, it can only take on discrete energy values known as energy levels since it is spatially restricted. Classical particles, on the other hand, can absorb any amount of energy. The word is most usually used to describe the energy levels of electrons bound by the electric field of the nucleus in atoms, ions, or molecules, but it can also apply to the energy levels of nuclei or vibrational or rotational energy levels in molecules. A quantized energy spectrum is defined as a system with a wide range of **energy levels.

**Quantized energy levels: The energy levels in the atoms are quantized, which means they can have only certain discrete energies. not continuous. This can be understood by comparing it with the rupee system is also a quantized system since it can have only certain values like 1 rupee, two rupee coins, or 100 rupee note, you cannot have a 1.5 rupee note or coin since the government has made the regulations that it can have only certain values. Likewise, electron energy levels are also quantized systems, they can only have certain discrete energy levels.

**Ground state: The lowest energy level in the atom is called the ground state.

**Excited state: Excited states are the state when an electron gets energy through thermal agitation or by absorbing photons, it goes to higher energy levels, and when it loses the energy, then it comes to the original state of the electron.

Energy Level Formula

**r(n) = n 2 + r(1)

**⁛ As a result, the formula for energy levels for radius is often known as Bohr's formula.

Electron Energy Level Formula

To compute the energy of an electron at the **n th level of the hydrogen atom, Bohr utilized electrons in circular and quantized orbits. This can be observed in the form of the electron energy level formula, which is shown below,

**E(n) = −1/n 2 × 13.6eV

⇒The least feasible energy constant value of a hydrogen electron is **13.6 eV.

⇒An electron absorbs energy in the form of photons, allowing it to be exciting to a higher energy level. After shifting to a higher energy level, also known as the excited state, the excited electron becomes less stable and will emit a photon to return to a lower and more stable energy level.

⇒The released energy for a specific transition is equal to the difference in energy between the two energy levels. By utilizing the following energy level equation, you can calculate the energy.

**hv = ΔE = (1/n 2 Low − 1/n 2 High )13.6eV

**The equation for the energy levels of hydrogen atoms is,

**E = E 0 /n 2

where,

E0 = 13.6 eV

n = 1,2,……and so on

Solved Examples

**1. In terms of electrons absorbing and emitting photons to shift energy levels, the hydrogen spectrum has the longest wavelength (n high ) of 4 and the shortest wavelength (n low ) of 2. Using the Bohr energy level formula, calculate the energy.

Given,
nhigh = 4
nlow = 2
By using the Bohr formula energy levels we get,
ΔE = (1/n2low − 1/n2high)13.6eV
ΔE = (1/22 − 1/42)13.6eV
ΔE = (0.25 - 0.0625) x 13.6eV
ΔE = 0.1875 x 13.6eV
ΔE = 2.55eV

**2. Find the photon's energy when an electron surrounding a hydrogen atom goes from the 4th to the 2nd level.

The energy of the photon is found by computing the difference in the energies of the fourth (n=4) and second (n=2) levels
For n=4
E4​=​ −13.6/ 42 = −13.6 /16 =−0.85eV
E42=​ −13.6/ 22 = −13.6 /4=−3.40eV
Now, the energy difference ΔE, which corresponds to the energy of the photon, is:
ΔE=E2​−E4​
ΔE=−3.40−(−0.85)
ΔE=−3.40+0.85
ΔE=−2.55eV

**3. In a hydrogen atom, an electron moves from the n=3 level to the n=1 level. What is the photon's energy when it is emitted?

Using,
ΔE = −2.18 x 10−18(1/n2f − 1/n2i)
Plugging in values:
ΔE = −2.18 x 10−18(1/12 − 1/32)
This will be the change in energy of the electron, which is the negative of the energy of the photon released.
ΔE = −1.94 x 10−18J
Thus, the energy of the photon is
E = 1.94 x 10−18J

4. How much energy would it take to lift an electron in a hydrogen atom from the n=1 to the n=4 energy level?

Using the formula for the energy of an electron in a hydrogen atom's nth energy level:
En = −13.6/n2 x eV
Plug in n=1 and n=4 then find the difference:
ΔE = (−13.6/1) − (−13.6/16)
ΔE = −12.75eV
Convert electronvolts to Joules:
ΔE = −12.75eV x 1.602 x 10−19J/1eV
ΔE = 2.04 x 10−18J

**5. Electrons in one mole of hydrogen atoms fall from the n=3 energy level to the n=2 energy level. Determine the amount of energy that has been released.

Using the following equation for the energy of an electron in Joules:
ΔE = −2.18 x 10−18(1/n2f − 1/n2i)
And
1mol = 6.02 x 1023 molecules
Combining equations and plugging in values:
ΔE1mole = 6.02 x 1023 x −2.18 x 10−18(1/22−1/32)
ΔE1mole = −182kJ
182kJ would be released

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