Energy Levels

Q: What is the significance of the principal quantum number 'n'?

The principal quantum number, $n$, is a positive integer (1, 2, 3, ...) that defines the main energy level or shell an electron occupies. In the Bohr model, it directly determines the radius of the electron's orbit ($r_n \propto n^2$) and, crucially, the energy of the electron ($E_n \propto 1/n^2$). Higher values of $n$ correspond to higher energy levels, larger orbits, and electrons that are less tightly bound to the nucleus. It essentially quantizes the electron's energy and angular momentum.

Updated 23 Mar 2026

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In the Bohr model of the hydrogen atom, energy levels refer to the discrete, quantized states of energy that an electron can occupy within the atom. These levels are not continuous but rather specific, fixed values, each corresponding to a particular stable orbit, or 'stationary state,' around the nucleus. An electron can transition between these levels by absorbing or emitting a photon of energy …

Quick Summary

Energy levels in the Bohr model describe the discrete, quantized amounts of energy an electron can possess within an atom. Instead of orbiting arbitrarily, electrons are restricted to specific 'stationary states,' each with a unique energy value.

The lowest energy state is the ground state ( $n=1$ ), and higher states are excited states ( $n=2, 3, ...$ ). These energy levels are negative, indicating the electron is bound to the nucleus, with zero energy representing a free electron.

The energy of an electron in the $n$ -th orbit of a hydrogen-like atom is given by $E_n = -\frac{13.6 Z^2}{n^2} \text{ eV}$ . Transitions between these levels involve the absorption or emission of photons with energy precisely equal to the energy difference between the levels, explaining the characteristic line spectra of atoms.

As the principal quantum number $n$ increases, the energy levels become less negative and are spaced more closely together, eventually converging to zero at the ionization limit.

Full explanation

The concept of energy levels is a cornerstone of atomic physics, particularly elucidated by Niels Bohr's model for the hydrogen atom. Before Bohr, classical physics struggled to explain the stability of atoms and the discrete nature of atomic spectra. Bohr's model, though superseded by more advanced quantum mechanics, provided a crucial stepping stone and remains highly relevant for understanding basic atomic structure and spectral phenomena, especially for hydrogen and hydrogen-like ions.

Conceptual Foundation: The Bohr Model's Postulates

Bohr's model is built upon three fundamental postulates:

Stationary Orbits: — Electrons revolve around the nucleus in certain definite, non-radiating orbits, called stationary states or non-radiating orbits. In these orbits, the electron does not emit electromagnetic radiation, contrary to classical electromagnetism.

Quantization of Angular Momentum: — The angular momentum of an electron in a stationary orbit is quantized. It can only take on discrete values that are integral multiples of

\frac{h}{2\pi}

, where

h

is Planck's constant. Mathematically, this is expressed as

L = mvr = n\frac{h}{2\pi}

, where

m

is the electron's mass,

v

is its speed,

r

is the radius of the orbit, and

n

is a positive integer (1, 2, 3, ...), known as the principal quantum number.

Energy Transitions: — An electron can jump from one stationary orbit to another. When it jumps from a higher energy orbit (

E_i

) to a lower energy orbit (

E_f

), it emits a photon whose energy is exactly equal to the energy difference between the two orbits:

h\nu = E_i - E_f

. Conversely, to jump from a lower to a higher energy orbit, the electron must absorb a photon of the same energy difference.

Key Principles and Derivations for Energy Levels

To derive the expression for the energy levels, we combine Bohr's postulates with classical mechanics and electrostatics.

Consider an electron of mass $m$ and charge $-e$ orbiting a nucleus of charge $+Ze$ (where $Z=1$ for hydrogen) in a circular orbit of radius $r$ with speed $v$ .

Centripetal Force and Electrostatic Force: — For a stable orbit, the electrostatic attractive force between the electron and the nucleus provides the necessary centripetal force.

F_{\text{centripetal}} = F_{\text{electrostatic}}

\frac{mv^2}{r} = \frac{1}{4\pi\epsilon_0} \frac{(Ze)(e)}{r^2}

\frac{mv^2}{r} = \frac{Ze^2}{4\pi\epsilon_0 r^2} \quad \text{(Equation 1)}

Quantization of Angular Momentum (Bohr's Postulate):

mvr = n\frac{h}{2\pi} \quad \text{(Equation 2)}

From Equation 2, we can express

v

v = \frac{nh}{2\pi mr}

. Substitute this into Equation 1:

m\left(\frac{nh}{2\pi mr}\right)^2 \frac{1}{r} = \frac{Ze^2}{4\pi\epsilon_0 r^2}

m\frac{n^2h^2}{4\pi^2 m^2 r^2} \frac{1}{r} = \frac{Ze^2}{4\pi\epsilon_0 r^2}

\frac{n^2h^2}{4\pi^2 m r^3} = \frac{Ze^2}{4\pi\epsilon_0 r^2}

We can cancel

r^2

from both sides and solve for

r

(the radius of the

n

-th orbit,

r_n

r_n = \frac{n^2h^2\epsilon_0}{\pi m Z e^2} \quad \text{(Equation 3 - Radius of n-th orbit)}

For hydrogen (

Z=1

) and

n=1

, this gives the Bohr radius, $a_0 = \frac{h^2\epsilon_0}{\pi m e^2} \approx 0.

529 \times 10^{-10} \text{ m} $. So,$ r_n = n^2 a_0$.

Total Energy of the Electron: — The total energy

E

of the electron in an orbit is the sum of its kinetic energy (KE) and potential energy (PE).

KE = \frac{1}{2}mv^2

PE = -\frac{1}{4\pi\epsilon_0} \frac{Ze^2}{r}

(The negative sign indicates an attractive force and that the electron is bound. Potential energy is zero when the electron is infinitely far from the nucleus).

From Equation 1, we have $mv^2 = \frac{Ze^2}{4\pi\epsilon_0 r}$ . Substitute this into the KE expression:

KE = \frac{1}{2} \left(\frac{Ze^2}{4\pi\epsilon_0 r}\right) = \frac{Ze^2}{8\pi\epsilon_0 r}

Now, sum KE and PE to get the total energy

E_n

for the

n

-th orbit:

E_n = KE + PE = \frac{Ze^2}{8\pi\epsilon_0 r_n} - \frac{Ze^2}{4\pi\epsilon_0 r_n}

E_n = -\frac{Ze^2}{8\pi\epsilon_0 r_n} \quad \text{(Equation 4)}

Notice that the total energy is negative and is half of the potential energy.

This is a characteristic of systems where the force is inversely proportional to the square of the distance (virial theorem).

Finally, substitute the expression for $r_n$ from Equation 3 into Equation 4:

E_n = -\frac{Ze^2}{8\pi\epsilon_0} \left(\frac{\pi m Z e^2}{n^2h^2\epsilon_0}\right)

E_n = -\frac{m Z^2 e^4}{8\epsilon_0^2 n^2 h^2} \quad \text{(Equation 5 - Energy of n-th level)}

For the hydrogen atom ( $Z=1$ ), the energy levels are:

E_n = -\frac{m e^4}{8\epsilon_0^2 h^2} \frac{1}{n^2}

The combination of constants

\frac{m e^4}{8\epsilon_0^2 h^2}

is known as the Rydberg constant for energy,

R_H

. Its value is approximately

2.18 \times 10^{-18}

J or

13.6

eV. Thus, the energy levels for hydrogen are:

E_n = -\frac{13.6}{n^2} \text{ eV}

Significance of Negative Energy:

The negative sign for energy levels indicates that the electron is bound to the nucleus. Energy must be supplied to remove the electron from the atom. An electron with zero energy is considered free, i.e., it has just enough energy to escape the atom's influence (ionization). Positive energy would correspond to a free electron with kinetic energy, not bound to the nucleus.

Real-World Applications:

Hydrogen Spectrum: — The most direct application is the explanation of the hydrogen line spectrum. When electrons transition between energy levels, they emit or absorb photons of specific energies, leading to distinct spectral lines (Lyman, Balmer, Paschen, Brackett, Pfund series). This was a major triumph of the Bohr model.

Lasers: — The principle of discrete energy levels is fundamental to the operation of lasers. Atoms are excited to higher energy levels, and then stimulated emission occurs as electrons drop to lower levels, releasing coherent light.

Spectroscopy: — The analysis of atomic and molecular spectra, based on energy level transitions, is a powerful tool in chemistry and physics for identifying substances and studying their properties.

Common Misconceptions:

Continuous Energy: — A common mistake is to think that electrons can have any energy value. Bohr's model explicitly states that energy is quantized, meaning only specific, discrete energy levels are allowed.

Ground State Energy is Zero: — The ground state (

n=1

) has the lowest (most negative) energy, not zero. Zero energy corresponds to an ionized atom (electron completely removed).

Bohr Model is Universally Applicable: — While revolutionary, the Bohr model is strictly applicable only to hydrogen and hydrogen-like ions (e.g., He

^+

, Li

^{2+}

) because it does not account for electron-electron repulsion or more complex quantum mechanical effects like electron spin or orbital shapes.

Electrons 'Orbit' like Planets: — While a useful analogy, electrons in reality do not follow well-defined classical orbits. Quantum mechanics describes them as probability distributions (orbitals).

NEET-Specific Angle:

For NEET, understanding the formula $E_n = -\frac{13.6 Z^2}{n^2} \text{ eV}$ is crucial. You should be able to:

Calculate the energy of an electron in a specific orbit for hydrogen or hydrogen-like ions.

Calculate the energy difference between two levels, which corresponds to the energy of an emitted or absorbed photon (

h\nu = \frac{hc}{\lambda} = E_i - E_f

Relate energy transitions to the different spectral series (Lyman, Balmer, Paschen, etc.) and their corresponding regions of the electromagnetic spectrum.

Determine ionization energy (energy required to remove an electron from the ground state to

n=\infty

Understand the relationship between energy levels and the principal quantum number

n

: as

n

increases, energy levels become less negative and closer together. The spacing between adjacent levels decreases rapidly with increasing

n

Key Concepts

Quantization of Energy

In the Bohr model, the quantization of energy means that an electron cannot possess just any arbitrary amount…

Negative Energy Levels

The energy levels in the Bohr model are calculated to be negative ( $E_n = -\frac{13.6 Z^2}{n^2} \text{ eV}$ ).…

Rydberg Constant and Spectral Series

The Rydberg constant ( $R_H$ ) is a fundamental constant that appears in the energy level formula and is…

Often confused with

Side-by-side differences the NEET paper likes to test.

Energy Levels vs Classical Model of Atom

Aspect	Energy Levels	Classical Model of Atom
Electron Orbits	Electrons can orbit at any radius and possess any energy (continuous spectrum).	Electrons can only exist in specific, discrete orbits with quantized energy levels (line spectrum).
Atomic Stability	Accelerating electrons should continuously radiate energy and spiral into the nucleus, making atoms unstable.	Electrons in stationary orbits do not radiate energy, ensuring atomic stability.
Energy Emission/Absorption	Atoms should emit/absorb light continuously across all frequencies.	Atoms emit/absorb photons only when electrons transition between specific energy levels, leading to discrete spectral lines.
Angular Momentum	Angular momentum of electron can take any value.	Angular momentum is quantized, $L = n\frac{h}{2\pi}$.

The classical model of the atom, based on Rutherford's planetary model and classical electromagnetism, predicted that electrons orbiting the nucleus should continuously radiate energy and eventually collapse into the nucleus, rendering atoms unstable.

It also failed to explain the observed discrete line spectra of elements. In stark contrast, Bohr's model introduced the revolutionary concept of quantized energy levels and angular momentum, proposing that electrons exist in stable, non-radiating 'stationary states.

' This explained atomic stability and the characteristic line spectra, marking a pivotal shift towards quantum understanding.

Why it is tested: For NEET, understanding the limitations of the classical model and how Bohr's model addressed these issues through the concept of quantized energy levels is fundamental. Questions often test the core differences and the implications for atomic stability and spectral characteristics.

Questions students ask

5 answered on this topic.

Why are the energy levels in the Bohr model negative?

The negative sign for the energy levels signifies that the electron is bound to the nucleus. It indicates an attractive force between the positively charged nucleus and the negatively charged electron.

By convention, the potential energy of an electron infinitely far from the nucleus is taken as zero. Since energy must be supplied to remove a bound electron from the atom (i.e., to move it to infinity), its initial energy within the atom must be less than zero.

The more negative the energy, the more tightly bound the electron is to the nucleus.

What is the significance of the principal quantum number 'n'?

The principal quantum number, $n$ , is a positive integer (1, 2, 3, ...) that defines the main energy level or shell an electron occupies. In the Bohr model, it directly determines the radius of the electron's orbit ( $r_n \propto n^2$ ) and, crucially, the energy of the electron ( $E_n \propto 1/n^2$ ). Higher values of $n$ correspond to higher energy levels, larger orbits, and electrons that are less tightly bound to the nucleus. It essentially quantizes the electron's energy and angular momentum.

What is ionization energy in the context of energy levels?

Ionization energy is the minimum energy required to completely remove an electron from an atom in its ground state, effectively taking it to an infinitely distant orbit where its energy is zero. For the hydrogen atom, this means transitioning the electron from the $n=1$ (ground state) energy level to the $n=\infty$ level. The ionization energy is therefore $E_{\infty} - E_1 = 0 - (-13.6 \text{ eV}) = 13.6 \text{ eV}$ for hydrogen.

Why does the Bohr model only work for hydrogen and hydrogen-like ions?

The Bohr model is successful only for single-electron systems like hydrogen (H), singly ionized helium (He $^+$ ), or doubly ionized lithium (Li $^{2+}$ ). This is because it does not account for the electrostatic repulsion between multiple electrons in multi-electron atoms. It also doesn't consider the fine structure of spectral lines, the Zeeman effect (splitting of lines in a magnetic field), or the wave nature of electrons, which are explained by more advanced quantum mechanics.

How does the spacing between energy levels change as 'n' increases?

The energy levels for hydrogen are given by $E_n = -13.6/n^2 \text{ eV}$ . As $n$ increases, the absolute value of $E_n$ decreases, meaning the energy levels become less negative and closer to zero. Consequently, the energy difference between adjacent levels, $\Delta E = E_{n+1} - E_n$ , decreases significantly as $n$ increases.

For example, the gap between $n=1$ and $n=2$ is much larger than the gap between $n=2$ and $n=3$ , or $n=3$ and $n=4$ . The levels become increasingly crowded as they approach the ionization limit ( $n=\infty$ ).

Revise in 30 seconds

Energy of $n$-th orbit (Hydrogen): —

E_n = -\frac{13.6}{n^2} \text{ eV}

Energy of $n$-th orbit (Hydrogen-like): —

E_n = -\frac{13.6 Z^2}{n^2} \text{ eV}

Radius of $n$-th orbit (Hydrogen-like): —

r_n = \frac{n^2 a_0}{Z}

, where

a_0 = 0.529 \mathring{A}

Angular Momentum (Bohr's Postulate): —

L = n\frac{h}{2\pi}

Photon Energy (Transition): —

\Delta E = E_i - E_f = h\nu = \frac{hc}{\lambda}

Rydberg Formula (Wavelength): —

\frac{1}{\lambda} = R Z^2 \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)

Ground State: —

n=1

First Excited State: —

n=2

Ionization Energy: — Energy to go from

n=1

n=\infty

(for H,

13.6 \text{ eV}

)

Spectral Series: — Lyman (

n_f=1

, UV), Balmer (

n_f=2

, Visible), Paschen (

n_f=3

, IR)

To remember the order of spectral series and their regions: Lazy Boys Play Baseball Professionally Lyman (n=1) - UltraViolet Balmer (n=2) - Visible Paschen (n=3) - InfraRed Brackett (n=4) - InfraRed Pfund (n=5) - InfraRed (Remember UV, Visible, IR for the first three, then all others are IR.)