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 ($n=\infty$).

Energy Levels

Physics

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Version 1Updated 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.

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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…

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.)