Principal Quantum Number (n)

In quantum mechanics, electrons in an atom do not orbit the nucleus in simple circular paths like planets around the Sun. Instead, their behavior is described by a set of four quantum numbers, each of which constrains a different property of the electron's quantum state. The most fundamental of these is the principal quantum number, symbolized by the letter n, which determines the overall energy level and the average distance of an electron from the nucleus — essentially defining which 'shell' the electron occupies. The principal quantum number can take any positive integer value: n = 1, 2, 3, 4, and so on. When n = 1, the electron is in the first shell, closest to the nucleus and at the lowest possible energy. As n increases, the electron occupies a larger, higher-energy shell. This directly explains why hydrogen's emission spectrum shows discrete lines: electrons can only absorb or emit photons with exactly the energy difference between two allowed shells, and no values in between are permitted. Understanding the principal quantum number is essential for making sense of the periodic table's structure. The period number of an element corresponds directly to the highest principal quantum number of its valence electrons. For example, sodium (Na) is in Period 3, and its outermost electron has n = 3. The concept also underpins chemical bonding, ionization energy trends, and atomic radii — all of which depend on how far electrons are from the nucleus and how strongly they are held.

Key takeaways
  • Allowed Values of n
  • Energy and the Bohr Model
  • Orbital Size and the n² Rule
  • Maximum Electrons per Shell
  • Connection to the Other Quantum Numbers

Allowed Values of n

The principal quantum number n must be a positive integer: n = 1, 2, 3, 4, … There is no upper theoretical limit, but in real neutral atoms, the highest occupied shell rarely exceeds n = 7 (as seen in Period 7 elements like oganesson). Each value of n defines an electron shell, historically labeled with letters: n = 1 is the K shell, n = 2 is the L shell, n = 3 is the M shell, and so on.

Energy and the Bohr Model

For the hydrogen atom, the energy of an electron in shell n is given exactly by Bohr's formula: E_n = −13.6 eV / n². The negative sign means energy is measured relative to a free electron at infinite distance (E = 0). The ground state (n = 1) has the lowest (most negative) energy at −13.6 eV. The n = 2 level sits at −3.4 eV. The energy difference between levels determines the frequency of light absorbed or emitted during transitions, following ΔE = hf.

Orbital Size and the n² Rule

The average radius of an electron's orbital scales approximately as n². For hydrogen, the average radius in shell n is r_n = n² × a₀, where a₀ = 0.529 Å is the Bohr radius. So the n = 1 orbital has average radius ≈ 0.529 Å, n = 2 gives ≈ 2.12 Å, and n = 3 gives ≈ 4.76 Å. This rapid growth explains why higher-shell electrons are shielded from the nucleus and held more loosely, leading to lower ionization energies for outer electrons.

Maximum Electrons per Shell

Each shell with principal quantum number n can hold a maximum of 2n² electrons. This arises from the combined constraints of all four quantum numbers. Shell n = 1 holds 2(1²) = 2 electrons, shell n = 2 holds 2(2²) = 8 electrons, shell n = 3 holds 2(3²) = 18 electrons, and shell n = 4 holds 2(4²) = 32 electrons. This rule directly explains the pattern of element counts in successive periods of the periodic table: 2, 8, 8, 18, 18, 32, 32.

Connection to the Other Quantum Numbers

The principal quantum number n also constrains the angular momentum quantum number l, which must satisfy 0 ≤ l ≤ n − 1. So for n = 1, only l = 0 (the s subshell) is allowed. For n = 2, l can be 0 or 1 (s and p subshells). For n = 3, l = 0, 1, or 2 (s, p, and d subshells). This hierarchy means that higher shells contain more subshells and therefore more orbitals and more electrons — consistent with the 2n² rule.

The Key Equation

En=−13.6 eVn2E_n = -\frac{13.6 \text{ eV}}{n^2}
  • E_n — Energy of the nth level (eV (electron volts)): The total mechanical energy of the electron when it occupies the shell defined by quantum number n. Negative values indicate a bound state.
  • 13.6 \text{ eV} — Rydberg energy (for hydrogen) (eV): The ionization energy of hydrogen from its ground state (n = 1). It equals 2.18 × 10⁻¹⁸ J and arises from fundamental constants: m_e e⁴ / (8ε₀²h²).
  • n — Principal quantum number (dimensionless (positive integer)): The main quantum number specifying the electron shell. n = 1, 2, 3, … As n increases, energy increases (becomes less negative) and orbital size grows.

Worked Example

Problem. An electron in a hydrogen atom transitions from the n = 3 shell to the n = 1 shell. Calculate the energy of the photon emitted during this transition, and determine whether this photon falls in the ultraviolet, visible, or infrared region of the electromagnetic spectrum. (1 eV = 1.60 × 10⁻¹⁹ J, h = 6.626 × 10⁻³⁴ J·s, c = 3.00 × 10⁸ m/s)

λ=hc∣ΔE∣=(6.626×10−34)(3.00×108)1.934×10−18≈1.03×10−7 m=103 nm\lambda = \frac{hc}{|\Delta E|} = \frac{(6.626 \times 10^{-34})(3.00 \times 10^{8})}{1.934 \times 10^{-18}} \approx 1.03 \times 10^{-7} \text{ m} = 103 \text{ nm}

Answer. The emitted photon has energy 12.09 eV (1.93 × 10⁻¹⁸ J) and wavelength ≈ 103 nm, placing it in the ultraviolet region (Lyman series).

Figure
Diagram
Hydrogen Energy Levels and Electron Transitionsn=1n=2n=3n=4n=5n=∞−13.6 eV−3.4 eV−1.51 eV−0.85 eV−0.54 eV0 eVLymanUV 103 nmBalmerVisiblePaschenIRElectrons emit photons when dropping to lower n; photon energy = ΔE between shells
Origins · 1 minNiels Bohr and how this was discovered
NB
Niels Bohr
1885–1962 · Danish

Niels Bohr proposed the Bohr model of the atom in 1913, which introduced the concept of discrete electron energy levels quantified by the principal quantum number n. His model successfully explained the hydrogen emission spectrum by postulating that electrons occupy specific shells and that light is emitted only when an electron transitions between shells. Although the Bohr model was later superseded by full quantum mechanics, the concept of the principal quantum number n remained central and unchanged. Bohr was awarded the Nobel Prize in Physics in 1922 for this work. Bohr once had a legendary scientific debate with Albert Einstein at the Solvay Conferences in the late 1920s over the nature of quantum mechanics. Einstein kept proposing thought experiments to disprove the uncertainty principle, and Bohr refuted every single one — often using Einstein's own theory of general relativity against him.

  • Azimuthal Quantum Number (l) — The azimuthal quantum number l is directly constrained by n (0 ≤ l ≤ n−1), determining subshell shape (s, p, d, f). Understanding n is a prerequisite to understanding l.
  • Electron Configuration and the Aufbau Principle — Electron configurations are built by filling shells in order of increasing n (and energy). The principal quantum number directly determines which shell is being filled.
  • Hydrogen Emission Spectrum and Spectral Series — The discrete spectral lines of hydrogen (Lyman, Balmer, Paschen series) arise from electrons transitioning between different values of n, making the principal quantum number essential to interpreting spectra.
  • Ionization Energy Periodic Trends — Ionization energy decreases down a group because valence electrons occupy higher-n shells, placing them farther from the nucleus and making them easier to remove.
  • Atomic Radius and Orbital Size — Orbital size scales as n², so atomic radius is directly connected to the principal quantum number of the valence electrons. This concept explains why atoms grow larger going down a group.