In Niels Bohr’s model of the atom, each electron shell holds a maximum number of electrons determined by a simple formula: 2n², where n is the shell number counting outward from the nucleus. That gives the first shell a capacity of 2, the second shell 8, the third 18, and the fourth 32. These numbers do a surprisingly good job of explaining why the periodic table looks the way it does, even though the model itself has been superseded by quantum mechanics for nearly a century.
Shell Capacities at a Glance
Each shell is traditionally labeled with both a number and a letter, starting from the shell closest to the nucleus. The maximum number of electrons each shell can hold follows directly from the 2n² rule:1Science Facts. Electron Shells and Orbitals
- Shell 1 (K): 2 × 1² = 2 electrons
- Shell 2 (L): 2 × 2² = 8 electrons
- Shell 3 (M): 2 × 3² = 18 electrons
- Shell 4 (N): 2 × 4² = 32 electrons
- Shell 5 (O): 2 × 5² = 50 electrons
- Shell 6 (P): 2 × 6² = 72 electrons
- Shell 7 (Q): 2 × 7² = 98 electrons
In practice, the last few shells never come close to filling. No known element has enough electrons to fill even the fifth shell completely. Oganesson, the heaviest element discovered so far, has 118 electrons spread across seven shells, and its outermost shell holds only 8. The formula tells you the theoretical maximum, not what nature actually populates.
Where the Formula Comes From
Bohr’s original 1913 model was built around a single bold idea: electrons orbiting the nucleus can only occupy certain allowed orbits, not just any orbit they please. The key rule was that an electron’s angular momentum had to be a whole-number multiple of a fundamental constant (Planck’s constant divided by 2Ï€). This quantization condition predicted the spectrum of hydrogen with remarkable accuracy, matching experimental measurements that had puzzled physicists for decades.2ChemTexts. The origin of the postulates in the Bohr model of the hydrogen atom
The 2n² rule wasn’t something Bohr himself derived in that first paper. It emerged as the model was extended to multi-electron atoms in the years that followed. Each shell labeled n turns out to contain n² distinct orbital states, and each state can hold two electrons (one spinning “up” and one spinning “down,” to put it loosely). Multiply n² states by 2 electrons per state and you get 2n². The factor of 2 comes from electron spin, a property that wasn’t even discovered until over a decade after Bohr’s original work. But the formula meshed so well with observed chemistry that it became one of the most useful takeaways from the Bohr model.
How Shell Capacities Shape the Periodic Table
The connection between the 2n² rule and the periodic table is the reason this model still gets taught. Each row of the periodic table roughly corresponds to filling a new shell. The first row has 2 elements (hydrogen and helium), matching the capacity of the first shell. The second row has 8 elements (lithium through neon), matching the second shell. So far, the pattern is clean.
Things get more interesting at the third row. The third shell can hold 18 electrons, but the third row of the periodic table has only 8 elements (sodium through argon). That’s because elements don’t fill the third shell completely before starting the fourth. The outermost electrons of potassium and calcium go into the fourth shell even though the third shell still has room. Only after those two elements do transition metals (scandium through zinc) begin backfilling the third shell’s remaining 10 slots. This is why the periodic table has that distinctive wide block of transition metals inserted in the middle.
Noble gases sit at the far right of each row, and their stability is tied to having completely filled outermost shells or, more precisely, completely filled outermost subshells. Helium has a full first shell with 2 electrons. Neon has a full second shell with 8. Argon has 8 electrons in its outermost occupied subshells of the third shell, even though the third shell isn’t technically full. The very low reactivity of noble gases comes from these filled configurations.3PubMed. Structure, stability, reactivity and bonding in noble gas compounds It’s one of the most direct pieces of evidence that shell structure isn’t just a theoretical convenience but reflects something real about how atoms behave.
Why Real Atoms Don’t Fill Shells in Strict Order
If electrons simply filled one shell before starting the next, chemistry would be simpler and the periodic table would be a straightforward grid. They don’t. In atoms with many electrons, the energy of a given shell depends not just on its number but also on its shape. Shells break into subshells (labeled s, p, d, and f), and the energies of these subshells can overlap between shells.
The practical result is that the 4s subshell fills before the 3d subshell, the 5s fills before the 4d, and the 4f subshell doesn’t begin filling until well into the sixth row of the periodic table. This filling order is sometimes called the Madelung rule or the n+â„“ rule. It captures the observation that electrons tend to occupy whichever available subshell has the lowest combined value of its shell number and shape number. The origin of this pattern has been debated for decades, with researchers describing it as a consequence of how d and f electrons interact with the atom’s overall structure rather than something derivable from first principles in a simple way.4Wiley Online Library / International Journal of Quantum Chemistry. The Löwdin challenge: Origin of the n+â„“, n (Madelung) rule for filling the orbital configurations of the periodic table
This overlap is the reason the Bohr model’s tidy shell picture starts to strain once you get beyond the first two rows. The model correctly predicts how many electrons each shell can hold in total, but it doesn’t explain why certain subshells of higher shells fill before lower shells are complete. For hydrogen and hydrogen-like ions (atoms with only one electron), Bohr’s shells map onto reality quite cleanly. For everything else, the shell capacities remain correct as upper limits, but the order of filling is a more complicated story.
The Octet Rule and the “Eight Electrons” Confusion
A common point of confusion, especially in introductory chemistry, is the idea that every shell holds exactly 8 electrons. This is wrong but understandable, because the octet rule (atoms tend to gain, lose, or share electrons to achieve 8 in their outer shell) gets so much emphasis. The octet rule is really about the outermost occupied subshells, not the shell as a whole. For the second shell, 8 happens to be both the shell capacity and the octet. For the third shell and beyond, the capacity is larger than 8, but the atoms at the end of each main-group row still have 8 electrons in their outermost s and p subshells.
So when you see that argon has 8 electrons in its outer shell and is a stable noble gas, that doesn’t mean the third shell maxes out at 8. The third shell can hold 18 electrons. Argon just fills the 3s and 3p portions (2 + 6 = 8) and leaves the 3d subshell empty. The transition metals that follow in the next row are the ones that fill those remaining 10 spots. This distinction trips up a lot of students who assume shells and octets are the same thing.
Where the Bohr Model Gets It Right and Where It Falls Short
The Bohr model was designed for hydrogen, and for hydrogen it works beautifully. Its predicted energy levels match experimental measurements of hydrogen’s spectral lines, confirming that Bohr’s quantization condition was onto something real.2ChemTexts. The origin of the postulates in the Bohr model of the hydrogen atom The 2n² shell capacities are also correct and carry over into the full quantum mechanical treatment. In that sense, the model was not wrong so much as incomplete.
Where it falls short is in treating electrons as particles orbiting the nucleus on neat circular paths, like planets around a star. Modern quantum mechanics replaces those orbits with probability clouds that describe where an electron is likely to be found, not where it definitely is at any given moment. The Bohr model also can’t explain fine details of atomic spectra (the splitting of spectral lines in magnetic fields, for instance), the behavior of atoms with more than one electron in any precise way, or chemical bonding. These limitations are why physicists moved past it in the 1920s.
Despite these shortcomings, research on how students actually understand atomic structure reveals something striking: the Bohr model’s mental image is remarkably sticky. In one study of chemistry students, even after learning more advanced models, the overwhelming majority continued to picture the atom as a miniature solar system with electrons on fixed circular orbits.5ResearchGate / International Journal for Innovation Education and Research. Misconceptions about Atomic Models Amongst the Chemistry Students Roughly 85% or more of the students sampled held misconceptions rooted in the Bohr picture. That persistence isn’t necessarily a problem when it comes to shell capacities, since 2n² is correct. The trouble arises when students assume electrons literally travel on circular tracks or that the shells are rigid boundaries rather than zones of probability.
Exceptions Worth Knowing About
A few specific elements break even the Madelung filling order. Chromium and copper are the classic examples. Chromium “should” have four electrons in its 3d subshell and two in 4s, but experimentally, it has five in 3d and one in 4s. Copper should have nine in 3d and two in 4s, but instead has ten in 3d and one in 4s. In both cases, a half-filled or completely filled d subshell turns out to be slightly more stable than the expected configuration.
These exceptions multiply as you move to heavier elements. The lanthanides and actinides, which fill f subshells, are riddled with deviations from the predicted filling order. The reason is that the energy differences between subshells become very small in heavy atoms, and subtle effects (electron-electron repulsion, relativistic effects on inner electrons) tip the balance in ways that simple filling rules can’t anticipate. The 2n² capacity of each shell remains valid, but the order in which subshells populate is more of a trend than a law.
Rydberg Atoms and the Bohr Model’s Comeback
There’s a corner of modern physics where the Bohr model’s picture of electrons on large, circular-like orbits becomes surprisingly accurate again. Rydberg atoms are atoms in which a single electron has been excited to an extremely high energy level, sometimes shell numbers in the hundreds. At those distances from the nucleus, the electron behaves almost classically. Its orbit is enormous compared to a normal atom, and the frequencies of light it emits as it drops between neighboring shells approach the orbital frequency you’d calculate from Bohr’s model, just as Bohr’s own correspondence principle predicted.6Advances in Atomic and Molecular Physics. Rydberg Atoms: High-Resolution Spectroscopy and Radiation Interaction—Rydberg Molecules
Rydberg atoms can be as large as a grain of sand, and they are exquisitely sensitive to electric fields. Researchers use them today in areas ranging from quantum computing to electric-field sensing. The fact that these exotic atoms obey Bohr’s century-old predictions so well is a nice reminder that the model was never entirely wrong. It captured something fundamental about quantized energy levels and shell structure. The 2n² capacities, the energy spacing between shells, and the general picture of electrons occupying distinct layers around the nucleus all survive in the modern framework. What changed was our understanding of what those “layers” actually look like up close.
Why These Numbers Matter in Everyday Chemistry
Knowing the shell capacities helps explain everyday chemical behavior without any math. Sodium has 11 electrons: 2 in the first shell, 8 in the second, and 1 in the third. That lone outermost electron is easy to lose, which is why sodium is so reactive (drop it in water and it practically explodes). Chlorine has 17 electrons: 2, 8, and 7. It’s one electron short of a filled outer subshell, so it aggressively grabs electrons from anything willing to donate. Put sodium and chlorine together and you get table salt, a compound held together by the electrostatic attraction between a sodium atom that lost its extra electron and a chlorine atom that gained one.
Carbon’s 6 electrons (2 in the first shell, 4 in the second) leave it four short of a filled second shell and four away from an empty one. That middle-ground position is why carbon forms four bonds and is the backbone of organic chemistry, including every molecule in your body. Oxygen’s 8 electrons (2 and 6) leave it two short, explaining why water is Hâ‚‚O: two hydrogen atoms each share an electron with oxygen to fill out its shell. These aren’t quirky coincidences. They’re direct consequences of the shell capacities the Bohr model first laid out.
None of this requires knowing the difference between a 3p and a 3d subshell. For understanding why elements react the way they do at a basic level, the Bohr model’s shell picture and the 2n² rule remain genuinely useful tools, not just historical artifacts kept alive by nostalgia. The model’s simplicity is its greatest strength for anyone who doesn’t need to calculate molecular orbitals for a living.