Quantum Mechanical Model of Atoms
Bohr's model gave electrons fixed, circular orbits — but the quantum mechanical model replaces orbits with orbitals, described by four quantum numbers.
Bohr's model gave electrons fixed, circular orbits — but that picture only works for hydrogen, and it directly contradicts a core principle of quantum mechanics. The quantum mechanical model replaces orbits with orbitals: probability regions described by four quantum numbers, filled according to three organizing rules, and expressed through electron configuration.
Key Takeaways
Bohr's model only works for hydrogen and contradicts the Heisenberg uncertainty principle (position and momentum can't both be known with perfect accuracy) — it was superseded by the wave-mechanical model.
Electrons occupy orbitals (probability regions), not fixed orbits.
Every electron is described by four quantum numbers: n (energy level/size, shell capacity = 2n²), l (shape/subshell: s, p, d, f, ranging 0 to n−1), ml (orientation, −l to +l, orbital count = 2l+1), and ms (spin, +1/2 or −1/2).
Shells (defined by n) contain subshells (defined by l), which contain orbitals (individual probability regions holding up to 2 electrons each).
Electrons fill orbitals according to Aufbau's principle (lowest energy first), the Pauli exclusion principle (max 2 electrons per orbital, opposite spins), and Hund's rule (singly fill degenerate orbitals before pairing).
Electron configurations use spectroscopic notation (energy level + subshell letter + electron-count superscript) and follow the Aufbau filling order 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p...
Chromium ([Ar]4s¹3d⁵) and copper ([Ar]4s¹3d¹⁰) are notable exceptions, favoring a half-filled or fully-filled 3d subshell over the standard predicted configuration.
Why Bohr's Model Wasn't Enough
Bohr's model accurately describes the electronic structure of hydrogen, which has a single electron orbiting the nucleus. But for any atom with more than one electron, the model fails — it can't account for the electron-electron interactions and more complex angular momentum considerations that multi-electron atoms introduce.
Bohr's model also has a deeper problem: its quantized, fixed orbits directly contradict the Heisenberg uncertainty principle, a fundamental result of quantum mechanics developed later. These limitations were ultimately resolved by the wave-mechanical model of the atom, developed by Erwin Schrödinger and Werner Heisenberg among others, which superseded Bohr's model by treating electrons as wave-like entities with probabilistic distributions around the nucleus, rather than particles in fixed orbits.
The Heisenberg Uncertainty Principle and Orbitals vs. Orbits
The Heisenberg uncertainty principle states that it is impossible to simultaneously determine, with perfect accuracy, both the position and the momentum of an electron. Pinning down the electron's position requires it to stop (removing momentum information); pinning down its momentum requires it to be moving (changing its position). Because of this, it's impossible to say exactly where an electron is at any given moment.
This is why the modern model replaces Bohr's fixed orbits with orbitals: a region of space around the nucleus defined by the probability of finding an electron there. Electrons move rapidly and are localized within these regions rather than following a clearly defined circular path at a fixed distance from the nucleus.
Orbitals, not orbits — that distinction is the single most important difference between Bohr's model and the modern quantum mechanical model.
The Four Quantum Numbers
To describe the distribution of electrons in an atom, quantum numbers were developed. Modern atomic theory holds that any electron in an atom can be completely described by four quantum numbers, each capturing a different property of the orbital it occupies:
Quantum Number | Symbol | Describes | Possible Values |
|---|---|---|---|
Principal | n | Energy level and size of the orbital | Any positive integer (1, 2, 3, ...) |
Angular momentum (azimuthal) | l | Shape of the orbital (subshell) | Integers from 0 to (n − 1) |
Magnetic | ml | Orientation of the orbital in space | Integers from −l to +l, including 0 |
Spin | ms | Spin of the electron | +1/2 or −1/2 |
The principal quantum number (n) defines the orbital's energy level and size — the larger n, the higher the energy and the larger the orbital. Each shell can hold a maximum number of electrons given by 2n².
The angular momentum quantum number (l) defines the orbital's shape and can range from 0 to (n − 1). Each value corresponds to a subshell: l = 0 is the s subshell, l = 1 is p, l = 2 is d, and l = 3 is f.
The magnetic quantum number (ml) defines the orbital's orientation in space, taking integer values from −l to +l (including 0). The number of orbitals in a subshell equals 2l + 1.
The spin quantum number (ms) has only two possible values, +1/2 or −1/2, reflecting the two possible orientations of an electron's spin. Two electrons in the same orbital must have opposite spins; electrons in different orbitals with the same ms value are said to have parallel spins.
Worked Examples: n = 1, 2, 3
n | Possible l values | Subshells | Possible ml values | Orbitals (orientations) |
|---|---|---|---|---|
1 | 0 | s | 0 | 1 (spherical) |
2 | 0, 1 | s, p | 0 (s); −1, 0, 1 (p) | 1 (s) + 3 (p: px, py, pz) |
3 | 0, 1, 2 | s, p, d | 0 (s); −1, 0, 1 (p); −2, −1, 0, 1, 2 (d) | 1 (s) + 3 (p) + 5 (d) |
For n = 1, the only possible subshell is s (l = 0), which has a single orientation (ml = 0) — a sphere. For n = 2, l can be 0 or 1, adding the p subshell, whose three ml values (−1, 0, 1) correspond to the three p orbitals: px, py, and pz. For n = 3, l can also be 2, adding the d subshell, whose five ml values (−2 through +2) correspond to five d orbitals.
Shells, Subshells, and Orbitals
These three terms describe a nested hierarchy:
Shells are the primary energy levels of an atom, defined by the principal quantum number n. The n = 1 shell is closest to the nucleus and lowest in energy; the n = 2 shell is farther out and higher in energy. Each shell can hold more electrons as n increases.
Subshells are categorized by the angular momentum quantum number l and specify the shape of the region where electrons are likely to be found. The n = 1 shell contains only an s subshell; the n = 2 shell contains both s and p subshells.
Orbitals are the individual regions within a subshell where electrons are most likely to be found, each holding up to two electrons with opposite spins. The 1s orbital is spherical, meaning there's an equal probability of finding an electron at any point around the nucleus at that distance. The 2p orbitals are dumbbell-shaped and oriented along the x, y, and z axes (2px, 2py, 2pz), meaning electrons at that energy level have a higher probability of being found along certain directions relative to the nucleus.
Aufbau's Principle, the Pauli Exclusion Principle, and Hund's Rule
Three rules govern how electrons actually fill orbitals:
Principle | Rule |
|---|---|
Aufbau's principle | The lowest-energy orbital is filled first. |
Pauli exclusion principle | Each orbital holds a maximum of 2 electrons, and those electrons must have opposite spins. |
Hund's rule | One electron is placed in each degenerate orbital (orbitals of equal energy within a subshell) before any orbital receives a second, paired electron. |
Writing Electron Configurations
An atom or ion's electron configuration describes the pattern in which its subshells are filled and how many electrons occupy each principal energy level and subshell. Electron configurations are written in spectroscopic notation: the first number gives the principal energy level, the letter gives the subshell, and the superscript gives the number of electrons in that subshell (for example, 2p⁴ means 4 electrons in the 2p subshell).
To write an electron configuration:
Start at hydrogen and add electrons one at a time, following the subshell filling order.
Apply Aufbau's principle to determine the order: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, and so on — electrons fill from lower- to higher-energy subshells, and each subshell fills completely before the next begins.
Apply the Pauli exclusion principle so that no orbital receives more than two electrons, and any two electrons sharing an orbital have opposite spins.
Apply Hund's rule so that every orbital in a subshell gets one electron before any orbital gets a second.
A useful shortcut: the periodic table itself encodes this filling order. The lowest s subshell is 1s, the lowest p subshell is 2p, the lowest d subshell is 3d, and the lowest f subshell is 4f.
Exceptions to the Aufbau Principle: Chromium and Copper
Some transition metals deviate from the predicted Aufbau filling order because a half-filled or fully-filled d subshell is more stable than the "expected" configuration, due to reduced electron-electron repulsion and increased exchange energy.
Chromium (atomic number 24) would be expected to have the configuration [Ar]4s²3d⁴. Instead, it adopts [Ar]4s¹3d⁵, promoting one electron from 4s to 3d to achieve a half-filled, more stable 3d subshell.
Copper (atomic number 29) would be expected to have the configuration [Ar]4s²3d⁹. Instead, it adopts [Ar]4s¹3d¹⁰, promoting one electron from 4s to 3d to achieve a fully-filled, more stable 3d subshell.
Common MCAT Mistakes
Mixing up which quantum number describes what. n gives energy level/size, l gives shape (subshell), ml gives orientation, and ms gives spin — each answers a different question about the electron.
Violating Hund's rule when filling degenerate orbitals. Electrons fill each orbital in a subshell singly, with parallel spins, before any orbital gets a second, paired electron — pairing electrons too early is a common error.
Forgetting the chromium and copper exceptions. Both deviate from the standard Aufbau order ([Ar]4s¹3d⁵ and [Ar]4s¹3d¹⁰, not the "expected" 4s²3d⁴ and 4s²3d⁹) because a half-filled or fully-filled d subshell is more stable.
Treating shells, subshells, and orbitals as interchangeable. They're a nested hierarchy — shells (n) contain subshells (l), which contain individual orbitals (ml), each holding up to two electrons.
MCAT-Style Concept Check
Question: What is the correct ground-state electron configuration of chromium (atomic number 24)?
A) [Ar]4s²3d⁴
B) [Ar]4s¹3d⁵
C) [Ar]4s²3d⁵
D) [Ar]3d⁶
Answer: B
Explanation: Chromium is an exception to the standard Aufbau filling order. Rather than the "expected" [Ar]4s²3d⁴, one electron shifts from 4s to 3d to give a half-filled 3d subshell, [Ar]4s¹3d⁵, which is more stable due to reduced electron-electron repulsion and increased exchange energy.
FAQ
What is the difference between an orbital and an orbit?
An orbit, from Bohr's model, is a fixed circular path an electron follows at a set distance from the nucleus. An orbital, from the quantum mechanical model, is a probability region describing where an electron is likely to be found — it doesn't specify an exact path, only a likely location.
What are the four quantum numbers?
The principal quantum number (n) gives energy level and size, the angular momentum quantum number (l) gives orbital shape (subshell), the magnetic quantum number (ml) gives orbital orientation, and the spin quantum number (ms) gives the electron's spin (+1/2 or −1/2).
What is Hund's rule?
Hund's rule states that electrons fill every orbital in a subshell singly, with parallel spins, before any orbital in that subshell receives a second, paired electron. It minimizes electron-electron repulsion within a subshell.
Why do chromium and copper deviate from the expected electron configuration?
Both achieve extra stability from a half-filled (chromium, 3d⁵) or fully-filled (copper, 3d¹⁰) d subshell, which lowers electron-electron repulsion and increases exchange energy enough to favor promoting one electron from 4s to 3d over the standard Aufbau-predicted configuration.
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