Synthesis: Quantum Mechanics Explains the Table
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You should be comfortable with the four quantum numbers (\(n, l, m_l, m_s\)) and understand that atomic orbitals are defined by these values. You should also recall the basic layout of the periodic table, distinguishing between periods (rows) and groups (columns).
Why this matters
The periodic table is not merely a sorting device; it is a direct map of quantum mechanical constraints.
In the nineteenth century, chemists arranged elements by mass and observed recurring patterns in reactivity, yet they lacked a fundamental explanation for why these patterns existed. The development of quantum mechanics provided the missing link: the structure of the atom, specifically how electrons occupy discrete energy levels and orbitals, dictates the chemical behaviour of every element. By understanding the rules that govern electron configuration, we can predict atomic size, ionisation energy, and bonding tendencies without relying solely on empirical observation. This lecture synthesises the quantum model with the periodic trends you have previously studied, demonstrating that the shape of the periodic table is a consequence of the Pauli exclusion principle and the energy ordering of atomic orbitals.
The Quantum Logic of the Table
Imagine the periodic table as a filling station for electrons. Each element represents a specific number of electrons that must be placed into the available "rooms" (orbitals) of the atom. The shape of the table—its rows and columns—is determined by how these rooms are organised and filled. The rows, or periods, correspond to the filling of a new principal energy level (\(n\)). The columns, or groups, correspond to elements that have the same number of valence electrons in their outermost shell. For example, all elements in Group 1 have one electron in their outermost \(s\) orbital, while Group 2 elements have two. This repetition of valence electron configurations is what gives elements in the same group similar chemical properties. The table is essentially a visual representation of the ground-state electron configurations of the elements, arranged in order of increasing nuclear charge.
Building Up the Atom
To predict the electron configuration of any element, we apply the building-up principle. This principle states that we fill atomic orbitals one electron at a time, starting from the lowest energy level and moving upwards. The order in which orbitals are filled is not simply by increasing principal quantum number \(n\); instead, it follows the Madelung rule (also known as the \(n+l\) rule). According to this rule, orbitals are filled in order of increasing \(n+l\). If two orbitals have the same \(n+l\) value, the one with the lower \(n\) value is filled first. This explains why the \(4s\) orbital is filled before the \(3d\) orbital, even though \(3d\) has a lower principal quantum number. The Pauli exclusion principle limits each orbital to a maximum of two electrons, which must have opposite spins. Finally, Hund's rule dictates that for degenerate orbitals (such as the three \(p\) orbitals), electrons occupy separate orbitals with parallel spins before pairing up. These three rules together determine the unique electron configuration of every element in the ground state.
Effective Nuclear Charge and Shielding
While the total nuclear charge \(Z\) increases across a period, the valence electrons do not feel the full force of the nucleus. This is because inner-shell electrons partially shield the valence electrons from the nuclear attraction. The net attraction felt by a valence electron is described by the effective nuclear charge, \(Z_{\text{eff}}\), defined as \(Z_{\text{eff}} = Z - S\), where \(S\) is the shielding constant. Slater's rules provide a method to calculate \(S\). For an electron in an \(ns\) or \(np\) orbital, the shielding contribution from other electrons in the same group is \(0.35\) each, from electrons in the \((n-1)\) shell is \(0.85\) each, and from electrons in shells \((n-2)\) and lower is \(1.00\) each. As we move from left to right across a period, the number of protons in the nucleus increases, but the shielding from inner electrons remains relatively constant because the valence electrons are added to the same shell. Consequently, \(Z_{\text{eff}}\) increases, pulling the valence electrons closer to the nucleus and reducing the atomic radius. This increase in effective nuclear charge is the primary driver of the periodic trends in atomic size and ionisation energy.
Connecting Configuration to Trends
The periodic trends in atomic radius and first ionisation energy are direct consequences of the changes in effective nuclear charge and the principal quantum number of the valence electrons. Moving left to right across a period, \(Z_{\text{eff}}\) increases while the principal quantum number \(n\) of the valence shell remains constant. This stronger attraction results in a smaller atomic radius and a higher first ionisation energy, as more energy is required to remove an electron from a more tightly bound system. Conversely, moving down a group, the principal quantum number \(n\) of the valence shell increases. Although \(Z\) increases significantly, the addition of new inner shells provides substantial shielding, causing \(Z_{\text{eff}}\) to remain roughly constant. The valence electrons are therefore further from the nucleus and experience less attraction, leading to larger atomic radii and lower first ionisation energies. This synthesis of quantum mechanical principles explains why the periodic table has its characteristic shape and why elements exhibit periodic chemical properties.
The results
The building-up principle
The ground-state electron configuration of an atom is obtained by filling the atomic orbitals of the previous result one electron at a time, subject to three rules: (1) the Madelung rule: orbitals fill in order of increasing \(n+l\), and for orbitals sharing the same \(n+l\), in order of increasing \(n\); (2) the Pauli exclusion principle: no orbital holds more than two electrons, and those two must have opposite spin (\(m_s=+\tfrac12\) and \(m_s=-\tfrac12\)); (3) Hund's rule: within a set of degenerate orbitals (equal \(n\) and \(l\), differing only in \(m_l\)), electrons occupy separate orbitals with parallel spin before any orbital is doubly occupied.
Periodic trends
Define the effective nuclear charge felt by an electron as \(Z_{\text{eff}}=Z-S\), where \(S\) (the shielding constant) is computed from Slater's rules: for an electron in an \(ns\) or \(np\) orbital, sum \(0.35\) for every other electron in the same \(n s,np\) group, \(0.85\) for every electron in the \(n{-}1\) shell, and \(1.00\) for every electron in shells \(n{-}2\) and below. Then, moving left to right across a period, \(Z_{\text{eff}}\) for the valence electron increases (atomic radius decreases, first ionisation energy increases); moving down a group, \(Z_{\text{eff}}\) stays roughly constant while the valence principal quantum number \(n\) increases (atomic radius increases, first ionisation energy decreases).
A common misconception is that the \(3d\) orbitals are always higher in energy than the \(4s\) orbitals. In reality, the energy ordering of orbitals is not fixed; it depends on the nuclear charge. For neutral atoms, \(4s\) is lower than \(3d\), but once electrons are added to the \(3d\) subshell, the \(3d\) orbitals become lower in energy than \(4s\). This is why transition metals lose their \(4s\) electrons before their \(3d\) electrons when forming cations. Always consider the specific context of the atom or ion when discussing orbital energies. C
Worked example
Problem. Use Slater's rules to find \(Z_{\text{eff}}\) for the valence electron of lithium (\(Z = 3\)), sodium (\(Z = 11\)) and potassium (\(Z = 19\)), and use the result to explain why atoms grow down Group 1.
- Li, \(1s^2\,2s^1\): no other \(n = 2\) electrons; two \(1s\) electrons in shell \(n-1\): \(S = 2 \times 0.85 = 1.70\), so \(Z_{\text{eff}} = 3 - 1.70 = 1.30\).
- Na, \([\mathrm{Ne}]\,3s^1\): eight \(n = 2\) electrons at 0.85 and two \(1s\) at 1.00: \(S = 6.80 + 2.00 = 8.80\), so \(Z_{\text{eff}} = 11 - 8.80 = 2.20\).
- K, \([\mathrm{Ar}]\,4s^1\): eight \(n = 3\) electrons at 0.85 and ten inner electrons at 1.00: \(S = 6.80 + 10.00 = 16.80\), so \(Z_{\text{eff}} = 19 - 16.80 = 2.20\).
- \(Z\) rises by 16 from Li to K, but \(Z_{\text{eff}}\) changes by less than 1, while the valence shell goes from \(n = 2\) to \(n = 4\).
Units check. \(Z_{\text{eff}}\) is a charge in units of \(e\), so it is dimensionless. With the pull nearly unchanged and the valence shell further out at each step, the radius grows and \(I_1\) falls down the group.
Three quick tests
- According to the Madelung rule, which orbital is filled first: \(3d\) or \(4s\)?
Answer
\(4s\), because \(n+l\) for \(4s\) is \(4+0=4\), while for \(3d\) it is \(3+2=5\). - Why does the atomic radius increase down a group despite the increase in nuclear charge?
Answer
The principal quantum number \(n\) increases, placing valence electrons further from the nucleus, and increased shielding from inner shells offsets the increased nuclear charge. - What is the maximum number of electrons that can occupy a single atomic orbital?
Answer
Two, with opposite spins, as per the Pauli exclusion principle.
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Every lecture and every result in this unit, with prerequisites and threads.
Lecture notes drafted with a local model and checked line by line by the editor; every number in the worked example was recomputed.