40-5 Building the Periodic Table

40-5 Building the Periodic Table#

Prompts

  • What is a shell? A subshell? How many states are in a subshell with quantum number \(\ell\)? In a shell with quantum number \(n\)?

  • Explain the subshell labeling (1s, 2p, 3d, …). Why does a closed subshell have zero net angular momentum and zero magnetic moment?

  • Describe the procedure for building the electron configuration of an atom: add electrons one by one, fill lowest-energy subshells first, obey the Pauli exclusion principle.

  • Compare neon, sodium, and chlorine: Why is neon inert? Why are sodium and chlorine chemically active? What role does the valence electron or the “hole” play?

  • Iron has configuration \(1s^2 2s^2 2p^6 3s^2 3p^6 3d^6 4s^2\). Why do the last two electrons go into 4s rather than 3d? What does this say about filling order?

Lecture Notes#

Overview#

  • The periodic table lists elements in order of increasing atomic number \(Z\); each neutral atom has \(Z\) electrons.

  • Electrons occupy shells (same \(n\)) and subshells (same \(n\) and \(\ell\)); a closed subshell holds the maximum allowed electrons and has zero net angular momentum and magnetic moment.

  • Atoms are built by adding electrons one by one into the lowest available subshells, obeying the Pauli exclusion principle.

  • Noble gases have closed shells and are inert; alkali metals and halogens have one valence electron or one vacancy and are chemically active.

  • Subshell energies depend primarily on \(n\) but also on \(\ell\); the filling order is not always the “logical” \(n\)-then-\(\ell\) sequence (e.g., 4s before 3d).

Shells and Subshells#

  • Shell: all states with the same principal quantum number \(n\).

  • Subshell: all states with the same \(n\) and orbital quantum number \(\ell\).

Subshells are labeled by \(n\) and a letter for \(\ell\):

\(\ell\)

0

1

2

3

4

5

…

Letter

s

p

d

f

g

h

…

Examples: \(1s\) (\(n=1\), \(\ell=0\)), \(2p\) (\(n=2\), \(\ell=1\)), \(3d\) (\(n=3\), \(\ell=2\)).

States per subshell: \(2(2\ell+1)\) (from \(m_\ell = -\ell,\ldots,+\ell\) and \(m_s = \pm\tfrac{1}{2}\)).

States per shell: \(2n^2\) (sum over \(\ell = 0, 1, \ldots, n-1\)).

Subshell energy depends primarily on \(n\), but also on \(\ell\); higher \(\ell\) is generally higher in energy for a given \(n\).

Closed Subshells#

A closed subshell contains the maximum number of electrons allowed by the Pauli principle.

  • All \(L_z\) and \(S_z\) contributions cancel (for every \(+m_\ell\) there is a \(-m_\ell\); spins pair).

  • Net angular momentum and net magnetic moment are zero.

  • The probability density is spherically symmetric.

Why closed structures matter

Atoms with only closed subshells (noble gases) have no “loosely dangling” electrons to drive chemical bonding. They are nearly chemically inert.

Building the Periodic Table#

Procedure: Add electrons one by one to the atom. Each electron goes into the lowest-energy subshell that still has an available state, obeying the Pauli exclusion principle.

In multielectron atoms, the potential seen by an electron includes the nuclear attraction and the repulsion from other electrons. Wave functions differ from hydrogen; energies are computed numerically. The filling order of subshells is determined by their energies, which do not always follow the simple \(1s \to 2s \to 2p \to 3s \to \cdots\) sequence.

Example: Neon (Z = 10)#

Configuration: \(1s^2\,2s^2\,2p^6\)

  • \(1s^2\): 2 electrons (closed)

  • \(2s^2\): 2 electrons (closed)

  • \(2p^6\): 6 electrons (closed)

All three subshells are closed. Neon has zero net angular momentum, zero magnetic moment, and is chemically inert — a noble gas.

Example: Sodium (Z = 11)#

Configuration: \(1s^2\,2s^2\,2p^6\,3s^1\)

  • The first 10 electrons form a closed neon-like core (zero angular momentum).

  • The 11th electron is in the \(3s\) subshell — the next lowest in energy.

This valence electron is loosely bound and largely outside the core. Sodium’s angular momentum and magnetic moment come from this single electron. Sodium readily donates its valence electron to form bonds — an alkali metal.

Example: Chlorine (Z = 17)#

Configuration: \(1s^2\,2s^2\,2p^6\,3s^2\,3p^5\)

  • Closed core: \(1s^2\,2s^2\,2p^6\,3s^2\) (10 electrons).

  • The \(3p\) subshell (\(\ell=1\)) holds 6 states; 5 are filled, 1 is a vacancy (hole).

Chlorine tends to accept one electron to fill that hole. It forms stable compounds (e.g., NaCl) — a halogen.

Example: Iron (Z = 26)#

Configuration: \(1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,3d^6\,4s^2\)

  • The \(3d\) subshell can hold 10 electrons; only 6 are present.

  • The last two electrons go into 4s rather than 3d.

Filling order

The \(4s\) subshell is lower in energy than \(3d\) for this range of \(Z\). The configuration \(3d^6\,4s^2\) has lower total energy than \(3d^8\). Except for the simplest elements, subshells are not always filled in the order of increasing \(n\) alone.

Noble Gases vs. Alkali Metals vs. Halogens#

Noble gases

Alkali metals

Halogens

Valence

Closed subshells

One extra electron

One vacancy

Net angular momentum

Zero

From valence electron

From hole

Ionization energy

High

Low

High (to remove electron)

Chemical behavior

Inert

Reactive (donate electron)

Reactive (accept electron)

Light Emission and Absorption#

For a transition between atomic energy levels (Section 40-1):

(422)#\[ hf = E_{\text{high}} - E_{\text{low}} \]

The photon frequency (or wavelength) is determined by the energy difference between the two levels.

Summary#

  • Shells (same \(n\)) and subshells (same \(n\), \(\ell\)) organize electron states; subshells hold \(2(2\ell+1)\) electrons.

  • Closed subshells have zero net angular momentum and magnetic moment; noble gases are inert.

  • Atoms are built by filling lowest-energy subshells first, obeying the Pauli principle.

  • Alkali metals have one valence electron; halogens have one vacancy; both are chemically active.

  • Filling order can differ from simple \(n\)-then-\(\ell\) (e.g., 4s before 3d).