40-1 Properties of Atoms#

Prompts

What are the four basic properties of atoms? How does quantum physics explain stability and bonding?
Explain ionization energy and its role in the periodic table. Why do the six periods have 2, 8, 8, 18, 18, and 32 elements?
For an atomic transition, how are photon energy, frequency, and wavelength related to the energy levels? When does an atom emit vs. absorb light?
Describe the Einstein–de Haas experiment. What does it demonstrate about the relationship between angular momentum and magnetism?
What are the four quantum numbers for an electron in an atom? How do orbital angular momentum and spin differ? Why does the spin magnetic moment have a factor of 2 compared to the orbital case?

Lecture Notes#

Overview#

Atoms have four basic properties: stability, bonding, systematic structure, and discrete light emission/absorption.
Ionization energy (energy to remove the most loosely bound electron) varies periodically with atomic number \(Z\), reflecting the structure of the periodic table.
Each electron has orbital angular momentum \(\vec{L}\) and spin angular momentum \(\vec{S}\), each with an associated magnetic dipole moment; these are coupled (opposite directions for electrons).
Quantum numbers \(n\), \(\ell\), \(m_\ell\), and \(m_s\) specify electron states; shells (same \(n\)) and subshells (same \(n\) and \(\ell\)) organize the allowed states.

Some Properties of Atoms#

Four basic properties of atoms:

Stability — Atoms persist essentially unchanged for billions of years.
Bonding — Atoms combine to form molecules and solids.
Systematic structure — Properties repeat periodically (e.g., ionization energy vs. \(Z\)).
Discrete light emission/absorption — Atoms jump between quantum states by emitting or absorbing photons.

Atomic number

The atomic number \(Z\) is the number of protons in the nucleus. For a neutral atom, \(Z\) is also the number of electrons.

Ionization Energy and the Periodic Table#

The ionization energy is the energy required to remove the most loosely bound electron from a neutral atom. A plot of ionization energy vs. atomic number \(Z\) shows periodic repetition:

Each period starts with a reactive alkali metal (Li, Na, K, …) and ends with an inert noble gas (Ne, Ar, Kr, …).
The numbers of elements in the six complete periods are

\[ 2,\quad 8,\quad 8,\quad 18,\quad 18,\quad 32. \]

Quantum physics predicts these numbers from the allowed electron states in shells and subshells.

Light Emission and Absorption#

An atom can make a transition between quantum states by emitting or absorbing light. The photon energy equals the energy difference between the levels:

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

Emission: atom jumps from higher to lower level; photon carries away the energy.
Absorption: atom jumps from lower to higher level; photon supplies the energy.

Finding the frequencies of atomic spectra reduces to finding the energies of the quantum states.

Angular Momentum and Magnetism#

In quantum physics, each electron state involves an orbital angular momentum \(\vec{L}\) and an orbital magnetic dipole moment \(\vec{\mu}_{\text{orb}}\) that are coupled — they point in opposite directions (because the electron is negatively charged).

Einstein–de Haas experiment

In 1915, Einstein and de Haas suspended an iron cylinder in a solenoid. When a magnetic field was turned on, the atomic magnetic moments aligned with the field. Because angular momentum is coupled to magnetic moment, the atomic angular momenta aligned as well. To conserve total angular momentum (initially zero), the cylinder began to rotate. This demonstrated that magnetization is due to electron angular momentum.

Orbital Angular Momentum#

The magnitude of the orbital angular momentum is quantized:

(402)#\[ L = \sqrt{\ell(\ell+1)}\,\hbar,\quad \ell = 0, 1, 2, \ldots, (n-1) \]

The z component (along a chosen measurement axis) is

(403)#\[ L_z = m_\ell \hbar,\quad m_\ell = 0, \pm 1, \pm 2, \ldots, \pm \ell \]

\(\vec{L}\) does not have a definite direction; only \(L_z\) can be measured.
\(\ell\) is the orbital quantum number; \(m_\ell\) is the orbital magnetic quantum number.

Orbital Magnetic Dipole Moment#

The orbital magnetic moment is related to orbital angular momentum:

(404)#\[ \vec{\mu}_{\text{orb}} = -\frac{e}{2m}\vec{L} \]

The minus sign reflects the electron’s negative charge. Quantized magnitude and z component:

(405)#\[ \mu_{\text{orb}} = \frac{e}{2m}\sqrt{\ell(\ell+1)}\,\hbar,\qquad \mu_{\text{orb},z} = -m_\ell \mu_B \]

where the Bohr magneton is

(406)#\[ \mu_B = \frac{e\hbar}{2m} = \frac{eh}{4\pi m} \approx 9.27 \times 10^{-24}\ \text{J/T} \]

Spin Angular Momentum#

Every electron has an intrinsic spin angular momentum \(\vec{S}\) with no classical counterpart (it is not \(\vec{r}\times\vec{p}\)). The magnitude is

(407)#\[ S = \sqrt{s(s+1)}\,\hbar,\quad s = \tfrac{1}{2} \]

The electron is a spin-½ particle. The z component is

(408)#\[ S_z = m_s \hbar,\quad m_s = +\tfrac{1}{2}\ \text{(spin up) or}\ -\tfrac{1}{2}\ \text{(spin down)} \]

Spin Magnetic Dipole Moment#

The spin magnetic moment is

(409)#\[ \vec{\mu}_s = -\frac{e}{m}\vec{S} \]

Magnitude and z component:

(410)#\[ \mu_s = \frac{e}{m}\sqrt{s(s+1)}\,\hbar,\qquad \mu_{s,z} = -2m_s \mu_B \]

Factor of 2

The spin magnetic moment has a factor of 2 compared to the orbital case: \(\mu_{s,z} = -2m_s\mu_B\) vs. \(\mu_{\text{orb},z} = -m_\ell\mu_B\). This is a relativistic quantum effect (Dirac equation).

Quantum Numbers Summary#

Quantum number	Symbol	Allowed values	Related to
Principal	\(n\)	1, 2, 3, …	Distance from nucleus
Orbital	\(\ell\)	0, 1, …, \(n-1\)	Orbital angular momentum
Orbital magnetic	\(m_\ell\)	\(0, \pm 1, \ldots, \pm \ell\)	\(L_z\)
Spin	\(s\)	\(\tfrac{1}{2}\) (always)	Spin angular momentum
Spin magnetic	\(m_s\)	\(\pm\tfrac{1}{2}\)	\(S_z\)

Shells and Subshells#

Shell: all states with the same \(n\).
Subshell: all states with the same \(n\) and \(\ell\).

For a given \(\ell\), there are \(2\ell+1\) values of \(m_\ell\) and 2 values of \(m_s\) (spin up/down). Thus:

(411)#\[ \text{States per subshell} = 2(2\ell+1) \]

Summing over \(\ell = 0, 1, \ldots, n-1\):

(412)#\[ \text{States per shell} = 2n^2 \]

Multielectron Atoms#

For an atom with many electrons, the total angular momentum \(\vec{J}\) is the vector sum of all orbital and spin angular momenta. The effective magnetic dipole moment \(\vec{\mu}_{\text{eff}}\) is the component of the total magnetic moment in the direction of \(\vec{J}\) (because of the spin factor of 2, the total moment is not parallel to \(\vec{J}\)).

In typical atoms, most orbital and spin contributions cancel; the net \(\vec{J}\) and \(\vec{\mu}_{\text{eff}}\) often come from a small number of electrons, e.g., a single valence electron.

Summary#

Atoms are stable, bond systematically, and emit/absorb light at discrete frequencies given by \(hf = E_{\text{high}} - E_{\text{low}}\).
Ionization energy varies periodically with \(Z\); periods have 2, 8, 8, 18, 18, 32 elements.
Orbital angular momentum \(\vec{L}\) and orbital magnetic moment \(\vec{\mu}_{\text{orb}}\) are coupled; spin \(\vec{S}\) and spin magnetic moment \(\vec{\mu}_s\) are coupled.
Quantum numbers \(n\), \(\ell\), \(m_\ell\), \(m_s\) specify electron states; shells (same \(n\)) and subshells (same \(n\), \(\ell\)) organize them.
The Einstein–de Haas experiment demonstrated that magnetization arises from electron angular momentum.