Atomic Structure & Periodic Trends

Atomic Structure & Periodic Trends

Part 1 of 5 — Quantum Numbers & Electron Configuration

Atomic structure is tested in virtually every MCAT general chemistry passage. Questions focus on quantum number rules, electron configurations, and predicting properties from position on the periodic table.

The Four Quantum Numbers

Every electron in an atom is described by a unique set of four quantum numbers:

Subshell labels: $\ell = 0 \to s$, $\ell = 1 \to p$, $\ell = 2 \to d$, $\ell = 3 \to f$

Number of orbitals in a subshell: $2\ell + 1$

• $s$: 1 orbital; $p$: 3 orbitals; $d$: 5 orbitals; $f$: 7 orbitals

Three Rules for Filling Orbitals

1. Aufbau Principle: Fill lowest-energy orbitals first.
   Order: $1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, \ldots$

2. Pauli Exclusion Principle: No two electrons in the same atom can have identical quantum numbers. Each orbital holds at most 2 electrons with opposite spins.

3. Hund's Rule: Within a subshell, place one electron in each orbital before pairing. All singly-occupied orbitals have the same spin (maximizes multiplicity).

Writing Electron Configurations — Examples

Carbon ($Z = 6$): $1s^2\, 2s^2\, 2p^2$

Iron ($Z = 26$): $[\text{Ar}]\, 3d^6\, 4s^2$
Noble-gas shorthand: write the preceding noble gas in brackets, then continue.

Fe²⁺: Remove electrons from the outermost shell first (4s before 3d).
$\text{Fe}^{2+}: [\text{Ar}]\, 3d^6$

Fe³⁺: $[\text{Ar}]\, 3d^5$
Note: $3d^5$ is a half-filled subshell — extra stability.

Exceptions to Aufbau

Two common MCAT exceptions:

• Chromium ($Z=24$): Expected $[\text{Ar}]\, 3d^4\, 4s^2$; Actual $[\text{Ar}]\, 3d^5\, 4s^1$ — half-filled $3d$ is stabilized.
• Copper ($Z=29$): Expected $[\text{Ar}]\, 3d^9\, 4s^2$; Actual $[\text{Ar}]\, 3d^{10}\, 4s^1$ — fully-filled $3d$ is stabilized.

Quantum Numbers & Configuration — Check Your Understanding 🎯

Worked Example: Identifying an Element from Configuration

A neutral atom has the electron configuration $[\text{Ne}]\, 3s^2\, 3p^4$.

Step 1 — Count total electrons:
Ne has 10 electrons. Add $3s^2\, 3p^4 = 6$ more → 16 total electrons.

Step 2 — Identify the element:
$Z=16$ is sulfur (S).

Step 3 — Determine unpaired electrons:
The $3p^4$ configuration: by Hund's rule, fill each of the 3 p-orbitals with one electron first → $3p$: ↑↓ ↑ ↑ → 2 unpaired electrons.

Step 4 — Common ion:
S typically gains 2 electrons to form $\text{S}^{2-}$, achieving the $[\text{Ar}]$ configuration ($3p^6$).

MCAT Trap: $4s$ vs. $3d$ Removal Order

Students often think $3d$ electrons (lower $n$) are removed before $4s$ when forming cations. This is wrong. The outermost shell electrons are always removed first (highest $n$). Once the $4s$ electrons are gone, the $3d$ orbitals become lower in energy than $4s$ due to reduced shielding.

Configuration Application Questions 🎯

Key Takeaways — Part 1

• $\ell$ ranges from $0$ to $n-1$; $m_\ell$ ranges from $-\ell$ to $+\ell$. Impossible combos are a common MCAT trap.
• Aufbau order: $1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p \ldots$ Note $4s$ fills before $3d$.
• Hund's rule: maximize unpaired electrons within a subshell.
• Cation formation: remove outermost-shell (highest-$n$) electrons first, always.
• Cr ($3d^5\, 4s^1$) and Cu ($3d^{10}\, 4s^1$): exceptions due to stability of half-filled and fully-filled $d$ subshells.
• Isoelectronic species share the same electron configuration; differ in nuclear charge.

Atomic Structure & Periodic Trends

Part 2 of 5 — Periodic Trends: Atomic Radius, Ionization Energy & Electronegativity

Periodic trends follow directly from effective nuclear charge ($Z_{eff}$) — the net positive charge an outer electron "feels" after inner electrons partially shield it.

$Z_{eff} = Z - \sigma$

where $Z$ = atomic number, $\sigma$ = shielding constant (approximately equal to the number of inner-shell electrons).

Atomic Radius

Trend: ↑ down a group, ↓ across a period (left to right).

Cation vs. anion radius:

• Cations are smaller than the neutral atom (fewer electrons, same or similar $Z_{eff}$, electrons pulled closer).
• Anions are larger than the neutral atom (more electrons → more electron-electron repulsion, expanded cloud).

Isoelectronic series (all 10 electrons): $\text{N}^{3-} > \text{O}^{2-} > \text{F}^- > \text{Ne} > \text{Na}^+ > \text{Mg}^{2+} > \text{Al}^{3+}$
As $Z$ increases across the series, same electron count is pulled harder → smaller radius.

Ionization Energy (IE)

Energy required to remove one electron from a gaseous atom: $\text{X}(g) \to \text{X}^+(g) + e^-$

Trend: ↓ down a group, ↑ across a period — opposite of atomic radius.

Successive ionization energies: Each successive IE is larger (removing from a more positive ion). A large jump between $\text{IE}_n$ and $\text{IE}_{n+1}$ means you crossed into a core shell.

Example — Sodium ($Z=11$, config $[\text{Ne}]\, 3s^1$):

• $\text{IE}_1 \approx 496$ kJ/mol (remove $3s^1$, easy)
• $\text{IE}_2 \approx 4{,}562$ kJ/mol (must break into the filled Ne core → huge jump)

This is how the MCAT can ask you to identify an element's group from a table of successive IEs.

Exceptions to the across-period IE trend:

• Group IIA → Group IIIA: $\text{IE}$ drops slightly (e.g., $\text{Mg} > \text{Al}$) because Al removes a $3p$ electron, which is shielded by the $3s$ pair.
• Group VA → Group VIA: $\text{IE}$ drops slightly (e.g., $\text{P} > \text{S}$) because S has a paired $3p$ electron with extra repulsion, making it easier to remove.

Electronegativity

Tendency to attract bonding electrons. Measured by the Pauling scale.

Trend: Increases across a period, decreases down a group.
Most electronegative: F (3.98)
Least electronegative (among metals): Cs/Fr

Electronegativity difference predicts bond type:

• $\Delta EN < 0.5$: nonpolar covalent
• $0.5 \leq \Delta EN < 1.7$: polar covalent
• $\Delta EN \geq 1.7$: ionic character

Electron Affinity (EA)

Energy change when a gaseous atom gains one electron: $\text{X}(g) + e^- \to \text{X}^-(g)$

Generally becomes more negative (more exothermic) across a period and up a group.

Exceptions:

• Group IIA (filled $s$ subshell) and Group VA (half-filled $p$ subshell) have near-zero or positive EA — extra stability makes electron gain unfavorable.
• Noble gases have positive EA (no tendency to gain electrons).

Periodic Trends — Check Your Understanding 🎯

Trend Summary Table

*With minor exceptions as discussed above.

Worked Example: Predicting Properties

Question: Without looking at a table, which has a higher first ionization energy — $\text{O}$ or $\text{F}$?

Reasoning: Both are in Period 2. F is one position to the right of O, so it has a higher $Z_{eff}$. However, O ($3p^4$ analogy: $2p^4$, paired electron) vs F ($2p^5$) — F has no extra pairing repulsion disadvantage in removing one $2p$ electron. F has higher $Z_{eff}$ AND no extra repulsion penalty → F has higher IE than O. ✓ (confirmed: $\text{IE}_1(\text{O}) = 1{,}314$ kJ/mol; $\text{IE}_1(\text{F}) = 1{,}681$ kJ/mol)

Electronegativity & Special Cases 🎯

Key Takeaways — Part 2

• $Z_{eff}$ is the engine behind all periodic trends. Higher $Z_{eff}$ → smaller radius, higher IE, higher EN.
• Atomic radius: largest at bottom-left (Cs), smallest at top-right (He/F among main-group).
• IE exceptions: Group IIA > Group IIIA; Group VA > Group VIA — memorize the explanations.
• Isoelectronic series: size decreases as $Z$ increases (same electrons, more protons).
• Cations are smaller, anions are larger than their neutral atoms.
• EA exceptions: Group IIA and Group VA have near-zero/positive EA due to subshell stability.
• On the MCAT, successive IE data is a reliable way to identify an element's group.

Atomic Structure & Periodic Trends

Part 3 of 5 — Emission Spectra, Photoelectron Spectroscopy & the Bohr Model

These topics appear frequently in MCAT passages because they connect quantum mechanics to experimental data. You won't need to derive the equations — but you must interpret graphs and predict relative values.

The Bohr Model (Hydrogen-Like Atoms)

Bohr proposed that electrons orbit the nucleus only at specific, quantized energy levels.

Energy of Level $n$ in Hydrogen:

$E_n = -\frac{13.6\text{ eV}}{n^2} = -\frac{2.18 \times 10^{-18}\text{ J}}{n^2}$

Negative sign: electron is bound to the nucleus (energy $= 0$ at $n = \infty$, ionized).

Ground state: $n=1$, most negative energy, most stable.

Excited state: $n > 1$; higher energy (less negative).

Energy of a Photon Emitted During a Transition:

$\Delta E = E_{\text{final}} - E_{\text{initial}} = -13.6\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)\text{ eV}$

For a downward transition ($n_i > n_f$): $\Delta E < 0$, photon is emitted.
For an upward transition ($n_f > n_i$): $\Delta E > 0$, photon is absorbed.

$E_{\text{photon}} = h\nu = \frac{hc}{\lambda}$

where $h = 6.626 \times 10^{-34}$ J·s, $c = 3.0 \times 10^8$ m/s.

Spectral Series in Hydrogen

MCAT rule: The largest energy photon in a series is the transition from $n = \infty$ (ionization) to $n_f$. The smallest energy photon is the adjacent transition ($n = n_f + 1$ to $n = n_f$).

Photoelectron Spectroscopy (PES)

PES bombards a sample with high-energy photons (UV or X-ray). Electrons are ejected, and their kinetic energy is measured.

$KE = h\nu - BE$

where $BE$ = binding energy (energy needed to remove the electron from that subshell).

Interpreting a PES Spectrum

• X-axis: Binding energy (increases left to right, or right to left depending on convention — read the axis label!)
• Y-axis: Relative number of electrons (peak area/height)
• Each peak corresponds to one subshell
• Core electrons (inner subshells) have higher binding energy than valence electrons
• Peak relative height ∝ number of electrons in that subshell

Example — Sodium (Na, $[\text{Ne}]\, 3s^1$):

• Peaks at high BE: $1s$ (2 electrons), $2s$ (2 electrons), $2p$ (6 electrons)
• Peak at low BE: $3s$ (1 electron)
• The $3s$ peak is half the height of $2s$ or $1s$ (only 1 electron vs. 2)
• The $2p$ peak is 3× the height of $2s$ (6 electrons vs. 2)

Emission Spectra & Bohr Model 🎯

Wave-Particle Duality & de Broglie Wavelength

Louis de Broglie proposed that matter has wave-like properties. For a particle with momentum $p = mv$:

$\lambda = \frac{h}{mv}$

MCAT implication: Electrons (small mass) have observable wavelengths. Macroscopic objects have negligible wavelengths.

Higher velocity → shorter wavelength (more momentum → smaller de Broglie wave).

Heisenberg Uncertainty Principle

$\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$

You cannot simultaneously know both position and momentum exactly. This is why quantum mechanics uses probability orbitals (regions where electrons are likely to be) rather than fixed orbits.

Worked Example: Photon Energy

Calculate the energy and wavelength of a photon emitted during the $n=3 \to n=1$ transition in hydrogen.

$\Delta E = 13.6\left(\frac{1}{1^2} - \frac{1}{3^2}\right) = 13.6 \times \frac{8}{9} = 12.09\text{ eV}$

Convert to joules: $12.09 \times 1.602 \times 10^{-19} = 1.94 \times 10^{-18}$ J

$\lambda = \frac{hc}{E} = \frac{(6.626 \times 10^{-34})(3.0 \times 10^8)}{1.94 \times 10^{-18}} = 1.02 \times 10^{-7}\text{ m} = 102\text{ nm}$

This is ultraviolet (Lyman series, $n_f = 1$).

PES Interpretation & Wave-Particle Duality 🎯

Key Takeaways — Part 3

• Bohr model: $E_n = -13.6/n^2$ eV for H. Downward transitions emit photons; upward absorb.
• Spectral series: Lyman (UV, $n_f=1$), Balmer (visible, $n_f=2$), Paschen (IR, $n_f=3$).
• PES peaks: lowest BE = valence electrons (outermost); highest BE = core electrons ($1s$).
• Peak area ∝ number of electrons in that subshell. Use it to identify elements.
• de Broglie: $\lambda = h/(mv)$; electrons have measurable wavelengths; macroscopic objects do not.
• Heisenberg: Position and momentum cannot both be known exactly — basis for orbital (probability cloud) model.

Atomic Structure & Periodic Trends

Part 4 of 5 — Nuclear Chemistry: Radioactive Decay & Half-Life

Nuclear chemistry is a distinct but related MCAT topic. You need to recognize decay types, balance nuclear equations, and calculate remaining quantity using half-life.

Types of Radioactive Decay

where $A$ = mass number (protons + neutrons), $Z$ = atomic number (protons).

Alpha decay example: $^{238}_{92}\text{U} \to ^{234}_{90}\text{Th} + ^4_2\text{He}$

Beta-minus decay example: A neutron converts to a proton. $^{14}_{6}\text{C} \to ^{14}_{7}\text{N} + ^0_{-1}e + \bar{\nu}$

Conservation Laws in Nuclear Reactions

1. Mass number ($A$) is conserved: sum of $A$ on left = sum on right.
2. Atomic number ($Z$) is conserved: sum of $Z$ on left = sum on right.
3. Charge is conserved.

These laws let you identify unknown products:

Example: $^{234}_{90}\text{Th}$ undergoes beta-minus decay. What is the product?

• $A$: $234 = 234 + 0$ ✓ (unchanged)
• $Z$: $90 = (Z_{\text{product}}) + (-1)$ → $Z_{\text{product}} = 91$
• $Z=91$ is Protactinium (Pa): $^{234}_{91}\text{Pa}$

Half-Life

The half-life $t_{1/2}$ is the time for half of a radioactive sample to decay.

$N(t) = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}} = N_0\, e^{-\lambda t}$

where $\lambda = \ln 2 / t_{1/2}$ is the decay constant.

MCAT shortcut: Work in units of half-lives.

Worked Example:
A 40 g sample of $^{131}\text{I}$ ($t_{1/2} = 8$ days) remains after 32 days. How much is left?

$32 \div 8 = 4$ half-lives.

$\text{Remaining} = 40 \times (1/2)^4 = 40/16 = \boxed{2.5\text{ g}}$

Binding Energy & Mass Defect

The nucleus is lighter than the sum of its individual protons and neutrons. This mass defect $\Delta m$ is the mass converted to energy that holds the nucleus together.

$E = \Delta m \cdot c^2 \quad (\text{Einstein's equation})$

Binding energy per nucleon is maximized around iron-56 ($^{56}\text{Fe}$) — this is the most stable nucleus. Elements heavier than Fe release energy by fission; lighter elements release energy by fusion.

Nuclear Decay & Balancing 🎯

Half-Life Applications in Medicine

Nuclear medicine relies on radioactive isotopes with appropriate half-lives:

• $^{99m}\text{Tc}$ ($t_{1/2} = 6$ h): Most common medical imaging isotope. Short half-life minimizes patient radiation dose. Emits gamma rays detected by SPECT scanners.
• $^{131}\text{I}$ ($t_{1/2} = 8$ d): Treats thyroid cancer and hyperthyroidism. Iodine concentrates in the thyroid gland; beta emission destroys tissue locally.
• $^{18}\text{F}$ ($t_{1/2} = 110$ min): Used in PET scanning (positron emission). Short half-life means it must be made on-site at a cyclotron.

Radiocarbon Dating

$^{14}\text{C}$ ($t_{1/2} = 5{,}730$ years) is used to date organic materials up to ~50,000 years old. Living organisms maintain a constant ratio of $^{14}\text{C}/^{12}\text{C}$ through metabolism. After death, $^{14}\text{C}$ decays. Measuring the remaining ratio gives the age.

MCAT Worked Problem:
A wood sample contains 12.5% of its original $^{14}\text{C}$. How old is the sample? ($t_{1/2} = 5{,}730$ years)

$12.5\% = (1/2)^n$ → $n = 3$ half-lives.
Age $= 3 \times 5{,}730 = \boxed{17{,}190\text{ years}}$

Half-Life Calculations 🎯

Key Takeaways — Part 4

• Alpha decay ($\alpha$): $A-4$, $Z-2$. Emits an $^4_2\text{He}$ nucleus.
• Beta-minus ($\beta^-$): $A$ unchanged, $Z+1$. Neutron → proton.
• Beta-plus ($\beta^+$) / Electron capture: $A$ unchanged, $Z-1$. Proton → neutron.
• Gamma ($\gamma$): No change in $A$ or $Z$; releases excess nuclear energy as photons.
• Balance nuclear equations using conservation of $A$ and $Z$.
• Half-life: $N = N_0(1/2)^{t/t_{1/2}}$. Count half-lives for quick calculations.
• Binding energy per nucleon peaks at $^{56}\text{Fe}$ — the most stable nucleus.
• Medical isotopes have short half-lives to minimize patient radiation exposure.

Atomic Structure & Periodic Trends

Part 5 of 5 — Mixed MCAT Review: Passage-Style Practice

This final part uses integrated questions that combine quantum numbers, periodic trends, spectra, and nuclear chemistry — exactly as the MCAT mixes topics within a single passage.

Quick-Reference Summary

Quantum Number Rules

• $n$: positive integer $\geq 1$
• $\ell$: $0$ to $n-1$
• $m_\ell$: $-\ell$ to $+\ell$
• $m_s$: $\pm 1/2$ only

Aufbau Exceptions

Periodic Trend Directions

$\text{Atomic radius: } \nearrow \text{ down-left}$ $\text{IE, EN: } \nearrow \text{ up-right}$

Half-Life Formula

$N = N_0 \left(\frac{1}{2}\right)^{n_{\text{half-lives}}}$

Bohr Model (Hydrogen)

$E_n = -\frac{13.6\text{ eV}}{n^2}$

Integrated MCAT-Style Questions — Set 1 🎯

Integrated MCAT-Style Questions — Set 2 🎯

Final Review: Atomic Structure High-Yield Points

Most tested MCAT topics in this module:

1. Quantum number validity — check $\ell \leq n-1$ and $|m_\ell| \leq \ell$
2. Electron configurations of ions — remove highest-$n$ electrons first
3. Unpaired electrons — apply Hund's rule to the $d$ subshell; know Cr and Cu exceptions
4. IE successive jumps → identify element's group
5. Periodic trends — Mg > Al and P > S in IE are common distractors
6. Isoelectronic series — same electrons, size decreases as $Z$ increases
7. PES spectrum reading — peak area = electron count, lowest BE = valence
8. Balancing nuclear equations — conserve $A$ and $Z$
9. Half-life countdown — $(1/2)^n$ for $n$ half-lives elapsed
10. Bohr model transitions — higher $n$ difference = higher energy photon (for same $n_f$)