4.1 Atomic Structure, Quantum Numbers & Periodic Table Trends

4.1 Atomic Structure, Quantum Numbers & Periodic Table Trends

Understanding atomic structure and periodic trends forms the foundational bedrock of FSc Chemistry for the Pakistan Army Medical Cadet (AMC) Initial Test. This section reviews subatomic discovery, Bohr's atomic model, quantum mechanical numbers, electron configurations, and periodic properties.

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Subatomic Particles & Early Atomic Models

The discovery of subatomic particles revolutionized chemical science in the late 19th and early 20th centuries:

1. Electron ($e^-$): Discovered by J.J. Thomson in 1897 using discharge tube experiments (cathode rays). Thomson measured the charge-to-mass ratio ($e/m$) of the electron as $1.7588 \times 10^{11} \text{ C/kg}$. R.A. Millikan subsequently determined the exact charge of an electron as $1.6022 \times 10^{-19} \text{ C}$ using his famous oil drop experiment, yielding an electron mass of $m_e = 9.1093 \times 10^{-31} \text{ kg}$.
2. Proton ($p^+$): Discovered by E. Goldstein (1886) as canal rays (positive rays) in a modified discharge tube. The $e/m$ ratio of positive rays depends on the nature of the gas inside the tube and is maximum when hydrogen gas is used ($m_p = 1.6726 \times 10^{-27} \text{ kg}$, approximately 1836 times heavier than an electron).
3. Neutron ($n^0$): Discovered by James Chadwick in 1932 by bombarding a Beryllium target with $\alpha$-particles: $^{9}{4}\text{Be} + ^{4}{2}\text{He} \rightarrow ^{12}{6}\text{C} + ^{1}{0}\text{n}$. Neutrons are neutral particles with mass $m_n = 1.6749 \times 10^{-27} \text{ kg}$.

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Bohr's Atomic Model & Hydrogen Spectrum

Niels Bohr (1913) proposed a quantized model for single-electron systems (such as $\text{H}$, $\text{He}^+$, $\text{Li}^{2+}$) based on Planck's quantum theory.

Key Postulates of Bohr's Model

1. Electrons revolve around the nucleus in specific non-radiating circular orbits called stationary states.
2. The angular momentum ($L$) of an electron in a stationary orbit is quantized in integral multiples of $h / 2\pi$: $m v r = \frac{n h}{2 \pi} \quad (n = 1, 2, 3, \dots)$
3. Energy is absorbed or emitted only when an electron jumps from one orbit to another: $\Delta E = E_2 - E_1 = h \nu = \frac{h c}{\lambda}$.

Mathematical Formulas in Bohr's Model

• Radius of $n$-th Orbit ($r_n$): $r_n = \frac{n^2 h^2 \varepsilon_0}{\pi m e^2} = 0.529 \times n^2 \text{ \u00c5} = 0.0529 \times n^2 \text{ nm}$
  • For the first orbit of Hydrogen ($n=1$), $r_1 = 0.529 \text{ \u00c5} = 5.29 \times 10^{-11} \text{ m}$ (Bohr radius).
• Energy of Electron in $n$-th Orbit ($E_n$): $E_n = -\frac{m e^4}{8 \varepsilon_0^2 h^2 n^2} = -\frac{2.18 \times 10^{-18}}{n^2} \text{ J} = -\frac{1312}{n^2} \text{ kJ/mol}$
  • The negative sign signifies that the electron is bound to the nucleus. As $n \rightarrow \infty$, $E_{\infty} = 0$.

Hydrogen Spectral Series

When excited hydrogen atoms return to lower energy states, they emit discrete spectral lines governed by the Rydberg equation: $\bar{\nu} = \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$ where $R_H = 1.0974 \times 10^7 \text{ m}^{-1}$ (Rydberg constant).

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Quantum Numbers & Orbital Shapes

Quantum mechanics replaces classical circular orbits with three-dimensional probability regions called orbitals. Four quantum numbers describe an electron fully:

1. Principal Quantum Number ($n$):
   • Indicates main energy shell, size of orbital, and average distance from nucleus.
   • Values: $n = 1, 2, 3, 4, \dots$ (K, L, M, N shells).
   • Maximum electrons in shell $= 2n^2$.
2. Azimuthal / Subsidiary Quantum Number ($l$):
   • Defines subshell shape and orbital angular momentum ($L = \sqrt{l(l+1)} \frac{h}{2\pi}$).
   • Values: $l = 0, 1, 2, \dots, (n-1)$.
     • $l = 0 \rightarrow s$ (spherical)
     • $l = 1 \rightarrow p$ (dumbbell)
     • $l = 2 \rightarrow d$ (double dumbbell)
     • $l = 3 \rightarrow f$ (complex rosette)
   • Maximum electrons in subshell $= 2(2l+1)$.
3. Magnetic Quantum Number ($m_l$):
   • Specifies spatial orientation of the orbital in a magnetic field.
   • Values: $m_l = -l, \dots, 0, \dots, +l$ (Total $2l+1$ orientations per subshell).
4. Spin Quantum Number ($m_s$):
   • Describes intrinsic electron spin orientation.
   • Values: $m_s = +1/2$ (clockwise / spin up) or $-1/2$ (counter-clockwise / spin down).

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Rules for Electronic Configuration

1. Aufbau Principle: Electrons fill available atomic orbitals in order of increasing energy levels. Energy order is determined by the $(n+l)$ rule:
   • Lower $(n+l)$ value means lower orbital energy.
   • If two orbitals have the same $(n+l)$ value, the orbital with lower $n$ has lower energy (e.g., $3d$ vs $4p$: both have $n+l=5$, but $3d$ fills first because $n=3 < 4$).
   • Filling order: $1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s$.
2. Pauli Exclusion Principle: No two electrons in the same atom can have the exact same set of all four quantum numbers. Consequently, an orbital can hold a maximum of 2 electrons with opposite spins.
3. Hund's Rule of Maximum Multiplicity: Orbitals of equal energy (degenerate orbitals, e.g., $2p_x, 2p_y, 2p_z$) are occupied singly with parallel spins before pairing occurs.

Anomalous Configurations in $d$-Block Elements

Extra stability associated with half-filled ($d^5$) and fully-filled ($d^{10}$) subshells causes electron promotion from $4s$ to $3d$:

• Chromium ($Z=24$): Expected $[Ar] 3d^4 4s^2 \rightarrow$ Actual $[Ar] 3d^5 4s^1$
• Copper ($Z=29$): Expected $[Ar] 3d^9 4s^2 \rightarrow$ Actual $[Ar] 3d^{10} 4s^1$

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Periodic Table Trends & Properties

Periodic trends depend on two major factors: Nuclear Charge ($Z$) and Shielding Effect ($S$), combining into Effective Nuclear Charge ($Z_{eff} = Z - S$).

Key Trends Summary Table

Notable Periodic Anomalies

1. Ionization Energy Anomalies:
   • Be ($1s^2 2s^2$) vs B ($1s^2 2s^2 2p^1$): $IE_1(\text{Be}) > IE_1(\text{B})$ because Be has a stable, fully-filled $2s$ subshell, whereas B loses an easily removable $2p$ electron.
   • N ($1s^2 2s^2 2p^3$) vs O ($1s^2 2s^2 2p^4$): $IE_1(\text{N}) > IE_1(\text{O})$ because Nitrogen possesses a half-filled, symmetrical $2p^3$ subshell with extra exchange energy stability.
2. Isoelectronic Species Radii:
   • For species with identical electron counts (e.g., $\text{O}^{2-}, \text{F}^-, \text{Na}^+, \text{Mg}^{2+}$ all with 10 electrons), radius decreases as nuclear charge ($Z$) increases: $\text{O}^{2-} (Z=8) > \text{F}^- (Z=9) > \text{Na}^+ (Z=11) > \text{Mg}^{2+} (Z=12)$

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Worked Numerical Examples

Example 1: Bohr Orbit Energy Calculation

Problem: Calculate the energy of an electron in the second orbit ($n=2$) of a hydrogen atom in $\text{kJ/mol}$.

Solution: Using Bohr's energy formula: $E_n = -\frac{1312}{n^2} \text{ kJ/mol}$ For $n=2$: $E_2 = -\frac{1312}{2^2} = -\frac{1312}{4} = -328 \text{ kJ/mol}$

Example 2: Quantum Number Set Identification

Problem: Specify the set of quantum numbers for the valence electron in a neutral Sodium atom ($Z=11$).

Solution:

1. Electronic configuration of Na ($Z=11$): $1s^2 2s^2 2p^6 3s^1$.
2. The outermost electron resides in the $3s$ orbital:
   • Principal quantum number $n = 3$
   • Azimuthal quantum number $l = 0$ (for $s$ subshell)
   • Magnetic quantum number $m_l = 0$
   • Spin quantum number $m_s = +1/2$ or $-1/2$

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s-, p-, and d-Block Snapshot (Descriptive Inorganic)

FSc descriptive inorganic rarely dominates the short AMC academic paper, but group-trend MCQs appear:

• s-block (Groups 1–2): alkali and alkaline-earth metals; low ionization energies; form ionic oxides/hydroxides; flame tests and strong bases are classic distractors.
• p-block (Groups 13–18): includes metalloids and non-metals; oxidation-state variety increases down a group; noble gases are largely inert; carbon/nitrogen/oxygen chemistry feeds organic and acid-base questions.
• d-block (transition metals): variable oxidation states, coloured ions, catalytic behaviour, and complex-ion formation. Remember that Zn/Cd/Hg are often taught with the d-block discussion even when their chemistry is less typically “transition-like.”

If an item asks why transition-metal compounds are coloured, answer in terms of d–d electronic transitions in partially filled d-orbitals (with the usual FSc-level wording), not nuclear chemistry.