Fundamentals of General Chemistry

Understanding the Electromagnetic Spectrum in Chemistry

Every form of electromagnetic radiation can be placed on a continuous spectrum that ranges from low‑frequency radio waves to high‑frequency gamma rays. In the laboratory, the type of radiation used determines how energy is transferred to matter.

Microwave Radiation and Everyday Applications

Microwave ovens, like the one used to pop popcorn, operate in the microwave region of the spectrum. These waves have wavelengths between 1 mm and 30 cm and frequencies from 300 MHz to 300 GHz. Domestic ovens typically use a frequency of 2.45 GHz, which falls comfortably within this range.

• Microwaves penetrate food quickly, causing polar molecules (especially water) to rotate and generate heat.
• Unlike infrared radiation, which heats surfaces, microwaves heat the interior of the material.

Memory tip: The word “microwave” itself tells you the region—think “micro‑waves = microwave region.”

Applying the Rydberg Formula to Hydrogen Spectra

The Rydberg equation predicts the wavelength of light emitted or absorbed when an electron transitions between energy levels in a hydrogen atom:

1/λ = R_H (1/n₁² – 1/n₂²), where R_H ≈ 1.097 × 10⁷ m⁻¹.

Example: Transition from n = 5 to n = 1

Plugging the quantum numbers into the formula gives:

• n₁ = 1 (final level), n₂ = 5 (initial level).
• 1/λ = R_H (1 – 1/25) = R_H·24/25.
• λ ≈ 9.49 × 10⁻⁸ m.

Mnemonic: “Rydberg Rápido: 1/λ = R (1 – 1/n²). The larger the initial n, the larger the wavelength.”

Quantum Numbers: Rules and Restrictions

Four quantum numbers describe every electron in an atom:

• Principal quantum number (n) – positive integer (1, 2, 3, …).
• Azimuthal quantum number (ℓ) – integer from 0 to n‑1.
• Magnetic quantum number (m_ℓ) – integer from –ℓ to +ℓ.
• Spin quantum number (m_s) – either +½ or –½.

Forbidden Combination Example

The set {4, 4, –1, +½} violates the rule that ℓ must be less than n. Since ℓ = 4 while n = 4, ℓ is not allowed (ℓ ≤ n‑1). Therefore this combination is forbidden.

Memory aid: Remember the order “N‑ℓ‑mℓ‑s”. First check n, then ensure ℓ ≤ n‑1, then verify mℓ lies within ±ℓ, and finally confirm spin is ±½.

Counting Orbitals in a Subshell

The number of orbitals in a subshell is determined by the azimuthal quantum number ℓ:

Number of orbitals = 2ℓ + 1.

Case Study: ℓ = 2 (d‑subshell)

For ℓ = 2, the calculation yields 2·2 + 1 = 5 orbitals. This is why the d‑subshell is often visualized as a five‑pointed star.

Mnemonic: “l + l + 1 = orbitals.” Plug in ℓ = 2 → 2 + 2 + 1 = 5.

Photon Energy Calculations

Energy of a photon can be expressed in two common ways:

• E = h·c/λ – useful when wavelength is known.
• E = q·e – where q is the energy in electron‑volts (eV) and e = 1.602 × 10⁻¹⁹ J/eV.

Gamma‑Ray Photon from Technetium‑99 Decay

A 140.511 keV gamma ray carries:

• 140.511 keV = 140 511 eV.
• E = 140 511 eV × 1.602 × 10⁻¹⁹ J/eV ≈ 2.25 × 10⁻¹⁴ J.

Mnemonic: “keV → multiply by 10³ to get eV, then *1.6e‑19 for joules.”

Energy from a Sodium Vapor Lamp

Each photon emitted at λ = 589 nm has energy:

• E_photon = (6.626 × 10⁻³⁴ J·s)(3.00 × 10⁸ m/s) / 5.89 × 10⁻⁷ m ≈ 3.38 × 10⁻¹⁹ J.
• 5.00 mg of Na corresponds to (5.00 × 10⁻³ g) / (23 g mol⁻¹) ≈ 2.17 × 10⁻⁴ mol, or ≈ 1.31 × 10²⁰ atoms.
• Total energy = 1.31 × 10²⁰ × 3.38 × 10⁻¹⁹ J ≈ 4.43 × 10¹ J (but the quiz asked for per‑photon energy, which is 3.38 × 10⁻¹⁹ J).

Memory tip: H c ÷ λ = energy. Convert mass → moles → atoms, then multiply by photon energy.

Interpreting Electron Configurations

Electron configurations reveal the distribution of electrons among shells and subshells. The notation [Kr]4d¹⁰5s²5p⁴ indicates:

• Core electrons equivalent to krypton (Z = 36).
• Full 4d subshell (10 electrons).
• 5s² – a filled s‑subshell.
• 5p⁴ – four electrons in the p‑subshell of the fifth shell.

Since the valence shell (n = 5) has six electrons (5s²5p⁴), the element belongs to group 16 (the chalcogens). The element with Z = 52 that fits this pattern is Tellurium (Te).

Mnemonic: “4d¹⁰5s²5p⁴ → think ‘Te‑lú‑rio’ – a group‑16 element in the 5th period.”

Ionic Radius Trends Across the Periodic Table

Ionic radius generally increases with added electrons (more negative charge) and decreases across a period due to increasing nuclear charge.

Comparing S²⁻, Cl⁻, and P³⁻

All three ions have the same electron configuration (isoelectronic) – 18 electrons – but differ in nuclear charge:

• Phosphorus (Z = 15) → P³⁻ has the smallest nuclear charge, so its radius is the largest.
• Chlorine (Z = 17) → Cl⁻ is intermediate.
• Sulfur (Z = 16) → S²⁻ falls between Cl⁻ and P³⁻.

Thus the order of increasing ionic radius is:

P³⁻ < Cl⁻ < S²⁻.

Quick recall: More protons pull the electron cloud tighter; fewer protons let it expand.

Key Takeaways for General Chemistry Mastery

• Identify the correct region of the electromagnetic spectrum for everyday devices (microwaves, infrared, etc.).
• Use the Rydberg formula to calculate wavelengths of hydrogen spectral lines.
• Apply the hierarchy of quantum numbers to determine allowed electron states.
• Remember that the number of orbitals in a subshell follows 2ℓ + 1.
• Convert photon energies between keV, eV, and joules with the factor 1 eV = 1.602 × 10⁻¹⁹ J.
• Translate electron configurations into element identities, focusing on valence‑shell patterns.
• Predict ionic radius trends by considering isoelectronic series and nuclear charge.

By mastering these concepts, students build a solid foundation for more advanced topics in chemistry, physics, and materials science.