Lecture 2 Notes: Light, Quantum Theory, and Bohr Hydrogen Model

Light and Quantum Theory

• Quantum Theory replaces classical atomic ideas due to problems with continuous energy predictions and instability of electron orbits.
• Core ideas: energy exchange occurs in quanta; radiation comes in discrete packets called photons; spectra reveal quantization of atomic energy levels.

Learning outcomes (from transcript)

• Discuss how the classical theory of the atom is replaced by quantum theory.
• Describe the difference between continuous and line spectra and how each is produced.
• Explain why classical views of the atom are unacceptable:
  • Electron motion in a classical model would produce a continuous spectrum.
  • Electron would continuously lose energy and spiral into the nucleus due to electrostatic attraction.
• Describe Bohr’s atom as a planetary model and use Bohr’s model to explain line spectra.
• Draw the line spectra of hydrogen (Lyman, Balmer, Paschen series).
• Calculate the energy of an electron in a given energy level of hydrogen and the energy difference between levels (ground and excited states).

Continuous vs. Line Spectra

• Question: What causes radiation in a line spectrum and why is it different from a continuous spectrum?
• Answer (as implied by slides): line spectra arise from transitions between discrete energy levels; only certain energies are allowed, leading to specific wavelengths.

Line Spectrum of Hydrogen

• Why heating hydrogen produces light: energy absorption excites electrons to higher energy levels.
• How line spectrum is produced: transitions between quantized energy levels emit photons at specific energies.

What does the line spectrum tell us?

• Each line corresponds to a specific wavelength λ and thus a specific frequency ν, hence a specific energy difference ΔE.
• There is a limited set of energy values available to excited gaseous atoms; spectral lines reveal quantum of energy (photon).
• This is evidence for quantization of energy levels (QUANTUM THEORY).

Quantum Theory (Key formulas and constants)

• Planck’s quantum hypothesis: the quantum of energy is ΔE = hν = hc/λ.
• ΔE is associated with absorption or emission of a photon.
• Planck’s constant: $h = 6.626\times 10^{-34}\ \text{J s}$
• ν = frequency; λ = wavelength; relationship: $ν = \frac{c}{λ}\quad\text{or}\quad λ = \frac{c}{ν}$
• Quantized energy changes: discrete allowed energy differences between levels.
• The Photoelectric Effect provides evidence for quantization of energy.

Quantum View of Atomic Structure

• Rutherford’s classical model is unsatisfactory because electrons would spiral into the nucleus as they lose energy, predicting a continuous spectrum.
• Bohr replaced Rutherford’s model for hydrogen with a quantum model.
• Planck’s relation: $ΔE = hν = \frac{hc}{λ}$
• Einstein proposed that angular momentum of the electron in a hydrogen atom is quantized; specific angular momentum values are allowed: $L = \mathbf{r} \times \mathbf{p} = r m v \sin\theta$ with quantization conditions to be introduced.
• L changes when r changes; this ties orbital motion to energy changes.

Bohr’s Atomic Model of Hydrogen

• Relationship between Planck and Einstein: combined framework leading to quantized energy levels.
• Key connection: energy difference is related to transitions between quantized orbits.
• Classical kinetic energy for circular motion and angular momentum concepts integrate to yield quantization of orbits.
• Result: Electron can occupy only certain radii (quantized r) and thus has quantized energy levels.

Bohr’s Atomic Model – Quantization of Angular Momentum

• Angular momentum is quantized: $L = m v r \sin\theta = \frac{h r}{λ} = \frac{n h}{2π}$
• n = whole number (principal quantum number).
• Consequence: Allowed orbits exist at specific radii (distance r is quantized).

Bohr’s Atomic Model of Hydrogen – Energy Quantization

• Energy is quantized: $E<em>n = -\frac{R</em>H}{n^2}$
• Rydberg constant for hydrogen in energy units: $R_H = 2.179 \times 10^{-18}\ \text{J}$
• Corresponding frequency version (from the slide): $\nu_R = 3.29 \times 10^{15}\ \text{Hz}$
• Energy levels E1, E2, E3 correspond to quantum numbers n = 1, 2, 3, …
• General energy difference between levels: $\Delta E = E<em>f - E</em>i = -R<em>H \left( \frac{1}{n</em>f^2} - \frac{1}{n_i^2} \right)$
• Ground state energy often cited as 0 eV (reference), with ionization energy of 13.6 eV.

Explaining the Line Spectrum of Hydrogen

• Using Bohr’s model, the line spectrum arises from transitions between quantized energy levels.
• The spectrum provides a quantum of energy measure corresponding to the transition: lines appear at specific energies (frequencies).
• Ionization energy: 13.6 eV; ground state energy: 0 eV; energy-level differences correspond to emitted/absorbed photons.
• Absorption occurs when nf > ni; Emission occurs when nf < ni.

Hydrogen Line Spectra (Series)

• Lyman Series (Ultraviolet): transitions to n1 = 1 from higher levels (nf = 2, 3, 4, …). Example wavelengths mentioned: 410.2 nm (line shown on slide).
• Balmer Series (Visible): transitions to n1 = 2; notable lines:
  • 656.3 nm (red) — Hα (n = 3 → 2)
  • 486.1 nm (blue-green) — Hβ (n = 4 → 2)
  • 434.1 nm (violet) — Hγ (n = 5 → 2)
  • 410.2 nm (violet) — Hδ (n = 6 → 2)
• Paschen Series (Infrared): transitions to n1 = 3 (nf = 4, 5, 6, …).
• Energy-level diagram (as shown in the slides) illustrates the Lyman, Balmer, and Paschen series and the wavelengths along a nanometer scale from about 750 nm to 400 nm in the plotted axis.
• Basic energy references on the diagram:
  • Ionization energy: 13.6 eV
  • Hydrogen ground state: 0 eV
  • Other labeled values on the diagram include 12.07 eV, 10.19 eV, and -12.73 eV (as shown in the provided figure)
• The spectral lines are associated with transitions between specific n levels (nf and ni).

Energy-level Diagram and Series Labels (from transcript visuals)

• Lyman Series (Ultraviolet): n1 = 1
• Balmer Series (Visible): n1 = 2; representative lines include 656.3 nm, 486.1 nm, 434.1 nm, 410.2 nm
• Paschen Series (Infrared): n1 = 3; representative lines
  a. Axis of wavelengths shown 750, 700, 650, 600, 550, 500, 450, 400 (nm)
• The energy values and the ionization point are depicted: 13.6 eV (ionization), 0 eV (ground state)

Concept Questions and Worked Examples (as per transcript)

• Worked Example 2.1 (Concept Question): A radio broadcast at 909 kHz (UK Radio 5 Live). What wavelength does this correspond to?

  • Strategy: Use equation $c = \lambda \nu$.
  • Convert frequency to Hz: $\nu = 909\times 10^3\ \text{s}^{-1}$.
  • Speed of light: $c = 2.998\times 10^{8}\ \text{m s}^{-1}$.
  • Solution (as shown): $\lambda = \frac{c}{\nu} = \frac{2.998\times10^8}{909\times10^3} \approx 3.29\times10^2\ \text{m}$
  • Therefore, wavelength is about 330 m (in the radio range).

• Worked Example 2.4 (Concept Question): For the Paschen series (n1 = 3) with nf = 4, 5, 6, find the frequencies of the first three lines.

  • Strategy: Use the Rydberg formula for hydrogen: $\Delta E<em>n = E</em>f - E<em>i = -R</em>H\left(\frac{1}{n<em>f^2} - \frac{1}{n</em>i^2}\right)$ and relate to frequency via $\nu = \frac{\Delta E}{h}$ (equivalently via $\nu = \frac{c}{\lambda}$).
  • Given constants from slides: $R_H = 2.179\times 10^{-18}\ \text{J}$ and the Paschen series has n1 = 3; higher levels nf = 4, 5, 6.
  • Slide notes give first line for Paschen nf = 4; second nf = 5; third nf = 6; (with a reference value listed as 2.998 x 10^8 Hz in the transcript, which appears to be inconsistent with typical Paschen frequencies — actual Paschen lines have IR wavelengths and correspondingly high frequencies ~10^14 Hz). The key takeaway is applying the Rydberg relation to compute ν for nf = 4, 5, 6 with n1 = 3.

• Concept Question summary: Use the Bohr model and Rydberg formula to determine wavelengths or frequencies for transitions between specified n levels; identify whether the line would be part of the Lyman, Balmer, or Paschen series based on n1.

Summary of key points

• Electromagnetic radiation from atoms is quantized: photons carry energy $ΔE = hν = \frac{hc}{λ}$.
• Atomic energy levels are quantized; electrons transition between allowed levels, emitting or absorbing photons with energy equal to the level difference.
• The Bohr model describes hydrogen with quantized angular momentum and discrete orbits, leading to quantized energy levels and specific spectral lines.
• Energy levels can be written as $E<em>n = -\frac{R</em>H}{n^2}$ with $R<em>H = 2.179\times10^{-18}\ \text{J}$ (Rydberg energy). The energy difference for a transition is $ΔE = E</em>f - E<em>i = -R</em>H\left(\frac{1}{n<em>f^2} - \frac{1}{n</em>i^2}\right)$, which relates to the emitted/absorbed photon energy.
• Hydrogen line series: Lyman (n1 = 1, UV), Balmer (n1 = 2, visible), Paschen (n1 = 3, infrared). Notable Balmer lines: 656.3 nm (red, Hα), 486.1 nm (blue-green, Hβ), 434.1 nm (violet, Hγ), 410.2 nm (violet, Hδ).
• Ionization energy of hydrogen is 13.6 eV; ground state energy is 0 eV.
• Absorption vs emission depends on whether nf > ni (absorption) or nf < ni (emission).
• Practice problems include converting between frequency and wavelength (c = λν) and applying the Rydberg formula to find wavelengths/frequencies of series lines.

Next session guidance (as per transcript)

• Review sections 2.2 and 2.3; read ahead Section 2.4: The nature of the electron.
• Work on Section 2.4 concept questions; practice problems 2.2, 2.4, 2.5.
• Concept Question: Determine the wavelength in hydrogen’s line spectrum for the transition from n = 5 to n = 2 and identify the line type.
• Box 2.1: Radiation from the Sun.