CHEM1100 Study Guide: Quantum Theory and the Electronic Structure of Atoms

Chapter 3: Quantum Theory and the Electronic Structure of Atoms

3.1 Energy and Energy Changes

• Definition of Energy: Capacity to do work or transfer heat.

• Forms of Energy:

  • Kinetic Energy ($E_k$): Energy of motion.

  • Formula: $E_k = \frac{1}{2} mu^2$

  • Where:

    • $m$ = mass of the object

    • $u$ = velocity of the object

  • Special Note: Thermal energy is a specific form of kinetic energy, related to the random motion of atoms and molecules.

  • Potential Energy: Energy possessed by an object due to its position.

  • Two key forms for chemists:

    • Chemical Energy: Energy stored in the structural units of chemical substances.

    • Electrostatic Energy: Potential energy arising from the interactions between charged particles.

    • Formula: $E<em>{el}$ is proportional to $\frac{Q</em>1 Q_2}{d}$

    • Where $Q<em>1$ and $Q</em>2$ are charges and $d$ is the distance between them.

• Interconvertibility of Energy: Kinetic and potential energy can be converted into one another. Total energy is constant (Law of Conservation of Energy).

3.2 Units of Energy

• SI Unit for Energy: Joule ($J$)

  • Defined as the amount of energy possessed by a $2 \text{ kg}$ mass moving at $1 \text{ m/s}$.

  • Alternatively, it's the energy exerted when a force of $1 \text{ newton}$ is applied over a distance of $1 \text{ meter}$ ($1 \text{ J} = 1 \text{ N} \cdot \text{m}$).

• Kilojoule ($kJ$): A larger unit commonly used due to the small magnitude of a joule ($1 \text{ kJ} = 1000 \text{ J}$).

3.3 Worked Examples

Example 3.1: Calculate the kinetic energy of a helium atom moving at $125 \text{ m/s}$.

• Given: Mass of helium atom = $4.003 \text{ amu}$ = $4.003 \times 1.661 \times 10^{-24} \text{ g}$

• Conversion:

  • Mass in kg = $4.003 \times 1.661 \times 10^{-24} \text{ g} \times \frac{1 \text{ kg}}{1000 \text{ g}} = 6.649 \times 10^{-27} \text{ kg}$

• Calculation:

  • $E_k = \frac{1}{2}(6.649 \times 10^{-27} \text{ kg})(125 \text{ m/s})^2 = 5.19 \times 10^{-23} \text{ J}$

Example 3.2: Attraction between different charges.

• Comparison of Attractions:

  • Attraction between charges $+2$ and $-2$ is four times greater than between $+1$ and $-1$ (under the same distance).

3.2 The Nature of Light

• Electromagnetic Spectrum: Range of all types of light, of which visible light is a small part.

• Speed of Light in Vacuum:

  • Constant speed: $c = 2.99792458 \times 10^8 \text{ m/s}$ (commonly approximated as $3.00 \times 10^8 \text{ m/s}$).

• Relationship among Speed, Frequency, and Wavelength:

  • Formula: $c = \nu \times \lambda$

  • Where:

  • $\nu$ (nu) = frequency in hertz ($Hz$)

  • $\lambda$ (lambda) = wavelength in meters.

3.3 Properties of Waves

• Characteristics of Electromagnetic Radiation:

  • Wavelength ($\lambda$): Distance between identical points on consecutive waves.

  • Frequency ($\nu$): Number of waves that pass a point per second.

  • Amplitude: Vertical distance from the midline of a wave to the peak or trough.

3.4 The Electromagnetic Spectrum

• Components of Electromagnetic Wave: Contains both electric and magnetic components that share the same frequency and wavelength.

3.5 The Double-Slit Experiment

• Interference Pattern: Occurs when light passes through two slits, demonstrating wave properties.

  • Constructive Interference: Occurs when waves are in phase.

  • Destructive Interference: Occurs with out-of-phase waves.

3.6 Worked Example 3.3

• Laser Frequency Calculation: For a Neodymium-doped Yttrium Aluminum Garnet (Nd:YAG) laser at $532 \text{ nm}$ wavelength:

  • Convert to meters: $532 \text{ nm} = 5.32 \times 10^{-7} \text{ m}$

• Frequency Calculation:

  • Rearranged formula: $\nu = \frac{c}{\lambda}$

  • Result: $\nu = 5.64 \times 10^{14} \text{ s}^{-1}$

3.7 Quantum Theory

• Historical Overview: Early physicists struggled to describe atoms with classical physics.

  • Classical laws do not apply to subatomic particles.

• Quantization of Energy:

  • Hot solids emit electromagnetic radiation in discrete wavelengths.

• Max Planck's Proposal: Energy is emitted/absorbed in packets called quanta.

• Energy of a Quantum: $E = h\nu$ where $h$ = Planck's constant, $h = 6.63 \times 10^{-34} \text{ J} \cdot \text{s}$.

  • Analogy: Like stepping on stairs (quantized) versus a ramp (continuous).

3.8 Photons and the Photoelectric Effect

• Einstein's Explanation of the Photoelectric Effect: Light can eject electrons from metal surfaces when it meets a certain threshold frequency.

  • Below threshold frequency, no electron ejection occurs.

  • Each photon carries energy as $E_{photon} = h\nu$.

  • The kinetic energy ($E_k$) of the ejected electron relates to the equation:

  • $E_k = h\nu - W$ where $W$ is the binding energy of the electron.

3.9 Worked Example 3.4

• Photon Energy Calculations:

  • (a) Photon with $50.0 \text{ nm}$ (infrared region): Use $E = h\nu$.

  • (b) $52 \text{ nm}$ (ultraviolet region): E calculation leads to number values via conversion to $J$.

  • (c) Max kinetic energy of an ejected electron from induced energy lesser than binding energy.

Conclusion: Summary of Key Points

• Forms of Energy.

• Law of Conservation of Energy.

• Quantum Theory concepts, including energy quantization, photon behavior, and specific phenomena like the photoelectric effect and atomic line spectra.