Quantum Chemistry: Newtonian Mechanics to Bohr Atom

• Quantum chemistry is introduced
• Dr. Aeshah Hassan Al-Amri is the presenter
• A reference is made of the text: Rao, P. M. S. (2022). Quantum Chemistry (1st ed.). CRC Press

The outlines

• Topics in the following order are:
  • Newtonian Mechanics
  • Black Body Radiation
  • Photoelectric Effect
  • Compton Effect
  • Atomic Spectra
  • Atomic Models
  • The Bohr Atom
  • Failure of the Old Quantum Theory

Outcomes of lecture will enable students to:

• Distinguish between classical and quantum mechanics
• Interpret Max Planck's quantum hypothesis, resolving the ultraviolet catastrophe
• Explain the photoelectric effect and its experimental observations
• Calculate wavelength shift of scattered photons using the Compton wavelength shift formula
• Apply Rydberg formula to predict wavelengths of spectral lines in the hydrogen atom for different series (Lyman, Balmer, Paschen, Brackett, and Pfund)
• Assess successes and limitations of each Atomic Model in explaining atomic structure and spectral lines

Classical Mechanics

• Formulated by Isaac Newton
• Describes the motion of macroscopic objects like planets, cars, and baseballs
• Based on deterministic laws, allowing precise prediction of future behavior if initial conditions are known
• Governed by Newton's Laws of Motion, that describes the relationship between forces, mass, and acceleration
• Conservation laws state that Energy, momentum, and angular momentum are conserved in closed systems

Classical Mechanics Limitations:

• Cannot describe phenomenona at very small, atomic, and subatomic scales
• Cannot describe phenomenona at very speeds, close to the speed of light
• Cannot explain quantum phenomena like wave-particle duality, and superposition

Quantum Mechanics

• Describes the behavior of particles at atomic and subatomic scales, such as electrons, photons, and atoms
• Introduces probabilistic behavior, meaning that outcomes are not deterministic but are described by probabilities
• Particles exhibit both wave-like and particle-like properties, which is known as Wave-Particle Duality
• Certain properties like energy, angular momentum are quantized, and can only take discrete values in a process known as Quantization
• The Uncertainty Principle is Formulated by Werner Heisenberg
• States that certain pairs of physical properties, like position and momentum, cannot be simultaneously measured with precision
• Superposition describes how A quantum system can exist in multiple states simultaneously until measured
• Is Governed by the Schrödinger equation: ih(dψ/dt) = Hψ, where h is the reduced Planck's constant and H is the Hamiltonian operator

The two mechanics compared

• Scale:
  • Macroscopic objects vs Atomic and subatomic particles
• Determinism:
  • Deterministic vs Probabilistic
• Energy:
  • Continuous vs Quantized
• Wave-Particle
  • Not applicable vs Fundamental
• Uncertainty:
  • No inherent uncertainty vs Heisenberg Uncertainty Principle
• Mathematical Tools:
  • Differential equations vs Wave functions, operators, and matrices

Black Body Radiation

• A black body is an idealized object that absorbs all incident radiation
• Emits radiation depending on the temperature of the emitting body
• Kirchhoff's law states that the ratio of emissive power (E) to absorptivity (A) is constant for a given temperature
  • Since A₀=1, E = AE₀, where E₀ = total emissive power of a black body
  • a black body then, is both the most efficient absorber and emitter of radiant energy

Stefan-Boltzmann Law

• In 1879, Stefan empirically discovered (later derived theoretically by Boltzmann) that the total emissive power of a black body is proportional to the fourth power of its absolute temperature
• Described as E= e σT⁴, where e is emissivity, σ is the Stefan-Boltzmann constant, and E is the emission rate of radiant energy per unit area

Wien's Displacement Law

• In 1894, Wien found that the wavelength corresponding to the maximum energy emission (λmax) is inversely proportional to the temperature (T)
• Is expressed as λmaxT = constant

Rayleigh-Jeans Law

• In 1900, Rayleigh and Jeans applied classical electromagnetic theory and the equipartition principle to derive an equation for black body radiation
• This law works well for long wavelengths but predicts ultraviolet catastrophe, since energy emission would theoretically becomes infinite at short wavelengths at short wavelengths

Planck's Quantum Hypothesis

• Max Planck proposed departure from classical physics
• Suggested that energy gets emitted or absorbed in discrete quanta, rather than continuously
• Introduced the concept of energy quanta
• E=hv, where h is Planck's constant and v is the frequency of radiation
• Used this hypothesis to derive the correct formula for black body radiation: E = 2πc²/λ⁵ * 1/(e^(hc/λkT) - 1)

Quantum Hypothesis and Black Body Radiation compared

• According to classical physics, atomic or molecular vibrations, oscillators could gets excited continuously
• Any amount of energy, no matter how small, could excite an oscillator to a higher energy state
• Planck's quantum hypothesis introduced the idea that energy is quantized oscillators can only absorb or emit energy in discrete amounts, or "quanta," proportional to their frequency
• E=hv is the equation of a single quantums energy, where
• h is Planck's constant and v is the frequency of the oscillator

Resolution of the Ultraviolet Catastrophe

• The classical theory Predicted that the energy radiated by a black body would increased without bound as the frequency increased
• This led to what became known as the "ultraviolet catastrophe".

Application to the Photoelectric Effect

• Planck's quantum hypothesis not only resolved the problem of black body radiation but also laid the groundwork for quantum mechanics
• Einstein's application of this hypothesis to the photoelectric effect further validated the idea that energy is quantized
• This led to better understanding of the interaction between light and matter
• In 1905, Albert Einstein applied the concept of quantization to explain the photoelectric effect
• Postulated light itself is quantized into packets of energy, aka, photons

Einsteins findings

• Each photon carries an energy with the formula E = hv
• When light shines on a metal surface, electrons are ejected only if the energy of the incoming photons is sufficient to overcome the work function of the metal which, is the minimum energy needed to eject an electron
• What Einstein Called as the Threshold Frequency

The Photoelectric Effect and Experimental Observations

• This groundbreaking explanation provided a clear, quantized mechanism for the photoelectric effect
• Classical wave theory of light could not adequately explain this
• The photoelectric effect refers to the ejection of electrons from a material (typically a metal) when it is irradiated with light (usually visible or ultraviolet)
• Experimental observations, known as laws of photoelectricity are described below

Proportionality of Photoelectric Current to Light Intensity

• the total photoelectric current which is the number of electrons ejected per second os proportional to the intensity of the incident light
• this sugests that more photons, with a higher intensity can cause more electrons to get ejected
• For each metal, a threshold frequency (v₀) exists, below which no electrons are emitted, no matter the light intensity
• This implies that energy of incident photons must exceed minimum value (work function W) to eject electrons
• Maximum Kinetic Energy of Electrons is Independent of Light Intensity and depends on the frequency of the incident light, not its intensity. It increases linearly with frequency

Further Photoelectric Information points

• From a classical perspective, higher intensity should theoretically increase the energy of the ejected electrons, which is unexpected
• The relationship is expressed as: K.E.max = hv - W
• Here h is Planck's constant, v is the frequency of the incident light, and W is the work function of the metal
• Classical electromagnetic theory, treats light as a continuous wave is not fit to explain the photoelectric effect
• According to this view, kinetic energy of the ejected electrons should depend on the intensity of the light and the photoelectric effect should occur at any frequency, provided the intensity is sufficiently high

More information points

• Einstein extended Planck's idea of quantization by stating: that light consists of discrete packets of energy called photons
• Each photon carries an energy E=hv.
• The key concept: that light behaves as both a wave and a particle (wave-particle duality, which contradicts classical electromagnetic theory, which treated light purely as a wave
• The photoelectric effect demonstrates that energy exchange between light and matter occurs in discrete quanta, not continuously
• Einstein's explanation was experimentally verified by Robert Millikan in 1916
• Verifying the quantum nature of light
• Einstein's quantum theory successfully explains the photoelectric effect by treating light as consisting of photons with energy E=hv

The Compton Effect:

• Is known as Compton scattering
• A fundamental phenomenon in quantum mechanics that showcases the particle-like nature of electromagnetic radiation, particularly X-rays and gamma rays
• Discovered by Arthur Holly Compton in 1923, for which he was awarded the Nobel Prize in Physics in 1927.
• Photon-Electron Collision:
  • When a high-energy photon (such as an X-ray or gamma ray) collides with a stationary electron, it transfers some of its energy and momentum to the electrons
• The result is that that the photon scatters at an angle relative to original direction with its wavelength increases (energy decreases)

Wavelength Shift

• The Compton wavelength shift formula is ∆λ = λ' − λ = h/(mₑc)(1 − cos θ)
  • λ is the initial wavelength of the photon, λ' is the wavelength after scattering
  • h is Planck's constant (6.626 × 10⁻³⁴ Js), me is the electron rest mass (9.109 × 10⁻³¹ kg), c is the speed of light (3 × 10⁸ m/s), θ is the scattering angle of the photon
  • The term h/(mₑc) is known as the Compton wavelength of the electron with a value of approximately 2.426 × 10⁻¹² m

Importance

• The effect provided conclusive evidence for the dual nature of light (wave-particle duality)
• It supported the quantum theory of radiation and contributed to development of quantum mechanics
• It showcases that photons carry momentum, which is key concept in modern physics

Atomic Spectra

• It has been observed that when an electric discharge is passed through an element in the gaseous state, light will be emitted.
• Analysis by a prism or grating spectrometer shows sharp lines of a a definite wavelength, that is characteristic of a element
• For elements such as hydrogen, this line spectrum turns out to gets more complex for heavier elements

More on Atomic Spectra

• In 1883, Liveing and Dewar realized that several possible series existed in the spectra of alkali and alkaline earth metals
• They could not, then, discover an empirical relation to present the order
• In 1885, Balmer discovered an equation whose values are extremely good in agreement to those that are observed in hydrogen

The general equation

• ν = R (1/n₁² - 1/n₂²)
  • n₁ = 1 is the Lyman series, in the ultraviolet spectrum
  • n₁ = 2 is the Balmer series, and covers the visible spectrum