CHEM10003: Lectures 1-8 Summary

Absorption vs Emission Spectroscopy

• Spectroscopy Overview:

  • Analyzes matter by observing light interaction (absorption or emission).
  • General principle: energy input, light output.

Absorption Spectroscopy

• Process: Sample absorbs specific wavelengths of incoming light.
• Energy Transition: Moves from a lower to a higher energy state.
• Output: Measures what light is absorbed by the sample (missing wavelengths).
• Quantization: Only discrete energy states are accessed, indicating quantization.

Emission Spectroscopy

• Process: Sample emits light of specific wavelengths after energy input (typically heat).
• Energy Transition: Moves from a higher to a lower energy state, emitting light.
  • Heating leads to transitions to higher energy states.
  • Emission occurs as the sample returns to a lower energy state (cold sample).
• Quantization: Light is emitted from discrete energy states, indicating quantization.

Relevance of Atomic Emission Spectra

• Astrophysics/Astrochemistry: Determines the composition of stars.
• Atomic Structure: Helps in understanding the electronic structure of atoms.
• Element Characterization: Each chemical element has a unique atomic emission spectrum.
  • Different elements emit light of different energies.

Emission Spectrum of Hydrogen Atom

• Electronic Structure: Emission lines correspond to different higher energy (electronic excited) states.
  • Hydrogen atom contains one electron which changes configuration in each excited state.
• Line Convergence: Emission lines are closer in the higher energy region.
• Energy Transition: Electron falls from a higher excited state ($E<em>2$) to a lower energy state ($E</em>1$), emitting a photon.
• Energy Difference: The energy difference ($\Delta E$) equals the energy of the emitted photon.
  • $\Delta E = E<em>2 - E</em>1 = h\nu$ where $h$ is Planck's constant and $\nu$ is frequency.
    *Video Recommendation: Quantum dots for real-world application of energy levels and emitted light.

Rydberg Equation

• Purpose: Predicts the positions of all lines in the hydrogen atom emission spectrum.
• Equation: $\nu = R<em>H \left( \frac{1}{n</em>{final}^2} - \frac{1}{n_{initial}^2} \right)$
  • $R_H$ is the Rydberg constant ($3.28984 \times 10^{15} s^{-1}$).
  • $n<em>{initial}$ and $n</em>{final}$ are initial and final energy levels (integers).
• Example Calculation: Electron falls from $n=2$ to $n=1$.
  • $\nu = R_H \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = 2.47 \times 10^{15} s^{-1} = 2.47 \times 10^{15} Hz$
  • Note: 1 Hz = 1 s-1

Energy of Emitted Light

• Equation: $\Delta E = -h\nu = -hR<em>H \left( \frac{1}{n</em>{final}^2} - \frac{1}{n<em>{initial}^2} \right) = -2.179 \times 10^{-18} J \left( \frac{1}{n</em>{final}^2} - \frac{1}{n_{initial}^2} \right)$.
• Sign Convention: Minus sign indicates energy release.
• Example Calculation: Electron falls from $n=2$ to $n=1$.
  • $\Delta E = -2.179 \times 10^{-18} J \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = -1.633 \times 10^{-18} J$

Emission Series

• Description: Emission lines observed in different regions of the electromagnetic spectrum based on the final energy level.
• Series and Regions:
  • Lyman: UV region, $n<em>{final} = 1$, $n</em>{initial} = 2, 3, 4, 5, 6, …$
  • Balmer: Visible region, $n<em>{final} = 2$, $n</em>{initial} = 3, 4, 5, 6, 7, …$
  • Paschen: Far IR region, $n<em>{final} = 3$, $n</em>{initial} = 4, 5, 6, 7, 8, …$
  • Brackett: Far IR region, $n<em>{final} = 4$, $n</em>{initial} = 5, 6, 7, 8, 9, …$
  • Pfund: Far IR region, $n<em>{final} = 5$, $n</em>{initial} = 6, 7, 8, 9, 10, …$

Rydberg Equation: Additional Points

• $n<em>{final}$ and $n</em>{initial}$ are quantum numbers (integers).
• Smallest possible value for $n$ is 1.
• For emission: $n<em>{final} < n</em>{initial}$ and \Delta E < 0.
• For absorption: $n<em>{final} > n</em>{initial}$ and \Delta E > 0.
• $Z$ is the atomic number (equation works for one-electron atoms/ions).
• Ionization energy can be calculated as a transition from $n<em>{initial} = 1$ to $n</em>{final} = \infty$.
• Equation: $\Delta E = -2.179 \times 10^{-18} J \left( \frac{1}{n<em>{final}^2} - \frac{1}{n</em>{initial}^2} \right) Z^2$

Bohr Model

• Quantization: Electrons in the hydrogen atom can only assume discrete energy levels.
• Model Description: Modified Rutherford model to include quantization of electronic energy.
  • Each orbit is associated with an energy level.
  • Higher energy levels correspond to orbits further from the nucleus.
• Quantum Number: Each orbit is associated with a quantum number ($n = 1, 2, 3, 4, …$).
  • Absorption increases $n$.
  • Emission decreases $n$.
• Limitations:
  • Fails to describe emission spectra of many-electron atoms.
  • Cannot explain the intensity of spectral lines.
  • Treats electrons as particles only, neglecting wave-like behavior.

Background Info: Momentum and Kinetic Energy

• Kinetic Energy: A moving particle has kinetic energy.
  • $E_{kin} = \frac{1}{2} mv^2$
• Momentum: A moving particle has momentum.
  • $p = mv$
  • Momentum is conserved during collisions.

Photoelectric Effect

• Process: Electromagnetic radiation strikes a metal surface, causing electron emission.
• Observations:
  • Electrons are only ejected if the light reaches a critical (threshold) frequency.
  • Kinetic energy of ejected electrons is proportional to the frequency of incoming light.
  • Above the threshold frequency, increasing light intensity ejects more electrons.

Einstein's Explanation of the Photoelectric Effect

• Quantization: Based on Planck’s concept of quantized energy.
• Light as Particles: Light consists of discrete bundles of energy called photons ($E = h\nu$).
  • Photons have wave nature and wavelength (color).
• Momentum Transfer: Photons carry momentum and transfer it to electrons during collisions.
• Critical Energy: Only photons with sufficient energy can eject electrons.
• Intensity: Higher intensity light contains more photons, leading to more collisions and emitted electrons.

Photoelectric Effect: Energy Considerations

• Energy Balance: Incoming photon energy must exceed the binding energy of the electron to the metal ($E_{bind}$).
• Kinetic Energy: Excess energy is transferred to the electron as kinetic energy ($E_{kin}$).
  • $E<em>{kin\,e^-} = h\nu</em>{incoming} - E<em>{bind} = h\nu</em>{incoming} - h\nu_0$

Wave-Particle Duality of Light

• Wave Properties: Light has characteristic wavelengths (or frequency) and amplitude.
• Particle Properties: Light has discrete energy (quantized).
  • Energy of one photon: $E = h\nu$
  • Number of photons in a light beam relates to intensity (square of amplitude).

Wave-Particle Duality of Particles

• Double Slit Experiment: Electrons, under certain circumstances, behave like waves, creating interference patterns.

De Broglie Wavelength

• Concept: Matter can be assigned a wavelength.
• Equation: $\lambda = \frac{h}{p} = \frac{h}{mv}$
  • $\lambda$: de Broglie wavelength
  • $h$: Planck's constant
  • $p$: momentum
  • $m$: mass
  • $v$: velocity

Why Heavy Particles Don't Behave Like Waves

• Macroscopic Example: A ball (mass = 0.164 kg, speed = 3.33 m/s).
  • $\lambda = \frac{h}{mv} = \frac{6.626 \times 10^{-34} Js}{0.164 kg \times 3.33 m/s} = 1.21 \times 10^{-33} m$
• Microscopic Example: An electron at the same speed.
  • $\lambda = \frac{h}{mv} = \frac{6.626 \times 10^{-34} Js}{9.1094 \times 10^{-31} kg \times 3.33 m/s} = 0.219 \times 10^{-3} m$

Heisenberg's Uncertainty Principle

• Principle: It is impossible to accurately determine both the exact position and momentum (velocity) of a particle simultaneously.
• Mathematical Expression:
  • $\Delta x \Delta p \geq \frac{h}{4\pi}$
  • $\Delta x m \Delta v \geq \frac{h}{4\pi}$
  • $\Delta x$: uncertainty in position
  • $\Delta p$: uncertainty in momentum
  • $\Delta v$: uncertainty in velocity
• Relevance: Significant for tiny, light particles, not for everyday observations.

Uncertainty Principle: Examples

• Car Example: Uncertainty in position is negligible.
  • $\Delta x \geq \frac{h}{4\pi m \Delta v} \approx \frac{6.626 \times 10^{-34}}{4 \pi \times 1000 \times 0.01} m = 5 \times 10^{-36} m$
• Electron Example: Uncertainty in position is significant.
  • $\Delta x \geq \frac{h}{4\pi m \Delta v} \approx \frac{6.626 \times 10^{-34}}{4 \pi \times 9.1094 \times 10^{-31} \times 0.01} m = 6 \times 10^{-3} m = 6 mm$

Implications for Atomic Models

• Bohr Model Limitations: Heisenberg's principle indicates that electrons do not move on well-defined orbits.
• Probabilistic Model: We can only determine probabilities for electron locations in an atom - regions of high and low probability.

Useful Equations

• Relationships between wavelength, frequency, and speed of light:
  • $c = \lambda \nu$
  • $c = \nu \tilde \nu$
  • $\tilde \nu = \frac{1}{\lambda}$
• Planck-Einstein Relation:
  • $E = h\nu$
• Rydberg Equation:
  • $\nu = R<em>H \left( \frac{1}{n</em>{final}^2} - \frac{1}{n_{initial}^2} \right)$
• Particle momentum:
  • $p = mv$
• Kinetic energy of emitted electron:
  • $E<em>{kin\,e^-} = h\nu</em>{incoming} - E<em>{bind} = h\nu</em>{incoming} - h\nu_0$
• de Broglie wavelength:
  • $\lambda = \frac{h}{p} = \frac{h}{mv}$
• Heisenberg’s Uncertainty Principle:
  • $\Delta x \Delta p \geq \frac{h}{4\pi}$
  • $\Delta x m \Delta v \geq \frac{h}{4\pi}$
• $\Delta E = -2.179 \times 10^{-18} J \left( \frac{1}{n<em>{final}^2} - \frac{1}{n</em>{initial}^2} \right) Z^2$

Requirements for a Better Model

• Describe quantized energy levels of electrons.
• Describe emission spectra of one-electron and many-electron atoms.
• Account for the wave nature of electrons.
• Incorporate the Heisenberg Uncertainty Principle (probabilities of electron locations).
• Solution: Quantum Mechanics and the Schrödinger Equation.

Schrödinger Equation

• Description: Mathematical formulation for treating the quantum world.
• Wavefunction Approach: Describes the standing wave for an electron with the wavefunction $\Psi$.
  • Wavefunctions can have positive and negative regions (phases).
  • Regions where the wavefunction is zero are called nodes.
• Schrödinger's Assumption: An electron bound to a nucleus behaves like a standing wave.
• Equation: $\hat{H} \Psi = E \Psi$
  • $\hat{H}$: Hamiltonian operator
  • $\Psi$: wavefunction
  • $E$: energy of the particle

Meaning of the Wavefunction

• Mathematical Construct: Wavefunctions are mathematical constructs that describe the state of a quantum-mechanical particle.
• Phases and Nodes: Wavefunctions have regions with positive and negative signs (phases), separated by nodes where the wavefunction is zero.
• Information Content: The wavefunction of an electron contains all its information.
• Probability: Square of the wavefunction ($\Psi^2$) is related to the probability of finding a particle at a particular point in space.
  • Areas under the curve tells us the probability of finding a particle in a specific region (aka: we perform an integration!)

Electron Density

• Calculation: Square of the electronic wavefunction allows calculation of electron density.
• Interpretation: Electron density indicates the probability of finding an electron around an atomic nucleus.
• Visualization: Typically plotted as isosurfaces (surfaces of constant electron density).

Schrödinger Equation Details

• Hamiltonian: Operator that acts on $\Psi$ to calculate energy ($E$) of the particle(s).
  • Contains kinetic and potential energy terms.
• Applications: Can be solved exactly for one-electron systems (e.g., H atom).
• Approximations: Requires approximations for many-electron systems; handled by Quantum Chemists.

Solving Schrödinger Equation for H Atom

• Components of Hamiltonian:
  • Kinetic energy of the electron.
  • Interaction between electron and proton (nucleus).
• Results: Several energy levels are obtained - ground state and excited states.
• Atomic Orbitals: Wavefunctions obtained for each energy level.
  • Orbital: A one-electron wavefunction in three-dimensional space.

Energy Levels in H-Atom

• Equation: From Schrödinger Equation matches Rydberg Equation.
  • $E_n = -2.179 \times 10^{-18} J \frac{1}{n^2}$
  • Electron bound to atom has negative energy.
  • Lower (more negative) the energy is, the more stable the chemical system.
• $n$: principal quantum number (integer values starting at 1).
• Electronic Shell: Each value of $n$ represents an electronic shell ($n = 1$: first shell, $n = 2$: second shell, etc.).

Quantum Numbers and Orbitals

• Degeneracy: Energetically higher shells (except $n = 1$) have multiple orbitals with the same energy (degenerate).
• New Quantum Numbers: Angular momentum quantum number ($l$) and magnetic momentum quantum number ($m_l$) are needed to account for these orbitals.

Meaning of Quantum Numbers

• Principal Quantum Number ($n$):
  • $n = 1, 2, 3, 4, 5, …$
  • Determines energy for that shell.
  • Determines size of orbital (more spatial expansion for larger $n$).
• Angular Momentum Quantum Number ($l$):
  • $l = 0, 1, 2, …, (n-1)$
  • Determines the type and shape of that orbital.
  • $l = 0$: s orbital, $l = 1$: p orbital, $l = 2$: d orbital, $l = 3$: f orbital.
• Magnetic Momentum Quantum Number ($m_l$):
  • $m_l = -l, (-l+1), …, -1, 0, 1, …, (l-1), l$
  • Determines the orientation of the orbital.

Shapes of Orbitals

• s Orbitals ($l = 0$): Spherical.
• p Orbitals ($l = 1$): Dumbbell-shaped with two lobes of different signs and a nodal plane (angular node) between the lobes.
• d Orbitals ($l = 2$), ml determines orientation, 5 orbitals in subshell.
• f Orbitals ($l = 3$), ml determines orientation, 7 orbitals in subshell.

Mathematical Functions

• An orbital depends on three variables.
  * Radius and two angles.
  * Can be described as product of two functions.

Radial and Angular Component

• Depends on $l$ and ${m_l}$. Shape and Orientation. Can contain angular nodes.
• Depends on $n$. Size of orbital. Can contain radial nodes.

Expression for the 1s Orbital

• (you do NOT have to memorise these mathematical expressions for the exam!)
• $\psi<em>{1s}(r,\theta,\phi) = A</em>{1s} e^{-\frac{r}{a_0}}$
• $A<em>{1s} = \frac{1}{\sqrt{\pi a</em>0^3}}$
• Where $a_0=0.53 \times 10^{-10} m=53$ pm (Bohr radius).
• The radial component decays exponentially. This tells me something about the angular part (shape).

Phases

• Nodes are areas where the phases change.
• The angular component of an orbital can have angular nodes
• The radial component can have radial/spherical nodes.

Radial Distribution Function

• At any distance $r$, the volume of a shell element is $4\pi r^2 dr$
• $R^2(r)4\pi r^2$
• dr. (you do NOT have to memorise this mathematical expression for the exam!)
  However, it is important you understand what the plots on the next slide mean and how to interpret them!

Boundary Surface

• Plot an orbital and define a "boundary surface”.
• The surface captures 90% of the electron probability.

Hydrogen Summary

• Three-dimensional functions.
• Contain a radial and angular function.
• Both radial and angular functions can have different phases (signs). Areas between the phases (the functions are zero) are called radial and angular nodes, respectively.
• Their squares give rise to probability distributions, which are connected to electron densities, which can be observed.

Atomic Orbitals Tool

• Orbitals are very powerful tools that help us understand Chemistry.
• They are regularly used to explain: The Periodic Table, Molecular shapes, Chemical bonding, Chemical reactivity.
• https://phet.colorado.edu/en/simulations/hydrogen-atom