Exam Study Notes

Electromagnetic Radiation

• Electromagnetic radiation is quantized.
• It can be viewed as a stream of particles called photons.
• The energy of one photon is given by: E_{photon} = hν, where:
  • h is Planck's constant.
  • ν is the frequency.
  • c is the speed of light.
  • \lambda is the wavelength.

Einstein's Equation and Photon Mass

• Einstein's equation: E = mc^2, indicating that energy has mass.
• The apparent mass of a photon can be calculated using E_{photon} = hν = mc^2.
• Therefore, the mass of a photon is: m = \frac{hν}{c^2} = \frac{h}{λc}.

Dual Nature of Light

• Electromagnetic radiation exhibits a dual nature:
  • Wave-like: Light travels as a wave.
  • Particulate: Light consists of photons (particles) with mass and energy.

Energy and Wavelength Relationship

• E_{photon} = hν, which means energy is directly proportional to frequency (E \propto ν).
• E_{photon} = \frac{hc}{λ}, which means energy is inversely proportional to wavelength (E \propto \frac{1}{λ}).

de Broglie's Equation

• For particles other than light, the de Broglie equation relates wavelength to momentum: \lambda = \frac{h}{mu}, where:
  • m is the mass of the particle.
  • u is the velocity of the particle.

Electromagnetic Spectrum

• The electromagnetic spectrum includes (from high to low energy):
  • Gamma rays
  • X-rays
  • Ultraviolet (UV) light
  • Visible light
  • Infrared radiation
  • Microwaves
  • Radio waves
• Visible light consists of a continuous spectrum of colors (ROYGBIV).

Atomic Line Emission Spectrum

• Continuous Spectrum: Contains light of all wavelengths and colors (e.g., sunlight through a prism).
  • Wavelength range: 400 nm - 700 nm
• Line Spectrum: Shows only certain colors or specific wavelengths of light (e.g., emission spectrum of hydrogen atom).
  • Obtained by exciting atoms (e.g., H_2(g) \rightarrow 2H(g)) and analyzing the emitted light.

Energy Absorption and Emission

• When atoms absorb energy (hν), they become excited (contain excess energy).
  • Ground State: Lowest energy state (stable).
  • Excited State: High energy state (unstable).
    • hν = ΔE, where ΔE is the energy difference between the ground and excited states.
• An electron in an atom is promoted to a higher energy level upon absorbing energy. The excess energy is released as light of various wavelengths, producing a line emission spectrum.

Quantized Energy

• Specific light of specific energy is emitted: ΔE = hν = \frac{hc}{λ}.
• The energy of the electron in the atom is quantized.

Analogy: Flying from Muscat to Dubai

• The process of an electron gaining and losing energy is analogous to an airplane flying from Muscat to Dubai.
  • Excited state: Ascending (gaining altitude).
  • Ground state: Descending (losing altitude).
  • Energy is gained or lost in specific amounts, similar to the quantized energy levels of an electron.

Hydrogen Emission Spectrum

• The line spectrum of hydrogen has only 4 visible lines, indicating only certain energies are allowed.
• Energy levels are quantized.
• Electronic transitions between energy levels result in the emission of specific wavelengths of light.
  • n = 6 \rightarrow n = 2 (violet)
  • n = 5 \rightarrow n = 2 (blue)
  • n = 4 \rightarrow n = 2 (green)
  • n = 3 \rightarrow n = 2 (red)

Bohr Model

• Energy Levels: n = 1, 2, 3, … , ∞
• Energy of an electron in the hydrogen atom: En = -R{hc} \frac{Z^2}{n^2}, where:
  • R_{hc} = 2.179 \times 10^{-18} J (Rydberg constant)
  • Z = nuclear charge (for hydrogen, Z = +1)

Energy Level Calculations

• Examples of energy levels:
  • E_1 = -2.179 \times 10^{-18} J
  • E_2 = -5.448 \times 10^{-19} J
  • E_3 = -2.421 \times 10^{-19} J
  • E_4 = -1.362 \times 10^{-19} J
• As energy increases, the energy levels get closer together.
• Energy change during a transition: ΔE = E{nf} - E{ni} = -2.179 \times 10^{-18} J (\frac{1}{nf^2} - \frac{1}{ni^2})

Electron Transitions

*In the ground state, the electron of the hydrogen atom occupies the energy level n = 1. When the hydrogen atom is in an excited state, the electron occupies higher energy levels (1 < n < ∞). When the hydrogen atom ionizes, the electron goes to n = ∞.

Examples of Energy and Wavelength Calculations

• Calculate the energy of light emitted during an electronic transition from n = 4 to n = 2.
• Calculate the wavelength of that light.
• Explain why the line spectrum does not contain a line corresponding to the transition n = 6 \rightarrow n = 1.

Example Calculation: Electronic Transition

Consider the electronic transition n = 4 \rightarrow n = 2 (emission):
(a) Calculate \lambda
(b) Calculate the energy in kJ/mol

(a)
\Delta E = -2.179 \times 10^{-18} J (\frac{1}{2^2} - \frac{1}{4^2}) = -4.086 \times 10^{-19} J/quantum

λ = \frac{hc}{ΔE} = \frac{6.626 \times 10^{-34} Js \times 2.9979 \times 10^8 m/s}{4.086 \times 10^{-19} J} = 4.861 \times 10^{-7} m = 486.1 nm

This corresponds to green light.

(b) Convert -4.086 × 10^{-19} J/photon to kJ/mol.

\frac{-4.086 \times 10^{-19} J}{photon} \times \frac{6.022 \times 10^{23} photons}{mol} \times \frac{kJ}{10^3 J} = -246.1 kJ/mol

Example Calculation: Energy Level Identification

Consider the electronic transition n = x \rightarrow n = 3, with -129.6 kJ/mol emitted.

(a) Can we see the radiation emitted?

(b) Identify the energy level n = X.

(a) Calculate the wavelength:

\frac{-129.6 kJ}{mol} \times \frac{10^3 J}{1 kJ} \times \frac{1 mol}{6.022 \times 10^{23} photons} = -2.152 \times 10^{-19} J/photon

λ = \frac{hc}{ΔE} = \frac{6.626 \times 10^{-34} Js \times 2.9979 \times 10^8 m/s}{2.152 \times 10^{-19} J} = 9.231 \times 10^{-7} m = 923.1 nm

Since λ is outside the range of 400-700 nm, we cannot see the light emitted.

(b) Identify the energy level n = X:

-2.152 \times 10^{-19} J = -2.179 \times 10^{-18} J (\frac{1}{3^2} - \frac{1}{X^2})

\frac{-2.152 \times 10^{-19} J}{-2.179 \times 10^{-18} J} = \frac{1}{9} - \frac{1}{X^2}

0.09876 = 0.1111 - \frac{1}{X^2}

\frac{1}{X^2} = 0.01235

X^2 = 80.97

X = 9

Ionization Energy Calculation

• Calculate \lambda and E_{photon} for the electromagnetic radiation with the minimum amount of energy required to ionize the hydrogen atom: H(g) \rightarrow H^+(g) + e^- ( n = 1 \rightarrow n = ∞ ).

Quantum Mechanics

Location of the Electron

• Bohr Model: When the hydrogen atom is in the ground state, its electron occupies the innermost circular orbit at a fixed distance of 52.92 pm from the nucleus. The location of the electron is assumed to be fixed.

• Heisenberg Uncertainty Principle: The exact position and momentum of the electron in the hydrogen atom are uncertain: \Delta x \cdot \Delta (mv) ≥ \frac{h}{4π}, where
  \Delta x
  is the uncertainty in position and
  \Delta (mv)
  is the uncertainty in momentum.
  $

• This means the more certain we are of the position of a particle, the less certain we are of its momentum, and vice versa, \Delta x \cdot \Delta v ≥ \frac{h}{4mπ}, where m is mass

  • Because electrons are small and have a small mass, exact possition of electron around the nucelus is unknown

Schrödinger Equation

• We can calculate the probability of finding an electron at a particular point at a given time.
• The solution to the Schrödinger equation is a wave function, which defines the region in space where an electron with a particular energy is likely to be found.
• ĤΨ(x, y, z) = EΨ(x, y, z)
  • Ĥ: Hamiltonian operator. Represents the total energy of the system, including kinetic and potential energy.
  • Ψ(x, y, z): Wave function. Describes the quantum state of the electron in terms of its spatial coordinates.
  • E: Energy of the electron.
• When the Schrondinger equation is analyzed, solutions are obtained. Each solution is called a wavefunction Ψ (x, y, z) and has its own value of E

Atomic Orbitals

• A specific wave function is called an orbital. Atoms have atomic orbitals.
• A wave function consists of a radial function and an angular function: Ψ(r, θ, φ) = R(r)Y(θ, φ)
• The square of the wave function, Ψ^2, defines the distribution of electron density in 3-D space around the nucleus, indicating the probability of finding the electron at a particular point or in a certain region in space outside the nucleus.
• An atomic orbital with a specific wave function, Ψ(x, y, z), has a characteristic energy and a characteristic distribution of electron density. It defines a 3-D region in space outside of the nucleus where the electron is most likely to be found.
• Atomic orbital is characterized with 3 quantum numbers: n, l, ml
  • n: energy of the orbital
  • l: type or shape of the orbital
  • ml: orientation or direction of the orbital

Electron Density

• Ψ^2 gives the locations of the electrons around the nucleus and the probability of finding an electron around a nucleus.

Radial Distribution Function

• The probability of finding an electron in thin spherical shells of uniform thickness at distances r from the nucleus.
  • 4πr^2R^2
• Quantum mechanics: the electron of the hydrogen atom in the ground state can be found anywhere outside the nucleus, but the most probable distance away from the nucleus is 52.92 pm (Bohr radius).

Comparison of Bohr and Quantum Mechanical Models

Quantum Numbers

• Quantum numbers characterize an atomic orbital in 3-dimensional space (x,y,z).
  • n, l, ml define a specific atomic orbital Ψ.

Principal Quantum Number (n)

• Values of n: 1, 2, 3, 4, 5, … ∞
• n is related to the size and energy of the atomic orbital.

Angular Momentum Quantum Number (l)

• Values of l: 0 → n-1 (integers)
• l is related to the shape or type of the atomic orbital.
  • n = 1 (1st energy level): l = 0
  • n = 2 (2nd energy level): l = 0, 1
  • n = 3 (3rd energy level): l = 0, 1, 2

| l | 0 | 1 | 2 | 3 | 4 | 5 |
| :---- | :-: | :-: | :-: | :-: | :-: | :-: |
| Letter | s | p | d | f | g | h |

Magnetic Quantum Number (ml)

• Values of ml: +l, …, 0, …, -l
• ml is related to the orientation or direction of the atomic orbital in space. ml= (2l+1)
  • l = 0 (s-orbital): ml = 0 (1 value)
  • l = 1 (p-orbital): ml = +1, 0, -1 (3 values)
  • l = 2 (d-orbital): ml = +2, +1, 0, -1, -2 (5 values)
    *

Shells and Subshells

• Shell: A set of atomic orbitals with the same value of n (e.g., n=3 includes 3s, 3p, 3d orbitals).
  • Number of subshells = number of different kinds of orbitals in an energy level.
• Subshell: A set of atomic orbitals with the same value of l (e.g., p-subshell has 3 orbitals: px, py, pz).

Quantum Number Summary for Atomic Orbitals

Orbital Shapes and Energies

s-orbitals

• l = 0; spherical shape.
• Size and energy increase as n increases.
• Number of nodes for s-orbitals: n - 1
  • Areas of zero electron density
• Number of radial nodes: n - l - 1

p-orbitals

• l = 1; dumbbell shape.
• Three p-orbitals: px, py, pz (same energy, degenerate).
• Two lobes separated by a node at the nucleus.
  • Number of radial nodes: n - l - 1

d-orbitals

• l = 2; more complex shapes.
• Five d-orbitals: dxy, dxz, dyz, dx2-y2, dz2.
  • Number of radial nodes: n - l - 1 (p316 textbook)

f-orbitals

• First occur in level n = 4; l = 3
• Seven f-orbitals.

Electron Spin

• When an electron spins, it produces a magnetic field.
• Electron spin quantum number (ms): +1/2, -1/2.
• Pauli Exclusion Principle: No two electrons in an atom can have the same set of four quantum numbers (n, l, ml, ms).

Chemical Bonding

Valence Electrons

• Electrons outside of the noble gas core which are used for chemical bonding.
• The number of valence electrons for a given element is indicated by the Group Number of the element.

Lewis Dot Symbols

• An arrangement of valence electrons around an atom using dots. E.g. H, He, Li, Be , B, C, N, O, F & Ne