Electronic Structure: Wave Nature of Light

Electronic Structure

• This chapter focuses on the arrangement and energy of electrons, which is the electronic structure.
• Extremely small particles have properties best explained through wave behavior.

The Wave Nature of Light

• Understanding electromagnetic radiation is crucial to grasping the electronic structure of atoms
• Electromagnetic radiation travels as waves at light speed.
• Wavelength (λ) is the distance between similar points on adjacent waves.
• Frequency (v) is the number of waves passing a point per unit of time.
• For waves at the same velocity, longer wavelengths mean smaller frequency.
• Electromagnetic radiation travels all at the same 2.998 × 10^8 m/s velocity of light (c).
• The relationship between wavelength and frequency: c = vλ

Electromagnetic Radiation

• Common Wavelength Units for Electromagnetic Radiation

• There are many types of electromagnetic radiation.
• Electromagnetic radiations possess different wavelengths and energies.
• Wavelength units depend on the lengths being measured.

Quantized Energy and Photons

• Three observations involving atoms and electromagnetic radiation that wave behavior cannot explain:
  • Light emission from hot objects (blackbody radiation).
  • Electron emission from light shining on metal surfaces (photoelectric effect).
  • Light Emission from electronically excited gas atoms (emission spectra)

Black Body Radiation

• An object glows when heated.
• The wave nature of light does not explain how an object can glow when its temperature increases.
• Classical physics predicted emission of UV, X-rays, and gamma rays, which was not observed.

Quanta

• Max Planck explained energy by proposing that energy comes in packets called quanta.
• Quantum is the singular version of quanta.

The Photoelectric Effect

• Einstein used quanta to explain the photoelectric effect.
• Each metal has a different energy at which it ejects electrons; at lower energy, electrons are not emitted.
• Energy is proportional to frequency E = hv
  • h, Planck's constant is 6.626 × 10^-34 J⋅s.

Supplemental Equations

• Previously seen equations: vλ = c and E = hv
• Combined equations solve as follows: E = hc/λ
• Work Function (Φ) is the minimum energy to remove an electron from a material's surface in a vacuum.
• Work functions vary for different metals.
• Threshold frequency is related to the work function: hv = Φ
• Threshold wavelength has the solution: hc/λ = Φ

Atomic Emission of Gas

• The emission spectra observed from energy emitted by atoms and molecules was another mystery in the early twentieth century.

Line Spectra and the Bohr Model

• Atoms and molecules do not produce a continuous spectrum; they produce line spectra of discrete wavelengths.
• Each element produces a unique line spectrum.

The Hydrogen Spectrum

• Johann Balmer (1885) found a simple formula relating the four emission lines to integers.
• Johannes Rydberg advanced this formula, where Rh is the Rydberg constant: 1/λ = (Rh) (1/(nf^2) - 1/(ni^2))
• Niels Bohr explained why this mathematical relationship works.

The Bohr Model

• Niels Bohr adopted Planck's assumption explaining these phenomena by these points:
  • Electrons in a hydrogen atom are only permitted in orbits of certain radii corresponding to specific energies.
  • Electrons in allowed orbits are in an "allowed" energy state, do not radiate energy, and do not spiral into the nucleus.
  • Energy is emitted or absorbed only when the electron transitions between energy states as a photon with energy E = hv.
• Electrons in the ground state are in the lowest energy and n = 1.
• Excited states are any n > 1.
• The transitions from one energy level to another can be calculated using ΔE = Ef - Ei = (-2.18 × 10^-18 J) (1/nf^2 - 1/ni^2)
• Positive ΔE means energy is absorbed, meaning a photon is absorbed when nf > n;.
• Negative ΔE means energy is released, meaning a photon is emitted when nf < n;.
• The Bohr model is limited because it only works for hydrogen, which only has one electron.
• Classical physics would show that an electron falls into the positively charged nucleus, however, Bohr assumed it would not.
• The Bohr model does not accommodate the wave-like properties of electrons.

Important Ideas from the Bohr Model

• Electrons exist only in certain discrete energy levels, which are described by quantum numbers.
• Energy is involved in the transition of an electron from one level to another.

The Wave Behavior of Matter

• Louis de Broglie theorized that matter exhibits wave properties if light can have material properties.
• The mass and wavelength relationship was demonstrated as λ = h/mv

The Uncertainty Principle

• Heisenberg proposed that the dual nature of matter (particle and wave) limits how precisely both momentum and position can be known.
• The more precisely momentum is known, the less precisely position is known as: (Δx)(Δmv)≥h/4π

Quantum Mechanics and Atomic Orbitals

• Erwin Schrödinger developed a mathematical treatment incorporating the wave and particle nature of matter, known as quantum mechanics.

Atomic Orbitals

• Solving Schrödinger's wave equation for hydrogen provides wave functions for the electron.
• The solution Schrödinger's wave equations electron density is the square of the electron density, the or probability of where an electron is likely to be at any given time.

Orbitals and Quantum Numbers

• Solving the wave equation provides a set of wave functions or orbitals and their energies
• Each orbital, a spatial distribution of electron density with set of three quantum numbers: n, l, m₁

Principal Quantum Number (n)

• The principal quantum number, n, defines the energy level on which the orbital resides.
• The principal quantum number are positive integral values: 1, 2, 3, ...
• Principal quantum numbers corresponds to the values in the Bohr model.
• Orbitals become larger as n increases and the electron spends more time away from the nucleus.
• The electron also has a higher energy and is less tightly bound to the nucleus as n increases.

Angular Momentum Quantum Number (l)

• This quantum number defines the shape of the orbital.
• The angular momentum quantum number takes integer values from 0 to n-1.
  • If n = 1, then l = 0
  • If n = 2, then l = 0, 1
  • If n = 3, then l = 0, 1, 2
• Letters are designated to distinct values of l:
  • l = 0 is designated s.
  • l = 1 is designated p.
  • l = 2 is designated d.
  • l = 3 is designated f.
• This defines orbital shapes.
• For n = 3, then l = 0, 1, 2 means energy level 3 as 3 possible sublevels designated as 3s, 3p, and 3d
• Energy Level to Possible Sublevels:

Magnetic Quantum Number (ml)

• The three-dimensional orientation of the orbital describes magnetic quantum number.

• Integer values of m₁ range from -/ to /, including 0.

• The number of orbitals at any given energy level include 1 s orbital, 3 p orbitals, 5 d orbitals, and 7 f orbitals

  • For s (l = 0), m₁ = 0
  • For p (l = 1), m₁ = -1, 0, 1
  • For d (l = 2), m₁ = -2, -1, 0, 1, 2

  Sublevel	l	Possible m1 (integers from -I to I)	Total Number of Orbitals per Subshell
  s	0	0	1
  p	1	-1, 0, 1	3
  d	2	-2, -1, 0, 1, 2	5
  f	3	-3, -2, -1, 0, 1, 2, 3	7

Quantum Numbers Summary

• Orbitals sharing the same value of n form an electron shell.
• Different orbital types form subshells within a shell.

Representation of Orbitals: s

• The value of / for s orbitals is 0.
• S Orbitals are spherical in shape.
• The spheres radius increases with the value of n.
• The # of peaks in ns orbital equal n.
• The number of nodes (zero probability of finding an electron) in an ns orbital is n - 1.
• Electron density spreads out as n increases, increasing the probability of finding an electron further from the nucleus.

Representation of Orbitals: p

• In p orbitals the value of the angular momentum quantum number is l = 1.
• Two lobes exist, with a node between them.

Representation of Orbitals: d

• The value angular momentum quantum is l = 2.
• Four of the five d orbitals have four lobes; the remaining one resembles a p orbital with a doughnut around the center.

f Orbitals

• Seven equivalent orbitals in a sublevel with complicated shapes, not shown.
• l = 3

Hydrogen Atom Orbital Energies

• Orbitals have the same energy at the same energy level in a one-electron hydrogen atom.
• Chemists call orbitals with equal energy, degenerate orbitals.

Many-Electron Atoms

• Repulsion increases between electrons as their number increases.
• Not all orbitals at the same energy level are degenerate in atoms with multiple electrons
• Orbital sets in the same sublevel are still degenerate.
• Energy levels begin to overlap, so 4s is lower in energy than 3d.

Spin Quantum Number (ms)

• Two electrons sharing the same orbital have different energies.
• An electrons spin describes its magnetic field, affecting its energy.
• +/- 1/2 are the only 2 allowed values of spin quantum number.

Pauli Exclusion Principle

• No two electrons in the identical atom possesses the same four quantum numbers.
• Thus, no two electrons can have the exact same energy.
• Each electron in an atom must differ by at least one of the four quantum number values, including n, l, m₁, and ms

Electron Configurations

• Electron configuration is how electrons are distributed within an atom.
• Ground state is the name for the organization that has the lowest possible energy and most stable.
• Each component is consists of these three things:
  • a number denoting the energy level, here n = 4
  • a letter denoting the type of orbital, here p.
  • a superscript denoting the number of electrons in those orbitals; in 4p^5 there are 5 electrons.

Orbital Diagrams

• Each box in the diagram represents one orbital.
• Half-arrows represent the electrons.
• The arrow's direction represents the electron's relative spin.

Aufbau Principle

• The aufbau principle, or building-up principle, involves filling subshells in the lowest energy up.

Hund's Rule

• Maximum spin is attained when filling degenerate orbitals, also yielding the lowest energy.
• Each orbital gets a one electron minimum during the process if possible.

Condensed Electron Configurations

• Valence Electrons: elements sharing the same group of the periodic table share the same # of electrons. Core electrons is the name for filled inner shells including nobble gases as well as the completely filled d and f sublevels.
• Noble gas symbols are placed in brackets listing only valence electrons.
  • Helium: 1s2 to Lithium: 1s22s1 can be written as [He]2s1

Transition Metals

• Argon (atomic number 18) finishes period 3 with electron configuration 1s^22s^22p^63s^23p^6.
• Potassium (atomic number 19) might have been expected electrons in 3d but 4s takes precedence.
• Therefore transition metals follow filling of 4s by filling 3d during the fourth period.

Lanthanides and Actinides

• Special and unique names given to the elements that fill the f orbitals.
• Lanthanide elements (atomic numbers 57 to 70) have electrons entering the 4f sublevel.
• Actinide series have electrons entering the 5f sublevel and include Uranium (atomic number 92) and Plutonium (atomic number 94).

Electron Configurations and the Periodic Table

• Orbitals fill with increasing order of energy.
• Different periodic table blocks match different orbital types: S= Blue P = Pink d = orange f= tan, * these orbitals include transition elements
• The s and p blocks are called main-group elements.
• The periodic table is followed directly when discovering the electron configuration among most elements.
• Selenium for the example: Se : [Ar] 4s^2 3d^10 4p^4

Some Anomalies.

• Anomalies or irregularities occur when there is adequate electrons to half-fill s and d orbitals sharing a given row. The element Chromium shows this for the reason that [Ar] 4s13d5 would be found rather than the predicted [Ar] 4s23d4 configuration.
• The 4s and 3d orbitals occur near enough in energy for all those that occur within f-block atoms.

Group Electron Configurations

Li =[He]2s1 Na =[Ne]3s1 K = [Ar]4s1 Rb = [Kr]5s11 Cs = [Xe]6s1 Fr = [Rn]7s1

Excited States

• Atoms can gain energy causing electrons in the ground state can gain energy and progress to an excited state.
• Ground state configuration can be expressed as: K 1s^22s^22p^63s^23p^64s^1 as an example compared to exited state expression as: Example: K 1s^22s^22p^63s^23p^65p^1

Electron Configurations for Ions

• Atoms will loose or gain electrons to become cations and anions. This reaction tend to make them adopt electron configurations that are common among noble gases. K Metal - 1s^22s^22p^63s^23p^64s^1 K+ ion – 1s^22s^22p^63s^23p^6 Ar atom – 1s^22s^22p^63s^23p^6
• Atoms and ions with 1s^22s^22p^6 are isoelectronic having the same EC. (more in chapter 7)