CH 6 (11/18) (PG 4-10)

Atom Line Spectra: unique patterns of bright lines produced by light emitted by excited atoms when their electrons transition from high to lower energy levels
Gases can be excited, meaning their electrons can be promoted to higher energy states.
As electrons transition from higher to lower energy states, light is emitted.
The emitted light has only specific wavelengths exclusive to each element.

Niels Bohr proposed that electrons in atoms exist only in specific discrete orbits called stationary states.
Electrons are confined to quantized energy states.
The energy of an electron in a hydrogen atom is described by the formula: $E = -\frac{C}{n^2}$
- where C is a constant, and n is the quantum number (values: 1, 2, 3, …).

Early atomic structure suggested that electrons orbit the nucleus similar to planets around the sun.
Any orbit should theoretically be possible; thus, any energy level should also be possible.
However, a charged particle in an electric field changing direction emits energy, leading to a predicted collapse of the atom as the electron would lose energy and spiral into the nucleus.

Bohr's Assertion: The line spectra of elements indicate specific energy states for electrons.
- The energy levels are quantized; thus only certain energy distances from the nucleus are permissible.
Formula for energy levels in hydrogen: $E = -\frac{C}{n^2}$
- C is a constant related to the energy of the hydrogen atom.
- n can take values of 1, 2, 3, …, leading to the characterization of energy levels such as ground (n=1) and excited states (n>1).
- Energy is zero when n = ∞, indicating that the electron is completely separated from the nucleus.

Determining Photon Wavelength: The transition between energy levels (n) can be determined with the energy formulas and constants:
- Where the constant for energy in transitions is given as
  $R_{hc} = 2.18 \times 10^{-18} J$
- Constants:
- R (Rydberg constant) = $1.0974 \times 10^{7} m^{-1}$
- h (Planck's constant) = $6.626 \times 10^{-34} J s$
- c (speed of light) = $2.998 \times 10^{8} m s^{-1}$
- The sign of change in energy ( $\Delta E$ ) indicates absorption (+) or emission (-).
- Photon wavelength: λ = (hc)/|E|

λ = (h)/(mv)
$h = 6.626 \times 10^{-34} J s$ (Planck's constant)
m = mass
v = velocity
mv = momentum
The wave-particle duality applies mainly to small particles like electrons; larger objects like golf balls do not exhibit observable wave properties.

States that it is impossible to simultaneously determine the exact position and momentum (mass times velocity) of an object.
Most relevant for electrons, which are treated with wave mechanics due to their unique behavior at small scales.

Building upon Bohr's, de Broglie's, and Heisenberg's theories, Erwin Schrödinger proposed a model where:
- Matter (especially electrons) behaves as both a wave and a particle.
- Wave Function (Ψ) describes the properties of the electron.
- Electrons have quantized energy levels expressed through solving the Schrödinger equation, where Ψ² provides the probability distribution of finding electrons in regions of space (orbitals).

Quantum numbers arise from Schrödinger's equation, describing the state of an electron.
- Principal Quantum Number (n): specifies the energy shell (n = 1, 2, 3,…).
- Angular Momentum Quantum Number (l): specifies the subshell (l = 0 to n-1).
- Magnetic Quantum Number (ml): values from -l to +l, defining individual orbitals.

Each orbital can hold up to 2 electrons defined by four quantum numbers.
The Pauli Exclusion Principle states that no two electrons can have the same set of four quantum numbers. This principle establishes each electron's unique identity within an atom.