Atomic Structure, Electron Configurations, and the Hydrogen Spectrum

Atomic Structure and the Photoelectric Effect

Photoelectric Effect

The photoelectric effect refers to the emission of electrons from a material (usually a metal) when it absorbs light of sufficient energy. The minimum energy required to remove an electron from the surface is called the work function.

• Threshold Frequency: The minimum frequency of light needed to cause electron emission.

• Einstein's Photoelectric Equation: $E_{photon} = h\nu = \phi + KE_{electron}$, where $\phi$ is the work function.

• Example: If the work function of potassium is $3.68 \times 10^{-19}$ J, and the incident light has energy greater than this, electrons will be emitted.

Hydrogen Spectrum and the Lyman Series

Lyman Series Equation

The Lyman series describes the set of spectral lines in the ultraviolet region resulting from electron transitions to the $n=1$ energy level in hydrogen.

• Equation: $\frac{1}{\lambda} = 1.097 \times 10^7 \left(1 - \frac{1}{n^2}\right)$, where $n = 2, 3, ...$

• Application: Used to calculate the wavelength of emitted photons when electrons fall to the ground state.

• Example: For $n=2$, $\lambda$ can be calculated for the first line of the Lyman series.

Atomic Orbitals and Electron Configurations

Electron Configuration Principles

Electron configuration describes the arrangement of electrons in an atom's orbitals. The ground state configuration follows the Aufbau principle, Pauli exclusion principle, and Hund's rule.

• Aufbau Principle: Electrons fill orbitals in order of increasing energy.

• Pauli Exclusion Principle: No two electrons in an atom can have the same set of quantum numbers.

• Hund's Rule: Electrons occupy degenerate orbitals singly before pairing.

• Example: The electron configuration for molybdenum (Mo) is [Kr] $5s^1 4d^5$ (due to stability from half-filled $d$ subshell).

Identifying Elements from Electron Configurations

Electron configurations can be used to identify elements:

• [Ar] 3d10 4s2 4p5: Bromine (Br)

• [Ne] 3s2 3p5: Chlorine (Cl)

• [Ar] 3d5 4s1: Chromium (Cr)

• [Kr] 5d5 6s1: Molybdenum (Mo)

• [Xe] 4f14 5d10 6s2 6p6: Radon (Rn)

Quantum Numbers and Atomic Orbitals

Quantum Numbers

Quantum numbers describe the properties of atomic orbitals and the electrons in them:

• Principal Quantum Number ($n$): Indicates the energy level.

• Angular Momentum Quantum Number ($l$): Indicates the shape of the orbital ($s$, $p$, $d$, $f$).

• Magnetic Quantum Number ($m_l$): Indicates the orientation of the orbital.

• Spin Quantum Number ($m_s$): Indicates the spin direction ($+\frac{1}{2}$ or $-\frac{1}{2}$).

Tables

Electron Configurations and Element Identification

Electron Configuration	Element Name
[Ar] 3d10 4s2 4p5	Bromine (Br)
[Ne] 3s2 3p5	Chlorine (Cl)
[Ar] 3d5 4s1	Chromium (Cr)
[Kr] 5d5 6s1	Molybdenum (Mo)
[Xe] 4f14 5d10 6s2 6p6	Radon (Rn)

Correct Orbital Diagram for Molybdenum

Subshell	Electron Distribution
5s	↑
4d	↑ ↑ ↑ ↑ ↑

Additional info: The correct configuration for Mo is [Kr] 5s1 4d5 due to the stability of half-filled d subshells.

Key Equations

• Photoelectric Effect: $E_{photon} = h\nu = \phi + KE_{electron}$

• Lyman Series: $\frac{1}{\lambda} = 1.097 \times 10^7 \left(1 - \frac{1}{n^2}\right)$