Introduction to Electronic Structure Atoms

Introduction to Electronic Structure

• Electronic structure involves the arrangement and energy of electrons in atoms.

• The chapter begins with discussing the concept of waves, which is crucial for understanding the electronic structure of extremely small particles.

Understanding Waves

Nature of Electromagnetic Radiation

• To grasp electronic structure, one must comprehend electromagnetic radiation.

• Wavelength (λλ): Defined as the distance between corresponding points on adjacent waves.

• Frequency (νν): The number of waves passing a given point per unit time.

• Relationship: In waves moving at the same speed, a longer wavelength corresponds to a smaller frequency.

Calculating Frequency

• Example: If the time period for waves is one second, then frequencies would be given as 2 s−12 s−1 and 4 s−14 s−1 respectively.

Speed of Light

• All electromagnetic radiation travels at the same velocity, defined as the speed of light (cc), which is approximately 3.00×1083.00×108 m/s.

• Relationship: c=λνc=λν.

Sample Exercises on Wavelength and Frequency

Solution to Exercise 6.1

• (a) Wave with longer wavelength has a lower frequency: ν=cλν=λc​.

• (b) Infrared radiation has a longer wavelength than visible light, thus can be identified accordingly.

Practice Exercises

1. Identify possible wavelengths of infrared light from given options.

2. Distinguish between blue and red light from described waves.

Energy and Light

The Nature of Energy

• The wave nature of light cannot explain the phenomenon of glowing objects at increased temperatures.

Introduction to Quanta

• Max Planck's Concept: Energy exists in discrete packets known as quanta (singular: quantum).

The Photoelectric Effect

• Albert Einstein's Contribution: Used the concept of quanta to explain the photoelectric effect where:

  • Different metals eject electrons at specific energy levels.

  • Energy (EE) is proportional to frequency: E=hνE=hν, where hh is Planck's constant, 6.626×10−346.626×10−34 J∙s.

Sample Exercise 6.3: Energy Calculations

• Given λ=589λ=589 nm, calculating energy of photon involves:

  1. Convert wavelength to frequency: ν=cλν=λc​ leading to ν=5.09×1014ν=5.09×1014 s−1−1.

  2. Use E=hνE=hν to find E=3.37×10−19E=3.37×10−19 J.

  3. For one mole of photons: Total Energy =(6.02×1023 photons/mol)×(3.37×10−19 J/photon)=2.03×105 J/mol=(6.02×1023 photons/mol)×(3.37×10−19 J/photon)=2.03×105 J/mol.

Practice Exercises

1. Formulate expressions for energy of a mole of photons.

2. Calculate energy of a laser emitting at frequency 4.69×10144.69×1014 s−1−1 and energy of a pulse with 5.0×10175.0×1017 photons.

Atomic Emissions

Spectra Observed in Atoms

• Emission spectra reveal energy emitted by atoms but do not show continuous spectra; instead, discrete line spectra appear unique to each element.

The Hydrogen Spectrum

• Johann Balmer (1885): Identified a formula relating hydrogen spectral lines to integers.

• Johannes Rydberg: Advanced the formula further.

• Niels Bohr's Explanation: Integrated Planck’s assumptions into his model.

The Bohr Model

Fundamental Principles

1. Electrons occupy specific orbits corresponding to certain energies.

2. Only specific energies are allowed for electrons, leading to no radiation in those states.

3. Energy Absorption/Emittance: Governed by E=hνE=hν, moving electrons between allowed states.

4. Change in energy, ΔE=−hcRH(1nf2−1ni2)ΔE=−hcRH(nf21​−ni21​) where RH=1.097×107RH=1.097×107 m−1−1.

Limitations of the Bohr Model

• This model only accurately describes hydrogen and does not account for electron dynamics from classical physics.

Important Ideas from the Bohr Model

1. Electrons exist only in discrete energy levels.

2. Electronic transitions involve energy changes.

The Wave Nature of Matter

Concept by Louis de Broglie

• Proposed that matter would have wave properties, summarizing it through the equation: λ=hmvλ=mvh​.

Calculating de Broglie Wavelength

• Example of an electron at speed 5.97×1065.97×106 m/s and its mass is 9.11×10−319.11×10−31 kg.

Heisenberg Uncertainty Principle

• More precise knowledge of particle momentum results in a less precise knowledge of its position; described mathematically as: (Δx)(Δmv)≥h4π(Δx)(Δmv)≥4πh​.

Quantum Mechanics Overview

Introduction by Erwin Schrödinger

• Developed a mathematical framework integrating the wave-particle duality, resulting in quantum mechanics.

Wave Functions and Orbitals

• The square of the wave function (ψ2ψ2) represents electron density, indicating the probability of an electron's presence.

Quantum Numbers

• A solution to Schrödinger's equation yields a set of quantum numbers that describe energy levels and spatial configurations of electrons.

Principal Quantum Number (nn)

• Specifies energy levels; integer values begin from 1.

Angular Momentum Quantum Number (ll)

• Determines orbital shape with values from 0 to n−1n−1.

• Designated as:

  • l=0l=0 (s)

  • l=1l=1 (p)

  • l=2l=2 (d)

  • l=3l=3 (f)

Magnetic Quantum Number (mlml​)

• Describes the 3D orientation of orbitals, with permissible integer values ranging from −l−l to ll.

Electron Shells and Subshells

• A collection of orbitals with the same value of nn constitutes an electron shell; orbitals sharing both nn and ll values form subshells.

Orbital Shapes and Properties

s Orbitals

• l=0l=0 values are spherical with increasing radius as nn increases. The peak count equals nn and nodes total to n−1n−1.

p Orbitals

• l=1l=1 yields two-lobed structures with a node.

d Orbitals

• l=2l=2 shapes include four-lobed orientations and resemble p orbitals with accompanying nodes.

f Orbitals

• Complex geometries with seven equivalent orbitals, located in a sublevel with l=3l=3.

Electron Configuration Basics

Definition

• The arrangement of electrons in an atom is termed electron configuration, with the lowest energy state being termed ground state.

Orbital Diagrams

• Visual representation with boxes for orbitals and half-arrows indicating electron spin.

Hund's Rule

• Stipulates that the lowest energy configuration is achieved when electrons with the same spin populate each degenerate orbital singly before pairing occurs.

Condensed Electron Configurations

Concept

• Group elements are characterized by having similar outer shell electron configurations, grouped by valence electrons.

Notable Patterns

• Similar patterns occur; for instance, halogens exhibit an ns2np5ns2np5 configuration.

Electron Configuration Anomalies

• Transitional irregularities may arise at the half-filled and fully filled conditions within the d-orbitals.

Example: Chromium's Configuration

• The expected configuration is altered to [Ar]4s13d5[Ar]4s13d5 due to energy proximity of the 4s and 3d orbitals.

Understanding Waves

Speed of Light

• Relationship: $c = \lambda \nu$.

Solution to Exercise 6.1

• (a) Wave with longer wavelength has a lower frequency: $\nu = \frac{c}{\lambda}$.

Energy and Light

The Photoelectric Effect

• Energy ($E$) is proportional to frequency: $E = h\nu$, where $h$ is Planck's constant, $6.626 \times 10^{-34}$ J$\cdot$s.

Sample Exercise 6.3: Energy Calculations

• Convert wavelength to frequency: $\nu = \frac{c}{\lambda}$ leading to $\nu = 5.09 \times 10^{14}$ s$^{-1}$.

• Use $E = h\nu$ to find $E = 3.37 \times 10^{-19}$ J.

• For one mole of photons: Total Energy $= (6.02 \times 10^{23} \text{ photons/mol}) \times (3.37 \times 10^{-19} \text{ J/photon}) = 2.03 \times 10^{5} \text{ J/mol}$.

The Bohr Model

Fundamental Principles

3. Energy Absorption/Emittance: Governed by $E = h\nu$, moving electrons between allowed states.

4. Change in energy, $\Delta E = -hcRH(\frac{1}{nf^2} - \frac{1}{ni^2})$ where $RH = 1.097 \times 10^7$ m$^{-1}$.

The Wave Nature of Matter

Concept by Louis de Broglie

• Proposed that matter would have wave properties, summarizing it through the equation: $\lambda = \frac{h}{mv}$.

Heisenberg Uncertainty Principle

• Described mathematically as: (Delta x)(Delta mv) ge h/4pi}.

Quantum Mechanics Overview

Wave Functions and Orbitals

• The square of the wave function (psi^2) represents electron density.

Quantum Numbers

Angular Momentum Quantum Number ($l$)

• Determines orbital shape with values from 0 to $n-1$.

• Designated as:

  • l=0 (s)

  • l=1 (p)

  • l=2 (d)

  • l=3 (f)

Magnetic Quantum Number ($m_l$)

• Permissible integer values ranging from $-l$ to $l$.

Orbital Shapes and Properties

s Orbitals

• l=0 values are spherical with increasing radius as n increases. The peak count equals n and nodes total to n-1.

p Orbitals

• l=1 yields two-lobed structures with a node.

d Orbitals

• l=2 shapes include four-lobed orientations and resemble p orbitals with accompanying nodes.

f Orbitals

• Seven equivalent orbitals, located in a sublevel with l=3.

Condensed Electron Configurations

Notable Patterns

• Halogens exhibit an ns^2, np^5 configuration.

Example: Chromium's Configuration

• The expected configuration is altered to [Ar]4s^1, 3d^5 due to energy proximity of the 4s and 3d orbitals.

Understanding Waves

• Relationship: $c = \lambda \nu$ .

• Frequency from wavelength: $\nu = \frac{c}{\lambda}$ .

Energy and Light

• Energy ($E$) is proportional to frequency (Photoelectric Effect): $E = h\nu$ , where $h$ is Planck's constant, $6.626 \times 10^{-34} \text{ J}\cdot\text{s}$ .

The Bohr Model

• Energy Absorption/Emittance: Governed by $E=h\nu$ .

• Change in energy: $\Delta E = -hcR<em>H\left(\frac{1}{n</em>f^2} - \frac{1}{n<em>i^2}\right)$ where $R</em>H=1.097 \times 10^7 \text{ m}^{-1}$ .

The Wave Nature of Matter

• de Broglie Wavelength: $\lambda = \frac{h}{mv}$ .

Heisenberg Uncertainty Principle

• Mathematically described as: $(\Delta x)(\Delta mv) \ge \frac{h}{4\pi}$ .

Quantum Mechanics Overview

• The square of the wave function: $\psi^2$ represents electron density.

Quantum Numbers

• Angular Momentum Quantum Number ($l$): values from 0 to $n-1$ .

  • $l=0$ (s)

  • $l=1$ (p)

  • $l=2$ (d)

  • $l=3$ (f)

• Magnetic Quantum Number ($m_l$): permissible integer values ranging from $-l$ to $l$ .

Orbital Shapes and Properties

• s Orbitals: $l=0$ , peaks equal $n$ , nodes total to $n-1$ .

• p Orbitals: $l=1$ .

• d Orbitals: $l=2$ .

• f Orbitals: $l=3$ .

Condensed Electron Configurations

• Halogens configuration: $ns^2np^5$ .

• Chromium's Configuration anomaly: $[\text{Ar}]4s^13d^5$.