Comprehensive Study Guide on Electron Configurations, Line Spectra, and Ionisation Energies

Electromagnetic Spectrum and Waves

• Electromagnetic Radiation & Spectrum Definition

  • The electromagnetic spectrum is defined as the entire range of frequencies or wavelengths of electromagnetic radiation.

  • Electromagnetic radiation moves in oscillating waves through space.

• Fundamental Wave Properties

  • Wavelength ($\lambda$): The distance measured between two consecutive crests in an oscillating wave. Expressed in meters ($m$).

  • Frequency ($f$ or $v$): The number of wave cycles that pass a given fixed point per second. Expressed in hertz ($Hz$) or reciprocal seconds ($s^{-1}$).

  • Speed of Light ($c$): The speed at which electromagnetic waves travel in a vacuum, given as $c = 3.00 \times 10^8 accession m\,s^{-1}$.

• Relationships Between Wavelength, Frequency, and Energy

  • Frequency and Wavelength: Inversely proportional to each other. As wavelength increases, frequency decreases.

    • Low frequency corresponds to long wavelength ($\lambda$).

    • High frequency corresponds to short wavelength ($\lambda$).

  • Energy and Frequency: Directly proportional to each other.

    • As radiation energy increases, frequency increases accordingly.

    • High energy radiation corresponds to high frequencies and short wavelengths.

    • Low energy radiation corresponds to low frequencies and long wavelengths.

• Regions of the Electromagnetic Spectrum

  • The spectrum is divided into seven distinct regions, arranged in order of wavelength, frequency, or energy:

    1. Radio waves: Longest wavelength ($10^0\,m$ to $10^4\,m$), lowest frequency ($10^4\,Hz$ to $10^8\,Hz$), least energy. Used to broadcast radio, telephone, and television signals.

    2. Microwaves: Wavelengths around $10^{-2}\,m$, frequency around $10^{10}\,Hz$. Used in cooking, radar, telephone, and other communication signals.

    3. Infrared (IR): Wavelengths from $10^{-6}\,m$ to $10^{-4}\,m$, frequency around $10^{12}\,Hz$. Transmits heat from the sun, fires, and radiators.

    4. Visible light: Wavelengths from $400\,nm$ ($4 \times 10^{-7}\,m$) to $700\,nm$ ($7 \times 10^{-7}\,m$), frequency around $10^{14}\,Hz$. Makes objects visible to the human eye.

    5. Ultraviolet (UV): Wavelengths around $10^{-8}\,m$, frequency around $10^{16}\,Hz$. Absorbed by the skin; used in fluorescent tubes.

    6. X-rays: Wavelengths around $10^{-10}\,m$, frequency around $10^{18}\,Hz$. Used to view internal structures of bodies and objects.

    7. Gamma rays ($\gamma$-rays): Shortest wavelength ($10^{-12}\,m$ to $10^{-16}\,m$), highest frequency ($10^{20}\,Hz$), most energy. Used in medicine for killing cancer cells.

• Mathematical Equations and Constants

  • Wave Equation:         $c = \lambda f$

    • Where $c$ is the speed of light ($3.00 \times 10^8\,m\,s^{-1}$), $\lambda$ is wavelength in meters ($m$), and $f$ is frequency in $s^{-1}$ or $Hz$

  • Planck Relation (Quantum Energy of a Photon):         $E = hf$         $E = hv = \frac{hc}{\lambda}$

    • Where $E$ is energy in Joules ($J$)

    • $h$ is Planck's constant = $6.626 \times 10^{-34}\,J\,s$ (or $4.136 \times 10^{-15}\,eV\,s$

    • $f$ (or $v$) is frequency in $s^{-1}$

    • $c$ is speed of light = $3.00 \times 10^8\,m\,s^{-1}$

    • $\lambda$ is wavelength in meters ($m$

• Visible Spectrum Details and Data Booklet Values

  • Wavelength region spans from $400\,nm$ to $700\,nm$ ($4 \times 10^{-7}\,m$ to $7 \times 10^{-7}\,m$

  • Spectral order (VIBGYOR):

    • Violet: $400\,nm$ (highest energy, highest frequency, shortest wavelength)

    • Blue: $424\,nm$

    • Green: $491\,nm$

    • Yellow: $575\,nm$

    • Orange: $585\,nm$

    • Red: $647\,nm$ to $700\,nm$ (lowest energy, lowest frequency, longest wavelength)

Emission and Absorption Spectra

• Continuous Spectrum vs. Line Spectrum

  • Continuous Spectrum: Displays all wavelengths or frequencies of light across the visible spectrum continuously without gaps, ranging seamlessly from red to violet.

  • Emission Line Spectrum: Displays only specific discrete wavelengths or frequencies of light emitted by energized atoms, appearing as isolated colored lines against a dark background.

  • Absorption Spectrum: Produced when light passes through a cold gas; a spectrometer compares transmitted radiation relative to incident radiation. Displays dark absorption lines at specific wavelengths against a continuous spectrum, representing radiation absorbed by atoms moving from lower to higher energy levels.

• Electron Transitions and Energy Absorption/Emission

  • Electrons are subatomic particles responsible for the absorption and emission of electromagnetic light in atoms.

  • Absorption: An electron absorbs a discrete, exact photon packet of energy to move (transition) from a lower energy level (ground state) to a higher energy level (excited state).

  • Excited State: An unstable atomic configuration where electrons reside in higher energy levels relative to the ground state.

  • Emission: The unstable electron returns to a lower energy level by emitting a photon containing an energy quantity equal to the energy absorbed.

  • The amount of energy released corresponds directly to the energy difference between the two energy levels involved in the transition.

• Atomic Fingerprinting via Spectroscopy

  • Every chemical element possesses a unique electron configuration, thereby producing its own distinct emission line spectrum.

  • Spectroscopy can be utilized to identify unknown elements by matching emission lines against reference spectra of known elements (e.g., Helium, Oxygen, Neon, Argon, Xenon).

• Hydrogen Emission Spectrum

  • When an electric discharge passes through a tube containing hydrogen gas, hydrogen molecules dissociate into atoms and emit light.

  • Passing this light through slits and a prism produces specific discrete lines in the visible region at exact wavelengths:

    • $656.2\,nm$ (Red)

    • $486.1\,nm$ (Blue-green)

    • $434.0\,nm$ (Blue)

    • $410.0\,nm$ (Violet-blue)

Bohr Model and Atomic Energy Levels

• Main Energy Levels (Principal Quantum Numbers)

  • Electrons reside in main energy levels denoted by the principal quantum number $n$ ($n = 1, 2, 3, 4, 5, 6, \dots$

  • $n = 1$ represents the energy level closest to the nucleus and lowest in energy.

  • Ground State: The lowest possible energy state of an atom where all electrons occupy the lowest available energy levels.

  • As $n$ increases, both the distance of the orbital from the nucleus and its total energy increase.

• Convergence of Energy Levels

  • Energy levels in an atom are not evenly spaced.

  • Main energy levels get progressively closer together (converge) at higher energy values.

  • Consequently, spectral lines in an emission spectrum converge at higher frequencies/energies.

• Hydrogen Spectral Series and Transitions

  • Ultraviolet (UV) Region: Transitions from higher energy levels ($n \ge 2$) dropping down to the ground state ($n = 1$). Represents the highest energy transitions.

  • Visible Light Region (Balmer Series): Transitions from higher energy levels ($n \ge 3$) dropping down to $n = 2$. Corresponds to visible emissions.

  • Infrared (IR) Region: Transitions from higher energy levels ($n \ge 4$) dropping down to $n = 3$. Represents lower energy transitions.

• Ionisation and the Convergence Limit

  • At $n = \infty$, the convergence limit is reached.

  • At $n = \infty$, the electron is completely removed from the electrostatic attraction of the nucleus, resulting in the formation of an ion (ionisation).

• Limitations of Bohr's Model

  • Does not account for electron sub-levels ($s, p, d, f$

  • Does not accurately account for the maximum capacity or arrangement of electrons per level in multi-electron atoms.

  • Only works accurately for one-electron hydrogenic systems (e.g., hydrogen atom).

Sub-Levels and Atomic Orbitals

• Definitions

  • Sub-levels ($s, p, d, f$): Sub-divisions within main energy levels based on the specific geometry of atomic orbitals.

  • Atomic Orbital: A specific region of three-dimensional space around the nucleus where there is a high probability of finding an electron.

  • Each atomic orbital can hold a maximum of 2 electrons.

• Orbital Shapes and Orientations

  • $s$ Orbitals: Spherical shape centered on the nucleus ($1s, 2s, 3s$

  • $p$ Orbitals: Dumbbell-shaped/lobed structures aligned along three mutually perpendicular spatial axes:

    • $p_x$ orbital along the x-axis

    • $p_y$ orbital along the y-axis

    • $p_z$ orbital along the z-axis

• Energy Relationships of Sub-levels

  • Within any main energy level, sub-level energies increase in the following order:         s < p < d < f

• Energy Level and Orbital Capacity Summary

  • Main Energy Level $n = 1$:

    • Sub-levels present: $1s$

    • Number of $s$ orbitals: 1

    • Number of $p$ orbitals: 0

    • Number of $d$ orbitals: 0

    • Number of $f$ orbitals: 0

    • Total orbitals: 1

    • Maximum electron capacity: 2

  • Main Energy Level $n = 2$:

    • Sub-levels present: $2s, 2p$

    • Number of $s$ orbitals: 1

    • Number of $p$ orbitals: 3

    • Number of $d$ orbitals: 0

    • Number of $f$ orbitals: 0

    • Total orbitals: 4

    • Maximum electron capacity: 8

  • Main Energy Level $n = 3$:

    • Sub-levels present: $3s, 3p, 3d$

    • Number of $s$ orbitals: 1

    • Number of $p$ orbitals: 3

    • Number of $d$ orbitals: 5

    • Number of $f$ orbitals: 0

    • Total orbitals: 9

    • Maximum electron capacity: 18

  • Main Energy Level $n = 4$:

    • Sub-levels present: $4s, 4p, 4d, 4f$

    • Number of $s$ orbitals: 1

    • Number of $p$ orbitals: 3

    • Number of $d$ orbitals: 5

    • Number of $f$ orbitals: 7

    • Total orbitals: 16

    • Maximum electron capacity: 32

Electron Configurations

• Fundamental Principles for Filling Orbitals

  • Aufbau Principle: Electrons occupy orbitals of lower energy levels first before filling higher energy orbitals.

  • Pauli Exclusion Principle: An atomic orbital can accommodate a maximum of two electrons, and they must possess opposite spin states.

  • Hund's Rule: When filling degenerate orbitals (orbitals belonging to the same sub-level with identical energy, such as $2p_x, 2p_y, 2p_z$), each orbital receives a single electron with parallel spin before any orbital receives a second electron.

• Sub-level Filling Order in Order of Increasing Energy

  • 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s

  • Note: $4s$ fills prior to $3d$ because $4s$ sits at a lower energy level in uncharged atoms.

• Full Electron Configurations for Elements $Z = 1$ to $11$

  • Hydrogen ($\text{H}$, $Z=1$): $1s^1$

  • Helium ($\text{He}$, $Z=2$): $1s^2$

  • Lithium ($\text{Li}$, $Z=3$): $1s^2\,2s^1$

  • Beryllium ($\text{Be}$, $Z=4$): $1s^2\,2s^2$

  • Boron ($\text{B}$, $Z=5$): $1s^2\,2s^2\,2p^1$

  • Carbon ($\text{C}$, $Z=6$): $1s^2\,2s^2\,2p^2$

  • Nitrogen ($\text{N}$, $Z=7$): $1s^2\,2s^2\,2p^3$

  • Oxygen ($\text{O}$, $Z=8$): $1s^2\,2s^2\,2p^4$

  • Fluorine ($\text{F}$, $Z=9$): $1s^2\,2s^2\,2p^5$

  • Neon ($\text{Ne}$, $Z=10$): $1s^2\,2s^2\,2p^6$

  • Sodium ($\text{Na}$, $Z=11$): $1s^2\,2s^2\,2p^6\,3s^1$

• Ground State vs. Excited State Examples

  • Sodium Ground State: $1s^2\,2s^2\,2p^6\,3s^1$

  • Sodium Excited State: $1s^2\,2s^2\,2p^6\,3p^1$ (the valence electron in $3s$ is promoted to $3p$

• Periodic Table Blocks

  • s-block: Groups 1A ( alkali metals) and 2A (alkaline earth metals), plus Helium.

  • p-block: Groups 3A to 8A (main group elements).

  • d-block: Transition metals (Groups 3B to 2B).

  • f-block: Lanthanides and Actinides.

• Condensed Noble Gas Core Configurations

  • $\text{[He]}$ represents $1s^2$

  • $\text{[Ne]}$ represents $1s^2\,2s^2\,2p^6$

  • $\text{[Ar]}$ represents $1s^2\,2s^2\,2p^6\,3s^2\,3p^6$

• Exceptions to the Aufbau Principle (Chromium and Copper)

  • Chromium ($\text{Cr}$, $Z=24$):

    • Expected configuration: $\text{[Ar]}\,,4s^2\,3d^4$

    • Actual condensed configuration: $\text{[Ar]}\,,4s^1\,3d^5$

    • Reason: A half-filled $d$ sub-level ($3d^5$) offers additional stability.

  • Copper ($\text{Cu}$, $Z=29$):

    • Expected configuration: $\text{[Ar]}\,,4s^2\,3d^9$

    • Actual condensed configuration: $\text{[Ar]}\,,4s^1\,3d^{10}$

    • Reason: A fully filled $d$ sub-level ($3d^{10}$) offers additional stability.

• Transition Metal Ion Configurations

  • When d-block transition metals form positive ions, electrons are lost from the $4s$ orbital before the $3d$ orbitals.

  • Iron ($\text{Fe}$, $Z=26$): $1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,4s^2\,3d^6$

  • $\text{Fe}^{2+}$ in $\text{FeCl}_2$: $1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,3d^6$ or $\text{[Ar]}\,,3d^6$

  • $\text{Fe}^{3+}$ ion: $1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,3d^5$ or $\text{[Ar]}\,,3d^5$

First Ionisation Energy and Calculations (HL)

• Definition of First Ionisation Energy (IE)

  • The minimum energy required to remove one mole of electrons from one mole of gaseous atoms to form one mole of gaseous $1+$ ions.

  • General chemical equation:         $\text{X}(g) \rightarrow \text{X}^+(g) + e^{-_1}$

• Factors Controlling Electrostatic Attraction

  1. Nuclear Charge: Number of protons in the nucleus. Higher nuclear charge increases attraction for outer electrons.

  2. Atomic Radius: Distance between the nucleus and outer valence electrons. Greater distance reduces electrostatic attraction.

  3. Shielding Effect: Repulsion caused by inner core electrons shielding the outer electrons from the nuclear charge.

• Periodic Trends across Periods and Groups

  • Across a Period (e.g., $\text{Li} \rightarrow \text{Ne}$ or $\text{Na} \rightarrow \text{Ar}$):

    • First ionisation energy generally increases.

    • Reason: Protons are added to the nucleus (increasing nuclear charge) while electrons are added to the same main energy level, maintaining relatively constant shielding.

  • Down a Group:

    • First ionisation energy decreases.

    • Reason: Valence electrons occupy higher principal energy levels further from the nucleus with increased inner-shell shielding.

• Discontinuities / Anomalies in First Ionisation Energy Trends

  • Decrease from Beryllium ($\text{Be}$) to Boron ($\text{B}$) [and $\text{Mg}$ to $\text{Al}$]:

    • $\text{Be}: 1s^2\,2s^2$ vs $\text{B}: 1s^2\,2s^2\,2p^1$

    • The removed electron in Boron comes from a $2p$ sub-level, which is higher in energy and slightly further from the nucleus than the $2s$ electron in Beryllium.

  • Decrease from Nitrogen ($\text{N}$) to Oxygen ($\text{O}$) [and $\text{P}$ to $\text{S}$]:

    • $\text{N}: 1s^2\,2s^2\,2p^3$ vs $\text{O}: 1s^2\,2s^2\,2p^4$

    • In Oxygen, the removed electron comes from a doubly occupied $2p$ orbital.

    • Inter-electron repulsion between the paired electrons in the same orbital reduces the energy required to remove one of them.

• Calculating Ionisation Energy from Spectral Data

  • Convergence Limit: The highest energy frequency in a line spectrum where lines merge into a continuum ($n = 1 \rightarrow n = \infty$

  • At the convergence limit, the electron has escaped nuclear attraction entirely.

  • Ionisation energy per mole is calculated using the Planck relation and Avogadro's constant ($N_A = 6.02 \times 10^{23}\,mol^{-1}$):         $E_{photon} = hf = \frac{hc}{\lambda}$         $E_{molar} = E_{photon} \times N_A$

• Practice Calculations and Applications

  • **Example 1: Sodium streetlight yellow emission ($\lambda = 589\,nm = 589 \times 10^{-9}\,m$

    • a) **Frequency ($f$             $f = \frac{c}{\lambda} = \frac{3.00 \times 10^8\,m\,s^{-1}}{589 \times 10^{-9}\,m} = 5.09 \times 10^{14}\,s^{-1}$

    • b) **Energy carried by one photon ($E$             $E = hf = (6.63 \times 10^{-34}\,J\,s)(5.09 \times 10^{14}\,s^{-1}) = 3.37 \times 10^{-19}\,J$

    • c) **Energy carried by one mole of photons ($kJ\,mol^{-1}$             $E_{mole} = (3.37 \times 10^{-19}\,J)(6.02 \times 10^{23}\,mol^{-1}) = 203000\,J\,mol^{-1} = 203\,kJ\,mol^{-1}$

  • **Example 2: Ionisation energy of Hydrogen from convergence limit ($f = 3.28 \times 10^{15}\,s^{-1}$

    • Energy of 1 photon:             $E = hf = (6.63 \times 10^{-34}\,J\,s)(3.28 \times 10^{15}\,s^{-1}) = 2.17 \times 10^{-18}\,J$

    • Energy per mole:             $E_{mole} = (2.17 \times 10^{-18}\,J)(6.02 \times 10^{23}\,mol^{-1}) = 1.31 \times 10^6\,J\,mol^{-1} = 1310\,kJ\,mol^{-1}$

  • Example 3: Ionisation threshold of Copper

    • Minimum photon wavelength to ionise copper: $\lambda = 1.57 \times 10^{-7}\,m$

    • Electromagnetic spectrum region: Ultraviolet (UV) region.