Study Notes on Atomic Structure and Periodicity

Atomic Structure and Periodicity

Table of Contents

• 7.1 Electromagnetic Radiation

• 7.2 The Nature of Matter

  • The Photoelectric Effect

• 7.3 The Atomic Spectrum of Hydrogen

• 7.4 The Bohr Model

• 7.5 The Quantum Mechanical Model of the Atom

  • The Physical Meaning of a Wave Function

• 7.6 Quantum Numbers

• 7.7 Orbital Shapes and Energies

• 7.8 Electron Spin and the Pauli Principle

• 7.9 Polyelectronic Atoms

• 7.10 The History of the Periodic Table

• 7.11 The Aufbau Principle and the Periodic Table

• 7.12 Periodic Trends in Atomic Properties

  • Ionization Energy

  • Electron Affinity

  • Atomic Radius

• 7.13 The Properties of a Group: The Alkali Metals

  • Information Contained in the Periodic Table

7.1 Electromagnetic Radiation

• Energy travels through space by electromagnetic radiation (EMR).

• Examples: Sunlight, microwaves, X-rays, and radiant heat.

• Properties of waves:

  • Wavelength ($λ$): Distance between consecutive peaks/troughs.

  • Symbol: Lowercase Greek letter lambda ($λ$).

  • Frequency ($ν$): Number of waves/cycles per second.

  • Symbol: Lowercase Greek letter nu ($ν$).

  • Units: Hertz (Hz) where $1 ext{ Hz} = 1 ext{ s}^{-1}$.

• Speed of Light ($c$): $c = 2.9979 imes 10^8 ext{ m/s}$.

• Relationship: $c = λν$ (Speed of light equals wavelength times frequency).

• Inverse relationship: As wavelength increases, frequency decreases, and vice versa.

7.2 The Nature of Matter

• Historical concept: Democritus and Leucippus introduced atomic theory around 400 B.C. with no experimental evidence until the 19th century.

• Key figures:

  • Lavoisier: Quantitative data on chemical reactions.

  • Dalton: First systematic atomic theory based on stoichiometric experiments.

• Atoms are not discrete; they were once thought to consist of particles (mass) and energy (waves). No intermingling was assumed until the early 20th century, necessitated a new theory (quantum mechanics) to explain experimental results.

  • Max Planck: Proposed quantization of energy to explain blackbody radiation. Energy can change in discrete amounts ($E = nhν$ where $n = 1, 2, 3, …$).

  • Einstein: Proposed electromagnetic radiation as quantized particles (photons).

• Photoelectric Effect: Electrons are emitted from metal surfaces when light strikes them under certain conditions.

The Photoelectric Effect

• Characteristics:

  1. Specific threshold frequency ($ν0$) exists below which no electrons are emitted.

  2. Frequencies below $ν0$ result in no electron emission, regardless of intensity.

  3. Frequencies above $ν0$, the number of emitted electrons increases with intensity.

  4. Kinetic energy increases linearly with frequency above threshold frequency.

• Explanation: Photons have energy proportional to their frequency; excess energy becomes kinetic energy of emitted electrons.

7.3 The Atomic Spectrum of Hydrogen

• Emission spectrum of hydrogen consists of discrete wavelengths, confirming electron energy levels are quantized.

• Continuous Spectrum: Produced by white light through a prism (all wavelengths).

• Line Spectrum: Consists of specific lines corresponding to the transitions between energy levels.

  • Example: Wavelength emissions like $410 ext{ nm}$, $434 ext{ nm}$, etc. by hydrogen atoms.

7.4 The Bohr Model

• Proposed by Niels Bohr (1913): Electrons exist in quantized orbits.

• Bohr's assumptions lead to the derivation of energy levels in hydrogen:

  • Energy level formula: $E_n = - rac{2.178 imes 10^{-18} ext{ J}}{n^2}, n = 1, 2, 3, …$

• The model fits experimental data, such as the quantization of hydrogen emission spectra.

7.5 The Quantum Mechanical Model of the Atom

• Emerged mid-20th century, emphasizing wave properties of electrons.

• Schrödinger Equation: A mathematical expression describing electron states in terms of wave functions (orbitals).

  • $ext{Wave function} (Ψ)$ corresponds to the probability density of finding an electron.

• Heisenberg Uncertainty Principle: Cannot precisely know both position and momentum of a particle.

  • $ΔxΔ(p) ≥ rac{h}{4 ext{π}}$

7.6 Quantum Numbers

• Quantum numbers describe orbitals:

  • Principal $(n)$: Size/energy of the orbital. Integer values $1, 2, …$.

  • Angular Momentum $(ℓ)$: Shape of orbitals. Values $0$ (s), $1$ (p), $2$ (d), $3$ (f), etc.

  • Magnetic $(m)$: Orientation in space, ranging $-ℓ$ to $ℓ$.

7.7 Orbital Shapes and Energies

• s orbitals: Spherical shape, size increases with $n$.

• p orbitals: Lobed shape, labeled as $p<em>x$, $p</em>y$, $p_z$.

• d orbitals: Complex shapes with multiple lobes, first appear at $n=3$.

• Orbitals of the same principal quantum number are typically degenerate.

7.8 Electron Spin and the Pauli Principle

• Fourth quantum number: Electron Spin $(m_s)$: Two values $ rac{1}{2}$ or $- rac{1}{2}$.

• Pauli Exclusion Principle: No two electrons in an atom can have the same four quantum numbers, thus each orbital can hold 2 electrons with opposite spins.

7.9 Polyelectronic Atoms

• Polyelectronic atoms are predicted to have electron configurations similar to hydrogen but are affected by electron-electron repulsions.

  • Electron correlation problem: Difficulties in finding exact solutions due to repulsions between multiple electrons.

7.10 The History of the Periodic Table

• Mendeleev and Meyer developed the first periodic tables.

• Mendeleev predicted properties of undiscovered elements based on their placements.

7.11 The Aufbau Principle and the Periodic Table

• Aufbau Principle: Electrons fill the lowest energy orbitals first.

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

7.12 Periodic Trends in Atomic Properties

• Ionization Energy: Energy required to remove an electron. General trend: increases across a period and decreases down a group.

• Electron Affinity: Energy change when adding an electron, often negative (exothermic).

• Atomic Radius: Size of an atom; increases down a group and decreases across a period.

7.13 The Properties of a Group: The Alkali Metals

• Elements: Lithium, sodium, potassium, rubidium, cesium.

• Properties include low ionization energies and high reactivity, with the general trend of decreasing ionization energy down the group.

• Example Reaction: Sodium with chlorine produces sodium chloride: $2 ext{Na}(s) + ext{Cl}_2(g) → 2 ext{NaCl}(s)$

Key Terms and Review Questions

• Electromagnetic Radiation: Includes wavelength, frequency, speed.

• Photoelectric Effect: Discrete energy levels for electrons in an atom.

• Bohr Model: Describing hydrogen atom's quantized energy levels.

• Quantum Mechanical Model: Focuses on the probabilities of electron locations versus definite orbits.

• Aufbau Principle, Pauli Exclusion Principle, and Hund’s Rule outlined.

Exercises

• Problems to calculate wavelengths, electron affinities, ionization energies, and configurations across groups and periods.

Atomic Structure and Periodicity

Table of Contents

7.1 Electromagnetic Radiation

7.2 The Nature of Matter

The Photoelectric Effect

7.3 The Atomic Spectrum of Hydrogen

7.4 The Bohr Model

7.5 The Quantum Mechanical Model of the Atom

The Physical Meaning of a Wave Function

7.6 Quantum Numbers

7.7 Orbital Shapes and Energies

7.8 Electron Spin and the Pauli Principle

7.9 Polyelectronic Atoms

7.10 The History of the Periodic Table

7.11 The Aufbau Principle and the Periodic Table

7.12 Periodic Trends in Atomic Properties

Ionization Energy

Electron Affinity

Atomic Radius

7.13 The Properties of a Group: The Alkali Metals

Information Contained in the Periodic Table

7.1 Electromagnetic Radiation

Energy travels through space as electromagnetic radiation (EMR), which exhibits wave-like properties.

Examples of EMR include sunlight (visible light), microwaves, X-rays, radio waves, and radiant heat (infrared radiation), all forming part of the electromagnetic spectrum.

Key properties of waves:

Wavelength ($λ$): Defined as the distance between two consecutive peaks or troughs in a wave. It is typically measured in meters (m) or nanometers (nm).

Symbol: Lowercase Greek letter lambda ($λ$).

Frequency ($ν$): Represents the number of waves or cycles that pass a given point in space per second.

Symbol: Lowercase Greek letter nu ($ν$).

Units: Hertz (Hz), where $1 \text{ Hz} = 1 \text{ s}^{-1}$. This means one cycle per second.

Speed of Light ($c$): The constant speed at which all electromagnetic radiation travels in a vacuum, approximately $c = 2.9979 \times 10^8 \text{ m/s}$. For practical purposes, often rounded to $3.00 \times 10^8 \text{ m/s}$ .

Relationship between wavelength and frequency: The speed of light is the product of its wavelength and frequency: $c = λν$. This fundamental equation shows their inverse relationship: as wavelength increases, frequency decreases, and vice versa, because their product ($c$) is constant.

Energy of a Photon: The energy of a single photon of electromagnetic radiation is directly proportional to its frequency, given by Planck's equation: $E = hν$, where $h$ is Planck's constant (6.626 \times 10^{-34} \text{ J\cdot s}).

7.2 The Nature of Matter

Historical context: The concept of indivisible particles called atoms was first introduced by Greek philosophers Democritus and Leucippus around 400 B.C., purely as a philosophical idea without experimental basis. It wasn't until the 19th century that experimental evidence began to support atomic theory.

Key scientific figures and their contributions:

Antoine Lavoisier: Contributed quantitative data on chemical reactions, leading to the law of conservation of mass.

John Dalton: Proposed the first systematic atomic theory in the early 19th century, based on stoichiometric experiments. His theory described atoms as tiny, indivisible, indestructible particles.

Challenge to Classical Physics: For a long time, matter was considered to consist of particles (possessing mass), and energy was considered to consist of waves. The early 20th century, however, presented experimental results (like blackbody radiation and the photoelectric effect) that classical physics could not adequately explain, necessitating a new theory: quantum mechanics.

Max Planck: To explain blackbody radiation (the emission of light by hot objects), Planck proposed in 1900 that energy is not continuous but is quantized, meaning it can only be absorbed or emitted in discrete packets called quanta. The energy of a quantum is given by $E = nhν$, where $n$ is an integer (1, 2, 3, …), $h$ is Planck's constant, and $ν$ is frequency.

Albert Einstein: Building on Planck's work, Einstein proposed in 1905 that electromagnetic radiation itself is quantized and consists of particles called photons. This explained phenomena like the photoelectric effect, further solidifying the wave-particle duality of light.

Wave-Particle Duality: This concept, developed by Louis de Broglie, posits that both light and matter (like electrons) exhibit properties of both waves and particles. The de Broglie wavelength for a particle is given by $λ = h/(mv)$, where $m$ is mass and $v$ is velocity.

Photoelectric Effect: The observation that electrons are emitted from the surface of certain metals when light of a sufficiently high frequency shines on them.

The Photoelectric Effect

Characteristics of the Photoelectric Effect:

Threshold Frequency ($ν_0$): For each metal, there exists a minimum frequency of light, known as the threshold frequency, below which no electrons are emitted, regardless of the light's intensity.

No Emission Below Threshold: If the frequency of incident light is below $ν_0$, no electrons will be emitted, even if the light source is very intense or shines for a long duration. This contradicts classical wave theory, which suggested that higher intensity (more energy) should eventually dislodge electrons.

Intensity and Number of Electrons: For frequencies above $ν_0$, the number of emitted electrons (and thus the photocurrent) increases proportionally with the intensity of the incident light. This means a brighter light releases more electrons, but not more energetic ones.

Kinetic Energy and Frequency: The kinetic energy of the emitted electrons increases linearly with the frequency of the incident light, once the threshold frequency is exceeded. The kinetic energy is independent of the light's intensity.

Explanation by Einstein: Einstein explained the photoelectric effect by applying Planck's quantum theory. He suggested that light consists of photons, each with energy $E = hν$. When a photon strikes the metal surface, it transfers its energy to an electron. If the photon's energy ($hν$) is greater than the work function ($Φ$) of the metal (the minimum energy required to remove an electron from the surface), an electron is emitted. The excess energy is converted into the kinetic energy of the emitted electron.

The equation for the kinetic energy of the emitted electron is: $KE_{electron} = hν - Φ$.

Importantly, one photon interacts with one electron. This explains the threshold frequency and the linear relationship between kinetic energy and frequency.

7.3 The Atomic Spectrum of Hydrogen

When gaseous hydrogen atoms are excited (e.g., by passing an electric current through them), they emit light. This emitted light, when passed through a prism, does not produce a continuous rainbow but rather a line spectrum, consisting of only a few specific wavelengths.

Continuous Spectrum: Produced when white light (e.g., from a heated solid) is passed through a prism, resulting in a continuous array of all visible wavelengths, like a rainbow.

Line Spectrum: This type of spectrum consists of discrete, distinct lines of specific wavelengths. For hydrogen, these lines (e.g., $410 \text{ nm}$, $434 \text{ nm}$, $486 \text{ nm}$, $656 \text{ nm}$ in the visible region) confirm that electrons in an atom can only occupy certain discrete energy levels, rather than a continuous range of energies. The distinct lines correspond to the energy differences between these quantized energy levels when electrons transition from a higher energy state to a lower one.

Rydberg Formula: Johannes Rydberg developed an empirical formula to describe the wavelengths of all the spectral lines of hydrogen: $\frac{1}{λ} = R<em>H (\frac{1}{n</em>1^2} - \frac{1}{n<em>2^2})$, where $R</em>H$ is the Rydberg constant ($1.097 \times 10^7 \text{ m}^{-1}$) and $n<em>1$ and $n</em>2$ are integers representing energy levels, with $n<em>2 > n</em>1$. This formula successfully predicts the observed spectral lines.

7.4 The Bohr Model

Proposed by Niels Bohr in 1913, this model was the first to explain the observed line spectrum of hydrogen, building upon Rutherford's nuclear model and Planck's quantum theory.

Bohr's Postulates for the Hydrogen Atom:

Electrons in an atom exist only in specific, stable, circular orbits or stationary states without radiating energy. Each state has a definite energy.

An electron in one of these orbits has a specific energy level, given by the formula:

$E_n = - \frac{2.178 \times 10^{-18} \text{ J}}{n^2}, n = 1, 2, 3, …$
where $n$ is the principal quantum number. The negative sign indicates that the electron is bound to the nucleus, and energy must be added to remove it. $n=1$ corresponds to the lowest energy (ground state).

Electrons can only transition between these allowed orbits by absorbing or emitting a photon whose energy exactly matches the difference in energy between the two orbits ($ΔE = E<em>{final} - E</em>{initial} = hν$).

Successes of the Bohr Model: It successfully predicted the line spectrum of hydrogen and hydrogen-like ions (one-electron systems like $He^+$ and $Li^{2+}$) and calculated their energy levels and radii.

Limitations of the Bohr Model: Despite its successes, the Bohr model had significant limitations:

It could not explain the spectra of polyelectronic atoms (atoms with more than one electron).

It could not explain the splitting of spectral lines in the presence of a magnetic field (Zeeman effect).

It treated electrons as particles orbiting in well-defined paths, which was later contradicted by the Heisenberg Uncertainty Principle.

7.5 The Quantum Mechanical Model of the Atom

Emerging in the mid-20th century, primarily through the work of Erwin Schrödinger, Werner Heisenberg, and Paul Dirac, the quantum mechanical model provides a more sophisticated and accurate description of atomic structure than the Bohr model. It emphasizes the wave properties of electrons and describes their probable locations rather than definite orbits.

Schrödinger Equation: This is a complex mathematical equation that describes the behavior of electrons in terms of wave functions ($Ψ$). Solving the Schrödinger equation for a specific atom yields a set of wave functions (also called orbitals), each corresponding to a specific energy level.

The equation is typically written as: $\hat{H}Ψ = EΨ$, where $\hat{H}$ is the Hamiltonian operator, $Ψ$ is the wave function, and $E$ is the energy of the system.

The Physical Meaning of a Wave Function ($Ψ$): A wave function itself does not have direct physical meaning (it can be positive or negative). However, the square of the wave function, $|Ψ|^2$, represents the probability density of finding an electron at a particular point in space. This means we can only determine the probability of an electron being in a certain region, not its exact location. An orbital is a region of space around the nucleus where there is a high probability (e.g., 90%) of finding an electron.

Heisenberg Uncertainty Principle: Formulated by Werner Heisenberg, this principle states that it is fundamentally impossible to precisely know both the position ($Δx$) and the momentum ($Δp$) of a particle (like an electron) simultaneously. The more accurately one is known, the less accurately the other can be known.

Mathematically expressed as: $ΔxΔp ≥ \frac{h}{4π}$ where $h$ is Planck's constant.

This principle implies that electrons do not follow predictable orbits like planets, reinforcing the probabilistic nature of their location in the quantum mechanical model.

7.6 Quantum Numbers

Quantum numbers are a set of four numbers that completely describe an electron in an atom and define the properties of atomic orbitals. They arise naturally from the solutions to the Schrödinger equation.

Principal Quantum Number ($n$): Defines the size and main energy level of the orbital. It can take on positive integer values: $1, 2, 3, …$. Higher values of $n$ correspond to larger orbitals and higher energy levels. All orbitals with the same $n$ value are said to be in the same electron shell.

Angular Momentum (or Azimuthal) Quantum Number ($ℓ$): Defines the shape of the orbital (and a subshell within a shell). Its values depend on $n$ and range from $0$ to $(n-1)$.

$ℓ = 0$ corresponds to an s orbital (spherical shape).

$ℓ = 1$ corresponds to a p orbital (dumbbell shape).

$ℓ = 2$ corresponds to a d orbital (more complex shapes, typically four lobes).

$ℓ = 3$ corresponds to an f orbital (even more complex shapes).

Magnetic Quantum Number ($m<em>ℓ$): Defines the orientation of the orbital in space. Its values depend on $ℓ$ and range from $-ℓ$ to $+ℓ$, including $0$. Thus, for a given $ℓ$, there are $(2ℓ + 1)$ possible values of $m</em>ℓ$ .

For $ℓ=0$ (s orbital), $m_ℓ = 0$ (1 orbital).

For $ℓ=1$ (p orbitals), $m<em>ℓ = -1, 0, +1$ (3 orbitals: $p</em>x, p<em>y, p</em>z$).

For $ℓ=2$ (d orbitals), $m_ℓ = -2, -1, 0, +1, +2$ (5 orbitals).

7.7 Orbital Shapes and Energies

s orbitals: These orbitals are spherically symmetrical. Their size increases with increasing principal quantum number ($n$). For example, a $2s$ orbital is larger than a $1s$ orbital, and the electron in a $2s$ orbital spends more time farther from the nucleus.

p orbitals: These orbitals have a dumbbell shape, consisting of two lobes on opposite sides of the nucleus. For a given $n$, there are three degenerate (equal energy) p orbitals, oriented along the three Cartesian axes: $p<em>x, p</em>y$, and $p_z$ .

d orbitals: These orbitals first appear at $n=3$ and have more complex, multi-lobed shapes. There are five degenerate d orbitals for a given $n$: $d<em>{xy}, d</em>{xz}, d<em>{yz}, d</em>{x^2-y^2}$, and $d<em>{z^2}$ . Four of these have four lobes, while the $d</em>{z^2}$ orbital has two lobes along the z-axis and a donut-shaped region in the xy-plane.

Orbital Energies: In a hydrogen atom (or any one-electron system), the energy of an orbital depends only on the principal quantum number ($n$). Therefore, orbitals of the same principal quantum number are typically degenerate (e.g., $2s$ and $2p$ have the same energy).

In polyelectronic atoms (atoms with multiple electrons), electron-electron repulsions cause the energies of orbitals within the same shell but different subshells to split, meaning that $s < p < d < f$ for a given $n$ (e.g., $2s$ is lower in energy than $2p$). This is due to shielding and penetration effects, where electrons in inner orbitals reduce the effective nuclear charge felt by outer electrons, and electrons in certain orbitals (like s) can penetrate closer to the nucleus.

7.8 Electron Spin and the Pauli Principle

Electron Spin Quantum Number ($m<em>s$): In addition to the three quantum numbers that describe an orbital, a fourth quantum number is needed to describe the electron itself. This is the electron spin quantum number, which accounts for the intrinsic angular momentum of an electron, often visualized as the electron spinning on its axis (though this is a classical analogy). There are only two possible values for $m</em>s$: $+\frac{1}{2}$ (spin up) or $-\frac{1}{2}$ (spin down).

Pauli Exclusion Principle: Formulated by Wolfgang Pauli, this fundamental principle states that no two electrons in an atom can have the exact same set of all four quantum numbers ($n, ℓ, m<em>ℓ, m</em>s$). This means that an atomic orbital (defined by $n, ℓ, m_ℓ$) can hold a maximum of two electrons, and these two electrons must have opposite spins (one $+\frac{1}{2}$ and one $-\frac{1}{2}$).

7.9 Polyelectronic Atoms

Polyelectronic atoms are atoms containing more than one electron. While the quantum mechanical model provides a framework, the exact solution of the Schrödinger equation for these atoms is extremely complex due to electron-electron repulsions. These repulsions lead to significant differences compared to the simple hydrogenic model.

Electron Correlation Problem: This refers to the difficulty in precisely accounting for the instantaneous interactions (repulsions) between multiple electrons in an atom. Because electrons are constantly moving and repelling each other, their motions are correlated, making it impossible to treat the motion of one electron independently of the others. Approximate methods are used to solve the Schrödinger equation for polyelectronic atoms.

Effective Nuclear Charge ($Z_{eff}$): In polyelectronic atoms, outer electrons are shielded from the full nuclear charge by inner electrons. The effective nuclear charge is the net positive charge experienced by an electron in a polyelectronic atom. It is always less than the actual nuclear charge ($Z$).

Shielding and Penetration: Electrons in inner shells (closer to the nucleus) or in orbitals that penetrate closer to the nucleus (e.g., s orbitals) are more effective at shielding outer electrons from the nuclear charge. This explains why, for a given principal quantum number $n$, the energy order of subshells is $s < p < d < f$. Electrons in $s$ orbitals penetrate closer to the nucleus than $p$ electrons, which penetrate more than $d$ electrons, experiencing a higher effective nuclear charge and thus having lower energy.

7.10 The History of the Periodic Table

The organization of elements into the periodic table is a cornerstone of chemistry, allowing for the prediction of chemical and physical properties.

Early Attempts: Chemists in the 19th century recognized patterns in elemental properties. Various scientists tried to organize elements, but the most successful early arrangements came from Dmitri Mendeleev and Lothar Meyer.

Mendeleev and Meyer: Both independently developed periodic tables in 1869. Mendeleev is generally given more credit because his table was more widely published and he made bolder and more accurate predictions.

Mendeleev's Contributions:

Arrangement by Atomic Mass: Mendeleev arranged the known elements primarily by increasing atomic mass.

Periodic Law: He observed that certain properties recurred periodically when elements were arranged this way.

Prediction of Undiscovered Elements: Crucially, Mendeleev left gaps in his table for elements that had not yet been discovered. He used the trends from neighboring elements to predict the properties of these missing elements (e.g., "eka-silicon" which was later discovered as germanium). The accurate prediction of these properties gave his periodic table immense credibility.

Moseley's Work: In the early 20th century, Henry Moseley determined that the fundamental property governing the arrangement of elements is atomic number (number of protons), not atomic mass. This explained some discrepancies in Mendeleev's table where elements were placed out of atomic mass order to fit chemical properties.

7.11 The Aufbau Principle and the Periodic Table

Electron Configuration: The distribution of electrons among the orbitals of an atom is called the electron configuration. It describes how electrons fill up the available orbitals.

Aufbau Principle ("Building Up"): This principle states that electrons fill atomic orbitals in order of increasing energy. Lower energy orbitals are filled first before higher energy orbitals. The order is generally $1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p$ .

Hund's Rule: This rule states that for degenerate orbitals (orbitals of the same energy, like the three $p$ orbitals or five $d$ orbitals), electrons will first occupy these orbitals singly with parallel spins before any orbital is doubly occupied. This maximizes spin multiplicity and minimizes electron-electron repulsion.

Pauli Exclusion Principle (revisited): Reinforces that each orbital can hold a maximum of two electrons, and these two electrons must have opposite spins.

Valence Electrons: These are the electrons in the outermost principal energy level of an atom (and sometimes include d-electrons in transition metals). They are involved in chemical bonding and largely determine an atom's chemical properties.

Electron Configurations and the Periodic Table: The structure of the periodic table is a direct consequence of the Aufbau principle and the way electrons fill orbitals.

s-block elements: Groups 1 and 2 (alkali metals and alkaline earth metals) have their valence electrons in an $s$ orbital.

p-block elements: Groups 13-18 have their valence electrons in $p$ orbitals (in addition to $s$ orbitals).

d-block elements: Transition metals (Groups 3-12) have their highest energy electrons filling $d$ orbitals.

f-block elements: Lanthanides and actinides have their highest energy electrons filling $f$ orbitals.

7.12 Periodic Trends in Atomic Properties

The periodic table allows us to predict how the properties of elements change across periods (rows) and down groups (columns). These trends are due to changes in effective nuclear charge, atomic size, and electron shielding.

Ionization Energy (IE): The minimum energy required to remove one electron from a gaseous atom or ion in its ground state. The first ionization energy ($IE<em>1$) is for removing the first electron, $IE</em>2$ for the second, and so on. $IE<em>1 < IE</em>2 < IE_3$ because it is harder to remove an electron from an increasingly positively charged ion.

Trend Across a Period: Ionization energy generally increases across a period from left to right. This is because the effective nuclear charge increases, pulling valence electrons more strongly towards the nucleus, making them harder to remove.

Trend Down a Group: Ionization energy generally decreases down a group. This is because the principal quantum number ($n$) increases, meaning valence electrons are farther from the nucleus and experience greater shielding from inner electrons, thus they are less strongly attracted and easier to remove.

Exceptions: Minor irregularities occur due to subshell energies and electron-electron repulsion (e.g., a dip at Group 13 and Group 16 elements).

Electron Affinity (EA): The energy change that occurs when an electron is added to a gaseous atom in its ground state. A negative value indicates an exothermic process (energy released), meaning the atom has an affinity for the electron. A positive value indicates an endothermic process (energy required).

General Trend Across a Period: Electron affinity generally becomes more negative (more exothermic) across a period from left to right, as the increasing effective nuclear charge makes atoms more attractive to incoming electrons.

General Trend Down a Group: Electron affinity generally becomes less negative (less exothermic) down a group as the increasing size of the atom and greater shielding make it less attractive to an added electron.

Halo gens (Group 17) typically have the most negative electron affinities because adding an electron completes their outer $p$ subshell, achieving a stable noble gas configuration.

Atomic Radius: The size of an atom. It is typically defined as half the distance between the nuclei of two identical atoms bonded together.

Trend Across a Period: Atomic radius generally decreases across a period from left to right. This is because the effective nuclear charge increases while the principal quantum number ($n$) remains the same, pulling the valence electrons closer to the nucleus.

Trend Down a Group: Atomic radius generally increases down a group. This is due to the increasing principal quantum number ($n$) of the valence electrons, meaning they occupy higher energy levels that are farther from the nucleus, and increased shielding by inner shells.

Ionic Radius: Cations are smaller than their parent atoms (loss of electrons reduces electron-electron repulsion and outer shell may be removed). Anions are larger than their parent atoms (gain of electrons increases electron-electron repulsion).

7.13 The Properties of a Group: The Alkali Metals

The Alkali Metals constitute Group 1 of the periodic table, including Lithium (Li), Sodium (Na), Potassium (K), Rubidium (Rb), Cesium (Cs), and Francium (Fr).

Electronic Configuration: All alkali metals have a single $s$ valence electron (e.g., $[He]2s^1$ for Li, $[Ne]3s^1$ for Na). This renders them highly reactive metals.

Properties:

Low Ionization Energies: They have the lowest first ionization energies in their respective periods because removing their single valence electron results in a stable Noble Gas configuration. Ionization energy decreases down the group.

High Reactivity: Their low ionization energies mean they readily lose their single valence electron to form $+1$ ions ($M^+$), making them very reactive. Reactivity increases down the group as ionization energy decreases.

Metallic Character: They are soft, silvery-white metals with low melting and boiling points, and excellent electrical and thermal conductors. Metallic character increases down the group.

Reactions: They react vigorously with water to produce hydrogen gas and a metal hydroxide (e.g., $2\text{Na}(s) + 2\text{H}<em>2\text{O}(l) \rightarrow 2\text{NaOH}(aq) + \text{H}</em>2(g)$). They also readily react with halogens to form ionic salts (e.g., $2\text{Na}(s) + \text{Cl}_2(g) \rightarrow 2\text{NaCl}(s)$).

Information Contained in the Periodic Table:

Elements are arranged by increasing atomic number.

Rows (periods) indicate the principal quantum number (or highest shell being filled).

Columns (groups) indicate the number of valence electrons and thus similar chemical properties.

Blocks (s, p, d, f) correspond to the type of orbital being filled by the highest energy electrons.

Key Terms and Review Questions

Electromagnetic Radiation: Includes wavelength, frequency, speed of light, and photon energy ($E = hν$).

Photoelectric Effect: Explains the quantized nature of light and the emission of electrons from metal surfaces.

Bohr Model: Describes hydrogen atom's quantized energy levels and electron transitions.

Quantum Mechanical Model: Focuses on the probabilistic locations of electrons via wave functions (orbitals) and the Heisenberg Uncertainty Principle.

Quantum Numbers: Definitions and roles of $n, ℓ, m<em>ℓ, m</em>s$.

Aufbau Principle, Pauli Exclusion Principle, and Hund’s Rule: Fundamental principles for determining electron configurations.

Periodic Trends: Ionization energy, electron affinity, and atomic radius, including their definitions and trends across periods and down groups.

Exercises

Problems to calculate wavelengths, frequencies, and photon energies using $c = λν$ and $E = hν$.

Problems involving energy level transitions in the Bohr model ($ΔE = E<em>{final} - E</em>{initial}$).

Determining electron configurations and orbital diagrams for various atoms and ions.

Comparing and predicting relative ionization energies, electron affinities, and atomic radii for different

Atomic Structure and Periodicity

Table of Contents

• 7.1 Electromagnetic Radiation

• 7.2 The Nature of Matter

  • The Photoelectric Effect

• 7.3 The Atomic Spectrum of Hydrogen

• 7.4 The Bohr Model

• 7.5 The Quantum Mechanical Model of the Atom

  • The Physical Meaning of a Wave Function

• 7.6 Quantum Numbers

• 7.7 Orbital Shapes and Energies

• 7.8 Electron Spin and the Pauli Principle

• 7.9 Polyelectronic Atoms

• 7.10 The History of the Periodic Table

• 7.11 The Aufbau Principle and the Periodic Table

• 7.12 Periodic Trends in Atomic Properties

  • Ionization Energy

  • Electron Affinity

  • Atomic Radius

• 7.13 The Properties of a Group: The Alkali Metals

  • Information Contained in the Periodic Table

7.1 Electromagnetic Radiation

• Energy travels via electromagnetic radiation (EMR), exhibiting wave properties.

• Examples: Sunlight, microwaves, X-rays.

• Wave Properties:

  • Wavelength ($λ$): Distance between consecutive peaks.

  • Frequency ($ν$): Number of waves/cycles per second ($1 ext{ Hz} = 1 ext{ s}^{-1}$).

  • Speed of Light ($c$): $2.9979 \times 10^8 \text{ m/s}$.

• Relationship: $c = λν$ (inverse relationship between $λ$ and $ν$).

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

7.2 The Nature of Matter

• Historical: Democritus (~400 B.C.), Lavoisier (quantitative data), Dalton (first systematic atomic theory).

• Classical physics couldn't explain early 20th-century experiments (blackbody radiation, photoelectric effect).

  • Max Planck: Energy is quantized ($E = nhν$).

  • Albert Einstein: EMR consists of quantized particles (photons).

  • Louis de Broglie: Wave-particle duality; particle wavelength $λ = h/(mv)$.

• Photoelectric Effect: Electrons emitted from metal when light strikes it under certain conditions.

The Photoelectric Effect

• Characteristics:

  1. Threshold frequency ($ν_0$): Minimum frequency required for electron emission; no emission below this, regardless of intensity.

  2. For $ν > ν_0$: number of emitted electrons increases with light intensity.

  3. For $ν > ν_0$: kinetic energy of emitted electrons increases linearly with frequency.

• Explanation (Einstein): One photon ($E=hν$) ejects one electron. If $hν > Φ$ (work function), excess energy is electron's kinetic energy: $KE_{electron} = hν - Φ$.

7.3 The Atomic Spectrum of Hydrogen

• Excited hydrogen atoms emit a line spectrum (specific wavelengths), not a continuous spectrum (all wavelengths, like a rainbow).

• Line spectra confirm that electron energy levels in atoms are quantized.

• Rydberg Formula: Describes hydrogen spectral lines: $\frac{1}{λ} = R<em>H (\frac{1}{n</em>1^2} - \frac{1}{n_2^2})$.

7.4 The Bohr Model

• Niels Bohr (1913): Proposed electrons exist in stable, quantized circular orbits with specific energies.

• Energy Level Formula (Hydrogen): $E_n = - \frac{2.178 \times 10^{-18} \text{ J}}{n^2}, n = 1, 2, 3, …$.

• Electrons transition between orbits by absorbing/emitting photons ($ΔE = E<em>{final} - E</em>{initial} = hν$).

• Successes: Explained hydrogen line spectrum, calculated its energy levels.

• Limitations: Failed for polyelectronic atoms, couldn't explain Zeeman effect, violated Heisenberg Uncertainty Principle by proposing fixed orbits.

7.5 The Quantum Mechanical Model of the Atom

• Developed mid-20th century by Schrödinger, Heisenberg, Dirac. Emphasizes wave properties of electrons; describes probable locations.

• Schrödinger Equation: Equation describing electron states via wave functions ($Ψ$): $\hat{H}Ψ = EΨ$.

The Physical Meaning of a Wave Function

• $|Ψ|^2$ represents the probability density of finding an electron at a point in space. Orbitals are regions of high electron probability.

• Heisenberg Uncertainty Principle: Impossible to precisely know both position ($Δx$) and momentum ($Δp$) of a particle simultaneously: $ΔxΔp ≥ \frac{h}{4π}$.

7.6 Quantum Numbers

• A set of four numbers describing an electron and its orbital:

  • Principal ($n$): Defines energy level and size. Positive integers ($1, 2, 3, …$).

  • Angular Momentum ($ℓ$): Defines orbital shape. Values from $0$ to $(n-1)$. ($0=\text{s}, 1=\text{p}, 2=\text{d}, 3=\text{f}$
    ).

  • Magnetic ($m_ℓ$): Defines orbital orientation. Values from $-ℓ$ to $+ℓ$ (including $0$); $(2ℓ+1)$
    orbitals for a given $ℓ$.

  • Electron Spin ($m_s$): Describes electron's intrinsic spin. Values are $+\frac{1}{2}$ or $-\frac{1}{2}$.

7.7 Orbital Shapes and Energies

• s orbitals: Spherical, size increases with $n$.

• p orbitals: Dumbbell-shaped, three degenerate ($p<em>x, p</em>y, p_z$) for a given $n$.

• d orbitals: Complex multiple-lobed shapes, five degenerate for $n ≥ 3$.

• Orbital Energies:

  • Hydrogen: Energy depends only on $n$ (e.g., $2s$ and $2p$ are degenerate).

  • Polyelectronic Atoms: Electron-electron repulsions cause energy splitting: $s < p < d < f$ for a given $n$, due to shielding and penetration effects.

7.8 Electron Spin and the Pauli Principle

• Electron Spin ($m_s$): Intrinsic property of electrons (spin up or spin down).

• Pauli Exclusion Principle: No two electrons in an atom can have the same four quantum numbers. Thus, an orbital holds a maximum of two electrons with opposite spins.

7.9 Polyelectronic Atoms

• Atoms with multiple electrons; their exact Schrödinger equation solution is complex due to electron-electron repulsions.

• Electron Correlation Problem: Difficulty in accounting for instantaneous electron interactions.

• Effective Nuclear Charge ($Z_{eff}$): Net positive charge experienced by an electron, less than actual nuclear charge ($Z$) due to shielding.

• Shielding and Penetration: Inner electrons shield outer electrons. Orbitals closer to nucleus (s orbitals) penetrate more, experiencing higher $Z_{eff}$ and lower energy (explains $s < p < d < f$ order).

7.10 The History of the Periodic Table

• Mendeleev (and Meyer) (1869): Organized elements by increasing atomic mass and recurring properties. Mendeleev predicted undiscovered elements.

• Moseley (early 20th century): Established atomic number as the fundamental organizing principle.

7.11 The Aufbau Principle and the Periodic Table

• Electron Configuration: Distribution of electrons in atomic orbitals.

• Aufbau Principle: Electrons fill lowest energy orbitals first ($1s, 2s, 2p, 3s, 3p, 4s, 3d$…).

• Hund’s Rule: For degenerate orbitals, electrons fill singly with parallel spins before pairing.

• Pauli Exclusion Principle: Each orbital holds max two electrons with opposite spins.

• Valence Electrons: Outermost electrons, determine chemical properties.

• Periodic Table Blocks: s-block (Groups 1-2), p-block (Groups 13-18), d-block (transition metals), f-block (lanthanides/actinides) reflect orbital filling.

7.12 Periodic Trends in Atomic Properties

• Ionization Energy (IE): Energy to remove an electron from a gaseous atom.

  • Trend: Increases across a period (stronger $Z_{eff}$), decreases down a group (larger $n$, more shielding).

• Electron Affinity (EA): Energy change when adding an electron to a gaseous atom (often exothermic, negative value).

  • Trend: Generally more negative across a period (increasing $Z_{eff}$). Less negative down a group (increasing size, more shielding). Halogens have most negative EA.

• Atomic Radius: Size of an atom.

  • Trend: Decreases across a period (increasing $Z_{eff}$ pulling electrons closer). Increases down a group (increasing $n$, more electron shells).

• Ionic Radius: Cations are smaller than parent atoms; anions are larger than parent atoms.

7.13 The Properties of a Group: The Alkali Metals

• Group 1: Li, Na, K, Rb, Cs, Fr. Have one $s$ valence electron ($[Noble Gas]ns^1$).

• Properties: Very reactive metals, low ionization energies (decrease down group), high metallic character (increases down group).

• Reactions: Vigorously react with water ($2\text{Na}(s) + 2\text{H}<em>2\text{O}(l) \rightarrow 2\text{NaOH}(aq) + \text{H}</em>2(g))$ and halogens ($2\text{Na}(s) + \text{Cl}_2(g) \rightarrow 2\text{NaCl}(s)$).

• Information Contained in the Periodic Table: Arranged by atomic number; periods indicate principal quantum number; groups indicate valence electrons and similar properties; blocks show orbital filling.

Key Terms and Review Questions

• Electromagnetic Radiation: Wavelength, frequency, speed of light ($c=λν$), photon energy ($E=hν$).

• Photoelectric Effect: Quantized light, electron emission.

• Bohr Model: Hydrogen's quantized energy levels.

• Quantum Mechanical Model: Electron probability, wave functions, Heisenberg Uncertainty Principle.

• Quantum Numbers: $n, ℓ, m<em>ℓ, m</em>s$
  definitions.

• Electron Configuration Principles: Aufbau, Pauli Exclusion, Hund’s Rule.

• Periodic Trends: Ionization energy, electron affinity, atomic radius.

Exercises

• Problems involving $c = λν$, $E = hν$.

• Energy level transitions in Bohr model ($ΔE$).

• Electron configurations and orbital diagrams.

• Comparing/predicting periodic trends.