Chemistry Eighth Edition Chapter 5: Periodicity and the Electronic Structure of Atoms
Chapter 5: Periodicity and the Electronic Structure of Atoms
Atomic Radius
Definition: Atomic radius refers to the distance from the nucleus to the outermost shell of an electron in an atom. It is typically defined as half the distance between the nuclei of two identical atoms that are bonded together.
Trends:
Across a Period (left to right): Atomic radius generally decreases. This is because as you move across a period, the effective nuclear charge (Z_{eff}) increases due to more protons in the nucleus, pulling the electrons closer to the nucleus, while the shielding effect remains relatively constant for electrons in the same principal energy level.
Down a Group (top to bottom): Atomic radius generally increases. This occurs because new principal energy levels (electron shells) are added, increasing the distance of the outermost electrons from the nucleus, and the inner electrons provide more shielding (Z_{eff} decreases slightly for valence electrons).
Maxima occur for atoms of group 1A (alkali metals) elements: Lithium (Li), Sodium (Na), Potassium (K), Rubidium (Rb), Cesium (Cs), Francium (Fr), due to their lowest effective nuclear charge and largest orbital size within their respective periods.
Minima occur for atoms of group 7A (halogens) elements: Fluorine (F), Chlorine (Cl), Bromine (Br), Iodine (I), due to their highest effective nuclear charge and smallest orbital size within their respective periods (excluding noble gases which are often not considered in covalent radius trends due to their lack of covalent bonding).
Data Representation:
Atomic radius is typically measured in picometers (1\text{ pm} = 10^{-12}\text{ m}).
Graphically represented by Figure 5.1, showing periodic variations.
Wave Properties of Radiant Energy
Key Characteristics: Electromagnetic waves are characterized by:
Wavelength (\lambda): The spatial period of a periodic wave, measured in meters (m), representing the distance between successive corresponding points of the same phase (e.g., crests or troughs).
Frequency (\nu): The number of oscillations or cycles that pass a point in one second, measured in Hertz (Hz or \text{s}^{-1} ). It is inversely proportional to wavelength.
Amplitude: The height of the wave from the origin to a crest (or trough), indicative of the intensity or brightness of the radiant energy. A higher amplitude means more intense light.
Electromagnetic Spectrum
Light: A form of electromagnetic energy, which exists as a continuous spectrum of all possible wavelengths and frequencies, ranging from low-frequency radio waves to high-frequency gamma rays. All components travel at the speed of light in a vacuum.
Speed of Light (c): The speed of all electromagnetic energy in a vacuum, a fundamental physical constant measured as:
c = 3.00 \times 10^8 \text{ m/s}
Numerical Relationship
The relationship among wavelength, frequency, and speed of light is given by the equation:
c = \nu \lambda
This equation shows that wavelength and frequency are inversely proportional given a constant speed of light.
Example Calculations
Example: For a certain light emitted with a frequency of 6.88 \times 10^{14} \text{ Hz} , the wavelength can be calculated as:
\lambda = \frac{c}{\nu}
\lambda = \frac{3.00 \times 10^8 \text{ m/s}}{6.88 \times 10^{14} \text{ Hz}} \approx 4.36 \times 10^{-7} \text{ m} \rightarrow 436 \text{ nm}
Wave Phenomena
Diffraction: The bending of waves as they pass around the edge of an obstacle or through a small opening. This phenomenon provides evidence for the wave-like nature of light.
Interference: The superposition of two or more waves, resulting in a new wave pattern. Can be constructive (when waves align crest-to-crest, increasing amplitude and intensity) or destructive (when waves align crest-to-trough, decreasing amplitude or canceling out).
Double-slit Experiment: A classic experiment that dramatically demonstrates the wave-like behavior of light (and matter), showing an interference pattern when light passes through two closely spaced slits.
Particle-like Properties of Radiant Energy
Photoelectric Effect
Definition: The phenomenon where electrons are ejected from a metal surface when light shines on it, but only if the light's frequency is above a certain minimum value, known as the threshold frequency (\nu_0), for that specific metal. It provided concrete evidence for the particle nature of light.
Observations:
A plot of ejected electrons (current) versus frequency reveals a threshold value (\nu_0) below which no electrons are ejected, regardless of the light's intensity.
Increasing the light intensity (i.e., increasing the amplitude, or number of photons) above the threshold frequency increases the number of electrons ejected per unit time, but does not change the kinetic energy of the individual ejected electrons or the threshold frequency itself.
The kinetic energy of the ejected electrons increases linearly with the frequency of the incident light above the threshold frequency.
Planck’s Postulate
Quantization of Energy: Max Planck proposed that electromagnetic energy is not continuous but is emitted and absorbed in discrete packets or "quanta." The energy of a single quantum (or photon for light) is directly proportional to its frequency:
E = h\nu
Where E is the energy of the photon, h is Planck's constant, and \nu is the frequency.
Constants:
Planck’s constant (h): h = 6.626 \times 10^{-34} \text{ J s}
Calculation Example
Emission of X-rays caused by bombarding copper with electrons, where the energy of the X-ray photons can be calculated using the combined equations ( E = h\nu and c = \nu \lambda ):
E = h\frac{c}{\lambda} where \lambda = 0.154 \text{ nm} = 0.154 \times 10^{-9} \text{ m}
Resulting energy: E = (6.626 \times 10^{-34} \text{ J s}) \frac{(3.00 \times 10^8 \text{ m/s})}{(0.154 \times 10^{-9} \text{ m})} \approx 1.29 \times 10^{-15} \text{ J}
Atomic Line Spectra
Line Spectrum: Unlike a continuous spectrum (like that of white light), a line spectrum is a series of discrete lines of specific wavelengths/frequencies, created when light emitted by excited atoms is passed through a prism or diffraction grating. Each element has a unique line spectrum, acting as a "fingerprint."
Niels Bohr's Model of Hydrogen Atom
Structure: Niels Bohr proposed a revolutionary model for the hydrogen atom, suggesting that electrons move around the nucleus in specific, stable orbits (stationary states) with quantized energy levels. Electrons do not radiate energy when in these stable orbits.
Electron Emission/Absorption: When an electron transitions from a higher energy level (n{initial}) to a lower energy level (n{final}), it emits a photon of electromagnetic radiation with energy equal to the difference between the two energy levels ( \Delta E = E{initial} - E{final} ). Conversely, an electron can absorb a photon to jump to a higher energy level. This model successfully explained the observed line spectrum of hydrogen. However, it failed for multi-electron atoms.
Mathematical Relationships
The relation between wavelengths of emitted light when electrons transition between principal energy levels in a hydrogen-like atom is given by the Rydberg formula:
\frac{1}{ \lambda } = R \times \left( \frac{1}{n{final}^2} - \frac{1}{n{initial}^2} \right)
Where n{initial} and n{final} are positive integers representing the principal quantum numbers of the initial and final energy levels, respectively (n{initial} > n{final} for emission).
Rydberg Constant: R = 1.097 \times 10^7 \text{ m}^{-1} (for hydrogen).
Wavelike Properties of Matter
De Broglie’s Hypothesis
Louis de Broglie proposed the concept of wave-particle duality for all matter, suggesting that particles (like electrons) can also exhibit wave-like properties. This hypothesis was a crucial step in the development of quantum mechanics.
De Broglie Equation: Determines the wavelength (\lambda) of a particle in motion:
\lambda = \frac{h}{mv}
Where h is Planck's constant, m is the mass of the particle (in kg), and v is its velocity (in m/s). This wavelength is only significant for particles with very small mass, like electrons.
Quantum Mechanics
Schrödinger's Contributions
Wave Function (\Psi): Austrian physicist Erwin Schrödinger developed a mathematical equation that describes the behavior of electrons in atoms in terms of wave-like properties. The wave function, symbolized as \Psi , contains all the information about a quantum mechanical system (e.g., an electron in an atom).
Probability Density (|\Psi|^2): The square of the wave function, |\Psi|^2 , provides the probability density of finding an electron at a particular point in space. This means atomic orbitals (regions where electrons are most likely to be found) are described not as fixed paths but as probability distributions.
Schrödinger’s Equation
A fundamental mathematical formulation in quantum mechanics that describes how the quantum state of a physical system changes over time. For stationary states, it is an eigenvalue equation:
\hat{H}\Psi = E\Psi (where \hat{H} is the Hamiltonian operator, E is energy).
Can only be solved exactly for simpler systems like the hydrogen atom (a single electron). For multi-electron systems, due to electron-electron repulsion, approximations and computational methods are used to solve it.
Quantum Mechanical Model of the Atom
This model replaces Bohr's fixed orbits with probabilistic orbital descriptions.
Quantum Numbers
A set of four numbers that describe the unique quantum state and location of each electron in an atom. These numbers arise naturally from the solution of Schrödinger's equation for the hydrogen atom.
Principal Quantum Number (n):
Indicates the electron's principal energy level and, predominantly, the size of the orbital. Higher n values mean higher energy and larger orbitals.
Angular Momentum Quantum Number (\ell) (also known as the azimuthal or orbital quantum number):
Describes the shape of the orbital and defines subshells within an energy level.
Magnetic Quantum Number (m_{\ell}):
Specifies the orientation of an orbital in space around the nucleus within a subshell.
Values of Quantum Numbers
Principal Quantum Number (n):
Can be any positive integer (e.g., 1, 2, 3, …). Determines the main energy shell.
Angular Momentum Quantum Number (\ell):
Allowed values range from 0 to (n - 1). For a given n, there are n possible values of \ell.
Orbital Types (subshells):
\ell = 0: s orbital (spherical shape)
\ell = 1: p orbital (dumbbell shape, three orientations)
\ell = 2: d orbital (more complex shapes, five orientations)
\ell = 3: f orbital (even more complex shapes, seven orientations)
Magnetic Quantum Number (m_{\ell}):
For a given \ell, values range from -\ell to +\ell (inclusive), including 0. There are (2\ell + 1) possible m_{\ell} values, corresponding to the number of orbitals within a subshell of a given shape.
Example: If \ell = 1 (p orbital), m{\ell} can be -1, 0, +1, representing the three p orbitals (px, py, pz).
Spin Quantum Number (m_s)
This fourth quantum number is not derived from the Schrödinger equation but describes an intrinsic angular momentum of the electron, often visualized as the electron "spinning" on its axis.
It determines the magnetic orientation of the electron spin, with only two possible values: +1/2 (spin up) or -1/2 (spin down).
Pauli Exclusion Principle: States that no two electrons in an atom can have the exact same set of four quantum numbers (n, \ell, m{\ell}, ms). This implies that an orbital can hold a maximum of two electrons, and if it does, these electrons must have opposite spins.
Heisenberg’s Uncertainty Principle
Formulated by Werner Heisenberg, this principle states that it is fundamentally impossible to precisely know both the exact position and the exact momentum (mass times velocity) of a quantum particle, such as an electron, simultaneously. The more precisely one quantity is known, the less precisely the other can be known. This principle underlies the probabilistic nature of electron location described by the quantum mechanical model.
Electron Configurations of Multielectron Atoms
Definitions
Electron Configuration: A notation that describes the distribution of electrons among the atomic orbitals of an atom in its ground state.
Degenerate Orbitals: Orbitals that have the same energy level. In a free atom, orbitals within the same subshell (e.g., the three p orbitals, the five d orbitals) are degenerate. However, in the presence of an external magnetic field or when considering electron-electron repulsion, degeneracy can be lifted.
Ground-State Configuration: The electron configuration that corresponds to the lowest possible total energy for the atom, meaning electrons occupy the lowest available energy orbitals.
Aufbau Principle
Filling Order: The "building-up" principle states that electrons occupy the lowest energy orbitals first before filling higher energy orbitals. This sequential filling order is governed by a combination of:
Pauli Exclusion Principle: No two electrons in an atom can have the same four quantum numbers. Therefore, each orbital can hold a maximum of two electrons, which must have opposite spins (m_s = +1/2 and -1/2).
Hund’s Rule: For degenerate orbitals (orbitals of the same energy), electrons will first occupy separate orbitals with parallel spins (same m_s value) before pairing up in any one orbital. This minimizes electron-electron repulsion and leads to a lower energy state.
The sequence of filling is generally determined by increasing values of n+\ell, and for equal n+\ell values, by increasing $$n