Unit 3: Atomic Structure

Introduction to Atomic Structure

Our understanding of the atom has evolved significantly over time, from early philosophical ideas to sophisticated quantum mechanical descriptions. This journey has been driven by experimental observations that challenged existing theories and paved the way for new, more accurate models. In this unit, we will explore the pivotal models that shaped our current view of the atom, examining their strengths, weaknesses, and the fundamental principles they introduced.

1. Rutherford's Atomic Model

Ernest Rutherford, building upon J.J. Thomson's "plum pudding" model, conducted a groundbreaking experiment that dramatically altered the perception of atomic structure.

The Gold Foil Experiment

Rutherford's famous gold foil experiment, performed by his students Hans Geiger and Ernest Marsden in 1911, involved firing positively charged alpha (α) particles at a very thin sheet of gold foil (approximately 1000 atoms thick).

• Setup: Alpha particles, emitted from a radioactive source, were directed through a lead shield to form a narrow beam. This beam was aimed at a thin gold foil, and a fluorescent screen was placed around the foil to detect the scattered alpha particles.
• Observations:
  • Most of the alpha particles passed straight through the gold foil without any deflection.
  • A small fraction of the alpha particles were deflected at large angles.
  • A very few (about 1 in 20,000) alpha particles bounced back, undergoing nearly 180-degree deflections.
• Conclusions: Based on these observations, Rutherford proposed the following:
  • Nucleus: Since a few alpha particles were strongly repelled or bounced back, it indicated the presence of a tiny, dense, positively charged region at the center of the atom, which he called the nucleus. The positive charge of the nucleus was responsible for deflecting the positively charged alpha particles.
  • Electrons Orbit Outside: The electrons, being negatively charged and much lighter, were believed to orbit this central nucleus, similar to planets orbiting the sun.
  • Mostly Empty Space: The fact that most alpha particles passed straight through suggested that the atom is largely empty space. The volume occupied by the nucleus is extremely small compared to the total volume of the atom.

Limitations of Rutherford's Atomic Model

Despite its revolutionary insights, Rutherford's model had significant limitations when viewed through the lens of classical physics:

• Instability of Atoms: According to classical electromagnetic theory, an accelerating charged particle (like an electron orbiting the nucleus) should continuously radiate energy. If electrons were to continuously lose energy, their orbits would shrink, and they would eventually spiral into the nucleus, causing the atom to collapse. This contradicts the observed stability of atoms.
• Inability to Explain Line Spectra: Classical physics predicted that an electron spiraling into the nucleus would emit a continuous spectrum of light as its energy decreased continuously. However, experiments showed that atoms emit light only at specific, discrete wavelengths, producing a line spectrum. Rutherford's model could not explain this phenomenon.
• Doesn't Explain Electron Distribution: The model did not specify how electrons are arranged around the nucleus or what their energies are, nor did it explain why they do not fall into the nucleus.

2. Bohr's Atomic Model

Niels Bohr, in 1913, proposed a new model for the hydrogen atom that incorporated quantum ideas to address the shortcomings of Rutherford's model. His model was based on a few fundamental postulates.

Postulates of Bohr's Theory

1. Fixed Energy Levels (Orbits/Shells): Electrons revolve around the nucleus in certain definite circular paths called stationary orbits or energy levels. Each orbit has a fixed amount of energy associated with it. Electrons do not radiate energy when they are in these stationary orbits. These orbits are designated as K, L, M, N... or by principal quantum numbers n=1, 2, 3, 4...
2. No Radiation in Stationary States: An electron moving in a particular orbit does not emit or absorb energy. The energy of an electron remains constant as long as it stays in that orbit.
3. Energy Absorption/Emission During Jumps: Energy is absorbed when an electron jumps from a lower energy orbit to a higher energy orbit (excitation), and energy is emitted when an electron jumps from a higher energy orbit to a lower energy orbit (de-excitation). The energy difference (ΔE) between the two orbits is emitted or absorbed as a photon of light, given by ΔE = E_higher - E_lower = hν = hc/λ, where h is Planck's constant, ν is the frequency, c is the speed of light, and λ is the wavelength.
4. Quantization of Angular Momentum: The angular momentum of an electron in a stationary orbit is quantized, meaning it can only take on certain discrete values. It must be an integral multiple of h/2π.
   mvr = n(h/2π)
   Where:
   • m = mass of the electron
   • v = velocity of the electron
   • r = radius of the orbit
   • n = principal quantum number (1, 2, 3...)
   • h = Planck's constant

Application of Bohr's Model

Bohr's model successfully derived expressions for the energy of an electron and the radius of its orbit in a hydrogen atom (and other one-electron species).

• Energy of Electron in nth Orbit (for H-like species):
  E_n = -R_H * (Z^2 / n^2)
  Where:
  • E_n = energy of the electron in the nth orbit
  • R_H = Rydberg constant (approx. 2.18 x 10^-18 J)
  • Z = atomic number (charge of the nucleus)
  • n = principal quantum number (1, 2, 3...)
  
  For a hydrogen atom (Z=1), E_n = -2.18 x 10^-18 J / n^2. The negative sign indicates that the electron is bound to the nucleus.
• Energy Levels of Hydrogen Atom:
  • n=1 (ground state): E₁ = -2.18 x 10^-18 J
  • n=2: E₂ = -0.545 x 10^-18 J
  • n=3: E₃ = -0.242 x 10^-18 J
  • As n increases, the energy becomes less negative (higher energy), and the orbits are further from the nucleus.
• Radius of Orbits (for H-like species):
  r_n = a_0 * (n^2 / Z)
  Where:
  • r_n = radius of the nth orbit
  • a_0 = Bohr radius (approx. 52.9 pm or 0.529 Å)
  • n = principal quantum number
  • Z = atomic number
  
  For a hydrogen atom (Z=1), r_n = 52.9 * n^2 pm. This shows that the radius of the orbit increases quadratically with n.

Spectrum of Hydrogen Atom

Bohr's model successfully explained the line spectrum of the hydrogen atom.

• Emission Spectrum: When excited hydrogen atoms return to their lower energy states, they emit photons of specific energies, resulting in a line spectrum with discrete wavelengths. Each line corresponds to an electron transition between two specific energy levels.
• Series: The various spectral lines in the hydrogen emission spectrum are grouped into series based on the final energy level (n₁) to which the electron transitions.
  • Lyman Series: Electron transitions from n > 1 to n₁ = 1. These lines fall in the Ultraviolet (UV) region.
  • Balmer Series: Electron transitions from n > 2 to n₁ = 2. These lines fall in the Visible region.
  • Paschen Series: Electron transitions from n > 3 to n₁ = 3. These lines fall in the Infrared (IR) region.
  • Brackett Series: Electron transitions from n > 4 to n₁ = 4. These lines fall in the Infrared (IR) region.
  • Pfund Series: Electron transitions from n > 5 to n₁ = 5. These lines fall in the Infrared (IR) region.
• Rydberg Equation: The wavelengths of the spectral lines in the hydrogen spectrum can be calculated using the Rydberg equation, which was empirically derived before Bohr's theory and later explained by it:
  1/λ = R_H * (1/n_1^2 - 1/n_2^2)
  Where:
  • λ = wavelength of the emitted/absorbed light
  • R_H = Rydberg constant (approx. 1.097 x 10⁷ m^-1)
  • n_1 = principal quantum number of the lower energy level (final state)
  • n_2 = principal quantum number of the higher energy level (initial state), where n_2 > n_1

3. Defects of Bohr's Theory

Despite its successes, Bohr's model had several limitations that led to its eventual replacement by the more sophisticated quantum mechanical model:

• Only Works for One-Electron Species: Bohr's model could accurately predict the spectra of hydrogen (H), helium ion (He⁺), and lithium ion (Li²⁺), which all have only one electron. However, it failed to explain the spectra of multi-electron atoms.
• Cannot Explain Splitting of Spectral Lines: When atomic spectra are observed in the presence of an external magnetic field (Zeeman effect) or an external electric field (Stark effect), the spectral lines are found to split into multiple finer lines. Bohr's theory could not account for this splitting.
• Doesn't Account for Sublevels and Orbital Shapes: The model treated electrons as particles orbiting in planar circles and did not explain the existence of subshells (s, p, d, f) or the complex three-dimensional shapes of orbitals.
• Violates Heisenberg's Uncertainty Principle: Bohr's model assumes that an electron's position and momentum can be precisely known simultaneously (i.e., it has a well-defined orbit). This contradicts Heisenberg's Uncertainty Principle, a fundamental tenet of quantum mechanics.
• Does Not Explain Chemical Bonding: Bohr's model offered no explanation for how atoms form molecules through chemical bonds.

4. Quantum Mechanical Model

The quantum mechanical model of the atom, developed by Erwin Schrödinger and others, provides a more accurate and comprehensive description of atomic structure, overcoming the limitations of earlier models. It is based on the wave nature of matter and the probabilistic nature of electron location.

de Broglie's Wave Equation

In 1924, Louis de Broglie proposed that, similar to light, matter also exhibits wave-particle duality. He suggested that all moving particles, especially subatomic particles like electrons, have wave-like properties.

λ = h / mv

Where:

• λ = wavelength of the particle (de Broglie wavelength)
• h = Planck's constant (6.626 x 10^-34 J·s)
• m = mass of the particle
• v = velocity of the particle

This equation implies that electrons, previously thought of only as particles, also possess wave characteristics. This wave nature is crucial to understanding their behavior within an atom.

Heisenberg's Uncertainty Principle

Werner Heisenberg, in 1927, formulated the Uncertainty Principle, which states that it is impossible to simultaneously determine with perfect accuracy both the position and the momentum (or velocity) of a subatomic particle like an electron.

Δx * Δp ≥ h / 4π

Where:

• Δx = uncertainty in position
• Δp = uncertainty in momentum (Δp = m * Δv, where Δv is uncertainty in velocity)
• h = Planck's constant

This principle fundamentally challenges the classical idea of electrons moving in fixed, well-defined orbits. If we know an electron's exact position, we cannot know its exact momentum, and vice versa. This led to the abandonment of the concept of definite electron paths.

Concept of Probability

Because of the wave-particle duality and the Uncertainty Principle, the quantum mechanical model describes the electron's location in terms of probability rather than precise orbits.

• Electron Cloud Model: Instead of fixed orbits, electrons are described as existing in regions of space around the nucleus where there is a high probability of finding them. This region is often visualized as an "electron cloud," with the density of the cloud representing the probability density.
• Probability Density: The square of the wave function (ψ²) provides the probability density of finding an electron at a particular point in space. It does not mean the electron is smeared out, but rather that its exact position cannot be known, only the likelihood of finding it in a given volume.
• Atomic Orbitals: These three-dimensional regions of space where the probability of finding an electron is high (typically 90-95%) are called atomic orbitals. Each orbital has a characteristic shape and energy.

5. Quantum Numbers

The quantum mechanical model uses a set of four quantum numbers to completely describe the state of an electron in an atom. These numbers specify the electron's energy, shape of its orbital, its orientation in space, and its spin.

• Principal Quantum Number (n):
  • Definition: It defines the main energy level or shell in which the electron resides.
  • Values: Positive integers (1, 2, 3, ...).
  • Significance: Larger 'n' values indicate higher energy levels, larger average distance from the nucleus, and larger orbital size.
  • Shells: n=1 (K shell), n=2 (L shell), n=3 (M shell), etc.
• Azimuthal (or Angular Momentum) Quantum Number (l):
  • Definition: It describes the shape of the electron orbital and the subshell within a principal energy level.
  • Values: Integers from 0 to (n-1).
  • Significance: For a given 'n', there are 'n' possible values of 'l'.
  • Subshells:
    • l=0 corresponds to an s subshell (spherical shape)
    • l=1 corresponds to a p subshell (dumbbell shape)
    • l=2 corresponds to a d subshell (more complex shapes)
    • l=3 corresponds to an f subshell (even more complex shapes)
• Magnetic Quantum Number (m_l):
  • Definition: It describes the orientation of the orbital in space relative to a set of coordinate axes.
  • Values: Integers from -l to +l, including 0.
  • Significance: For a given 'l', there are (2l + 1) possible values of m_l, which means there are (2l + 1) orbitals of a particular type in a subshell.
    • If l=0 (s subshell), m_l=0 (1 s orbital)
    • If l=1 (p subshell), m_l=-1, 0, +1 (3 p orbitals: p_x, p_y, p_z)
    • If l=2 (d subshell), m_l=-2, -1, 0, +1, +2 (5 d orbitals)
• Spin Quantum Number (m_s):
  • Definition: It describes the intrinsic angular momentum of an electron, often referred to as its "spin."
  • Values: Only two possible values: +1/2 (spin up) or -1/2 (spin down).
  • Significance: An electron behaves as if it's spinning on its axis, creating a tiny magnetic field. This spin is quantized and can be in one of two opposite directions.

6. Orbitals and Shapes

Atomic orbitals are the three-dimensional regions around the nucleus where there is the highest probability of finding an electron. Their shapes are determined by the azimuthal quantum number (l).

• s orbital (l=0):
  • Shape: All s orbitals are spherical. The electron probability density is spherically symmetric around the nucleus.
  • Number of Orbitals: There is only 1 s orbital per subshell (because m_l=0).
  • Size: The size of the s orbital increases with increasing principal quantum number (1s < 2s < 3s...).
  • Nodes: Higher energy s orbitals (2s, 3s, etc.) have radial nodes, which are spherical regions of zero electron probability.
• p orbital (l=1):
  • Shape: Each p orbital has a dumbbell shape, consisting of two lobes on opposite sides of the nucleus. There is a nodal plane passing through the nucleus where the probability of finding the electron is zero.
  • Number of Orbitals: There are 3 p orbitals per subshell (because m_l=-1, 0, +1).
  • Orientations: These three p orbitals are oriented perpendicular to each other along the x, y, and z axes, and are denoted as p_x, p_y, and p_z. They are degenerate, meaning they have the same energy in an isolated atom.
  • Existence: p orbitals first appear in the n=2 shell (2p, 3p, etc.).
• d orbital (l=2):
  • Shape: Most d orbitals have a more complex cloverleaf shape (four lobes), except for d_z², which has two lobes along the z-axis and a donut-shaped ring in the xy-plane.
  • Number of Orbitals: There are 5 d orbitals per subshell (because m_l=-2, -1, 0, +1, +2).
  • Orientations: d_xy, d_yz, d_xz, d_x²-y², d_z². They are degenerate.
  • Existence: d orbitals first appear in the n=3 shell (3d, 4d, etc.).
• f orbital (l=3):
  • Shape: f orbitals have even more complex shapes with eight lobes.
  • Number of Orbitals: There are 7 f orbitals per subshell (because m_l=-3, -2, -1, 0, +1, +2, +3).
  • Existence: f orbitals first appear in the n=4 shell (4f, 5f, etc.).

7. Rules for Electronic Configuration

Electronic configuration describes the distribution of electrons among the various atomic orbitals in an atom. This distribution follows specific rules:

Aufbau Principle

The Aufbau principle (from the German "building up") states that in the ground state of an atom, electrons fill atomic orbitals in order of increasing energy. Lower energy orbitals are filled before higher energy orbitals.

• Energy Order: The approximate order of filling orbitals is:
  1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s < 5f < 6d < 7p
• This order can be easily remembered using the Madelung rule or (n+l) rule, where orbitals with a lower (n+l) value are filled first. If two orbitals have the same (n+l) value, the one with the lower 'n' value is filled first (e.g., 4s (4+0=4) before 3d (3+2=5)).

Pauli's Exclusion Principle

Wolfgang Pauli's Exclusion Principle states that no two electrons in the same atom can have the identical set of all four quantum numbers (n, l, m_l, m_s).

• Implication: This means that an atomic orbital can hold a maximum of two electrons, and these two electrons must have opposite spins (one m_s = +1/2 and the other m_s = -1/2). If two electrons share the same n, l, and m_l values, their m_s values must be different.

Hund's Rule of Maximum Multiplicity

Hund's Rule states that for degenerate orbitals (orbitals within the same subshell that have the same energy), electrons will fill them singly first, with parallel spins, before any orbital is doubly occupied.

• Implication: This rule maximizes the total spin of the electrons in degenerate orbitals, leading to a more stable configuration. For example, when filling the three p orbitals (p_x, p_y, p_z) with three electrons, each orbital gets one electron with parallel spin first (e.g., all spin up) before any pairing occurs.

8. Electronic Configurations (up to atomic number 30)

Applying the Aufbau principle, Pauli's exclusion principle, and Hund's rule, we can write the electronic configurations for elements.

Exceptions to Aufbau Principle

While the Aufbau principle provides a general guideline, there are notable exceptions, particularly for transition metals. These exceptions arise due to the extra stability associated with half-filled or completely filled subshells.

• Chromium (Cr, Z=24):
  • Expected configuration: [Ar] 4s² 3d⁴
  • Actual configuration: [Ar] 4s¹ 3d⁵
  • Reason: Moving one electron from the 4s orbital to the 3d orbital results in a half-filled 3d subshell (3d⁵) and a half-filled 4s subshell (4s¹). Both half-filled and fully-filled subshells have enhanced stability due to symmetrical distribution of electrons and exchange energy.
• Copper (Cu, Z=29):
  • Expected configuration: [Ar] 4s² 3d⁹
  • Actual configuration: [Ar] 4s¹ 3d¹⁰
  • Reason: Moving one electron from the 4s orbital to the 3d orbital results in a completely filled 3d subshell (3d¹⁰) and a half-filled 4s subshell (4s¹). A completely filled subshell offers significant extra stability.

These exceptions highlight that while the Aufbau principle is a useful rule of thumb, the actual electron configuration is determined by the lowest overall energy state of the atom, which sometimes favors the stability gained from half-filled or fully-filled subshells.