Atomic Structure: Bohr Model, Quantum Mechanics, and Electron Configuration

Atomic Structure and Models

Bohr Model of the Atom

The Bohr model was an early attempt to describe the structure of the atom, focusing on quantized energy levels for electrons. It successfully explained the hydrogen atom's emission spectrum but failed for multi-electron atoms.

• Energy Levels: Electrons occupy discrete energy levels (n = 1, 2, 3, ...).

• Energy Change Formula: The energy difference between two levels is given by: $\Delta E = E_{\text{final}} - E_{\text{initial}} = R_H \left( \frac{1}{n_{\text{final}}^2} - \frac{1}{n_{\text{initial}}^2} \right)$ where $R_H$ is the Rydberg constant ($2.1799 \times 10^{-18}$ J).

• Photon Emission: When an electron drops from a higher to a lower energy level, a photon is emitted: $\Delta E = h \nu = \frac{hc}{\lambda}$ Example: For $n=3 \rightarrow n=2$ in H atom, $\Delta E = -2.1799 \times 10^{-18} \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = -3.03 \times 10^{-19}$ J, corresponding to $\lambda = 656$ nm (red line in Balmer series).

• Limitations: The Bohr model cannot accurately describe multi-electron atoms due to electron-electron interactions and fails to incorporate the uncertainty principle.

Uncertainty Principle

The Heisenberg uncertainty principle states that it is impossible to simultaneously know both the exact position and momentum of an electron. This principle contributed to the downfall of the Bohr model for complex atoms.

• Mathematical Expression: $\Delta x \Delta p \geq \frac{h}{2}$

• Implication: Electrons cannot be described as particles in fixed orbits; instead, their behavior is probabilistic.

Quantum Mechanical Model of the Atom

Schrödinger Equation

The quantum mechanical model, developed by Erwin Schrödinger, describes electrons as wavefunctions, providing a probabilistic interpretation of their location and energy.

• Time-Independent Schrödinger Equation: $\hat{H} \Psi = E \Psi$ where $\hat{H}$ is the Hamiltonian operator, $\Psi$ is the wavefunction, and $E$ is the energy.

• Wavefunction ($\Psi$): Represents the amplitude of an electron's wave at a given position. The probability of finding an electron in a region is proportional to $|\Psi|^2$.

• Energy Quantization: Only certain energy values are allowed (quantized), not continuous.

• Probability Density: $\text{Probability} \propto |\Psi|^2$ This gives the likelihood of finding an electron in a particular region of space.

• Hydrogen Atom Solutions: The equation can be solved exactly for hydrogen, yielding quantized energy levels and orbital shapes.

Quantum Numbers

Quantum numbers describe the properties of atomic orbitals and the electrons in them.

• Principal Quantum Number ($n$): $n = 1, 2, 3, ...$ Determines the size and energy of the orbital.

• Angular Momentum Quantum Number ($l$): $l = 0, 1, ..., n-1$ Determines the shape of the orbital:

  • $l = 0$: s-orbital

  • $l = 1$: p-orbital

  • $l = 2$: d-orbital

  • $l = 3$: f-orbital

• Magnetic Quantum Number ($m_l$): $m_l = -l, ..., 0, ..., +l$ Determines the orientation of the orbital in space.

Atomic Orbitals and Electron Density

Atomic orbitals are regions in space where electrons are likely to be found. The shapes and orientations are determined by quantum numbers.

• Radial Probability Density: $D(r) = \Psi^2 4 \pi r^2$ Describes the probability of finding an electron at a distance $r$ from the nucleus.

• Orbital Shapes: s-orbitals ($l=0$): Spherical p-orbitals ($l=1$): Dumbbell-shaped d-orbitals ($l=2$): Cloverleaf-shaped f-orbitals ($l=3$): Complex shapes

Multi-Electron Atoms and Electron Correlation

Electron Correlation and Shielding

Multi-electron atoms cannot be described by the Bohr model due to electron-electron repulsions and the inability to simultaneously know all positions and momenta.

• Effective Nuclear Charge ($Z_{\text{eff}}$): $Z_{\text{eff}} = Z - \sigma$ Where $Z$ is the atomic number and $\sigma$ is the shielding parameter (accounts for electron repulsion and screening).

• Energy of an Electron in Multi-Electron Atom: $E_{\text{elec}} = R_H \left( \frac{Z_{\text{eff}}^2}{n^2} \right)$

• Electron Correlation Problem: Approximations are needed for multi-electron atoms, often treating each electron as if it were in a hydrogen-like orbital with $Z_{\text{eff}}$.

Summary Table: Quantum Numbers and Orbital Properties

Quantum Number	Symbol	Possible Values	Physical Meaning
Principal	n	1, 2, 3, ...	Energy level, size of orbital
Angular Momentum	l	0 to n-1	Shape of orbital (s, p, d, f)
Magnetic	ml	-l to +l	Orientation of orbital

Key Equations

• Bohr Energy Levels: $E_n = -R_H \frac{Z^2}{n^2}$

• Energy Change: $\Delta E = R_H \left( \frac{1}{n_{\text{final}}^2} - \frac{1}{n_{\text{initial}}^2} \right)$

• Photon Wavelength: $\Delta E = \frac{hc}{\lambda}$

• Schrödinger Equation: $\hat{H} \Psi = E \Psi$

• Probability Density: $|\Psi|^2$

Additional info:

• These notes cover foundational concepts in atomic structure, quantum mechanics, and electron configuration, suitable for a General Chemistry college course.

• Visuals referenced include energy level diagrams, orbital shapes, and quantum number tables.