Electronic Structure of Atoms: The Bohr Model and Quantum Theory

Electronic Structure of Atoms

The Bohr Atom

The Bohr model, developed by Niels Bohr in 1913, was a pioneering theory describing the arrangement of electrons in atoms, particularly hydrogen. It introduced the concept of quantized energy levels for electrons.

• Specific Energy Levels: Electrons exist in specific, quantized energy levels at various distances from the nucleus.

• Orbits: Electrons revolve in orbits around the nucleus, similar to planets around the sun.

• Quantization: An electron must be in one of these energy levels; it cannot exist between them.

• Ground State: The lowest energy state is called the ground state (most stable).

• Excited State: An electron in a higher energy level than the ground state is in an excited state.

Bohr’s Model of the Atom (1913)

Bohr postulated that electrons in atoms can only occupy certain allowed orbits with fixed energies. Energy is emitted or absorbed when an electron moves between these orbits.

• Quantized Energies: The energy of an electron in a given orbit is given by:

$E_n = -R_H \left( \frac{1}{n^2} \right)$ where $n$ is the principal quantum number ($n = 1, 2, 3, ...$) and $R_H$ is the Rydberg constant ($2.18 \times 10^{-18}$ J).

• Photon Emission: When an electron falls from a higher energy orbit ($n_i$) to a lower one ($n_f$), a photon is emitted with energy equal to the difference between the two levels:

$E_{\text{photon}} = \Delta E = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$

• Energy and Frequency: The energy of the photon relates to its frequency ($\nu$) by:

$E = h \nu$

• Transitions: Energy is emitted when an electron moves to a lower energy level, and absorbed when moving to a higher one.

Key Features of the Bohr Model

• Only certain orbits with specific energies are allowed for electrons in a hydrogen atom.

• An electron in a permitted orbit has an "allowed" energy state.

• Energy is emitted or absorbed only as the electron transitions between these states, corresponding to the emission or absorption of a photon.

Energy Calculations and Example

To calculate the wavelength of a photon emitted when an electron transitions between energy levels:

1. Calculate the energy difference:

2. $E_{\text{photon}} = \Delta E = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$

3. Relate energy to wavelength:

4. $E = h \nu = \frac{h c}{\lambda}$

5. Solve for wavelength ($\lambda$):

6. $\lambda = \frac{h c}{E_{\text{photon}}}$

Example: Calculate the wavelength (in nm) of a photon emitted by a hydrogen atom when its electron drops from $n = 5$ to $n = 3$.

• $E_{\text{photon}} = 2.18 \times 10^{-18} \text{ J} \times (1/9 - 1/25) = -1.55 \times 10^{-19} \text{ J}$ (energy is emitted)

• $\lambda = \frac{6.63 \times 10^{-34} \text{ J} \cdot \text{s} \times 3.00 \times 10^8 \text{ m/s}}{1.55 \times 10^{-19} \text{ J}} = 1280 \text{ nm}$ (Infrared region)

Limitations of the Bohr Model

• Accurately predicts energy levels for hydrogen but fails for multi-electron atoms.

• Does not explain why electron energies are quantized or why electrons are restricted to certain orbits.

The Wave-Particle Duality and Quantum Mechanics

de Broglie Hypothesis

Louis de Broglie proposed that all matter exhibits both particle and wave properties. For electrons, this wave nature becomes significant.

• The wavelength ($\lambda$) of a particle is given by:

$\lambda = \frac{h}{mv}$ where $h$ is Planck's constant, $m$ is mass, and $v$ is velocity.

• Wave properties are only observable for submicroscopic particles due to the small value of $h$.

The Heisenberg Uncertainty Principle

Werner Heisenberg formulated the uncertainty principle, which states that it is impossible to know both the position and momentum of a particle with absolute precision.

$\Delta x \cdot \Delta (mv) \geq \frac{h}{4\pi}$

• The more precisely the position is known, the less precisely the momentum is known, and vice versa.

Quantum Mechanics and the Schrödinger Equation

Erwin Schrödinger developed a mathematical model (the Schrödinger equation) that treats electrons as wave-like entities. The solutions to this equation are called wave functions ($\psi$), which describe the probability distribution of an electron in an atom.

• The square of the wave function, $|\psi|^2$, gives the probability density of finding an electron at a particular location.

Quantum Numbers and Atomic Orbitals

Quantum Numbers

Quantum numbers arise from the solutions to the Schrödinger equation and describe the properties of atomic orbitals and the electrons in them.

• Principal Quantum Number ($n$): Indicates the main energy level or shell. $n = 1, 2, 3, ...$

• Angular Momentum Quantum Number ($l$): Defines the shape of the orbital. $l = 0, 1, 2, ..., n-1$

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

Summary Table: Quantum Numbers and Orbitals

Designation	n	l	Possible $m_l$ values	Number of Orbitals
1s	1	0	0	1
2s	2	0	0	1
2p	2	1	-1, 0, 1	3
3s	3	0	0	1
3p	3	1	-1, 0, 1	3
3d	3	2	-2, -1, 0, 1, 2	5
4f	4	3	-3, -2, -1, 0, 1, 2, 3	7

Shapes of Atomic Orbitals

• s Orbitals ($l = 0$): Spherical in shape. The size of the sphere increases with $n$.

• p Orbitals ($l = 1$): Dumbbell-shaped with two lobes and a node at the nucleus. There are three p orbitals per energy level ($m_l = -1, 0, 1$).

• d Orbitals ($l = 2$): Four have cloverleaf shapes, and one (the $d_{z^2}$ orbital) has a doughnut shape around the center. There are five d orbitals per energy level.

• f Orbitals ($l = 3$): More complex shapes, with seven f orbitals per energy level.

Example: The 2p orbitals have $l = 1$ and $m_l = -1, 0, 1$, corresponding to the $p_x$, $p_y$, and $p_z$ orientations.

Additional info: The above content is foundational for understanding atomic structure, electron configurations, and the periodic properties of elements in General Chemistry.