Electronic Structure of Atoms: The Bohr Model and Quantum Theory

Electronic Structure of Atoms

The Bohr Atom

The Bohr model was an early attempt to describe the arrangement of electrons in atoms, particularly hydrogen. It introduced the concept of quantized energy levels for electrons.

• Electrons exist in specific energy levels at various distances from the nucleus.

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

• An electron must be in one energy level; it cannot exist between levels.

• The energy of the electron is quantized, meaning it can only have certain discrete values.

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

• An electron at a higher energy level than its ground state is in an excited state.

Bohr’s Model of the Atom (1913)

Bohr proposed that electrons can only occupy certain allowed orbits with specific (quantized) energies. Energy is emitted or absorbed as electrons move between these orbits.

• Energy of an electron in the nth orbit:

$E_n = -R_H \left( \frac{1}{n^2} \right)$ Where:

• n = principal quantum number (n = 1, 2, 3, ...)

• RH = Rydberg constant = $2.18 \times 10^{-18}$ J

• When an electron in a higher energy orbit (e.g., n = 3) falls to a lower energy orbit (e.g., n = 2), a photon (particle of light) is emitted with energy equal to the difference between the two levels.

• Energy of the photon:

$E = h \nu$ Where:

• h = Planck's constant ($6.63 \times 10^{-34}$ J·s)

• \nu = frequency of the photon

Key Features of the Bohr Model

• Only orbits of certain radii, corresponding to specific energies, are permitted for the hydrogen atom.

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

• Energy is emitted or absorbed by the electron only as it moves from one energy state to another. This energy is emitted or absorbed as a photon with energy $E = h\nu$.

Ground State and Excited States

Electrons in the lowest energy orbit are in the ground state. Any energy higher than this is called an excited state.

• Each orbit has a specific energy, so transitions from one energy level to another can be calculated:

$\Delta E = E_f - E_i = (-2.18 \times 10^{-18} \text{ J}) \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$

• If $\Delta E$ is positive, energy is absorbed (photon absorbed).

• If $\Delta E$ is negative, energy is released (photon emitted).

Example Calculation: Wavelength of Emitted Photon

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

• Energy change:

$E_{\text{photon}} = \Delta E = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$ $E_{\text{photon}} = 2.18 \times 10^{-18} \text{ J} \times (1/9 - 1/25)$ $E_{\text{photon}} = -1.55 \times 10^{-19} \text{ J}$ (energy is emitted)

• Wavelength calculation (omit the minus sign):

$\lambda = \frac{hc}{E_{\text{photon}}}$ $\lambda = \frac{6.63 \times 10^{-34} \text{ J·s} \times 3.00 \times 10^8 \text{ m/s}}{1.55 \times 10^{-19} \text{ J}}$ $\lambda = 1280 \text{ nm}$ (Infrared)

Limitations of the Bohr Model

• Bohr's model accurately predicts the energy levels for hydrogen but fails for atoms with more than one electron.

• It does not explain why energies are quantized or why electrons are restricted to certain orbits.

Wave-Particle Duality and Quantum Mechanics

De Broglie Hypothesis

De Broglie proposed that all matter has wave-like properties. This concept is known as wave-particle duality.

• If light waves can behave like particles, then particles (such as electrons) can behave like waves.

• The wavelength associated with a particle is given by:

$\lambda = \frac{h}{mv}$ Where:

• m = mass of the particle

• v = velocity of the particle

• h = Planck's constant

• Wave properties are significant only for submicroscopic particles due to the small value of Planck's constant.

Heisenberg Uncertainty Principle

The Heisenberg uncertainty principle states that it is impossible to know both the momentum and position of a particle with absolute certainty.

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

$\Delta x \cdot \Delta p \geq h$

• This principle is fundamental to quantum mechanics and limits the accuracy of measurements at the atomic scale.

Quantum Mechanics and the Schrödinger Equation

Schrödinger developed a mathematical model (the Schrödinger equation) that incorporates both the wave and particle nature of electrons. This forms the basis of quantum mechanics.

• The solutions to the Schrödinger equation are called wave functions ($\psi$), which describe the probability distribution of an electron's position.

• 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.

• Principal Quantum Number (n): Indicates the energy level and relative size of the orbital. Values: n = 1, 2, 3, ...

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

• Magnetic Quantum Number (ml): Describes the orientation of the orbital in space. Values: ml = -l, ..., 0, ..., +l

Relationship Among Quantum Numbers

Designation	n	l	ml	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
4s	4	0	0	1
4p	4	1	-1, 0, 1	3
4d	4	2	-2, -1, 0, 1, 2	5
4f	4	3	-3, -2, -1, 0, 1, 2, 3	7

Types of Atomic Orbitals

• s Orbitals (l = 0): Spherical in shape; only one s orbital per energy level.

• p Orbitals (l = 1): Dumbbell-shaped with two lobes and a node between them; three p orbitals per energy level (ml = -1, 0, 1).

• d Orbitals (l = 2): More complex shapes, often with four lobes; five d orbitals per energy level (ml = -2, -1, 0, 1, 2).

• f Orbitals (l = 3): Very complex shapes; seven f orbitals per energy level (ml = -3, -2, -1, 0, 1, 2, 3).

Summary Table: Number of Orbitals per Subshell

Subshell	Number of Orbitals
s	1
p	3
d	5
f	7

Additional info: The quantum mechanical model replaces the concept of fixed orbits with orbitals, which are regions of space where electrons are most likely to be found. The shapes and orientations of these orbitals are determined by the quantum numbers.