Bohr Model, Quantum-Mechanical Model, and Atomic Orbitals: Study Notes

Bohr Model and Atomic Energy Levels

Bohr Model of the Atom

The Bohr model describes the atom as a nucleus surrounded by electrons in discrete energy levels. Electrons can move between these levels by absorbing or emitting energy.

• Energy Levels: Electrons occupy specific orbits (energy levels) with quantized energies.

• Energy of an Electron: The energy of an electron in the nth level of a hydrogen atom is given by:

$E_n = -2.18 \times 10^{-18} \left( \frac{1}{n^2} \right) \text{J}$

• Energy Change (ΔE): When an electron transitions between levels:

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

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

• Application: Used to calculate the energy absorbed or emitted when an electron moves between orbits.

Example: Calculate the energy of an electron in the n = 5 level of a hydrogen atom using the formula above.

Quantum-Mechanical Model of the Atom

Quantum Numbers

Electrons in atoms are described by four quantum numbers, which define their energy, shape, orientation, and spin.

Quantum Number	Symbol	Allowed Values	Property
Principal	n	1, 2, 3, ...	Energy level, size of orbital
Angular Momentum	l	0 to n-1	Shape of orbital
Magnetic	m_l	-l to +l	Orientation of orbital
Spin	m_s	+1/2, -1/2	Spin direction of electron

• Principal Quantum Number (n): Indicates the main energy level.

• Angular Momentum Quantum Number (l): Defines the shape (s, p, d, f) of the orbital.

• Magnetic Quantum Number (m_l): Specifies the orientation of the orbital in space.

• Spin Quantum Number (m_s): Describes the spin direction of the electron.

Example: For n = 3, l can be 0, 1, or 2. For l = 2, m_l can be -2, -1, 0, 1, 2.

Atomic Orbitals: Shape and Orientation

Orbital Types and Quantum Numbers

Atomic orbitals are regions in space where the probability of finding an electron is high. The shape and orientation depend on quantum numbers.

• s-orbitals (l = 0): Spherical shape, one orientation (m_l = 0).

• p-orbitals (l = 1): Dumbbell shape, three orientations (m_l = -1, 0, +1).

• d-orbitals (l = 2): Cloverleaf shape, five orientations (m_l = -2, -1, 0, +1, +2).

Energy Levels: Orbitals are grouped into shells (n) and subshells (l). The number of orbitals in a subshell is given by (2l + 1).

Pauli Exclusion Principle

Electron Configuration Rules

No two electrons in the same atom can have the same set of four quantum numbers. This principle limits the number of electrons in each orbital and shell.

• Maximum Electrons per Orbital: 2 (with opposite spins)

• Maximum Electrons per Shell: $2n^2$

Example: For n = 3, the maximum number of electrons is $2 \times 3^2 = 18$.

Summary Table: Quantum Numbers and Orbitals

n	l	Orbital Type	Number of Orbitals	Maximum Electrons
1	0	1s	1	2
2	0, 1	2s, 2p	1 (s), 3 (p)	2 (s), 6 (p)
3	0, 1, 2	3s, 3p, 3d	1 (s), 3 (p), 5 (d)	2 (s), 6 (p), 10 (d)

Additional info: The number of possible quantum numbers for an electron in a given orbital is determined by the allowed values of n, l, m_l, and m_s.