Quantum Theory and Electronic Structure of Atoms: Study Notes

Quantum Theory and the Electronic Structure of Atoms

Quantum Jumps and Energy Levels

Electrons in atoms absorb or emit radiant energy by making discrete transitions, or quantum jumps, between allowed energy levels. Each jump corresponds to a specific, quantized amount of energy.

• Absorption: Electron moves from a lower to a higher energy level (e.g., n = 1 to n = 2, 3, 4, etc.).

• Emission: Electron returns from an excited state to a lower energy state, releasing energy as a photon (e.g., n = 3 to n = 2, or n = 3 to n = 1).

• Energy Difference: The energy required for a transition depends on the difference between the initial and final energy levels.

Formula for Energy Change:

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

• ni: Initial principal quantum number

• nf: Final principal quantum number

Example: Electron drops from n = 6 to n = 2:

$\Delta E = -2.18 \times 10^{-18} \text{ J} \left( \frac{1}{2^2} - \frac{1}{6^2} \right) = -4.84 \times 10^{-19} \text{ J}$

This is an exothermic process (energy is released).

Relationship to Wavelength:

$\Delta E = h\nu = \frac{hc}{\lambda}$

• h: Planck's constant ($6.626 \times 10^{-34}$ J·s)

• c: Speed of light ($3.00 \times 10^8$ m/s)

• \lambda: Wavelength (m)

Since wavelength must be positive, use the absolute value of $\Delta E$.

Example Calculation:

$\lambda = \frac{hc}{|\Delta E|} = \frac{(6.626 \times 10^{-34} \text{ J·s})(3.00 \times 10^8 \text{ m/s})}{4.84 \times 10^{-19} \text{ J}} = 4.11 \times 10^{-7} \text{ m} = 411 \text{ nm}$ (visible light)

Endothermic Transitions

When an electron is promoted from a lower to a higher energy state, energy must be supplied (endothermic process).

Example: Energy required to promote an electron from n = 1 to n = 4:

$\Delta E = -2.18 \times 10^{-18} \text{ J} \left( \frac{1}{4^2} - \frac{1}{1^2} \right) = 2.044 \times 10^{-18} \text{ J}$

Corresponding wavelength:

$\frac{1}{\lambda} = \frac{\Delta E}{hc} = \frac{2.044 \times 10^{-18} \text{ J}}{(6.626 \times 10^{-34} \text{ J·s})(3.00 \times 10^8 \text{ m/s})} = 1.028 \times 10^7 \text{ m}^{-1}$

$\lambda = 9.725 \times 10^{-8} \text{ m} = 97.2 \text{ nm}$ (ultraviolet region)

The Wave-Particle Duality of Matter

The de Broglie Hypothesis

Louis de Broglie proposed that particles such as electrons can exhibit wave-like properties, just as light can behave as both a wave and a particle. This is known as wave-particle duality.

• Electrons in atoms behave like standing waves (analogous to a vibrating guitar string).

• Standing waves have nodes (points of zero amplitude).

• For an electron to exist in a stable orbit, its wavelength must fit exactly into the circumference of the orbit: $2\pi r = n\lambda$ (where n is an integer).

de Broglie Wavelength Formula:

$\lambda = \frac{h}{mu}$

• \lambda: Wavelength of the particle (m)

• h: Planck's constant

• m: Mass of the particle (kg)

• u: Velocity of the particle (m/s)

Example 1: Calculate the de Broglie wavelength of a 3000-lb car traveling at 60 mi/hr.

• Result: $\lambda \approx 3.74 \times 10^{-39}$ m (extremely small, undetectable)

Example 2: Wavelength of an electron (mass = $9.11 \times 10^{-31}$ kg) traveling at $2.65 \times 10^6$ m/s:

$\lambda = \frac{6.63 \times 10^{-34} \text{ Js}}{(9.11 \times 10^{-31} \text{ kg})(2.65 \times 10^6 \text{ m/s})} = 2.74 \times 10^{-10} \text{ m}$

Example 3: Velocity of an electron with a de Broglie wavelength of $1.5 \times 10^{-10}$ m (typical C–C bond length):

$u = \frac{h}{\lambda m} = \frac{6.63 \times 10^{-34} \text{ Js}}{(9.11 \times 10^{-31} \text{ kg})(1.5 \times 10^{-10} \text{ m})} = 4.85 \times 10^6 \text{ m/s}$

Conclusion: Only submicroscopic particles (like electrons) have measurable wavelengths; macroscopic objects do not.

Electron Diffraction

Experiments showed that electrons can be diffracted (like waves) when passed through thin metal foils, confirming their wave-like nature.

Development of Quantum Theory

Limitations of the Bohr Model

• Bohr's model accurately describes the hydrogen atom but fails for atoms with more than one electron.

• It cannot explain spectral line splitting in magnetic fields or the dual wave-particle behavior of electrons.

The Heisenberg Uncertainty Principle

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

Mathematical Expression:

$\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$

• \Delta x: Uncertainty in position

• \Delta p: Uncertainty in momentum

This principle means that electrons do not travel in well-defined orbits as Bohr suggested.

The Schrödinger Equation and Quantum Mechanics

Erwin Schrödinger developed an equation that describes the behavior and energy of electrons in atoms, incorporating both their particle and wave nature.

• The wave function ($\psi$) describes the probability amplitude for an electron's position.

• The probability of finding an electron in a region is given by $\psi^2$ (electron density).

• This approach forms the basis of quantum mechanics (or wave mechanics).

The Schrödinger equation can be solved exactly for hydrogen but only approximately for multi-electron atoms.

Quantum Numbers and Atomic Orbitals

Quantum Numbers

Quantum numbers describe the distribution and properties of electrons in atoms. There are four quantum numbers:

Quantum Number	Symbol	Allowed Values	Physical Meaning
Principal	n	1, 2, 3, ...	Size and energy of orbital
Angular Momentum	l	0 to (n-1)	Shape of orbital
Magnetic	ml	-l to +l	Orientation in space
Spin	ms	+1/2, -1/2	Spin direction of electron

Principal Quantum Number (n)

• Determines the size and energy of the orbital.

• n = 1, 2, 3, ...

• Higher n means larger orbital and higher energy (for hydrogen).

Angular Momentum Quantum Number (l)

• Determines the shape of the orbital.

• l = 0, 1, ..., (n-1)

• Names of orbitals by l value:

  • l = 0: s orbital

  • l = 1: p orbital

  • l = 2: d orbital

  • l = 3: f orbital

  • l = 4: g orbital (rare, for very high n)

Magnetic Quantum Number (ml)

• Describes the orientation of the orbital in space.

• ml = -l, ..., 0, ..., +l (total of 2l + 1 values)

• Examples:

  • l = 0 (s): ml = 0 (1 orbital)

  • l = 1 (p): ml = -1, 0, +1 (3 orbitals: px, py, pz)

  • l = 2 (d): ml = -2, -1, 0, +1, +2 (5 orbitals)

  • l = 3 (f): ml = -3, -2, -1, 0, +1, +2, +3 (7 orbitals)

Spin Quantum Number (ms)

• Describes the spin of the electron.

• ms = +1/2 or -1/2

• Each orbital can hold two electrons with opposite spins.

Atomic Orbitals and Their Shapes

• s Orbitals: Spherical in shape; size increases with n (e.g., 1s, 2s, 3s).

• p Orbitals: Dumbbell-shaped; three orientations (px, py, pz); start at n = 2.

• d Orbitals: Four-lobed shapes; five orientations; start at n = 3.

• f Orbitals: More complex shapes; seven orientations; start at n = 4.

Radial Probability Distribution: Shows the probability of finding an electron at a certain distance from the nucleus. For s orbitals, the maximum probability for 1s is at 52.9 pm (Bohr radius).

Counting Orbitals

The total number of orbitals for a given principal quantum number n is $n^2$.

Example: For n = 4:

• l = 0 (s): 1 orbital

• l = 1 (p): 3 orbitals

• l = 2 (d): 5 orbitals

• l = 3 (f): 7 orbitals

• Total: 1 + 3 + 5 + 7 = 16 orbitals

General formula: $n^2$ orbitals for each value of n.

Examples of Quantum Numbers

Example: List the four quantum numbers for two possible orbitals in n = 2:

• n = 2, l = 1, ml = 0, ms = -1/2

• n = 2, l = 1, ml = +1, ms = -1/2

Other combinations are possible by varying ml and ms within allowed values.

Summary Table: Quantum Numbers and Orbitals

n	l	Orbital Type	ml Values	Number of Orbitals
1	0	1s	0	1
2	0	2s	0	1
2	1	2p	-1, 0, +1	3
3	0	3s	0	1
3	1	3p	-1, 0, +1	3
3	2	3d	-2, -1, 0, +1, +2	5
4	0	4s	0	1
4	1	4p	-1, 0, +1	3
4	2	4d	-2, -1, 0, +1, +2	5
4	3	4f	-3, -2, -1, 0, +1, +2, +3	7

Additional info: The notes above expand on the original lecture content by providing definitions, formulas in LaTeX, and structured tables for clarity. The quantum mechanical model is foundational for understanding atomic structure, periodic trends, and chemical bonding, which are covered in subsequent chapters.