chapter 6 lec 2

Quantum Theory and Electronic Structure of Atoms

Quantum Jumps and Energy Levels

Electrons in atoms occupy discrete energy levels, and transitions between these levels involve the absorption or emission of specific, quantized amounts of energy. These transitions are called quantum jumps.

• Absorption: When an electron absorbs energy, it moves from a lower energy level (e.g., n = 1) to a higher one (e.g., n = 2, 3, 4, ...).

• Emission: When an electron returns to a lower energy state, it emits energy as a photon.

• The energy difference between two levels is given by: $\Delta E = E_{\text{final}} - E_{\text{initial}}$ For hydrogen-like atoms: $E_n = -2.18 \times 10^{-18} \text{ J} \left( \frac{1}{n^2} \right)$ So, $\Delta E = -2.18 \times 10^{-18} \text{ J} \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$

• 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).

• Since wavelength ($\lambda$) must be positive, use the absolute value of $\Delta E$:

• Relationship between energy and wavelength: $\Delta E = h \nu = \frac{hc}{\lambda}$ Where $h$ is Planck's constant ($6.626 \times 10^{-34} \text{ J s}$), $c$ is the speed of light ($3.00 \times 10^8 \text{ m/s}$).

• Example: $\lambda = \frac{hc}{|\Delta E|} = 4.11 \times 10^{-7} \text{ m} = 411 \text{ nm}$ (visible light)

• Endothermic process: Promoting an electron from n = 1 to n = 4 requires energy input. $\Delta E = 2.044 \times 10^{-18} \text{ J}$ Corresponding wavelength: $\lambda = 97.2 \text{ nm}$ (ultraviolet region)

The de Broglie Hypothesis: Wave-Particle Duality

Louis de Broglie proposed that particles such as electrons exhibit wave-like properties, just as light (waves) can behave like particles (photons). This is known as wave-particle duality.

• Electrons in atoms behave like standing waves (similar 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 (1, 2, 3, ...).

• de Broglie wavelength: $\lambda = \frac{h}{mu}$ where $m$ is mass, $u$ is velocity.

• Example (macroscopic object): A 3000-lb car at 60 mi/hr has $\lambda \approx 3.74 \times 10^{-39} \text{ m}$ (undetectably small).

• Example (electron): Electron with $u = 2.65 \times 10^6 \text{ m/s}$: $\lambda = 2.74 \times 10^{-10} \text{ m}$ (comparable to atomic dimensions)

• Application: Only submicroscopic particles (like electrons) have measurable wave properties.

Electron Diffraction: Evidence for Wave Nature

Experiments showed that electrons can be diffracted (bent) by thin metal foils, producing patterns similar to those made by X-rays. This confirms the wave-like behavior of electrons.

• Electron diffraction patterns (concentric rings) are direct evidence of electron wave properties.

Development of Quantum Theory

The realization that matter can have wavelike properties led to the development of quantum mechanics. Bohr's model explained the hydrogen atom but failed for more complex atoms.

• Electrons sometimes behave as waves, sometimes as particles.

• Additional spectral lines (e.g., in magnetic fields) could not be explained by Bohr's model.

• It is impossible to specify the exact position of a wave-like electron.

The Heisenberg Uncertainty Principle

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

• For an electron, the more precisely its momentum is known, the less precisely its position can be known, and vice versa.

• This principle invalidates the idea of electrons moving in well-defined orbits (as in Bohr's model).

The Schrödinger Equation and Quantum Mechanical Model

Erwin Schrödinger developed an equation that describes the behavior and energies of submicroscopic particles, incorporating both their wave and particle nature.

• The wave function ($\psi$) describes the probability amplitude of finding an electron in a particular region.

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

• An atomic orbital is a region in space where there is a high probability of finding an electron.

• The Schrödinger equation can be solved exactly only for the hydrogen atom; for multi-electron atoms, approximations are used.

Quantum Numbers and Atomic Orbitals

Quantum numbers describe the distribution and properties of electrons in atoms. Four quantum numbers are required:

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

• Angular Momentum Quantum Number (l): Indicates the shape of the orbital. Values: 0 to (n-1).

  • l = 0: s orbital (spherical)

  • l = 1: p orbital (dumbbell-shaped)

  • l = 2: d orbital (cloverleaf-shaped)

  • l = 3: f orbital (complex shapes)

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

• Magnetic Quantum Number (ml): Indicates the orientation of the orbital in space. Values: -l to +l (including 0).

• Spin Quantum Number (ms): Indicates the spin of the electron. Values: +½ or -½.

Summary Table: Quantum Numbers and Orbitals

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

Shapes and Properties of Atomic Orbitals

• s Orbitals: Spherical in shape. The size increases with increasing n (1s, 2s, 3s, ...).

• p Orbitals: Dumbbell-shaped, oriented along x, y, and z axes (px, py, pz). Begin at n = 2.

• d Orbitals: Four-lobed shapes (except dz2), begin at n = 3. Five orientations.

• f Orbitals: Complex shapes, begin at n = 4. Seven orientations.

• Nodes: Regions where the probability of finding an electron is zero.

• Radial Probability Distribution: Shows the probability of finding an electron at a certain distance from the nucleus. For 1s, the maximum 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: 1 orbital (ml = 0)

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

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

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

  • Total: 1 + 3 + 5 + 7 = 16 orbitals

Examples of Quantum Numbers

• For n = 2, possible quantum numbers for two different orbitals:

  • n = 2, l = 1, ml = 0, ms = -½

  • n = 2, l = 1, ml = +1, ms = -½

• All combinations of n, l, ml, and ms describe the unique quantum state of an electron in an atom.

Additional info: The above notes expand on the original lecture content by providing definitions, context, and examples for each quantum number and orbital type, as well as summarizing the key equations and principles of quantum theory relevant to general chemistry.