Chapter 8: Electronic Structure of Atoms – Study Notes

Electronic Structure of Atoms

How to Think About Light

The study of atomic structure is closely linked to the properties of light, which can be described as a stream of energy packets called photons. Photons exhibit both wave-like and particle-like behavior, a concept known as wave-particle duality. Understanding light is essential for probing the electronic structure of atoms, as electrons themselves share similar properties.

• Photons are "wave-particles" that carry energy.

• The wavelength ($\lambda$) and frequency ($\nu$) of light are inversely related: longer wavelength means lower frequency and lower energy; shorter wavelength means higher frequency and higher energy.

• Light is used to study atomic structure because electrons interact with light in similar ways.

Example: The photoelectric effect demonstrates the particle nature of light, where photons can eject electrons from a metal surface.

Characteristics of Light

Light is a form of electromagnetic radiation and can be described by its speed, wavelength, and frequency. The energy of light is quantized and can be calculated using fundamental constants.

• Speed of light: $c = \lambda \nu$

• Energy of light: $E = h \nu$

• Constants:

  • $c$ = speed of light ($2.998 \times 10^8$ m/s)

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

Example: Calculate the energy of a photon with a frequency of $5 \times 10^{14}$ Hz: $E = (6.626 \times 10^{-34}) \times (5 \times 10^{14}) = 3.313 \times 10^{-19}$ J.

The Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, from radio waves to gamma rays. The visible region is only a small part of the spectrum.

• Electromagnetic radiation can be depicted as a sine wave.

• Different regions of the spectrum correspond to different wavelengths and frequencies.

• Visible light ranges from approximately 400 nm (violet) to 750 nm (red).

Example: Infrared radiation has longer wavelengths than visible light and is used in remote controls and thermal imaging.

Quantitization of Energy Levels in Atoms

The Bohr model describes the atom in terms of quantized energy levels. Electrons can only occupy certain energy levels, similar to steps on a staircase. This quantization is most straightforward for hydrogen.

• Energy levels are given by: $E_n = -\frac{2.18 \times 10^{-18} \text{ J}}{n^2}$

• Where $n$ is the principal quantum number (energy level).

• For atoms other than hydrogen, energy levels are more complex.

Example: The energy of an electron in the $n=2$ level of hydrogen is $-0.545 \times 10^{-18}$ J.

Energy Levels of the Hydrogen Atom

Hydrogen's energy levels are well-defined and transitions between them result in absorption or emission of photons. Excitation involves moving an electron to a higher energy level, while ionization removes the electron from the atom.

• Ground state: lowest energy level ($n=1$).

• Excited states: higher energy levels ($n>1$).

• Energy transitions: $\Delta E = E_f - E_i$

Example: An electron moving from $n=2$ to $n=1$ emits a photon with energy equal to the difference between the two levels.

Ground States and Excited States

Atoms typically exist in their ground state, with electrons in the lowest possible energy levels. Absorption of energy can promote electrons to excited states, while emission returns them to lower energy levels.

• Light absorption: Atoms absorb photons matching the energy difference between levels.

• Light emission: Excited atoms emit photons as electrons return to lower energy levels.

Example: The emission spectrum of hydrogen shows distinct lines corresponding to electron transitions.

Atomic (Emission) Line Spectra

When atoms are excited, they emit light at specific wavelengths, producing line spectra unique to each element. These spectra are used to identify elements and study atomic structure.

• Emission spectra consist of discrete lines, not a continuous spectrum.

• Each line corresponds to a specific electron transition.

Example: Neon signs emit characteristic red-orange light due to electron transitions in neon atoms.

Energy of Absorption/Emission

The energy change associated with electron transitions can be calculated using the initial and final energy levels.

• Initial energy: $E_i = -\frac{2.18 \times 10^{-18} \text{ J}}{n_i^2}$

• Final energy: $E_f = -\frac{2.18 \times 10^{-18} \text{ J}}{n_f^2}$

• Energy difference: $\Delta E = 2.18 \times 10^{-18} \left( \frac{1}{n_i^2} - \frac{1}{n_f^2} \right)$

Example: Calculate the energy of a photon emitted when an electron drops from $n=3$ to $n=2$ in hydrogen.

Properties of Electrons

Electrons possess mass, charge, and magnetic properties. They also exhibit wave-like behavior, described by equations similar to those for light.

• Energy: $E = h \nu$ (photon), $E = \frac{mv^2}{2}$ (kinetic)

• Wavelength: $\lambda = \frac{hc}{E}$ (photon), $\lambda = \frac{h}{mv}$ (de Broglie)

• Velocity: $v = \left( \frac{2E}{m} \right)^{1/2}$ or $v = \frac{h}{\lambda m}$

Example: The de Broglie wavelength of an electron moving at $1 \times 10^6$ m/s is $\lambda = \frac{6.626 \times 10^{-34}}{9.11 \times 10^{-31} \times 1 \times 10^6} \approx 7.27 \times 10^{-10}$ m.

Heisenberg’s Uncertainty Principle

The wave nature of electrons means that it is impossible to know both their exact position and velocity simultaneously. This principle sets a fundamental limit on measurements in quantum mechanics.

• If position is known, velocity is uncertain.

• If velocity is known, position is uncertain.

Example: The more precisely we measure an electron's position, the less precisely we can know its momentum.

Where are the Electrons in an Atom?

Electrons occupy regions of space called orbitals, which are defined by probability distributions. The mathematical description of these regions is given by Schrödinger's Equation.

• Each electron is found in one orbital.

• Orbitals have specific shapes and volumes.

• Energy levels are quantized and unique for each atom.

Example: The 1s orbital is spherical and closest to the nucleus.

Atomic Properties of Electrons: Quantum Numbers

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

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

• Azimuthal quantum number ($l$): $l = 0, 1, ..., n-1$ (orbital shape: s, p, d, f, g)

• Magnetic quantum number ($m_l$): $m_l = -l, ..., 0, ..., +l$ (orbital orientation)

• Spin quantum number ($m_s$): $m_s = +1/2, -1/2$ (electron spin)

Example: For $n=2$, $l=1$ (p orbital), $m_l$ can be -1, 0, or +1, and $m_s$ can be +1/2 or -1/2.

Assigning Quantum Numbers

Quantum numbers are assigned based on the energy level and type of orbital. Each set of quantum numbers corresponds to a unique electron in an atom.

• For $n=1$: $l=0$, $m_l=0$, $m_s=+1/2$ or $-1/2$ (two electrons in 1s orbital)

• For $n=2$: $l=0$$or$1$;$m_l=0$(for$l=0$),$m_l=-1,0,+1$(for$l=1$);$m_s=+1/2$or$-1/2$

Example: The 2p subshell can hold six electrons, each with a unique set of quantum numbers.

Assigning Quantum Numbers: Table

The following table summarizes the relationship among $n$, $l$, and $m_l$ for $n=1$ to $n=4$:

n	Subshell Designation	Possible Values of l	Possible Values of ml	Number of Orbitals in Subshell	Total Number of Orbitals in Shell
1	1s	0	0	1	1
2	2s, 2p	0, 1	0; -1, 0, +1	1 (s), 3 (p)	4
3	3s, 3p, 3d	0, 1, 2	0; -1, 0, +1; -2, -1, 0, +1, +2	1 (s), 3 (p), 5 (d)	9
4	4s, 4p, 4d, 4f	0, 1, 2, 3	0; -1, 0, +1; -2, -1, 0, +1, +2; -3, -2, -1, 0, +1, +2, +3	1 (s), 3 (p), 5 (d), 7 (f)	16

s Orbitals (l = 0)

s orbitals are spherical in shape and centered around the nucleus. The probability density is highest at the nucleus and decreases with distance.

• 1s, 2s, 3s orbitals increase in size with increasing $n$.

• Probability density is proportional to $|\psi|^2$.

Example: The 1s orbital is the only orbital in the first shell and can hold two electrons.

p Orbitals (l = 1)

p orbitals have a dumbbell shape and are oriented along the x, y, and z axes. Each p subshell contains three orbitals ($p_x$, $p_y$, $p_z$).

• Each p orbital can hold two electrons.

• p orbitals appear in shells with $n \geq 2$.

Example: The 2p subshell contains three p orbitals, each oriented differently in space.

d Orbitals (l = 2)

d orbitals have more complex shapes, often described as cloverleaf or donut-shaped. There are five d orbitals in each d subshell.

• d orbitals appear in shells with $n \geq 3$.

• Each d orbital can hold two electrons.

Example: The 3d subshell contains five d orbitals, important for transition metals.

Quantum Numbers and Energy Levels

Quantum numbers define the arrangement and energy of electrons in atoms. Each shell and subshell contains a specific number of orbitals, which in turn determine the electron configuration.

• 1st shell ($n=1$): 1 orbital (1s)

• 2nd shell ($n=2$): 4 orbitals (1s, 3p)

• 3rd shell ($n=3$): 9 orbitals (1s, 3p, 5d)

• 4th shell ($n=4$): 16 orbitals (1s, 3p, 5d, 7f)

Example: The electron configuration of carbon is 1s2 2s2 2p2.

Additional info: These notes expand on the original slides by providing definitions, examples, and context for each concept, ensuring a comprehensive understanding suitable for exam preparation.