Electronic Structure of Atoms: Light, Quantization, and Atomic Models

Chapter 6

Electronic Structure of Atoms

Introduction

The electronic structure of atoms is fundamental to understanding chemical properties and behavior. This topic explores how light interacts with matter, the quantization of energy, and the development of atomic models.

Light and Electromagnetic Radiation

Nature of Light

Light is a form of electromagnetic radiation, which consists of both electric and magnetic waves oscillating perpendicular to each other. Electromagnetic radiation travels through a vacuum at the speed of light, denoted as c.

• Wavelength (λ): The distance between successive crests of a wave.

• Amplitude: The height of the wave crest.

• Frequency (ν): The number of wave cycles that pass a given point per second. Unit: hertz (Hz) or s-1.

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

• Relationship: $c = \lambda \nu$

Example: If a wave has a wavelength of 2.0 m and a frequency of $1.5 \times 10^8$ cycles/s, then $c = \lambda \nu$.

Electromagnetic Spectrum

Overview

The electromagnetic spectrum arranges electromagnetic radiation in order of increasing wavelength or frequency. It includes radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays.

• Continuous spectrum: Contains all wavelengths.

• Line spectrum: Contains only specific wavelengths.

• Visible light: A small portion of the electromagnetic spectrum.

Type	Wavelength Range	Example
Radio	>1 m	FM/AM signals
Microwave	1 mm – 1 m	Microwave ovens
Infrared	700 nm – 1 mm	Heat lamps
Visible	400–700 nm	Human vision
Ultraviolet	10–400 nm	Sunlight
X-ray	0.01–10 nm	Medical imaging
Gamma ray	<0.01 nm	Nuclear reactions

Quantized Energy and Photons

Planck's Hypothesis and Photons

Energy is emitted or absorbed in discrete units called quanta (photons). Max Planck proposed that the energy of a photon is proportional to its frequency.

• Planck's equation: $E = h \nu$

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

• Photoelectric effect: Electrons are ejected from metal surfaces when light of sufficient frequency shines on them.

• Photoelectric equation: $h \nu = KE + W$ Where KE is the kinetic energy of the ejected electron and W is the work function (energy required to remove the electron).

Example: Calculate the energy of a photon with frequency $5.0 \times 10^{14}$ Hz. $E=\frac{6.626\cdot10^{-34}\left(J\cdot s\right)\cdot3.00\cdot10^8\left(\frac{m}{s}\right)}{0.154\cdot10^{-9}\left(m\right)}=1.29\cdot10^{-15}$J

Line Spectra and the Bohr Model

Atomic Emission Spectra

Atoms emit light at specific wavelengths, producing a line spectrum. The hydrogen atom's emission spectrum was explained by the Bohr model.

• Rydberg equation: $\frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$

• Rydberg constant ($R_H$): $1.096776 \times 10^7$ m-1

• $n_1$ and $n_2$: Positive integers, $n_2 > n_1$

The Bohr Model

Energy States in Hydrogen Atom

Niels Bohr proposed that electrons occupy specific orbits with quantized energies. The energy of each orbit is given by:

• Energy of nth orbit: $E_n = - (hcR_H) \frac{1}{n^2}$

• For hydrogen: $E_n = -2.18 \times 10^{-18} \frac{1}{n^2}$ J

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

• The energy is negative, indicating a bound electron.

Energy Transitions

When an electron jumps between energy levels, it absorbs or emits a photon. The energy change is:

• $\Delta E = E_f - E_i = h \nu$

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

• Visible spectrum: Balmer series ($n_f = 2$)

• Lyman series: Ultraviolet emissions ($n_f = 1$)

Example: Calculate the wavelength of a photon emitted when an electron drops from $n = 3$ to $n = 1$ in hydrogen.

$\Delta E=\left(-2.18\cdot10^{-18}j\right)\left(\frac{1}{1^2}-\frac{1}{3^2}\right)=-1.94\cdot10^{-18}J$

$\lambda=\frac{c}{v}=\frac{hc}{\Delta E}=\frac{\left(6.626\cdot10^{-34}Js\right)\left(3.00\cdot10^8\left(\frac{m}{s}\right)\right)}{1.94\cdot10^{-18}J}=1.03\cdot10^{-7}m$

The Wave Behavior of Matter

de Broglie Hypothesis

Louis de Broglie proposed that matter exhibits wave-like properties. The wavelength associated with a particle is:

• $\lambda = \frac{h}{mv}$

• h: Planck's constant

• m: Mass of the particle

• v: Velocity of the particle

Example: Calculate the de Broglie wavelength of a 2.5 g Ping-Pong ball traveling at 15.6 m/s. $\lambda = \frac{6.63 \times 10^{-34}}{2.5 \times 10^{-3} \times 15.6} = 1.7 \times 10^{-32}$ m$ This wavelength is extremely small and unobservable for macroscopic objects.

The Uncertainty Principle

Heisenberg's Uncertainty Principle

Werner Heisenberg established that it is impossible to simultaneously know both the exact position and momentum of a particle. The uncertainty principle is expressed as:

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

• $\Delta x$ = uncertainty in position

• $\Delta p$ = uncertainty in momentum

Example: For an electron (mass $9.11 \times 10^{-31}$ kg) with an average speed of $5 \times 10^6$ m/s, if the uncertainty in speed is 1%, then $\Delta x = \frac{h}{4\pi m \Delta v}$.

Additional info: These notes cover the foundational concepts of atomic structure, including the nature of light, quantization of energy, atomic spectra, and the development of quantum theory. They are suitable for introductory college-level General Chemistry.