Premed · Premed · General Chemistry 1
Lecture 7: Electronic Structure of Atoms
General Chemistry I
Learning Objectives
By the end of this lecture, students will be able to:
- Describe the properties of electromagnetic radiation (wavelength, frequency, speed)
- Apply the relationship c = lambda nu and E = h nu
- Explain the photoelectric effect and its significance for the particle nature of light
- Describe the Bohr model of the hydrogen atom and calculate energy levels
- Explain how atomic emission and absorption spectra arise
- Relate the Bohr model to the concept of quantized energy levels
Lecture Content
I. Electromagnetic Radiation
Electromagnetic radiation (EMR) is a form of energy that propagates through space as oscillating electric and magnetic fields, which are perpendicular to each other and to the direction of travel. Several properties characterize EMR. The wavelength (lambda) is the distance between successive crests or troughs, measured in meters, nanometers (10^-9 m), or picometers (10^-12 m). The frequency (nu) is the number of complete wave cycles passing a given point per second, expressed in hertz (Hz = s^-1). The amplitude, the height from the midline to the crest, determines the intensity or brightness of the radiation. All electromagnetic radiation travels at the speed of light in a vacuum: c = lambda * nu = 3.00 x 10^8 m/s. Because c is constant, wavelength and frequency are inversely proportional: lambda = c / nu.
II. The Electromagnetic Spectrum
The electromagnetic spectrum arranges radiation by wavelength, frequency, or energy. From longest wavelength and lowest energy to shortest wavelength and highest energy, the order is: radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. Visible light occupies the narrow window from approximately 400 nm (violet) to 700 nm (red), with the familiar colors arranged as ROY G BIV -- red, orange, yellow, green, blue, indigo, and violet. The key relationship to remember is that higher frequency corresponds to shorter wavelength and higher energy.
<image>A diagram of the electromagnetic spectrum displayed as a horizontal bar. From left to right (increasing frequency, decreasing wavelength): radio waves, microwaves, infrared, visible light (expanded into a rainbow band showing red at 700 nm through violet at 400 nm), ultraviolet, X-rays, gamma rays. Above the bar, an arrow labeled "Increasing frequency (nu)" points right. Below the bar, an arrow labeled "Increasing wavelength (lambda)" points left. A second arrow below labeled "Increasing energy" points right. Typical wavelength values are marked at each region boundary.</image>
III. Quantization of Energy: Planck's Hypothesis
Classical physics predicted that heated objects should emit infinite energy at short wavelengths, a problem known as the "ultraviolet catastrophe." Experimental observations flatly contradicted this prediction. In 1900, Max Planck resolved the discrepancy by proposing that energy is emitted and absorbed not continuously but in discrete packets called quanta. The energy of a quantum is given by E = h nu = h c / lambda, where h is Planck's constant (6.626 x 10^-34 Js). Energy is therefore quantized: it can only take on certain discrete values that are integer multiples of hnu. This means that higher-frequency radiation carries more energy per quantum.
IV. The Photoelectric Effect
When light strikes a metal surface, electrons are ejected -- but only if the light's frequency exceeds a certain threshold value (nu_0), regardless of how intense the light is. Classical wave theory could not explain this observation, since increasing the intensity should eventually supply enough energy to liberate electrons at any frequency. In 1905, Albert Einstein resolved the puzzle by proposing that light consists of particles called photons, each carrying a discrete energy E = hnu. For an electron to be ejected, a single photon must carry enough energy to overcome the electron's binding energy, known as the work function (phi). The kinetic energy of the ejected electron is then KE = hnu - phi. Below the threshold frequency, no electrons are ejected no matter how bright the light. Above the threshold, more intense light means more photons striking the surface and hence more electrons ejected, while higher-frequency light gives each ejected electron greater kinetic energy. This phenomenon established the wave-particle duality of light.
<image>A diagram of the photoelectric effect experiment and results. Left panel: a metal surface being illuminated by photons (shown as wavy arrows). Electrons are being ejected from the surface. Labels indicate: incoming photon energy (E = hv), work function (phi), and kinetic energy of ejected electron (KE = hv - phi). Right panel: a graph with x-axis "Frequency (nu)" and y-axis "Kinetic energy of ejected electrons." The graph shows a straight line with slope = h (Planck's constant) that begins at the threshold frequency (nu_0) on the x-axis (where KE = 0). Below nu_0, no electrons are ejected (indicated by a dashed line at KE = 0). The x-intercept is labeled nu_0 and the y-intercept is labeled -phi.</image>
V. Atomic Emission Spectra
When atoms are excited by heat or electrical discharge, they emit light. White light dispersed through a prism produces a continuous spectrum containing all wavelengths, appearing as a smooth rainbow. Excited atoms, however, emit only specific wavelengths, producing a line (emission) spectrum of discrete colored lines. Each element has a unique emission spectrum that serves as an atomic fingerprint. Hydrogen's emission spectrum features a particularly well-known set of lines called the Balmer series, which falls in the visible region at 410, 434, 486, and 656 nm. An absorption spectrum arises when white light passes through a gas and specific wavelengths are absorbed, producing dark lines at the same positions where emission lines would appear. The existence of line spectra provided powerful evidence that electron energy in atoms is quantized.
VI. The Bohr Model of the Hydrogen Atom (1913)
Niels Bohr developed a model for the hydrogen atom that successfully explained its line spectrum. His model rested on three key postulates. First, electrons orbit the nucleus in circular paths of fixed radius, called orbits or energy levels. Second, each orbit has a fixed energy, and electrons do not radiate energy while remaining in a stationary orbit. Third, electrons can transition between orbits by absorbing or emitting a photon whose energy exactly matches the difference between the two energy levels.
The energy of an electron in orbit n is given by E_n = -2.18 x 10^-18 J (Z^2 / n^2). For hydrogen (Z = 1), this simplifies to E_n = -2.18 x 10^-18 / n^2 J, where n is the principal quantum number (1, 2, 3, ...). The ground state (n = 1) has the lowest and most negative energy. Excited states correspond to n > 1, and when n reaches infinity, the energy reaches zero, meaning the electron is free and the atom is ionized. The energy of a transition is delta_E = E_final - E_initial = -2.18 x 10^-18 (1/n_f^2 - 1/n_i^2). For emission, n_i > n_f, delta_E is negative, and a photon with energy |delta_E| is released. For absorption, n_f > n_i, delta_E is positive, and a photon is absorbed. The wavelength of the photon involved is given by lambda = hc / |delta_E|.
VII. Hydrogen Spectral Series
Different families of spectral lines correspond to transitions ending at different energy levels. The Lyman series comprises transitions to n = 1 and falls in the ultraviolet region. The Balmer series consists of transitions to n = 2 and appears in the visible region. The Paschen series involves transitions to n = 3 and lies in the infrared. The Brackett and Pfund series (transitions to n = 4 and n = 5, respectively) extend into the far infrared. In general, a greater energy difference between levels produces a photon of shorter wavelength.
<image>An energy level diagram for the hydrogen atom showing the Bohr model spectral series. Horizontal lines represent energy levels n = 1 through n = 6 and n = infinity (ionization limit at E = 0). Energy values (in eV or x 10^-18 J) are labeled beside each level. Downward arrows between levels represent photon emission, color-coded by series: Lyman series (arrows ending at n = 1, colored purple/UV), Balmer series (arrows ending at n = 2, colored with visible light colors -- red for n=3 to 2, blue-green for n=4 to 2, violet for n=5 to 2, n=6 to 2), and Paschen series (arrows ending at n = 3, colored red/IR). Each series is labeled. The spacing between levels decreases as n increases, illustrating the convergence of energy levels.</image>
VIII. Limitations of the Bohr Model
The Bohr model works beautifully for hydrogen and other one-electron species such as He+ and Li2+, but it fails for multi-electron atoms, unable to predict their spectra accurately. It incorrectly assumes that electrons travel in fixed circular orbits, when in reality electrons do not follow classical trajectories. The model does not account for the wave nature of the electron, nor can it explain the fine structure of spectral lines or the effects of magnetic fields. Despite these shortcomings, Bohr's central insight -- that energy levels are quantized -- survives in the quantum mechanical model that replaced it.
IX. de Broglie's Hypothesis (1924) -- Preview
Louis de Broglie reasoned that if light can exhibit particle properties (as photons), then perhaps particles can exhibit wave properties. He proposed that every moving particle has an associated wavelength: lambda = h / (m * v), where h is Planck's constant, m is mass, and v is velocity. For large, macroscopic objects the wavelength is immeasurably small and unobservable. For electrons, however, the small mass produces a significant wavelength, making wave behavior an essential consideration. Davisson and Germer confirmed de Broglie's hypothesis in 1927 by observing electron diffraction patterns. This wave-particle duality of matter laid the conceptual foundation for quantum mechanics.
