Chapter 7: ELECTRONS IN ATOMS AND PERIODIC PROPERTIES

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Chapter 7: ELECTRONS IN ATOMS AND PERIODIC PROPERTIES
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Electromagnetic (EM) Spectrum: a continuum of the different forms of electromagnetic radiation or radiant energy

A galaxy imaged in the visible spectrum. Radio telescopes. This is the Very Large Array (VLA) in NM. The same galaxy imaged in the radio spectrum at the VLA.

Much of a person’s energy is radiated away from the body in the form of infrared (IR) energy. You can produce more IR energy to warm yourself by moving around…this is why you shiver when you go outside in the cold with no coat on. This process is called thermogenesis. Why does your mother insist you wear a hat in the winter?

“IR” Photography Image obtained with “IR” film, which is really film that is activated by light at nm.

Electromagnetic radiation or “light” is a form of energy. Has both electric (E) and magnetic (H) components. Characterized by: Wavelength (λ) Amplitude (A)

Wavelength (l): The distance between two consecutive peaks in the wave. Increasing Wavelength l1 > l2 > l3 Unit: length (m)

Frequency (n): The number of waves (or cycles) that pass a given point in space per second. Decreasing Frequency n1 < n2 < n3 Units: 1/time (1/sec) or, Hertz (Hz)

The product of wavelength (l) and frequency (n) is a constant. (ν)(λ) = c Speed of light c = 3 x 108 m/s c is a constant, independent of λ

Properties of Waves Wavelength (λ) is the distance from peak-to-peak in a wave. Frequency (ν) is the number of waves in a specific time frame (usually per second = Hz) As wavelength goes up, frequency goes down (and vice versa) Electromagnetic waves travel at the speed of light: λ ν = c c = speed of light = 3 x 108 m/s

What statement is true when comparing red light to blue light?
A. Red light travels at a greater speed than blue light. B. Blue light travels at greater speed than red light. C. The wavelength of blue light is longer. D. The wavelength of red light is longer.

Behavior of Waves Waves refract, diffract and interact…
Refraction: The bending of light as it passes from one medium to another of different density Light is also bent by liquids, causing the straw to appear disconnected. Refraction throught a prism separates white light into its separate components (light of different wavelengths) without changing the light itself.

What causes the bright and dark spots?
Behavior of Waves Diffraction: The bending of electromagnetic radiation as it passes around an edge of an object or through a narrow opening. What causes the bright and dark spots?

Behavior of Waves Waves interact by adding together or cancelling each other out…

Light is a Wave, Right? Back in the old days…
It was generally agreed that matter and light were distinct. Matter was particulate in nature, light could be described using waves. Physicists circa 1900 had it all figured out… One famous physicist asserted that within ten years or so all the major problems in physics would be solved. The only thing left, really, was this niggling little problem with black-body radiation…

The State of Physics Before 1900
Newton discovered light could be separated by a prism in 170 Heat, electricity and phlogiston were weightless, “imponderable fluids” responsible for most observed processes It wasn’t until the early 1800s that the following, revolutionary ideas were put forth: light was a wave, atoms existed, and air could trap heat (the “greenhouse effect”).

Physics in the early 1900s The end of the 1800s saw an explosion of real scientific progress. Thermodynamics, electromagnetism and the kinetic molecular theory were well-developed. There were still some problems, though, that needed explaining… Blackbody radiation, new particle discoveries, what was the “ether”?, radioactivity, the instability of the atom, the photoelectric effect. There had to be one theory to explain them all…

The big problem Black body radiation: When a metal object is heated, it begins to glow. If it is heated hot enough, it glows “white hot”, emitting all the wavelengths of visible light – but little to none in the UV range or higher. Current (at the time) theories by Maxwell could not explain this common phenomenon. Max Planck proposed a new theory of light – that it behaved as a wave, but with particle-like properties. Light traveled in particle-like packets he called “quanta” (a single one is called a “quantum”). Each quantum was the smallest amount of energy found in nature.

h = Planck’s constant = 6.626 x 10-34 J•s
Light as Energy Planck found that in order to model this behavior, one has to envision that energy (in the form of light) is lost in integer values according to: DE = nhn frequency Energy Change n = 1, 2, 3 (integers) h = Planck’s constant = x J•s

Light as Energy (cont.) Ephoton = hν
In general the relationship between frequency and “photon” energy is h = x J•s c = x 108 m/s 1 Hz = 1 s-1 Ephoton = hν • Example: What is the energy of a 500 nm photon? ν = c/λ = (3x108 m/s)/(5.0 x 10-7 m) ν = 6 x /s E = hν =(6.626 x J•s)(6 x /s) = 4 x J

Energy Quantization The student can stop at any point on the ramp. Her distance from the ground changes continuously. The student can stop only at certain points on a flight of stairs. Her distance from the ground is quantized. Similarly, atomic energy levels are like steps…the energies available to an atom do not form a continuum, they are quantized.

Evidence of Quantization
Black-body Radiation (Planck) A system can transfer energy in “packets” of size hn. These packets are called quanta Prior to this discovery it was thought that systems could absorb or emit any amount of energy. Other observations supported this “quantum” view: Atomic Emission spectra (Balmer, Rydberg, Bohr) Light emitted from excited atoms occurs in discrete lines rather than a continuum. Photoelectric Effect (Einstein) Energy itself is actually quantized into packets called photons. This means energy has a particle-like nature as well as a wave-like nature. Electron Diffraction Patterns (Davisson & Germer) Matter also has a wave-like nature!!

Atomic Emission When we heat a sample of an element, the atoms become excited. When the atom relaxes it emits visible light. The color of the light depends on the element. Li Na K Ca Sr H Li Ba When the light emitted from excited atoms is passed through a prism, we see discrete bands of color at specific wavelengths.

Photon Emission ΔE = hc/λ  λ = hc/ΔE
An excited atom relaxes from high E to low E by emitting a photon. We can determine the energy difference (ΔE) between levels by measuring the wavelength of the emitted photon. Emission of photon ΔE = hc/λ  λ = hc/ΔE If λ = 410 nm, ΔE = 4.52 x J

The Photoelectric Effect
Observed by Albert Einstein in 1905: Light shining on a clean metal surface causes the emission of electrons but only when the light has a minimum threshold frequency (ν0) When ν<ν0  no electrons are emitted When ν>ν0  electrons are emitted, more e– emitted with greater intensity of light ν < ν0 ν > ν0 Einstein applied Planck's quantum theory to light: light exists as a stream of "particles" called photons.

The Photoelectric Effect (cont)
Frequency: determines whether e- are ejected, and their KE (velocity). Intensity: determines the number of e- that are ejected…but they will all have the same velocity!!

The Photoelectric Effect (cont.)
As frequency of incident light is increased, kinetic energy of emitted e- increases linearly. = hn0 Workfunction: energy needed to release e- Light apparently behaves as a particle.

The Photoelectric Effect (cont.)
For Na with Φ = 4.4 x J, what wavelength corresponds to νo? hν = Φ = 4.4 x J hc/λ = 4.4 x J λ = 4.52 x 10-7 m = 452 nm

Electron Diffraction Patterns
Light is shined through a crystal, and its waveforms are “scattered.” When they come out the other side, they create interference patterns on a detector plate. Diffraction can only be explained by treating light as a wave instead of a particle.

Diffraction of Particles?
Turns out we can get similar interference patterns by bombarding crystals with beams of high energy electrons also… This can only be explained by treating matter as a wave.

de Broglie Wavelength Ephoton = hν c = λν  ν = c/λ
If matter exhibits wave-like properties, we should be able to determine the wavelength of a particle. Recall the energy of a photon, and the definition of the speed of light: Substituting, E = hc/λ Using Einstein’s relationship, E = mc2, Ephoton = hν c = λν  ν = c/λ hc λ = mc λ = hc mc2 = h mc λ= h mc

You are a wave, too, but you have a VERY small wavelength!!
de Broglie Wavelength We can generalize this relationship to any velocity (v): What is the de Broglie wavelength of an electron traveling at the speed of light? (melectron = 9.31 x kg) What is the de Broglie wavelength of a 80 kg student walking across campus at 3 m/s? λ= h mc λ= h mv λ = (6.626 x J•s)/[(9.31x10-31 kg)(3x108 m/s)] = 2.37 x m λ = (6.626 x J•s)/[(80 kg)(3 m/s)] = 2.76 x m You are a wave, too, but you have a VERY small wavelength!!

What does all this mean for matter?
Scientists needed to find a way to explain how light, usually thought of as a wave, could behave like a particle AND how matter, which was usually thought of as a particle, could behave as a wave. They started with the hydrogen emission spectrum… Balmer and Rydberg developed equations that predicted where these lines should appear (even before anyone had observed them).

The Bohr Model Balmer and Rydberg didn’t know why their equations worked. Niels Bohr used their equations and observations to develop a quantum model for H. Central idea: electron circles the “nucleus” in only certain allowed circular orbitals. Bohr postulated that there is Coulombic attraction between e- and nucleus. However, classical physics is unable to explain why an H atom doesn’t simply collapse. Why doesn’t the electron just spiral into the nucleus?

The Bohr Model of the atom
Principle Quantum number: n An “index” of the energy levels available to the electron. 410 434 486 656

The Bohr Model (cont.) Bohr model for the H atom is capable of reproducing the energy levels given by the empirical formulas of Balmer and Rydberg. Z = atomic number (1 for H) n = integer (1, 2, ….) Ry x h = x J

The Bohr Model (cont.) Energy levels get closer together as n increases At n = infinity, E = 0  the electron is free (not a part of the atom)

The Bohr Model (cont.) We can use the Bohr model to predict what DE is for any two energy levels

The Bohr Model (cont.) Example: At what wavelength will emission from n = 4 to n = 1 for the H atom be observed? 1 4

The Bohr Model (cont.) Example: What is the longest wavelength of light that will result in removal of the e- from H? 1

Extension to Higher Z The Bohr model can be extended to any single electron system….must keep track of Z (atomic number). Z = atomic number n = integer (1, 2, ….) Examples: He+ (Z = 2), Li+2 (Z = 3), etc.

Extension to Higher Z (cont.)
Example: At what wavelength will emission from n = 4 to n = 1 for the He+ atom be observed? 2 1 4

So what happened to the Bohr Model?
Although it successfully described the line spectrum of hydrogen and other one-electron systems, it failed to accurately describe the spectra of multi-electron atoms. The Bohr model was soon scrapped in favor of the Quantum Mechanical model, although the vocabulary of the Bohr model persists. However, Bohr pioneered the idea of quantized electronic energy levels in atoms, so we owe him big. Thanks Niels Bohr!

Quantum Concepts The Bohr model was capable of describing the discrete or “quantized” emission spectrum of H. But the failure of the model for multielectron systems combined with other issues (the ultraviolet catastrophe, workfunctions of metals, etc.) suggested that a new description of atomic matter was needed.

Quantum Concepts (cont.)
This new description was known as wave mechanics or quantum mechanics. Recall, photons and electrons readily demonstrate wave-particle duality. The idea behind wave mechanics was that the existence of the electron in fixed energy levels could be though of as a “standing wave”.

Quantum Concepts (cont.)
What is a standing wave? • A standing wave is a motion in which translation of the wave does not occur. • In the guitar string analogy (illustrated), note that standing waves involve nodes in which no motion of the string occurs. • Note also that integer and half- integer values of the wavelength correspond to standing waves.

Quantum Concepts (cont.)
Louis de Broglie suggested that for the e- orbits envisioned by Bohr, only certain orbits are allowed since they satisfy the standing wave condition. not allowed

Quantum Concepts (cont.)
Erwin Schrodinger developed a mathematical formalism that incorporates the wave nature of matter: H, the “Hamiltonian,” is a special kind of function that gives the energy of a quantum state, which is described by the wavefunction, Y.

Quantum Concepts (cont.)
What is a wavefunction? = a probability amplitude Probability of finding a particle in space: The probability distribution for the hydrogen 1s orbital in three-dimensional space Probability = With the wavefunction, we can describe spatial distributions. The probability of finding the electron at points along a line drawn from the nucleus outward in any direction for the hydrogen 1s orbital.

Hydrogen’s Electron Cross section of the hydrogen 1s orbital probability distribution divided into successive thin spherical shells (b) The radial probability distribution The surface contains 90% of the total electron probability (the size of the orbital, by definition)

Quantum Concepts (cont.)
Another limitation of the Bohr model was that it assumed we could know both the position and momentum of an electron exactly. Werner Heisenberg observed that there is a fundamental limit to how well one can know both the position and momentum of a particle. where… Uncertainty in position Uncertainty in momentum

Quantum Concepts (cont.)
Example: What is the uncertainty in velocity for an electron in a 1 Å radius orbital in which the positional uncertainty is 1% of the radius. (1 Å = m) Δx = (1 Å)(0.01) = 1 x m HUGE!

Quantum Concepts (cont.)
Example (you’re quantum as well): What is the uncertainty in position for a 80 kg student walking across campus at 1.3 m/s with an uncertainty in velocity of 1%. Δp = mΔv = (80kg)(0.013 m/s) = 1.04 kg.m/s Very small……we know where you are.