1. Wave Particle Duality – Light and Subatomic Particles
In high school physics light was treated as having both wave like and particle like character. Diffraction and refraction of light both exemplify its wave like properties. Particle like properties of light can also be demonstrated readily. One way to do this is to consider the photoelectric effect (see text) which implies, surprisingly, that light photons have momentum.

2. Subatomic Particles – Wave Character
A number of experiments show that small particles have observable wave like properties. Such wave like properties become increasingly important as one moves to particles of smaller mass. The electron is the most important of these particles. Interesting diffraction and refraction experiments have been conducted with electrons.

3. Mathematical Description of Electrons
The fact that electrons exhibit wave like behavior suggested that equations used to describe waves, and light waves in particular, might be modified to describe electrons. We will see some familiar mathematical functions used to describe the electron (e.g. cos θ, sin θ, eiθ). We will use so-called wave functions (Ψ’s) to gain insight into the behavior of electrons in atoms.

4. De Broglie’s Contribution
De Broglie used results/equations from classical physics to rationalize experimental results which proved that subatomic particles (and some atoms and molecules) had wave like properties. He proposed, in particular, that particles with a finite rest mass had a characteristic wave length – as do light waves.

5. 8-5 Two Ideas Leading to a New Quantum Mechanics
Chemistry 140 Fall 2002
8-5 Two Ideas Leading to a New Quantum Mechanics
Wave-Particle Duality
Einstein suggested particle-like properties of light could explain the photoelectric effect.
Diffraction patterns suggest photons are wave-like.
deBroglie, 1924
Small particles of matter may at times display wavelike properties.
To explain the photoelectric effect, Einstein suggested that light has particle-like properties, which are displayed through photons. Other phenomena, however, such as the dispersion of light into a spectrum by a prism, are best understood in terms of the wave theory of light. Light, then, appears to have a dual nature.
Louis de Broglie
Nobel Prize 1918
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General Chemistry: Chapter 8

6. Wave-Particle Duality
Chemistry 140 Fall 2002
Wave-Particle Duality
E = mc2
h = mc2
h/c = mc = p
p = h/λ
In order to use this equation for a material particle, such as an electron, de Broglie substituted for the momentum, p, its equivalent—the product of the mass of the particle, m, and its velocity, u. When this is done, we arrive at de Broglie’s famous relationship.
De Broglie called the waves associated with material particles “matter waves.” If matter waves exist for small particles, then beams of particles, such as electrons, should exhibit the characteristic properties of waves, namely diffraction.
λ = h/p = h/mu
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General Chemistry: Chapter 8

7. Wave properties of electrons demonstrated
Chemistry 140 Fall 2002
(a) Diffraction of X-rays by metal foil.
(b) Diffraction of electrons by metal foil,
confirming the wave-like properties of electrons.
In 1927, C. J. Davisson and L. H. Germer of the United States showed that a beam of slow electrons is diffracted by a crystal of nickel. In a similar experiment in that same year, G. P. Thomson of Scotland directed a beam of electrons at a thin metal foil. He obtained the same pattern for the diffraction of electrons by aluminum foil as with X-rays of the same wavelength (Fig. 8-16).
Thomson and Davisson shared the 1937 Nobel Prize in physics for their electron diffraction experiments. George P. Thomson was the son of J.J. Thomson, who had won the Nobel Prize in physics in 1906 for his discovery of the electron. It is interesting to note that Thomson the father showed that the electron is a particle, and Thomson the son showed that the electron is a wave. Father and son together demonstrated the wave–particle duality of electrons.
FIGURE 8-16
Wave properties of electrons demonstrated
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General Chemistry: Chapter 8

8. Probabalistic Description of Electrons
Classical physics suggests that we should be able (given enough information) to describe the behaviour of any body – changes in velocity, kinetic energy, potential energy and so on over time. Classical physics suggests that all energies are continuously variable – a result which very clearly is contradicted by experimental results for atoms and molecules (line spectra/quantized energies).

9. Uncertainty Principle
The quantum mechanical description used for atoms and molecules suggests that for some properties only a probabalistic description is possible. Heisenberg suggested that there is a fundamental limitation on our ability to determine precise values for atomic or molecular properties simultaneously. The mathematical statement of Heisenberg’s so-called Uncertainty Principle is given on the next slide.

10. The Uncertainty Principle
Chemistry 140 Fall 2002
The Uncertainty Principle h
Δx Δp ≥ 4π
A collection of waves with varying wavelengths (left) can combine into a “wave packet” (right). The superposition of the different wavelengths yields an average wavelength (lav) and causes the wave packet to be more localized (Δx) than the individual waves. The greater the number of wavelengths that combine, the more precisely an associated particle can be located, that is, the smaller Δx . However, because each of the wavelengths corresponds to a different value of momentum according to the de Broglie relationship, the greater is the uncertainty in the resultant momentum.
Heisenberg and Bohr
FIGURE 8-17
The uncertainty principle interpreted graphically
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General Chemistry: Chapter 8

11. Atomic “Diagrams”
Many simple diagrams of atoms/atomic structure have limitations. In class we’ll consider some limitations of the C atom diagram shown on the next slide.
Representations of molecules are even more challenging – at least if we want to consider the electrons!

12. The “Carbon Atom”
Limitations?

13. A Molecular Model

14. Wave Functions and Standing Waves
If we wish to describe the behaviour of a system changing over time in three dimensional space we would need four variables - three coordinates and time. Classically, we might wish to describe the velocity of a satellite re-entering the Earth’s atmosphere and write, for example, v(x,y,z,t).
Copyright © 2011 Pearson Canada Inc.
General Chemistry: Chapter 8

15. Form of Wave Functions
For wave functions used to describe the behaviour of electrons in atoms we could, by analogy, write Ψ(x,y,z,t). For charged particles and Coulombic interactions, using spherical polar coordinates simplifies the mathematical work greatly. We then write wave functions as Ψ(r,ϴ,φ,t). It turns out that in many cases the properties of an electron, atom, or molecule do not change over time.

16. Wave Functions and Time
If the behaviour of an electron, atom or molecule does not change with time we can write simpler wave functions, such as Ψ(x,y,z) or Ψ(r,ϴ,φ). The classical analogy of a system with wave like behaviour not changing over time is any object that features standing waves. In such cases there is no destructive wave interference. Then wave like properties should be constant over time.

17. Standing Waves in Newtonian Mechanics
In macroscopic systems with “constant” wave like properties one sees nodes which do not move over time. An example would be a guitar string vibrating with, in the simplest case, two nodes occurring at each end of the string. More complex waves are possible for real systems – which again reflect constructive rather than destructive wave interference. Similar considerations apply to electrons.

18. General Chemistry: Chapter 8
Chemistry 140 Fall 2002
Wave Mechanics
Standing waves.
Nodes do not undergo displacement.
λ = , n = 1, 2, 3… 2L n
The string can be set into motion by plucking it. The blue boundaries outline the range of displacements at each point for each standing wave. The relationships between the wavelength, string length, and the number of nodes—points that are not displaced—are given by equation (8.12). The nodes are marked by bold dots.
FIGURE 8-18
Standing waves in a string
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General Chemistry: Chapter 8

19. The electron as a matter wave
Chemistry 140 Fall 2002
These patterns are two-dimensional cross-sections of a much more complicated three-dimensional wave. The wave pattern in (a), a standing wave, is an acceptable representation. It has an integral number of wavelengths (five) about the nucleus; successive waves reinforce one another. The pattern in (b) is unacceptable. The number of wavelengths is nonintegral, and successive waves tend to cancel each other; that is, the crest in one part of the wave overlaps a trough in another part of the wave, and there is no resultant wave at all.
FIGURE 8-19
The electron as a matter wave
Copyright © 2011 Pearson Canada Inc.
General Chemistry: Chapter 8

20. Dihydrogen Oxide Waves

21. Sample Term Test 1 Example
B4. Consider the following reaction
2 C(s) + O2(g) + 4 H2(g) → 2 CH3OH(g) ∆Hº = – kJ
[2] (a) Calculate a ΔUº value for the reaction at 25ºC.
   
[1] (b) How much heat would be released if 10.0 g of CH3OH(g) is formed from this reaction under constant volume conditions?
Reminder: The enthalpy change for the reaction as written gives the amount of heat released when two moles of gaseous methanol are formed at constant pressure. The internal energy change for the reaction as written gives the amount of heat released at constant V.

22. Sample Term Test 1 Example
[2] A5. A 16.0 L tank contains a mixture of helium gas at a partial pressure of kPa and nitrogen gas at a partial pressure of 75.0 kPa. If neon gas is added to the tank, what must the partial pressure of neon be to reduce the mole fraction of nitrogen to 0.200?

23. Sample Term Test 1 Example
[4] B5. A sample of zinc metal reacts completely with an excess of hydrochloric acid:
Zn(s) + 2 HCl(aq) → ZnCl2(aq) + H2(g)
The hydrogen gas produced is collected over water at 25.0ºC. The volume of the gas is 7.80 L, and the pressure is atm. Calculate the amount of zinc metal in grams consumed in the reaction. (Vapor pressure of water at 25ºC = 23.8 mmHg.)