Quantum Theory of the Atom

1. Quantum Theory of the Atom Chapter 7 Dr. Victor Vilchiz

2. Photoelectric Effect • Photoelectric Effect • Einstein extended Planck’s work to include the structure of light itself. • If a vibrating atom changed energy from 3hn to 2hn, it would decrease in energy by hn. • He proposed that this energy would be emitted as a bit (or quantum) of light energy. • Einstein postulated that these “bits” were the fundamental components of light. Those “bits” are now called photons and are considered particles of electromagnetic energy.

3. Quantum Effects and Photons • Photoelectric Effect • The energy of the photons proposed by Einstein would be proportional to the observed frequency, and the proportionality constant would be Planck’s constant. • In 1905, Einstein used this concept to explain the “photoelectric effect.” He was later awarded the Nobel Prize for his experimental proof and explanation of the effect. (Not for Relativity).

4. Photoelectric Effect • Photoelectric Effect Experiment • In the experiment electrons are ejected from a metal sheet when light shines on it. (see Figure 7.6) • However, the electrons are ejected only if the light exceeds a certain “threshold” frequency. • Violet light, for example, will cause potassium to eject electrons, but no amount of red light (which has a lower frequency) will lead to ejected electrons.

5. Photoelectric Effect • If Classical Theory was correct and apply to this experiment then when the red light was shined onto potassium then all we had to do was wait long enough (induction time) for energy to accumulate before the electron was ejected. • And there should not be any threshold frequency/energy.

6. Photoelectric Effect • Photoelectric Effect • Einstein’s assumption that an electron is ejected when struck by a single photon implies that light behaves like aparticle. The accepted view was that light was made out of waves. • When the photon hits the metal, its energy,hnis transferred to the electron. • The photon ceases to exist as a particle; it is said to be “absorbed.”

7. Photoelectric Effect 0 • Photoelectric Effect • The “wave” and “particle” pictures of light should be regarded as complementary views of the same physical entity. • This is called thewave-particle dualityof light. • The equation E = hn displays this duality; E is the energy of the “particle” photon, and n is the frequency of the associated “wave.”

8. Radio Wave Energy • What is the energy of a photon corresponding to radio waves of frequency 1.255 x 10 6 s-1? Solve for E, using E = hn, and four significant figures for h. (6.626 x 10-34 J.s) x (1.255 x 106 s-1) = 8.3156 x 10-28 = 8.316 x 10-28 J

9. Einstein’s Contribution In summary: He said, if energy is quantized maybe so is light. Light consists of small massless particles called photons. Ephoton=hn=DEatom This equation can explain both the lack of an induction time and the need of a threshold energy.

10. Theory of the Atom • In 1913, thanks to the previous work of Einstein and Planck, Niels Bohr in the search to explain the emission spectrum of hydrogen improves on a previous model of the atom introduced by Rutherford. • Prior to their work the stability of the atom could not be explained using the then-current theories. • Before looking at Bohr’s theory, we must first examine the “line spectra” of atoms.

11. Atomic Spectra 0 • Atomic Line Spectra • When a heated metal filament emits light, we can use a prism to spread out the light to give a continuous spectrum-that is, a spectrum containing light of all wavelengths. Blackbody radiation. • The light emitted by a heated gas, such as hydrogen, results in aline spectrum-a spectrum showing only specific wavelengths of light. (see Figure 7.2)

12. Atomic Spectra 0 • Atomic Line Spectra • In 1885, J. J. Balmer showed that the wavelengths, l, in the visible spectrum of hydrogen could be reproduced by a simple formula. • The known wavelengths of the four visible lines for hydrogen correspond to values of n = 3, n = 4, n = 5, and n = 6. (see Figure 7.2)

13. Bohr’s Atomic Model 0 • Bohr’s Postulates • Bohr’s goal was to explain the 3 sets of lines (UV, visible, and IR). In the process Bohr set down postulates to account for (1) the stability of the hydrogen atom and (2) the line spectrum of the atom. • 1.Energy level postulate An electron can have only specific energy levels in an atom. (Energy of the atom is quantized). • 2. Transitions between energy levels An electron in an atom can change energy levels by undergoing a “transition” from one energy level to another. (see Figures 7.10 and 7.11)

14. Bohr’s Atomic Model • Bohr’s Postulates • Bohr derived the following formula for the energy levels of the electron in the hydrogen atom. • Rh is a constant (expressed in energy units) with a value of 2.18 x 10-18 J.

15. Bohr’s Atomic Model • Bohr’s Postulates • When an electron undergoes a transition from a higher energy level to a lower one, the energy is emitted as a photon. • From Postulate 1,

16. Bohr’s Atomic Model • Bohr’s Postulates • If we make a substitution into the previous equation that states the energy of the emitted photon, hn, equals Ei - Ef, • Rearranging, we obtain

17. Bohr’s Atomic Model • Bohr’s Postulates • Bohr’s theory explains not only the emission of light, but also the absorbtion of light. • When an electron falls from n = 3 to n = 2 energy level, a photon of red light (wavelength, 685 nm) is emitted. • When red light of this same wavelength shines on a hydrogen atom in the n = 2 level, the energy is gained by the electron that undergoes a transition to n = 3.

18. A Problem to Consider • Calculate the energy of a photon of light emitted from a hydrogen atom when an electron falls from level n = 3 to level n = 1.

19. Bohr’s Atomic Model • There is one major flaw with the model. • It only explains the line spectrum of Hydrogen. It cannot be applied to any other element or system • About the same time Rydberg, a theoretician, came up with an empirical formula that expands on Borh’s model.

20. Rydberg’s Formula • He expanded the work of Balmer to include any levels not just n=2 • RH is Rydberg’s constant, 1.096x107m-1 • This formula can be applied to systems other than hydrogen. However, it is limited to systems with only one electron.

21. Quantum Mechanics • Bohr’s theory established the concept of atomic energy levels, energy staircase, but did not thoroughly explain the “wave-like” behavior of the electron. • Current ideas about atomic structure depend on the principles of quantum mechanics, a theory that applies to subatomic particles such as electrons.

22. Bohr’s Physical Model • As mentioned before Bohr’s model was an improvement to Rutherford’s proposed model. • Rutherford was the first to proposed a solar system like model with the nucleus as the center and fixed electrons around it. • Bohr used the same model but allowed the electrons to move from one orbit to another.

23. Quantum Mechanics • The first true development of quantum theory came with de Broglie’s contribution. • In 1923, Louis de Broglie reasoned that if light exhibits particle aspects, perhaps particles of matter show characteristics of waves. • He postulated that a particle with mass m and a velocity v has an associated wavelength. • The equation l = h/mv is called thede Broglie relation.

24. Quantum Mechanics • If matter has wave properties, why are they not commonly observed? • The de Broglie relation shows that a baseball (0.145 kg) moving at about 60 mph (27 m/s) has a wavelength of about 1.7 x 10-34 m. • This value is so incredibly small that such waves cannot be detected.

25. Quantum Mechanics • If matter has wave properties, why are they not commonly observed? • Electrons have wavelengths on the order of a few picometers (1 pm = 10-12 m). • Under the proper circumstances, the wave character of electrons should be observable.

26. Quantum Mechanics • If matter has wave properties, why are they not commonly observed? • In 1927, it was demonstrated that a beam of electrons, just like X rays, could be diffracted by a crystal. • The German physicist, Ernst Ruska, used this wave property to construct the first “electron microscope” in 1933. (see Figure 7.16)

27. Quantum Mechanics • Quantum mechanics is the branch of physics that mathematically describes the wave properties of submicroscopic particles. • We can no longer think of an electron as having a precise orbit in an atom. • To describe such an orbit would require knowing its exact position and velocity. • In 1927, Werner Heisenberg showed (from quantum mechanics) that it is impossible to know both simultaneously.

28. Uncertainty