Quantum Theory of the Atom

1. Quantum Theory of the Atom .

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6. The Wave Nature of Light • A wave is a continuously repeating change or oscillation in matter or in a physical field. Light is also a wave. • It consists of oscillations in electric and magnetic fields that travel through space. • Visible light, X rays, and radio waves are all forms of electromagnetic radiation. (See Animation: Electromagnetic Wave) .

7. The Wave Nature of Light • A wave can be characterized by its wavelength and frequency. • The wavelength,l (lambda), is the distance between any two adjacent identical points of a wave. (See Figure 7.3) • The frequency,n (nu), of a wave is the number of wavelengths that pass a fixed point in one second. Next Slide .

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9. In a vacuum, the speed of light, c, is 3.00 x 108 m/s. Therefore, The Wave Nature of Light • The frequency of a wave, n (waves/sec) multiplied by its wavelength, l (m/wave) gives the speed of the wave in m/s. • So, given the frequency of light, its wavelength can be calculated, or vice versa. l = c/n or n = c/l .

10. The Wave Nature of Light • What is the wavelength of yellow light with a frequency of 5.09 x 1014 s-1? (Note: s-1, commonly referred to asHertz (Hz) is defined as “cycles or waves per second”.) • Since c =nl,then rearranging, we obtainl = c/n .

11. The Wave Nature of Light • What is the frequency of violet light with a wavelength of 408 nm? (See Figure 7.5) • If c =nl,then rearranging, we obtainn = c/l. .

12. The Wave Nature of Light • The range of frequencies or wavelengths of electromagnetic radiation is called the electromagnetic spectrum. (See Figure 7.5) • Visible light extends from the violet end of the spectrum at about 400 nm to the red end with wavelengths about 800 nm. • Beyond these extremes, electromagnetic radiation is not visible to the human eye. .

13. Quantum Effects and Photons • Planck’s Quantization of Energy (1900) • According to Max Planck, the atoms of a solid oscillate with a definite frequency, n. .

14. Blackbody RadiationMax Planck (1900); Nobel Prize in 1918 (8pn2/c3) rn = kT Planck’s Constant h = 6.626 x 10-34 J *sec = Classical Theory [8ph(n/c)3] rn = e(hn/kT) - 1 Planck’s Theory E = nhn n = 1, 2, etc .

15. He proposed that an atom could have only certain energies of vibration that corresponded to a specific amount of energy (E) allowed by the formula • where h (Planck’s constant) is assigned a value of 6.63 x 10-34 J. s and n must be an integer. Max Planck’s Theory of Quantization of Energy .

16. Quantum Effects and Photons • Planck’s Quantization of Energy. • Thus, the only energies a vibrating atom can have are hn, 2hn, 3hn, and so forth. • The numbers symbolized by n are quantum numbers. • The vibrational energies of the atoms are said to be quantized. .

17. Quantum Effects and Photons • By the early part of twentieth century, the wave theory of light Seemed to be well entrenched. • In 1905, Albert Einstein proposed that light behaved as a wave and a particle using the photoelectric effect. (See Figure 7.6 and Animation: Photoelectric Effect) .

18. Quantum Effects and Photons • 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 postulated that this energy would be emitted as a photon (or quantum) of light energy. • Einstein postulated that light consists of waves (quanta or photons), or particles (electrons) of electromagnetic energy. E = hn .

19. 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 is Planck’s constant. • In 1905, Einstein used this concept to explain the “photoelectric effect.” .

20. Quantum Effects and Photons • Photoelectric Effect • The photoelectric effect is the ejection of electrons from the surface of a metal when light shines on it. (See Figure 7.6) • 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) has any effect. .

21. Quantum Effects and Photons • Photoelectric Effect • Einstein’s concluded that the ejected electron now behaves like a particle. • When the photon hits the metal, its energy, hnis taken up by the electron. • The photon ceases to exist as a particle; it is said to be “absorbed.”, and the energy is absorbed by the electron. .

22. Quantum Effects and Photons • Photoelectric Effect • The “wave” and “particle” pictures of light should be regarded as complementary views of the same physical entity. • This is called the wave-particle duality of light. • The equation E = hndisplays this duality; E is the energy of the “particle” photon, andnis the frequency of the associated “wave.” Electromagnetic Radiation .

23. 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. .

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

25. The Bohr Theory of the Hydrogen Atom • Prior to the work of Niels Bohr,the stability of the atom could not be explained using the then-current theories. • In 1913, Neils Bohr applied Planck’s and Einstein’s theories to explain the line spectrum of hydrogen. .

26. The Bohr Theory of the Hydrogen Atom • Atomic Line Spectra • A prism can spread out white light to give a continuous spectrum-that is, a spectrum containing light of all wavelengths. • The light emitted by a heated gas, such as hydrogen, produces a line spectrum (ie., only specific wavelengths of light are seen.) • (See Figure 7.2 and Animation: H2 Line Spectrum) .

27. The Bohr Theory of the Hydrogen Atom • Atomic Line Spectra • In 1885, J. J. Balmer showed that the wavelengths (l), in the visible spectrum of hydrogen could be calculated using: • 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) .

28. Line spectrum of the H atom See H-atom Line Spectrum .

29. The Bohr Theory of the Hydrogen Atom • Bohr’s Postulates • Bohr postulates accounted 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. 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) .

30. The Bohr Theory of the Hydrogen Atom • 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. .

31. From Postulate 1, The Bohr Theory of the Hydrogen Atom • 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. .

32. The Bohr Theory of the Hydrogen Atom • Bohr’s Postulates • If we make a substitution into the previous equation that states the energy of the emitted photon, DE = hn = Ef - Ei, Rearranging, we obtain .

33. The Bohr Theory of the Hydrogen Atom • Bohr’s Postulates • Bohr’s theory explains not only the emission of light, but also the absorbtion of light. • A transition from the n = 3 to n = 2 energy level, produces a photon of red light (wavelength, 685 nm) is emitted. • An electron in the n = 2 level, of the hydrogen atom absorbs a photon of red light the electron undergoes a transition to n = 3. .

34. 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. • Note that the energy of a bound e- is negative, but the energy emitted when an electron falls from a higher shell to a lower shell is positive because energy is emitted. .

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37. Quantum Mechanics • Bohr’s theory established the concept of atomic energy levels but did not thoroughly explain the “wave-like” behavior of the electron. • Quantum mechanics, a mathematical theory developed to help explain the wave-particle nature of electrons on an atom, or molecule in order to calculate the energetics of atomic and molecular systems. .

38. Quantum Mechanics • The first clue in the development of quantum theory came with the discovery of the de Broglie relation. • In 1923, Louis de Broglie believed that if light behaves as a particle, that particles could behave as a wave. • The resulting equation: • l = h/mv • is called the de Broglie relation. • The de Broglie relation can also be derived from combining Einstein’s equation with Planck’s equation (hn = mc2). .

39. Quantum Mechanics • If matter has wave properties, why are they not commonly observed? • The de Broglie relation can be used to calculate the wavelength of a baseball (0.145 kg) moving at about 60 mph (27 m/s). • This value is so incredibly small that such waves cannot be detected. • Remember: l is inversely proportional to n (ie., la 1/n), so the energy of a macroscopic particle produces an undetectable wavelength. .

40. 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. .

41. Quantum Mechanics • If matter has wave properties, why are they not commonly observed? • In 1927, a beam of electrons exhibited diffraction in a crystal (ie., X-ray diffraction; Bragg Equation). nl = 2d sin q, n = 1, 2, 3, ….. (p. 463; Figure 11.49,50) • The German physicist, Ernst Ruska, used this wave property to construct the first “electron microscope” in 1933. (See Figure 7.16) .

42. 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 that it is impossible to know both simultaneously. .

43. Quantum Mechanics • Heisenberg uncertainty principle states that the uncertainty in position (Dx) and momentum (px or mDvx) of a particle can be no larger than h/4p. • When the mass “m”is large (eg., a baseball) the uncertainties are small, but for small particles (ie., electrons), high uncertainties prevent defining an exact orbit for an electron around an atom. .

44. Quantum Mechanics • Although we cannot precisely define an electron’s orbit, we can obtain the probability of finding an electron at a given point around the nucleus. • Erwin Schrodinger defined this probability in a mathematical expression called a wave function, denotedy(psi). • The probability of finding a particle in a region of space is defined byy2. (See Figures 7.18 and 7.19) .

45. Quantum Numbers and Atomic Orbitals • Quantum mechanics allows one to identify any electron on any atom using four quantum numbers. • Principal quantum number (n) • Angular momentum quantum number (l) • Magnetic quantum number (ml) • Spin quantum number (ms) • The first three Q#’s define the wave function for a particular electron. • The fourth quantum number refers to the magnetic property of electrons (ie. spin). .

46. Quantum Numbers and Atomic Orbitals • The principal quantum number “n” represents the “shell number” in which an electron “resides.” • n = 1, 2, 3, …. to represent the shell # .

47. Quantum Numbers and Atomic Orbitals • The angular momentum quantum number (l) distinguishes “sub shells” within a given shell that have different shapes. • Each main “shell” is subdivided into “sub shells.” • “l” ranges from n – 1, ……..to 0 • The different subshells are denoted by letters. • Letter s p d f g … • l 0 1 2 3 4 …. .

48. Quantum Numbers and Atomic Orbitals • The magnetic quantum number (ml) distinguishes orbitals within a given sub-shell that have different shapes and orientations in space. • Each sub shell is subdivided into “orbitals,” each capable of holding a pair of electrons. • ml ranges from-lto +l. • Each orbital within a given sub shell ideally has the same energy. .

49. Quantum Numbers and Atomic Orbitals • The spin quantum number (ms) refers to the two possible spin orientations of the electrons residing within a given orbital. • Each orbital can hold only two electrons whose spins must oppose one another. • The possible values of ms are +1/2 and –1/2. • Summary of Quantum Numbers (See Table 7.1 and Figure 7.23 andAnimation: Orbital Energies) .

50. Quantum Numbers and Atomic Orbitals • Using calculated probabilities of electron “position,” the shapes of the orbitals can be described. • The s sub shell orbital (there is only one) is spherical. (See Figures 7.24 and 7.25 and Animation: 1s Orbital) • The p sub shell orbitals (there are three) are dumbbell shape. (See Figure 7.26 and Animation: 2px Orbital) • The d sub shell orbitals (there are five ) are a mix of cloverleaf and dumbbell shapes. (See Figure 7.27 andAnimations: 3dxy Orbital and 3dz2 Orbital) .