Chapter 7 Quantum Theory of the Atom

1. Chapter 7Quantum Theory of the Atom

2. The photoelectric effect is the ejection of an electron from the surface of a metal or other material when light shines on it.

3. Einstein proposed that light consists of quanta or particles of electromagnetic energy, called photons. The energy of each photon is proportional to its frequency: • E = hn • h = 6.626 × 10-34 J  s (Planck’s constant)

4. Einstein used this understanding of light to explain the photoelectric effect in 1905. • Each electron is struck by a single photon. Only when that photon has enough energy will the electron be ejected from the atom; that photon is said to be absorbed.

5. Light, therefore, has properties of both waves and matter. Neither understanding is sufficient alone. This is called the particle–wave duality of light.

6. The blue–green line of the hydrogen atom spectrum has a wavelength of 486 nm. What is the energy of a photon of this light? E = hn and c = nl so E = hc/l l = 4.86 nm = 4.86 × 10-7 m c = 3.00 × 108 m/s h = 6.63 × 10-34 J  s l = 4.09 × 10-19 J

7. In the early 1900s, the atom was understood to consist of a positive nucleus around which electrons move (Rutherford’s model). • This explanation left a theoretical dilemma: According to the physics of the time, an electrically charged particle circling a center would continually lose energy as electromagnetic radiation. But this is not the case—atoms are stable.

8. In addition, this understanding could not explain the observation of line spectra of atoms. • A continuous spectrum contains all wavelengths of light. • A line spectrum shows only certain colors or specific wavelengths of light. When atoms are heated, they emit light. This process produces a line spectrum that is specific to that atom. The emission spectra of six elements are shown on the next slide.

9. Light and Atoms • When an atom gains a photon, it enters an excited state. • This state has too much energy - the atom must lose it and return back down to its ground state, the most stable state for the atom. • Line spectra indicate light emitted when excited electrons lose energy. • An energy level diagram is used to represent these changes.

11. In 1913, Neils Bohr, a Danish scientist, set down postulates to account for • 1. The stability of the hydrogen atom • 2. The line spectrum of the atom

12. Energy-Level Postulate • An electron can have only certain energy values, called energy levels. Energy levels are quantized. • For an electron in a hydrogen atom, the energy is given by the following equation: • RH = 2.179 x 10-18 J • n = principal quantum number

13. Transitions Between Energy Levels • An electron can change energy levels by absorbing energy to move to a higher energy level or by emitting energy to move to a lower energy level.

14. Energy Level Diagram • Energy • Excited States • photon’s path • Ground State Light Emission Light Emission Light Emission

15. For a hydrogen electron the energy change is given by RH = 2.179 × 10-18 J, Rydberg constant

16. The energy of the emitted or absorbed photon is related to DE: • We can now combine these two equations:

17. Light is absorbed by an atom when the electron transition is from lower n to higher n (nf > ni). In this case, DE will be positive. • Light is emitted from an atom when the electron transition is from higher n to lower n (nf < ni). In this case, DE will be negative. • An electron is ejected when nf = ∞.

18. Energy-level diagram for the hydrogen atom.

19. Electron transitions for an electron in the hydrogen atom.

20. What is the wavelength of the light emitted when the electron in a hydrogen atom undergoes a transition from n = 6 to n = 3? ni = 6 nf = 3 RH = 2.179 × 10-18 J = -1.816 x 10-19 J 1.094 × 10-6 m

22. n = 3 n = 2 n = 1 • The red line corresponds to the smaller energy difference in going from n = 3 to n = 2. The blue line corresponds to the larger energy difference in going from n = 2 to n = 1. A minimum of three energy levels are required.

23. Planck • Vibrating atoms have only certain energies: E = hn or 2hn or 3hn • Einstein • Energy is quantized in particles called photons: E = hn • Bohr • Electrons in atoms can have only certain values of energy. For hydrogen:

24. Light has properties of both waves and particles (matter). What about matter?

25. In 1923, Louis de Broglie, a French physicist, reasoned that particles (matter) might also have wave properties. • The wavelength of a particle of mass, m (kg), and velocity, v (m/s), is given by the de Broglie relation:

26. Compare the wavelengths of (a) an electron traveling at a speed that is one-hundredth the speed of light and (b) a baseball of mass 0.145 kg having a speed of 26.8 m/s (60 mph). Electron me = 9.11 × 10-31 kg v = 3.00 × 106 m/s Baseball m = 0.145 kg v = 26.8 m/s

27. Electron me = 9.11 × 10-31 kg v = 3.00 × 106 m/s 2.43 × 10-10 m Baseball m = 0.145 kg v = 26.8 m/s 1.71 × 10-34 m

30. Building on de Broglie’s work, in 1926, Erwin Schrödinger devised a theory that could be used to explain the wave properties of electrons in atoms and molecules. • The branch of physics that mathematically describes the wave properties of submicroscopic particles is called quantum mechanics or wave mechanics.

31. Quantum mechanics alters how we think about the motion of particles. • In 1927, Werner Heisenberg showed how it is impossible to know with absolute precision both the position, x, and the momentum, p, of a particle such as electron. • Because p = mv this uncertainty becomes more significant as the mass of the particle becomes smaller.

32. Quantum mechanics allows us to make statistical statements about the regions in which we are most likely to find the electron. • Solving Schrödinger’s equation gives us a wave function, represented by the Greek letter psi, y, which gives information about a particle in a given energy level. • Psi-squared, y2, gives us the probability of finding the particle within a region of space.

33. The wave function for the lowest level of the hydrogen atom is shown to the left. • Note that its value is greatest nearest the nucleus, but rapidly decreases thereafter. Note also that it never goes to zero, only to a very small value.

34. Two additional views are shown on the next slide. • Figure A illustrates the probability density for an electron in hydrogen. The concentric circles represent successive shells. • Figure B shows the probability of finding the electron at various distances from the nucleus. The highest probability (most likely) distance is at 50 pm.

36. According to quantum mechanics, each electron is described by four quantum numbers: • 1.Principal quantum number (n) • 2. Angular momentum quantum number (l) • 3. Magnetic quantum number (ml) • 4. Spin quantum number (ms) • The first three define the wave function for a particular electron. The fourth quantum number refers to the magnetic property of electrons.

37. A wave function for an electron in an atom is called an atomic orbital (described by three quantum numbers—n, l, ml). It describes a region of space with a definite shape where there is a high probability of finding the electron. • We will study the quantum numbers first, and then look at atomic orbitals.

38. Principal Quantum Number, n • This quantum number is the one on which the energy of an electron in an atom primarily depends. The smaller the value of n, the lower the energy and the smaller the orbital. • The principal quantum number can have any positive value: 1, 2, 3, . . . • Orbitals with the same value for n are said to be in the same shell.