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144 Chapter 7 The Quantum-Mechanical Model of the Atom 6.626 X 10⁻³⁴ Solution: = 1.055 X 10⁻²⁰ J E = (6.626 X X 10⁸ = 2.75 X J 722 2.75 X 10⁻¹⁹ J - 1.06 X = 2.64 X 10⁻¹⁹ (6.626 X X = 751.8 nm = 7.5 X 10² nm (2.64 X Check: The units of the answer (nm) are correct. The magnitude of the answer is reasonable because it is a longer wavelength, but it is close to the original wavelength. 7.103 Given: r = 1.8 m Find: \ Conceptual Plan: r C C = 2πr Solution: (2)(3.141)(1.8m) = 11.3 m = the circumference of the orbit. So the largest wavelength that would fit the orbit would be 11 m. Check: The units of the answer (m) are correct. The magnitude of the wave is about the circumference of the orbit. Conceptual Problems 7.105 In the Bohr model of the atom, the electron travels in a circular orbit around the nucleus. It is a two-dimensional model. The electron is constrained to move only from one orbit to another orbit. But the electron is treated as a particle that behaves according to the laws of classical physics. The quantum-mechanical model of the atom is three-dimensional. In this model, we treat the electron, an absolutely small particle, differently than we treat particles with classical physics. The electron is in an orbital, which gives us the probability of finding the electron within a volume of space. Because the electron in the Bohr model is constrained to a circular orbit, it would theoretically be possible to know both the position and the velocity of the electron simultaneously. This contradicts the Heisenberg uncertainty principle, which states that position and velocity are complementary terms neither of which can be known with precision. So in the quantum-mechanical model we can only know the position or the velocity of an electron at any given time, never both. 7.107 (a) Because the interference pattern is caused by single electrons interfering with themselves, the pattern remains the same even when the rate of the electrons passing through the slits is one electron per minute. It will simply take longer for the full pattern to develop. (b) When a light is placed behind the slits, it flashes to indicate which hole the electron passed through, but the in- terference pattern is now absent. With the laser on, the electrons hit positions directly behind each slit, as if they were ordinary particles. (c) Diffraction occurs when a wave encounters an obstacle of a slit that is comparable in size to its wavelength. The wave bends around the slit. The diffraction of light through two slits separated by a distance comparable to the wavelength of the light results in an interference pattern. Each slit acts as a new wave source, and the two new waves interfere with each other, which results in a pattern of bright and dark lines. (d) Because the mass of the bullets and their particle size are not absolutely small, the bullets will not produce an interference pattern when they pass through the slits. The de Broglie wavelength produced by the bullets will not be sufficiently large to interfere with the bullet trajectory, and no interference pattern will be observed. Questions for Group Work 7.109 Light is electromagnetic radiation, a type of energy embodied in oscillating electric and magnetic fields. Light in a vacuum travels at 3.00 X 10⁸ m/s. Copyright © 2017 Pearson Education, Inc.