Modern Physics

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PHY251 Homework Set 4 Reading: Chapter 5 Homework: Chapter 5, Questions 1,3,4 Problems 6,8,9,16,18 Hints and Solutions Question V.1 In studying the Bohr model in this chapter, we ignored special relativity. Why, then, does the speed of light appear in our expression for the Bohr radius, Eq. (5-9)? (Hint: If the speed of light were 10 times what it is, by how much would the numerical value of the Bohr radius change?) Hints: No hints Solution: The Bohr radius does not depend on the speed of light c : the appearance of c in the formula for the Bohr radius is purely due to the presence of the dimensionless constant α , which itself introduces c in its definition α = e ²/(4 πε 0 ) / hc ; i.e. the product α c appearing in the Bohr radius, the orbital speed, the energy, and so on, do not depend on c . Question V.3 Given that the atom's mass is roughly the mass of the nucleus, that the atom's size - about the Bohr radius - is determined by the electronic structure of the atom, and that the density of matter is in the range 10 3 kg/m 3 to 10 4 kg/m 3 , what would you estimate the nuclear mass to be compared with the electron mass? Outline your assumptions and approach before you actually make any calculations. Hints: No hints Solution: We are asked to estimate the average mass of an atomic nucleus in matter, using the value of the atom's size as determined by the Bohr radius a 0 , and the density of ordinary matter 10 3 kg/m 3 . First, we will estimate the typical volume of an atom, and then determine how many atoms would fit in an m 3 . Using the typical mass of a m 3 of matter, we can then estimate the average mass of an individual atom. Because the atomic mass is largely determined by its nucleus, we thus find the typical nuclear mass. The typical atomic volume is ~(2 a 0 ) 3 = (0.1 nm) 3 = 10 -30 m 3 phy251_hw04 file:///C|/DOCUME~1/PHY_CO~1/PHY251/PHY251_hw04.HTML (1 of 4) [2/25/2001 8:21:45 PM]
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Thus, per m 3 , there will be about 10 30
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