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Unformatted text preview: CHAPTER 42 MOLECULAR AND SOLIDSTATE PHYSICS Section 42 2: Molecular Energy Levels Problem 1. Find the energies of the first four rotational states of the HCl molecules described in Example 421. Solution The energies of rotational states (above the j = 0 state) are given by Equation 422, where for the HCl molecule, 2 2 63 = I = . meV (from Example 421). Thus, E j j I j j rot meV. = + = + ( ) ( ) . 1 2 1 2 63 2 1 2 = For j = 1 2 , , , and 3 2 63 7 89 15 78 , , . , . , . . meV meV and meV rot E = Problem 2. Find the wavelength of electromagnetic radiation needed to excite oxygen molecules (O 2 ) to their first rotational excited state. The rotational inertia of an oxygen molecule is 1 95 10 46 2 . .  kg m Solution The difference in energy between the j = 1 and j = 0 states is E I I = + = 1 1 1 2 2 2 ( )( ) . = = The photon wavelength corresponding to this transition is l p p = = = =  hc E hcI cI = = = = 2 8 46 2 2 2 3 10 1 055 ( / ) ( . m s)(1.95 10 kg m 10 3 34 = J s mm. ) .48 Problem 3 A molecule drops from the j = 2 to the j = 1 rotational level, emitting a 2.50meV photon. If the molecule then drops to the rotational ground state, what energy photon will be emitted? Solution The energy of a photon emitted in a transition between rotational levels with j =  1 is shown in Example 421 to be E j I j j  = ( ) . 1 2 = Using the given data for the first transition, j j = = 2 1 to , we find 2 3 2 50 10 2 = = I = = . eV 1 25 10 3 .  eV. This is also equal to the energy of a photon in the j j = = 1 to transition. Problem 4. Calculate the wavelength of a photon emitted in the j = 5 to j = 4 transition of a molecule whose rotational inertia is 1 75 10 47 2 . .  kg m Solution The energy difference between adjacent rotational levels is proportional to the upper jvalue (see Example 421 and the solution to Problem 6 below), so l p p = = = = hc E cI j = = = 2 2 3 10 1 75 10 5 1 055 10 8 47 34 ( / . ) ( . ) m s)( kg m J s 2 62 5 . m. m Problem 5. Photons of wavelength 1.68 cm excite transitions from the rotational ground state to the first rotational excited state in a gas. What is the rotational inertia of the gas molecules? Solution The energy of the absorbed photon equals the difference in energy between the j = 1 and j = 0 rotational levels, which is (see Example 421) 2 1 5 23 1240 1 68 7 38 10 118 10 = = = I E hc = = = = =  l eV nm cm eV J. . . . Therefore, I = =  ( . ( . ) .41 . 1 055 10 118 10 9 10 34 2 23 46 J s) J kg m 2 = CHAPTER 42 979 Problem 6. A molecule absorbs a photon and jumps to the next higher rotational state. If the photon energy is three times what would be needed for a transition from the rotational ground state to the first rotational excited state, between what two levels is the transition?...
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This note was uploaded on 10/19/2010 for the course PHY 78647 taught by Professor Miller during the Spring '09 term at Albany College of Pharmacy and Health Sciences.
 Spring '09
 Miller
 Physics, Energy

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