Physics for Scientists and Engineers with Modern Physics
10th Edition
ISBN: 9781337553292
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
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Chapter 42, Problem 10P
To determine
To show that the moment of inertia is
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Let's consider the three atoms composing the molecule now have different masses and coordinate, while the axis of rotation is still z axis that is perpendicular to the xy plane. The first atom has a mass of 142.54 kg, with x coordinate at 3 m and y coordinate at 6 m. The second atom has a mass of 82.55 kg, with x coordinate at 1 m and y coordinate at 6 m. The third atom has a mass of 8 kg, with x coordinate at 5 m and y coordinate at 9 m. What is the moment of inertia in unit of kg m2 with respect to the x axis?
The CO molecule makes a transition from the J = 1 to the J = 2 rotational state when it absorbs a photon of frequency 2.30 x 1011 Hz. (a) Find the moment of inertia of this molecule from these data.
Show that the moment of inertia of a diatomic molecule composed of atoms of masses mA and mB and bond length R is equal to meffR2, where meff = mAmB/(mA+mB).
Chapter 42 Solutions
Physics for Scientists and Engineers with Modern Physics
Ch. 42.1 - For each of the following atoms or molecules,...Ch. 42.2 - Prob. 42.2QQCh. 42.2 - Prob. 42.3QQCh. 42 - Prob. 1PCh. 42 - Prob. 2PCh. 42 - Prob. 3PCh. 42 - Prob. 4PCh. 42 - Prob. 5PCh. 42 - The photon frequency that would be absorbed by the...Ch. 42 - Prob. 8P
Ch. 42 - Prob. 9PCh. 42 - Prob. 10PCh. 42 - (a) In an HCl molecule, take the Cl atom to be the...Ch. 42 - Prob. 12PCh. 42 - Prob. 13PCh. 42 - Prob. 14PCh. 42 - Prob. 15PCh. 42 - Prob. 16PCh. 42 - Prob. 17PCh. 42 - Prob. 19PCh. 42 - Prob. 21PCh. 42 - Prob. 22PCh. 42 - Prob. 23PCh. 42 - Prob. 24PCh. 42 - Prob. 25PCh. 42 - Prob. 26PCh. 42 - Prob. 27PCh. 42 - Prob. 28PCh. 42 - Prob. 29PCh. 42 - Prob. 30PCh. 42 - Prob. 32PCh. 42 - Prob. 33PCh. 42 - Prob. 35PCh. 42 - Prob. 36APCh. 42 - Prob. 37APCh. 42 - Prob. 39APCh. 42 - Prob. 40APCh. 42 - As an alternative to Equation 42.1, another useful...
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- The v = 0 to v = 1 vibrational transition of the HI molecule occurs at a frequency of 6.69 × 1013 Hz. The same transition for the NO molecule occurs at a frequency of 5.63 × 1013 Hz. Calculate (a) the effective force constant and (b) the amplitude of vibration for each molecule. (c) Explain why the force constant of the NO molecule is so much larger than that of the HI molecule.arrow_forwardConsider the HCl molecule, which consists of a hydrogen atom of mass 1 u bound to a chlorine atom of mass 35 u. The equilibrium separation between the atoms is 0.128 nm, and it requires 0.15 eV of work to increase or decrease this separation by 0.01 nm. (a) Calculate the four lowest rotational energies (in eV) that are possible, assuming the molecule rotates rigidly. (b) Find the molecules spring constant and its classical frequency of vibration. (Hint: Recall that U=12Kx2.) (c) Find the two lowest vibrational energies and the classical amplitude of oscillation corresponding to each of these energies. (d) Determine the longest wavelength radiation that the molecule can emit in a pure rotational transition and in a pure vibrational transition.arrow_forwardAs an alternative to Equation 42.1, another useful model for the potential energy of a diatomic molecule is the Morse potential U(r)=B[ea(rr0)1]2 where B, a, and r0 are parameters used to adjust the shape of the potential and its depth. (a) What is the equilibrium separation of the nuclei? (b) What is the depth of the potential well, defined as the difference in energy between the potentials minimum value and its asymptote as r approaches infinity? (c) If is the reduced mass of the system of two nuclei and assuming the potential is nearly parabolic about the well minimum, what is the vibrational frequency of the diatomic molecule in its ground state? (d) What amount of energy needs to be supplied to the ground-state molecule to separate the two nuclei to infinity?arrow_forward
- (a) In an HCl molecule, take the Cl atom to be the isotope 35Cl. The equilibrium separation of the H and Cl atoms is 0.127 46 nm. The atomic mass of the H atom is 1.007 825 u and that of the 35Cl atom is 34.968 853 u. Calculate the longest wavelength in the rotational spectrum of this molecule. (b) What If? Repeat the calculation in part (a), but take the Cl atom to be the isotope 37Cl, which has atomic mass 36.965 903 u. The equilibrium separation distance is the same as in part (a). (c) Naturally occurring chlorine contains approximately three parts of 35Cl to one part of 37Cl. Because of the two different Cl masses, each line in the microwave rotational spectrum of HCl is split into a doublet as shown in Figure P42.11. Calculate the separation in wavelength between the doublet lines for the longest wavelength.arrow_forwardTo determine the equilibrium separation of the atoms in the HCl molecule, you measure the rotational spectrum of HCl. You find that the spectrum contains these wavelengths (among others): 60.4 mm, 69.0 mm, 80.4 mm, 96.4 mm, and 120.4 mm. (a) Use your measured wavelengths to find the moment of inertia of the HCl molecule about an axis through the center of mass and perpendicular to the line joining the two nuclei. (b) The value of l changes by +-1 in rotational transitions. What value of l for the upper level of the transition gives rise to each of these wavelengths? (c) Use your result of part (a) to calculate the equilibrium separation of the atoms in the HCl molecule. The mass of a chlorine atom is 5.81 * 10-26 kg, and the mass of a hydrogen atom is 1.67 * 10-27 kg. (d) What is the longest-wavelength line in the rotational spectrum of HCl?arrow_forward2(6) Calculate the fundamental vibrational wavenumber (in cm-1) for HI molecule, if its angular vibrational frequency is 4.394×1014 s-1. Calculate the vibrational energy of the molecule in the ground state and the force constant. Assume the mass is the mass of a proton.arrow_forward
- The coordinates of unit a, of the atoms in a body-centered cubic lattice are (0,0,0), (0,1,0), (0,0,1), (0,1,1), (1,0,0), (1,1,0), (1,0,1), (1,1,1) and (1/2;1/2,1/2). Calculate the structure factors Fhkl when all the atoms are identicalarrow_forwardAlthough an ordinary H2 molecule consists of two identical atoms, this is not the case for the molecule HD, with one atom of deuterium (Le., heavy hydrogen, 2H). Because of its small moment of inertia, the HD molecule has a relatively large value of E: 0.0057 eV. At approximately what temperature would you expect the rotational heat capacity of a gas of HD molecules to "freeze out," that is, to fall significantly below the constant value predicted by the equipartition theorem?arrow_forwardHowever, the molecule we can encounter everyday continuously vibrates and interact with the surrounding causing its bond vector to vary slightly. According to a new spectroscopy analysis, the adjacent bond vectors was found to be A = 0.82i + 0.99j + 0.84k B = 1.09i + -1.01j + -0.97k What is the angle (in degrees) between the bonds based on this new data?arrow_forward
- According to a new spectroscopy analysis, the adjacent bond vectors was found to be A = 1.07i + 0.83j + 0.81k B = 1.03i + -0.99j + -0.85k What is the angle (in degrees) between the bonds based on this new data?arrow_forwardMost of the mass of an atom is in its nucleus. Model the mass distribution in a diatomic molecule as two spheres of uniform density, each of radius 2.00 x 10-15 m and mass 1.00 x 10-26 kg, located at points along the y axis as in 42.5a, and separated by 2.00 x 10-10 m. Rotation about the axis joining the nuclei in the diatomic molecule is ordinarily ignored because the first excited state would have an energy that is too high to access. To see why, calculate the ratio of the energy of the first excited state for rotation about the y axis to the energy of the first excited state for rotation about the x axis.arrow_forwardHow do you calculate the over turning moment of a cube that is 12inx12inx12in?arrow_forward
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