4.13 The oscillation frequencies of a diatomic molecule are typically1012 Hz-1014 Hz. Derive an order of magnitude estimate for theharmonic oscillator constant K for such molecules.

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Answered: 4.13 The oscillation frequencies of a…

Physics for Scientists and Engineers with Modern Physics

10th Edition

ISBN:9781337553292

Author:Raymond A. Serway, John W. Jewett

Publisher:Raymond A. Serway, John W. Jewett

Chapter42: Molecules And Solids

Section: Chapter Questions

Problem 21P

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Problem 1: This problem concerns a collection of N identical harmonic oscillators (perhaps an Einstein solid) at temperature T. The allowed energies of each oscillator are 0, hf, 2hf, and so on. a) Prove =1+x + x² + x³ + .... Ignore Schroeder's comment about proving 1-x the formula by long division. Prove it by first multiplying both sides of the equation by (1 – x), and then thinking about the right-hand side of the resulting expression. b) Evaluate the partition function for a single harmonic oscillator. Use the result of (a) to simplify your answer as much as possible. c) Use E = - дz to find an expression for the average energy of a single oscillator. z aB Simplify as much as possible. d) What is the total energy of the system of N oscillators at temperature T?
The potential energy of two atoms in a diatomic molecule is approximated by U(r) = a/r12-b/r6, where r is the spacing between atoms and a and b are positive constants. Suppose the distance between the two atoms is equal to the equilibrium distance found in part A. What minimum energy must be added to the molecule to dissociate it - that is, to separate the two atoms to an infinite distance apart? This is called the dissociation energy of the molecule. Express your answer in terms of the variables a and b. For the molecule CO, the equilibrium distance between the carbon and oxygen atoms is 1.13\times 10-10m and the dissociation energy is 1.54\times 10-18J per molecule. Find the value of the constant a. Express your answer in joules times meter in the twelth power. Find the value of the constant b. Express your answer in joules times meter in the sixth power.
The effective spring constant describing the potential energy of the HBr molecule is 410 N/m and that for the NO molecule is 1530 N/m. (a) Calculate the minimum amplitude of vibration for the HBr molecule. (b) Calculate the minimum amplitude of vibration for the NO molecule.
3.66. Show that the density-series second virial coefficients can be derived from isothermal volumetric data via the expression: B = lim (Z-1)/p p-0 p (molar density) = 1/V
(a) The probability that a molecule in the atmosphere is at a height between z and z + dz above the surface of the Earth is proportional to e-z/A dz (capital Greek lambda). Determine the constant of proportionality. Sketch this distribution. (b) What is the mean molecular height, i.e. the mean height of a molecule? (c) What is the variance in the molecular height? (d) What is the standard deviation in the molecular height?
Calculate the threshold frequency of Zinc with 6.9x10^-19 J Work Function.
For the simple cubic crystal described above, i.e. a = 0.7\: nma=0.7nm, calculate the surface density of atoms (i.e. number of atoms per unit area) on the (100) plane in unit of cm^{-2}cm−2. Values within 5% error will be considered correct.
The vibrational frequency of IBr is 269 cm'. (a) Calculate the vibrational partition function of IBr at 298 K. (b) Calculate the vibrational partition function of IBr at 1000 K. (c) Calculate the vibrational partition function of IBr at 1000 K using the formula in the limit of high temperature. Calculate the percent error compared to the true value in part b.
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For the face centered cubic crystal described above, i.e. a = 0.7\: nma=0.7nm, calculate the surface density of atoms (i.e. number of atoms per unit area) on the (100) plane in unit of cm^{-2}cm−2. Values within 5% error will be considered correct.
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4.13 The oscillation frequencies of a diatomic molecule are typically 1012 Hz-1014 Hz. Derive an order of magnitude estimate for the harmonic oscillator constant K for such molecules.