The rigid rotor model is often used to study the rotation of diatomic molecules. In this case, there is often a spectroscopic transition from a state with angular momentum quantum number, J, to one with J+ 1 (absorption) or to one with J-1 (emission). (a) Using the expression for rigid rotor energy levels, derive an expression for the energy difference corresponding to an emission transition J→J-1 and equate this to hv, the energy of a photon. Report your result as a positive number (in emission, energy is lost). (On the quiz, you will be given values for J and, and asked to compute the value of v.) 21 (b) In spectroscopic experiments, one typically observes a series of absorption lines corresponding to the transitions originating in states with successive values of J. For one diatomic, one observes a set of lines equally spaced at Av = 1.1338 × 1012sec1. What is the moment of inertia for this molecule in units of kg-m?? (Note that since molecules have masses on the order of 10-27 kg and bond lengths on the order of 100 pm, this quantity should be about 10-47 kg-m². Report the number multiplying 10-47.)

The rigid rotor model is often used to study the rotation of diatomic molecules. In this case, there is often a spectroscopic transition from a state with angular momentum quantum number, J, to one with J+ 1 (absorption) or to one with J-1 (emission). (a) Using the expression for rigid rotor energy levels, derive an expression for the energy difference corresponding to an emission transition J→J-1 and equate this to hv, the energy of a photon. Report your result as a positive number (in emission, energy is lost). (On the quiz, you will be given values for J and, and asked to compute the value of v.) 21 (b) In spectroscopic experiments, one typically observes a series of absorption lines corresponding to the transitions originating in states with successive values of J. For one diatomic, one observes a set of lines equally spaced at Av = 1.1338 × 1012sec1. What is the moment of inertia for this molecule in units of kg-m?? (Note that since molecules have masses on the order of 10-27 kg and bond lengths on the order of 100 pm, this quantity should be about 10-47 kg-m². Report the number multiplying 10-47.)

Principles of Instrumental Analysis

7th Edition

ISBN:9781305577213

Author:Douglas A. Skoog, F. James Holler, Stanley R. Crouch

Publisher:Douglas A. Skoog, F. James Holler, Stanley R. Crouch

Chapter6: An Introduction To Spectrometric Methods

Section: Chapter Questions

Problem 6.1QAP

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Consider the rotational spectrum of a linear molecule at 298 K with a moment of inertia of 1.23×10−461.23\times10^{-46}1.23×10−46 kg m2 . (a) What is the frequency for the transition from J = 2 to J = 3? (b) What is the most populated rotational level for this molecule? Would the transition in (a) give the most intense signal in the rotational spectrum?
An H2 molecule is in its vibrational and rotational ground states. It absorbs a photon of wavelength 2.211 2 μm and makes a transition to the υ = 1, J = 1 energy level. It then drops to the υ = 0, J = 2 energy level while emitting a photon of wavelength 2.405 4 mm. Calculate (a) the moment of inertia of the H2 molecule about an axis through its center of mass and perpendicular to the H–H bond, (b) the vibrational frequency of the H2 molecule, and (c) the equilibrium separation distance for this molecule.
A diatomic molecule containing 35Cl and another atom has a rotational transition from J=0 to J=1 corresponding to a frequency of 7.70×109 Hz. The bond length is 267 pm. Use this information to calculate the reduced mass of the molecule, then find the atomic mass number of the other atom.
Rotational spectra are affected slightly by the fact that different isotopes have different masses. Suppose a sample of the common isotope 1H35Cl is changed to 1H37Cl. (a) By what fraction is the molecule’s rotational inertia different? (The bond length is 0.127 nm in each case.) (b) What is the change in energy of theℓ = 1 to theℓ = 0 transition if the isotope is changed?
Calculate the energies of the first four rotational levels of 1H127I free to rotate in three dimensions; use for its moment of inertia I = μR2, with μ = mHmI/(mH + mI) and R = 160 pm. Use integer relative atomic masses for this estimate.
4) Calculate the energies E of the first three rotational levels (those giving the minimal energy) of 1H35Cl molecule that is free to rotate in three dimensions. For its moment of inertia I = μeffR2 use the effective mass with μeff = mH * mCl / (mH + mCl) and bond length R = 127 pm.
(a) Calculate the moment of inertia of (i) 1H2, (ii) 2H2, (iii) 12C16O2, (iv) 13CO2 . (b) Calculate the corresponding rotational constants, expressing your answers as a frequency in hertz (Hz) and as a wavenumber in reciprocal centimetres (cm-1).
Using the rigid rotor model, calculate the energies in Joules of the first three rotational levels of HBr, using for its moment of inertia I = μR2, with μ = mHmX/(mH + mX) and equilibrium internuclear distance = 1.63 Å. To put these energies into units that make sense to us, convert energy to kJ/mol. (Simply estimate atomic masses from the average atomic weights of the elements given in the periodic table).
E rotational is 2.777×10−20 J
Consider the diatomic molecule AB modeled as a rigid rotor (two masses separated by a fixed distance equal to the bond length of the molecule). The rotational constant of the diatomic AB is 25.5263 cm-1. (a) What is the difference in energy, expressed in wavenumbers, between the energy levels of AB with J = 10 and J = 6? (b) Consider now a diatomic A'B', for which the atomic masses are ma 0.85 mA and mB' 0.85 mB and for its bond length ra'B' = 0.913 rAB. What is the difference in energy, expressed in wavenumbers, between the energy levels of the A'B' molecule with J = 9 and J = 7?
Q/ The bond length of the C0O molecule is 112.8 pm. Calculate the following (a) The reduced mass. (b) The rotational constant of CO when moment of inertia (I) is 1.4486 x 1046 kg.m² (c) Calculate the wavelength of the photon absorbed when a CO molecule initially in the J=2 level. makes a transition to the J=3 level.
The spacing of lines in the microwave spectrum of 27Al1H is 12.604 cm−1; calculate the moment of inertia and bond length of the molecule.