Please answer this question! Make sure its hand written and legible!! 12. (McQuarrie 6-44) In this problem, we'll calculate the fraction of diatomic molecules in aparticular rotational level at a temperature T using the rigid rotor approximation. Thisfraction is governed by the Boltzmann distribution, which says that the number of moleculeswith energy EJ is proportional to е-Eл/kBT, where kÅ is the Boltzmann constant and T isthe temperature in Kelvin. Because the J-th rotational level has degeneracy (2J + 1), wewrite,NJ = c(2J + 1)e¯−BJ(J+1)/kTwhere we have used EJ = BJ(J+1). Plot NJ/No versus J for H35 C1 (В :127135 Cl (B = 0.114 cm-1) at 300 K.=10.60 cm-1) andAt approximately what rotational state J is the population ratio NJ/No a maximum in eachcase? Explain how this distribution of rotational state populations leads to the spectroscopicband structure of the sort seen in McQuarrie Figure 6.4 or in lecture.

1. Boltzmann Distribution:The Boltzmann distribution describes the distribution of particles (in…

Answered: 12. (McQuarrie 6-44) In this problem,…

Physical Chemistry

2nd Edition

ISBN:9781133958437

Author:Ball, David W. (david Warren), BAER, Tomas

Publisher:Ball, David W. (david Warren), BAER, Tomas

Chapter1: Gases And The Zeroth Law Of Thermodynamics

Section: Chapter Questions

Problem 1.75E

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(c) When a gas is expanded very rapidly, its temperature can fall to a few degrees Kelvin. At these low temperatures, unusual molecules like ArHCl (Argon weakly bonded to HCl) can form on mixing. For the isotopic species Ar H$CI, the following rotational transitions were observed: J (1 → 2): 6714.44 MHz J (2 → 3): 10068.90 MHz Assume the molecule can be treated as a linear diatomic molecule (ArCl). (i) Calculate the rotational constant (B) and centrifugal distortion (D) constant for this molecule.
(c) Consider the following rotational temperatures of diatomic molecules: qr(N2) = 2.9K, qr(HD) = 64.7K Assuming classical behaviour (i.e. continuum approximation): (i) Estimate the number of accessible rotational energy levels at 290 K for both molecules
Consider a rotating molecule of ©Li'H. At T = 300 K, the rotational kinetic energy in the eighth excited state is equivalent to kgT, where kB is the Boltzmann constant. What is the energy of the first excited state relative to the ground state?
5. For carbon monoxide at 298K, determine the fraction of molecules in the rotational levels for J=0, 5, 10, 15, and 20. The rotational constant (B) is 3.83x10^-23 Joules.
Calculate the vibrational and rotational temperatures (Ovib and Orot) for the diatomic molecular Lithium Hydride (LiH). The Li-H bond length is 0.1595 nm and the vibrational frequency (VL¡H) is 1405 cm'.
Which of the following statements are true about the relationship between C p,m and C v,m ?
3. At T = 300K, 1bar of ¹60¹80 in a 1m³ box (lengths ax ay = az = 1m) can be considered as an ideal gas. In that case, the average translational energy in each dimension for a molecule is given by: Ex = Ex = Ex = 1kT, where k = 1.38 x 10-23 J/K is the Boltzmann constant. The average rotational energy about an axis perpendicular to the O=O bond is: Erot=kT, Evib = KT. and the average vibrational energy is: Given that the fundamental vibrational frequency for ¹60¹80 is w = 4.741 x 10¹³ Hz, find the values of the quantum numbers nx, J, and u for an average ¹60¹80 molecule in this system.
14.44) The lowest triplet state in naphthalene (C10H8) is about 11,000 cm-1 below then lowest excited singlet electronic level at 77 K. Calculate the ratio of the population in these two states. [Hint: The Boltzmann equation is given by N2/N1 = (g2/g1 exp(-ΔE/kBT), where g1 and g2 are the degeneracies for level 1 and 2.]
The 14 N160 molecule undergoes a transition between its rotational ground state and its rotational first excited state. Approximating the diatomic molecule as a rigid rotor, and given that the bond length of NO is 1.152 Angstroms, calculate the energy of the transition. As your final answer, calculate the temperature T in Kelvin, such that Ethermal = kBT equals the %3D energy of the transition between NO's rotational ground state and fırst excited state.
Derive an expression for the mean energy of a collection of molecules that have three energy levels at 0, ε, and 3ε with degeneracies 1, 5, and 3, respectively.
The vibrational temperature of a molecule prepared in a supersonic jet can be estimated from the observed popula- tions of its vibrational levels, assuming a Boltzmann distri- bution. The vibrational frequency of HgBr is 5.58 × 1012 s-1, and the ratio of the number of molecules in the n = 1 state to the number in the n = 0 state is 0.127. Estimate the vibra- tional temperature under these conditions.
4. Spectroscopic measurements indicate that the rotational constants of SO2, a non-linear molecule, are 2.02736, 0.34417, and 0.29354 cm-1. Note: kg = 0.69503476 cm-1/K. (a) Compute zrot at T= 350 K assuming its symmetry number is o= 2. Hint: Using the rotational temperatures for the molecule might make this easier. (b) At what temperature would the partition function equal 2.0 × 104?

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