Answered: A molecule can have various types of…

Principles of Modern Chemistry

8th Edition

ISBN:9781305079113

Author:David W. Oxtoby, H. Pat Gillis, Laurie J. Butler

Publisher:David W. Oxtoby, H. Pat Gillis, Laurie J. Butler

Chapter4: Introduction To Quantum Mechanics

Section: Chapter Questions

Problem 39P: Chapter 3 introduced the concept of a double bond between carbon atoms, represented by C=C , with a...

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A). A molecule can have various types of energies (translational, rotational, vibrational, and electronic), the sum of which is the molecule's total energy.

?trans=(?^2?+?^2?+?^2?)(ℎ^2/8??^2/3)

?rot=?(?+1)ℎ^2/8?2?

?vib=(?+1/2)(ℎ?)

In the equations, ??, ??, ??, ?, and ? are quantum numbers, ℎ is Planck's constant, ? is the mass of the molecule, ? is the volume of the container, ? is the moment of inertia of the molecule, and ? is the fundamental vibration frequency.

For carbon monoxide, CO , the moment of inertia is ?=1.45×10−46 kg⋅m2, and the fundamental vibration frequency is ?=2130 cm−1. Let ?=12.8, and let all the quantum numbers be equal to 11 .

Calculate the translational, rotational, and vibrational energies per mole of CO for these conditions.

?trans= J/mol

?rot= J/mol

?vib= J/mol

B). If the electronic energy of CO is 9.14 eV per molecule, calculate the total energy of CO per mole.

?total= J/mol

C). Which types of energy are negligible compared to the total energy? That is, which energies could you omit from the sum without changing the total value (to 3 sig. figs.)?

a. translational

b. rotational

c. vibrational

d. electronic

First, calculate each energy in joules, then scale to joules per mole using Avogadro's number.

This problem requires several unit conversions. Note that a joule (J) is equivalent to kg⋅m2/s2.

Also, consider that a frequency in wavenumbers (inverse length) can be converted to inverse time using the speed of light, ?=2.998×10−8 m/s.

A molecule can have various types of energies (translational, rotational, vibrational, and electronic), the sum of which is
the molecule's total energy.
Etrans
(n? + n, + n?)
8mV (2/3)
h?
Erot = J (J + 1)
87² I
1
Evib =
v +
hv
In the equations, nỵ , ny, n̟ , J, and v are quantum numbers, h is Planck's constant, m is the mass of the molecule, V is
the volume of the container, I is the moment of inertia of the molecule, and v is the fundamental vibration frequency.
For carbon monoxide, CO, the moment of inertia is I = 1.45 × 10-46 kg-m² , and the fundamental vibration frequency is
v = 2130 cm-1. Let V = 12.8 L, and let all the quantum numbers be equal to 1.
Calculate the translational, rotational, and vibrational energies per mole of CO for these conditions.

If the electronic energy of CO is 9.14 eV per molecule, calculate the total energy of CO per mole.
Etotal =
J/mol
Which types of energy are negligible compared to the total energy? That is, which energies could you omit from the
sum without changing the total value (to 3 sig. figs.)?
translational
rotational
vibrational
electronic
O O O

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