# Homework 5

- Page ID
- 143072

Name: ______________________________

Section: _____________________________

Student ID#:__________________________

## Q1

Calculate the standard deviation of the bond length \(\sigma_X\) of the diatomic molecule \(\ce{^1H^{19}F }\) when it is in the ground state and first excited state using the quantum harmonic oscillator wavefunctions. The fundamental harmonic vibrational frequency of \(\ce{HF}\) is 4,460 \(cm^{-1}\) and the equilibrium bond length is 0.091nm. How do you interpret the change in the ratio of average bond length to \(\sigma_X\) as a function of energy in the vibration?

## Q2

What are two requirements for a molecule to absorb IR radiation (through vibration)?

## Q3

(a) Using the relevant transition moment integrals, demonstrate that the probability of a vibration described by a harmonic oscillator in absorbing IR radiation form the \(v=0\) to the \(v=2\) state is forbidden.

(b) Is the \(v=1\) to \(v=0\) transition also forbidden?

## Q4

Which of the following molecules absorb in the IR?

- \(F_2\)
- \(N_2\)
- \(O_3\)
- \(Ar\)
- \(Br_2\)
- \(HF\)
- \(H_2 O\)
- \(CD_2\)
- \(CO_2\)
- \(CH_4\)

## Q5

What do the presence of overtones in IR spectra reveal about the anharmonicity of the vibration?

## Q6

What is the energy in cm^{-1} of a photon of 600 nm energy that may be observed in electron (UV-VIS) spectroscopy? What is the energy of a 3-micron photon typical in IR spectroscopy? What is the energy of a photon absorbed in a typical CO rotation microwave line (\(0.11519 THz\))?

## Q7

Fill in this table.

Spectroscopic Signature |
Degree of Freedom |
||||
---|---|---|---|---|---|

Type EM Range |
Typical Wavelength of Transition |
Typical Energy of Transition |
sensitive to electronic transition (yes/no) |
Sensitive to vibrational transition (yes/no) |
sensitive to rotational transitions (yes/no) |

UV-Visible | |||||

Infrared | |||||

Microwave |

If any spectroscopy is sensitive to more than one degree of freedom, explain why.

## Q8

The moment of inertia of \(\ce{^1H^{19}F }\) is \(1.34 \times 10^{-47} \;kg\times m^2\). What is the energy for rotation for \(\ce{^1H^{19}F }\) in the \(J=4\) and \(J=15\) states? For a molecule to be thermally excited, the energy of the eigenstate must be comparable to \(k_bT\), with \(k_b\) as the Boltzmann's constant and \(T\) is absolute temperature. What temperature is needed for the \(J=4\) and \(J=15\) rotational states of \(\ce{^1H^{19}F }\) to be thermally occupied? (Hint: assuming the term "comparable" is "equal" for this problem).

## Q9

\(\ce{^1H^{19}F }\) has a bond length of 0.091 nm and fundamental stretching vibration at 4,460 cm^{-1}. What is the temperature required for the \(v=1\) mode to be thermally excited? (Hint: assuming the term "comparable" is "equal" for this problem).

## Q10

\(\ce{^1H^{19}F}\) has an equilibrium bond length of 91.7 pm and a spring constant of 970 N/m. The molecule rotates freely in a three-dimensional space as a gas.

- What is the zero point energy associated with this rotation? Will this differ if you were considering only vibration?
- What is the lowest energy microwave transition observed absorbed \(\ce{^1H^{19}F}\) ascribed to rotational motion (assuming a rigid rotor described the rotation)?