# Homework 5

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?

1. $$F_2$$
2. $$N_2$$
3. $$O_3$$
4. $$Ar$$
5. $$Br_2$$
6. $$HF$$
7. $$H_2 O$$
8. $$CD_2$$
9. $$CO_2$$
10. $$CH_4$$

## 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.

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