Homework 5

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Name: ______________________________

Section: _____________________________

Student ID#:__________________________

Q1

Calculate the standard deviation of the bond length \(\sigma_X\) of the diatomic molecule \(\ce{^1H^{35}Cl }\) when it is in the ground state and first excited state using the quantum harmonic oscillator wavefunctions. The fundamental harmonic vibrational frequency of \(\ce{HCl }\) is 2989.6 \(cm^{-1}\) and the equilibrium bond length is 0.127 nm. 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 (via its vibrations)?

Q3

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

Q4

Which of the following molecule absorb in the IR?

\(I_2\)
\(O_2\)
\(O_3\)
\(HBr\)
\(HF\)
\(H_2 O\)
\(CD_2\)
\(CO_2\)
\(CH_4\)

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 500 nm energy that may be observed in electron (UV-VIS) spectroscopy? What is the energy of a 6-micron photon typical in IR spectroscopy? What is the energy of a photon absorbed in a typical CO rotation microwave line (\(6 \times 10^{11} Hz\))?

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 \(\ce{^1H^{35}Cl }\) is \(2.6 \times 10^{-47} \;Kg\times m^2\). What is the energy for rotation for \(\ce{^1H^{35}Cl }\) in the \(J=5\) and \(J=20\) 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 \(J=5\) and \(J=20\) rotational states of \(\ce{^1H^{35}Cl }\) to be thermally occupied? (Hint: assuming the term "comparable" is "equal" for this problem).

Q9

\(\ce{^1H^{35}Cl }\) has a bond length of 0.12746 nm and fundamental stretching vibration at 2,886 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)?