Molecular Pair Potential The vibrational properties of a diatomic molecule can often be described by Mie's pair potential 3 U (r) = ¤ [(;)* - (;)*] . C (4) where U(r) is the potential energy between the two atoms, r is the distance between the two atoms, C, o are positive constants, and λ > 6 is a constant. The case with λ 12 is the Lennard-Jones potential and was covered in Lecture #2. For the purposes of this problem, assume there is a mass ‘m' that represents the dynamical mass of the molecule. 4 = TLTR: (a) Find a combination of C, m, o that has the units of frequency. (b) Derive the ratio of the equilibrium positions and of the frequencies of small oscillations for X = 10, 14. Show that the units for the equilibrium position and for the frequency of the oscillations are consistent.
Molecular Pair Potential The vibrational properties of a diatomic molecule can often be described by Mie's pair potential 3 U (r) = ¤ [(;)* - (;)*] . C (4) where U(r) is the potential energy between the two atoms, r is the distance between the two atoms, C, o are positive constants, and λ > 6 is a constant. The case with λ 12 is the Lennard-Jones potential and was covered in Lecture #2. For the purposes of this problem, assume there is a mass ‘m' that represents the dynamical mass of the molecule. 4 = TLTR: (a) Find a combination of C, m, o that has the units of frequency. (b) Derive the ratio of the equilibrium positions and of the frequencies of small oscillations for X = 10, 14. Show that the units for the equilibrium position and for the frequency of the oscillations are consistent.
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![**Molecular Pair Potential**
The vibrational properties of a diatomic molecule can often be described by Mie’s pair potential:
\[ U(r) = C \left[ \left( \frac{\sigma}{r} \right)^\lambda - \left( \frac{\sigma}{r} \right)^6 \right], \]
where \( U(r) \) is the potential energy between the two atoms, \( r \) is the distance between the two atoms, \( C, \sigma \) are positive constants, and \( \lambda > 6 \) is a constant. The case with \( \lambda = 12 \) is the Lennard-Jones potential and was covered in Lecture #2. For the purposes of this problem, assume there is a mass ‘m’ that represents the dynamical mass of the molecule.
**TLTR:**
(a) Find a combination of \( C, m, \sigma \) that has the units of frequency.
(b) Derive the ratio of the equilibrium positions and of the frequencies of small oscillations for \( \lambda = 10, 14 \). Show that the units for the equilibrium position and for the frequency of the oscillations are consistent.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd104ea5e-59f4-47a3-a0d3-8c9f1ad3b57c%2F68704900-d327-42b0-96a3-359658124517%2F14wjbl_processed.png&w=3840&q=75)
Transcribed Image Text:**Molecular Pair Potential**
The vibrational properties of a diatomic molecule can often be described by Mie’s pair potential:
\[ U(r) = C \left[ \left( \frac{\sigma}{r} \right)^\lambda - \left( \frac{\sigma}{r} \right)^6 \right], \]
where \( U(r) \) is the potential energy between the two atoms, \( r \) is the distance between the two atoms, \( C, \sigma \) are positive constants, and \( \lambda > 6 \) is a constant. The case with \( \lambda = 12 \) is the Lennard-Jones potential and was covered in Lecture #2. For the purposes of this problem, assume there is a mass ‘m’ that represents the dynamical mass of the molecule.
**TLTR:**
(a) Find a combination of \( C, m, \sigma \) that has the units of frequency.
(b) Derive the ratio of the equilibrium positions and of the frequencies of small oscillations for \( \lambda = 10, 14 \). Show that the units for the equilibrium position and for the frequency of the oscillations are consistent.
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