U(r = = c [G)`-)], 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 > 6 is a constant. The case with X 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. In the following, try to simplify the expressions as much as possible 5, and consider the frequency whose SI units are Hz. (a) Find a combination of C,m, o, that has the units of frequency. Does the combination depend on λ?
U(r = = c [G)`-)], 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 > 6 is a constant. The case with X 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. In the following, try to simplify the expressions as much as possible 5, and consider the frequency whose SI units are Hz. (a) Find a combination of C,m, o, that has the units of frequency. Does the combination depend on λ?
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![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. In the following, try to simplify the expressions as much as possible, and consider the frequency whose SI units are Hz.
(a) Find a combination of \( C, m, \sigma, \lambda \) that has the units of frequency. Does the combination depend on \( \lambda \)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F335d6cef-e110-4df1-b3c9-cf74ce65a19c%2F6364754a-0bbb-4d1f-8b50-6585bab75bfc%2F995ik4_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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. In the following, try to simplify the expressions as much as possible, and consider the frequency whose SI units are Hz.
(a) Find a combination of \( C, m, \sigma, \lambda \) that has the units of frequency. Does the combination depend on \( \lambda \)?
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