Question

Please do it step by step because I really want to understand the HW??

3. Vibrational excitations in a diatomic molecule can be described with a potential
energy function of the form
U (r) = D(e-2a(r-re) – 2e¬a(r-re))
where r is the interatomic distance, re is the equilibrium bond distance, D is the
dissociation energy (a constant), and a is a positive constant.
a) What are the dimensions of re, D, and a? Express your answer in terms of
products LªT'M°, where L is length, T is time, M is mass, and a, b, and c are
rational numbers.
b) What is the value of the potential at arbitrarely large separations (r > re).
c) Find the value of r for which U is a minimum.
d) What is the minimum value of U?
e) Show that for very small displacements x =
ergy can be approximated to
(r – re) << 1, the potential en-
U (x) – U (0)
ka
where k
2a? D.
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Transcribed Image Text

3. Vibrational excitations in a diatomic molecule can be described with a potential energy function of the form U (r) = D(e-2a(r-re) – 2e¬a(r-re)) where r is the interatomic distance, re is the equilibrium bond distance, D is the dissociation energy (a constant), and a is a positive constant. a) What are the dimensions of re, D, and a? Express your answer in terms of products LªT'M°, where L is length, T is time, M is mass, and a, b, and c are rational numbers. b) What is the value of the potential at arbitrarely large separations (r > re). c) Find the value of r for which U is a minimum. d) What is the minimum value of U? e) Show that for very small displacements x = ergy can be approximated to (r – re) << 1, the potential en- U (x) – U (0) ka where k 2a? D.

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