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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=rc)) 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 LaTbM°, 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? (r – re) << 1, the potential en- e) Show that for very small displacements x = ergy can be approximated to 1 U (x) – U (0) kx? where k = 20²D.

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=rc)) 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 LaTbM°, 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? (r – re) << 1, the potential en- e) Show that for very small displacements x = ergy can be approximated to 1 U (x) – U (0) kx? where k = 20²D.

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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=rc)) 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 LªTM°, 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- 1 U (x) – U (0) ka? where k = 2a² D.
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=rc)) 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 LªTM°, 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- 1 U (x) – U (0) ka? where k = 2a² D.
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