College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Please give the answers of c d step by step
![**Vibrational Excitations in Diatomic Molecules**
Vibrational excitations in a diatomic molecule can be described with a potential energy function of the form:
\[ U(r) = D(e^{-2a(r - r_e)} - 2e^{-a(r - r_e)}) \]
where \( r \) is the interatomic distance, \( r_e \) is the equilibrium bond distance, \( D \) is the dissociation energy (a constant), and \( a \) is a positive constant.
### Questions
**a)** What are the dimensions of \( r_e, D, \) and \( a \)? Express your answer in terms of products \( L^a T^b M^c \), 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 arbitrarily large separations (\( r \gg r_e \))?
**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 \( x = (r - r_e) \ll 1 \), the potential energy can be approximated to
\[ U(x) - U(0) = \frac{1}{2}kx^2 \]
where \( k = 2a^2D \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f6d2f58-ebc5-487b-abb4-91cd16bfc1c1%2Fc47ecb64-021c-4fe9-8945-d0a9b117b6e6%2Flegghg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Vibrational Excitations in Diatomic Molecules**
Vibrational excitations in a diatomic molecule can be described with a potential energy function of the form:
\[ U(r) = D(e^{-2a(r - r_e)} - 2e^{-a(r - r_e)}) \]
where \( r \) is the interatomic distance, \( r_e \) is the equilibrium bond distance, \( D \) is the dissociation energy (a constant), and \( a \) is a positive constant.
### Questions
**a)** What are the dimensions of \( r_e, D, \) and \( a \)? Express your answer in terms of products \( L^a T^b M^c \), 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 arbitrarily large separations (\( r \gg r_e \))?
**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 \( x = (r - r_e) \ll 1 \), the potential energy can be approximated to
\[ U(x) - U(0) = \frac{1}{2}kx^2 \]
where \( k = 2a^2D \).
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