help me with part e please The vibrational properties of a diatomic molecule can often be described by Mie'spair potential 3U(r) = c [(;-) ¹ − (;)],Cwhere U(r) is the potential energy between the two atoms, r is the distance betweenthe two atoms, C, o are positive constants, and > 6 is a constant. The case withX = 12 is the Lennard- Jones potential and was covered in Lecture #2. For thepurposes of this problem, assume there is a mass 'm' that represents the dynamicalmass of the molecule. In the following, try to simplify the expressions as much aspossible, and consider the frequency whose SI units are Hz.(a) Find a combination of C, m, o, that has the units of frequency. Does thecombination depend on λ?(b) Derive the equilibrium position, reg of the system for any >> 6(c) Derive the frequency of the small oscillations of the system, fosc for any λ > 6.Simplify the expression until a multiplicative factor of √X-6 appears. Fullcredit requires getting this factor because it shows the relevance of having > 6.Now, you know the role of λ in the formula of the frequency: Sometimes,dimensional analysis has a limited scope.(d) Show that the units for the equilibrium position and for the frequency of theoscillations are consistent based on the units of C, m, o, X.'(e) Find the ratio of the equilibrium positions and the frequencies of the smalloscillations between the cases λ = 10 and λ = 14. That is req(λ = 14)/reg(λ =10), and fosc (X= 14)/ fosc (λ = 10). Hint: the ratio of the equilibrium positionsdiffers from 1 by about 2%, whereas the the ratio of the two frequencies differsfrom 1 by about 54%!=

Answered: Image /qna-images/answer/d9ee9d24-9b3a-4da9-b778-8118c4e44a49.jpg

(e) Find the ratio of the equilibrium positions and the frequencies of the small oscillations between the cases λ = 10 and X= 14. That is req(λ = 14)/reg(λ: 10), and fosc (X= 14)/ fosc (λ = 10). Hint: the ratio of the equilibrium positions differs from 1 by about 2%, whereas the the ratio of the two frequencies differs from 1 by about 54%!

(e) Find the ratio of the equilibrium positions and the frequencies of the small oscillations between the cases λ = 10 and X= 14. That is req(λ = 14)/reg(λ: 10), and fosc (X= 14)/ fosc (λ = 10). Hint: the ratio of the equilibrium positions differs from 1 by about 2%, whereas the the ratio of the two frequencies differs from 1 by about 54%!