(12 marks) For CO, the bond length is 1.13x10-10 m and the force constant is 1906 Nm-1:

a) Calculate the energy difference between the ground rotational state and the first excited rotational state.

b) Calculate the energy difference between the ground vibrational state and the first excited vibrational state.

c) Rationalize these energy values with the following expectation:

d)

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Consider the probability of finding a molecule in its groundstate, P(e0), or first excited state, P(e1), given that energyseparation btwn the 2 states is e1-e0=deltae=2.5 kJ/mol. You mayassume that the states are nondegenerate (g0=g1=1), unless statedotherwise.

A. What is the ratio of the excited and ground statepopulations, P(e1)/P(e0), at room temp, T~300K? ( NOte thatRT=NakbT~2.5 kJ/mol at 300 K)

B. What would happen to the ratio, P(e1)/P(e0) if the temp wereincreased by a factor of 2? why does this make sense?

C. GIven that diatomic molecule Cl2 has a vibrational frequency( in wavenumber units) of 565 cm -1, calculate the ratio of itsexcited to ground state vibrational population, P(e1)/P(e0) at T~300 K. ( NOte that the energy of the transition is the same as thatof a photon of the same frequency, v, and kbT/hc~ 209 cm-1 at 300K)

D. GIven that the first rotational transition of Cl2 has afrequency of ~ 0.5 cm-1 and a degeneracy of g0=1 for the groundstate and g1=3 for the first excited state, calculate the ratio ofthe excited to ground state rotational populations P(e1)/P(e0) atT~ 300 K.

I need help ASAP. I want step-by-step solution.

a) Use the Boltzmann equation (see C h. 8, equation 8 - 1: For these calculations you may assume the degeneracy, g j /g 0 = 1) to calculate the relative population of the E = 0 and E = 1 electronic state s at room temperature (25 o C) if the transition from E = 0, v = 0, J = 0 to E = 1, v = 0, J = 0 occurs at a wavelength of 454 nm. b) Repeat the calculation in (a) and determine the relative populations of the v = 0 and v = 1 vibrational state s at room temperature if the transition to the first excited vibrational state occurs at a wavelength of 4. 54 microm. c) Repeat the calculation in (a) and determine the relative population of the J = 0, J = 1, J = 2 and J = 3 rotational states at room temperature if the transitions occur at 15 microm , 7.5 microm, and 3.75 microm. d) Now consider an absorption transition from the populated states of the molecular ground state to the E = 1, v = 0, J = 0 excited state. If the wavelength separation between rotational states cannot be resolved by the absorption spectrometer what would the minimum width in nm of the absorption transition be?

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