Next quarter in statistical mechanics you will learn that for a system in therma

Question

Next quarter in statistical mechanics you will learn that for a system in thermal equilibrium that the relative population of states is given by: n_i/n_j = g_i/g_j e^-(epsilon_i - epsilon_j)/k_B T Where n is the number in state i or j, g is the degeneracy of state i or j, epsilon is the energy of state i or j, k_B is the Boltzman constant and T is the temperature in Kelvin. Let's return to the^1H^127 I molecule in problem 2g. This molecule has a force constant of 172 Nm^-1. Consider rotational states only. Calculate n_2/n_0 and n_10/n_0 at 300K where the subscripts refer to the quantum number J. Consider vibrational states only. Calculate n_2/n_0 and n_10/n_0 where the subscripts refer to the quantum number v. Calculate the temperature where n_2/n_0 = 0.5 for both rotation and vibration. Can n_v/n_0 ever be greater than I for any value of v? Why or why not.

Explanation / Answer

a)0.9518417368645089

b)0.4900512789456213

c) 21.361913681937647 K

d)Yes, nv /n0 can be greater than 1 for any value of v.

The intensities of lines in a spectrum depend upon a linestrength factor related to the overlap between the wavefunctions of the two states, the light intensity, and the population of the lower state.  Within a given rotational or vibrational spectrum, the lower state populations are usually the dominant factor.  Under thermal conditions, these are given by the Boltzmann distribution.

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