Consider a gas of diatomic molecules at temperature T, each with moment of inertia I. If...

Consider a gas of diatomic molecules at temperature T, each with moment of inertia I. If Eg is the ground-state energy and Eex is the energy of an excited state, then the Maxwell-Boltzmann distribution predicts that the ratio of the number of molecules in the two states is nex ng = e −(Eex−Eg)/kBT . (1) a) Suppose we consider the excited state to be the ℓth rotational energy level, and the ground state to be ℓ = 0. Show that the ratio in equation (1) is equal to nℓ n0 = (2ℓ + 1)e −[ℓ(ℓ+1)¯h 2 ]/(2IkBT) . (Hint: the ℓth level contains multiple different states with different values of mℓ, all with the same energy. How many different values of mℓ are there for a given value of ℓ?) b) The moment of inertia of the carbon monoxide (CO) molecule is 1.449×10−46 kg·m2 . Use this to determine the numerical value of the ratio nℓ/n0 for a gas of CO molecules at a temperature of 300 K, for the following cases: (i) ℓ = 1 (ii) ℓ = 2 (iii) ℓ = 10 (iv) ℓ = 20 (v) ℓ = 50 c) Comment on your results from part (b). What do you see happening to the ratio as ℓ increases? Why does this happen? What does it tell us about the physical system?

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genius_generous answered 1 year ago

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