## Problem 15.10: Determine properties of rotating molecules from rotational spectrum

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The spectrum above is an approximation to the *P* and *R* branches of the rotational-vibrational absorption spectrum of CO. Restart. This is a spectrum for the case of a transition from the vibrational ground state to the first excited vibrational state. Each vibrational energy level, however, contains a number of rotational energy levels because the energy between rotational energy levels is in the 10^{−4} eV range while the energy difference between vibrational levels is 10^{−1} eV. So, as the molecule goes from ω_{0} to ω_{1} (ground to first excited vibrational states), there is also a transition between rotational states (governed by the selection rule Δ*J* = +/−1 where *J* is the angular momentum quantum number for the molecule. The *P* branch are transitions where Δ*J* = −1 and the R branch, Δ*J* = +1. The intensity depends on the initial population of the states. The rotational energy of the molecule is given classically by *E* = *L*^{2}/2*I* (kinetic energy of rotation where I is the moment of inertia of the molecule and L is the classical angular momentum). The rotational levels are quantized and we will use *J* as the angular momentum quantum number so that the energy of rotation is given by ε_{J }= *J*(*J *+ 1)*ħ*^{2}/2*I*.

- Why is the density of states,
*g*(ε) given by*g*(ε) = 2*J*+ 1? - Each of the molecules is considered indistinguishable, so we can use Maxwell-Boltzmann statistics:
*n*_{i }=*g*_{i}/exp(α + β*ε*_{i}). Let e^{−α }=*n*_{0}. Write an expression for*n*_{J}, the number of particles with a given value of*J*. - This distribution has a maximum at a value of
*J*that depends on the moment of inertia*I*and temperature*T*. Find an expression for the value of*J*as a function of*I*and*T*that corresponds to the maximum value of*n*_{J}. From the spectrum, what value of*J*corresponds to the most intense peak (and thus had the highest number of particles in it prior to absorption of energy). - Determine, finally, the value of the moment of inertia of CO and compare it with the value you get simply by noting the energy difference between the different peaks.

Problem based on Example 9.2, Beiser, *Modern Physics* 6th ed., McGraw Hill, 2003, p. 300.

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