Rotational Partition Function
And
Thermal Populations
An FTIR spectrum recovers information about the energy between rotational and
vibrational states. As shown in the following illustration, IR radiation may produce a
transition from one vibration-rotation state to another if and only if Δv = ± 1 and ΔJ = ±
1.
Knowing the energy of these transitions one should be able to calculate the relative
energies of individual rotational states.
Consider, then, the FTIR spectrum of HCl, shown below.
1) Determine the energies of the first nine rotational states of HCl relative to the ground
state energy.
2) Using these energies, approximate the rotational partition function using the formula,
where i runs over rotational quantum numbers and n takes on values from 0 to 8. Thus
you will find nine functions, each is a finite sum approximation of the infinite sum. For
example, when n = 1 you would include only the first rotational energy in the expansion,
n=2 implies you would sum over the first two rotational energy levels, etc. Recall that
each rotational level is (2J+1) degenerate. In other words, for J = 1 there are three
rotational energy levels with this energy. Now compare your sums—do they appear to be
converging?
3) Using what you think is your best approximation to the partition function, calculate the
probability of finding a molecule in each of the first nine rotational states at 300 K; at
1000 K. Use the formula
4) Use some other reference state to form your partition function and recalculate the
probabilities you found in problem (3). What do you find and why?
5) Use the probabilities you calculated to explain the relative amplitudes of the transitions
seen in the FTIR spectrum of HCl.