Aim of the Experiment: Study of the rotational probability distribution for several hetero-nuclear diatomic gases at two different temperatures, comparison of  their rotational partition function by direct calculation and by the simplest formula q=T/q_rot

Theory: The probability distribution of heteronuclear diatomic rotational levels is an interesting simple case in the study of molecular-level occupation patterns, as here the number of significantly occupied levels is limited, the energy-level degeneracies vary with levels so as to cause maximum occupation at a non-ground level, the degeneracies and energies have manageable analytical expressions and  the partition function series converge manageably fast. The relations to be used here are (rotational level quantum number J = 0,1,2,3,....;  q_rot = characteristic rotational temperature; T = Absolute temperature; k = Boltzmann constant)

Degeneracy, g (g_j, function of J) = 2 J + 1

Energy, E (E_J, function of J) = J (J+ 1) k q_rot

Partition function, q = g(0)exp[–E(0) / (kT)] + g (1) exp [–E(1) / (kT)] +  ............................

Probability, p (function of J) = [ g (J) exp {–E (J)/ (kT)}] / q

The probability of the levels first increases and then decreases with the increase in J value as shown in the figure below:

Procedure: Using the RotaProb  program available as a part of the Computers in Chemistry Experiments Set from www.oocities.org/riturajkalita/compu_chemi.htm, the probability of occupations of the various rotational energy levels are obtained for two heteronuclear diatomic molecules (HCl and CO) at two different system temperatures ( 25⁰C and 100⁰C). In the program screen, the characteristic rotational temperature (q_rot) of the diatomic molecules and the system temperatures (T) are to be entered as directed, and then on pressing the Continue button the computational investigation proceeds and the results will be obtained in the file 'RotaProb.txt'.

Results: The results for the two heteronuclear molecules at two temperatures are attached herewith as the photocopies of print-outs.

    The probability along with the rotational level number (J value) is tabulated as shown below:

Conclusion: It is seen that the maximum probability occurs at an intermediate excited level (not the ground level), as is expected. For a given molecule, the most probable level has a tendency to shift towards higher level as temperature increases  This is because of the fact that as temperature increases the higher levels get more populated.