Equipartition of EnergyRole of Degeneracy in the Boltzmann DistributionBoltzmann DistributionEquipartition of EnergyAt a given temperature, the energy of a collection of molecule is distribution among the various modes of motion (that is, translational, rotational, vibrational, electronic, …). All distributions are equally likely; however, the particles in the distributions are indistinguishable. Therefore, some distributions are more probable than others. This principle is known as the equipartition of energy.Illustration 1: Consider the following illustration putting two balls into three different boxes.If each ball is equally likely to be in any box, what is the most likely arrangement if we examine the box after shaking it?a) Two balls in one box.b) Two balls in separate boxes.Each ball is equally likely to be in any box so let’s look at all the possibilities.By looking at all the possibilities, we see that finding the two balls in separate boxes is twice as likely finding two ball in one box.Illustration 2: Consider the following illustration putting three balls into three different boxes.If each ball is equally likely to be in any box, what is the most likely arrangement if we examine the box after shaking it?a) Three balls in one box.b) Two balls in one box and one ball is a separate box.c) Three balls in separate boxes.1ProbabilityThree balls in one box. 3/27 = 11%Two balls in one box and one ball is a separate box. 18/27 = 67%Three balls in separate boxes. 6/27 = 22%Boltzmann DistributionAs the numbers get larger, that is 1023, the most probable distribution for a given amount of macroscopic energy among microscopic energy states is given by the Boltzmann distribution.ijEkTiEkTjn eNeNote: R = kNALet us examine how the energy of 106 HCl molecules is distributed among the different vibrational states at T = 298 K.      34 10 1e201 1E hc 6.626 10 J s 2.997 10 cm / s 2991cm2 215.617 10 J2                         %  23 21kT 1.381 10 J / K 298K 4.118 10 J     EE/kT exp(-E/kT)0 2.802  10-20 J 6.82 1.09  10-31 8.425  10-20 J 20.46 1.31  10-92 1.404  10-19 J 34.10 1.55  10-153 1.966  10-19 J 47.74 1.85  10-214 2.528  10-19 J 61.38 2.20  10-276.82 36 60 006 6.82 20.46 34.10 47.74 61.38 3n ne 1.09 101 n 10 1 10N 10 e e e e e 1.09 10                 20.46 961 16 6.82 20.46 34.10 47.74 61.38 36 61n n e 1.31 101.21 10N 10 e e e e e 1.09 10n 10 1.21 10 1                  The vibrational energy is large compared to the thermal energy; therefore, approximately 999,999 molecules out of a million are in the ground vibrational state at 298 K and 1 molecule is in the first excited state.2ni – number of molecules in the ith energy state.N – total number of molecules.Ei – energy of the ith energy state.k – Boltzmann’s constant: 1.38  10-23 J/KLet us consider what happens to the energy distribution at a higher temperature such as 2000 K?  23 20kT 1.381 10 J / K 2000K 2.762 10 J    EkT0 1.02 8689981 3.04 1138432 5.08 149113 7.11 19504 9.15 2605 11.17 346 13.21 4Role of Degeneracy in the Boltzmann DistributionThe Boltzmann distribution includes all degenerate states as equally probable as well. Therefore the more precise formulation of the Boltzmann distribution isijEkTi iEkTjjn g eNg eDegeneracy is important when partitioning the rotational energy of sample of molecules. Remember that each rotational state J can have an M value that ranges from J, …, -J. Thus each rotational state J has 2J +1 M values; thus the degeneracy for rotational energy levels is 2J +1.3EkT6EkTen 10egi – degeneracy of the ith energy state.Example: What is the distribution of rotational energy in a sample of HCl at 298 K? The rotational constant of HCl is 10.44 cm-1Recall that J g E(2J + 1) exp(EJ/kT)cumulativesum% ofpopulation0 1 0 J 1 1 4.941 3 4.146  10-22 J 2.713 3.713 13.412 5 1.244  10-21 J 3.696 7.409 18.273 7 2.488  10-21 J 3.826 11.235 18.914 9 4.146  10-21 J 3.288 14.523 16.255 11 6.219  10-21 J 2.430 16.953 12.016 13 8.707  10-21 J 1.569 18.522 7.767 15 1.161  10-20 J 0.895 19.417 4.428 17 1.493  10-20 J 0.493 19.910 2.449 19 1.866  10-20 J 0.205 20.115 1.0110 21 2.283  10-20 J 0.083 20.198 0.4811 23 2.736  10-20 J 0.030 20.228 0.15Most of the molecules are in the J = 3 state, with the J = 2 and J = 4 states also well populated.4         34 10 1J34E hcBJ J 1 6.626 10 J s 2.997 10 cm / s 10.44cm J J 12.073 10 J J J 1         The fact that most of our molecules are not in the rotational ground state accounts for the shape of the rotovibrational spectrum. The intensities of the lines are not the highest for the transitions involving the J = 0 state. Below is the acetylene rotovibrational spectrum. The highest intensities involve the J = 10 states. Therefore we surmise that the J = 10 states are the most populated. (The most populated states will yield the most transitions; therefore, the highest intensities.)(Taken from L. Willard Richards, J. of Chemical Education, Vol 43, pp. 644-647