1Boltzmann Distribution Equipartition of Energy At 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.2 Probability Three 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 Distribution As 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. ijEkTiEkTjneNe−−=∑ Note: R = kNA Let us examine how the energy of 106 HCl molecules is distributed among the different vibrational states at T = 298 K. ( )( )( )( )34101e2011Ehc6.62610Js2.99710cm/s2991cm2215.61710J2−−ν−=νν+=×⋅×ν+=×ν+% ()()2321kT1.38110J/K298K4.11810J−−=×=× ν Eν Eν/kT exp(-Eν/kT) 0 2.802 × 10-20 J 6.82 1.09 × 10-3 1 8.425 × 10-20 J 20.46 1.31 × 10-9 2 1.404 × 10-19 J 34.10 1.55 × 10-15 3 1.966 × 10-19 J 47.74 1.85 × 10-21 4 2.528 × 10-19 J 61.38 2.20 × 10-27 6.8236600066.8220.4634.1047.7461.383nn e1.09101n10110N10eeeee1.0910−−−−−−−−×====⇒=⋅=++++× 20.46961166.8220.4634.1047.7461.383661nne1.31101.2110N10eeeee1.0910n101.21101−−−−−−−−−−×====×++++×⇒=⋅×= 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. ni – 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/K3Let us consider what happens to the energy distribution at a higher temperature such as 2000 K? ()()2320kT1.38110J/K2000K2.76210J−−=×=× ν EkTν 0 1.02 868998 1 3.04 113843 2 5.08 14911 3 7.11 1950 4 9.15 260 5 11.17 34 6 13.21 4 Role of Degeneracy in the Boltzmann Distribution The Boltzmann distribution includes all degenerate states as equally probable as well. Therefore the more precise formulation of the Boltzmann distribution is ijEkTiiEkTjjngeNge−−=∑ 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. EkT6EkTen10eνν−ν−ν=∑gi – degeneracy of the ith energy state.4Example: What is the distribution of rotational energy in a sample of HCl at 298 K? The rotational constant of HCl is 10.44 cm-1 Recall that J g E (2J + 1) × exp(EJ/kT) cumulative sum % of population 0 1 0 J 1 1 4.94 1 3 4.146 × 10-22 J 2.713 3.713 13.41 2 5 1.244 × 10-21 J 3.696 7.409 18.27 3 7 2.488 × 10-21 J 3.826 11.235 18.91 4 9 4.146 × 10-21 J 3.288 14.523 16.25 5 11 6.219 × 10-21 J 2.430 16.953 12.01 6 13 8.707 × 10-21 J 1.569 18.522 7.76 7 15 1.161 × 10-20 J 0.895 19.417 4.42 8 17 1.493 × 10-20 J 0.493 19.910 2.44 9 19 1.866 × 10-20 J 0.205 20.115 1.01 10 21 2.283 × 10-20 J 0.083 20.198 0.48 11 23 2.736 × 10-20 J 0.030 20.228 0.15 Most of the molecules are in the J = 3 state, with the J = 2 and J = 4 states also well populated. ()()()()()( )( )34101J22EhcBJJ16.62610Js2.99710cm/s10.44cmJJ12.07310JJJ1−−−=+=×⋅×+=×+5The 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