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School of Aerospace Engineering Boltzmann Limit • • When can we assume that the Boltzmann limit is “accurate” (i.e., gi >> Ni or many more available quantum states then particles)? Number of particles P.G. @ SATP ~ 1019 molecules/cm3 (from pV=NkT ) – • Degeneracies of different energy modes (motions) 1) electronic: magnetic quantum number (g=2l+1) and spin (g=2s+1) – gel < O(10) 2) vibration: harmonic oscillator has no degeneracy – gvib = 1 3) rotation: rigid rotor (2J+1) – grot < O(100) • So how can gi >> Ni ? – translational energy? AE/ME 6765 Boltzmann Limit-1 Copyright © 2009 by Jerry M. Seitzman. All rights reserved. School of Aerospace Engineering Translational Energy QM States • Consider particle in a “box” (cube) – recall Q.M. solution of Sch. eq’n. for (kinetic) energy of single translating particle h2 1 L ε x ,nx = nx2 or nx = 8mε x ,nx 2 h 8m L L =1 cm • At SATP for air in a “small” box (1cm), how many QM states have energy less than the average KE of a molecule in x-direction (=1/2 k T… will show this later) ⇒ nx , KEavg ,x ≈ 2 mkT nx ,avgKE ≈ 4.2 ×108 L h J Boltzmann's Constant K h = 6.626 × 10 −34 J s TSATP = 298K , mN 2 ≅ 4.7 ×10 − 26 kg k = 1.38 ×10 − 23 • But this was just 1-D of cube # States with KE < KEavg = nx , KEavg n y , KEavg nz , KEavg = nx3, KEavg ≈ 7 × 10 25 Boltzmann Limit-2 Copyright © 2009 by Jerry M. Seitzman. All rights reserved. AE/ME 6765 1 School of Aerospace Engineering Boltzmann Limit Validity • So in our 1 cm box, if all our molecules were to be found in just these states # States " accessible" ≈ 10 26 1019/cm3 – actual number must be greater since some molecules must be in even higher energy states for the average to be correct • Number of particles in 1 cm box is L =1 cm ~1019 # QM States that could be populated 10 26 > 19 = 107 # Particles 10 ⇒each state is sparsely populated ; Boltzmann limit usually reasonable assumption unless densities are very high (or T’s very low) • For translations, we could “lump” energy states having energies between εi and Δεi into one energy level (large box) with approximately same εi and degeneracy gi Extended Degeneracies Boltzmann Limit-3 Copyright © 2009 by Jerry M. Seitzman. All rights reserved. AE/ME 6765 2