R T / Θvib - Chem.hope.edu

εtrans , r=2h8ma2R2Let’s try to think about degeneracy now. Picture our sphere... all the combinations ofx,y,z values that reside on the surface of the sphere give the same energy. Let’s sketchthis out in two dimensions. (sketch 1 quadrant with dots at integer x,y. then draw anannulus (e.g. area of r + dr)).So, let’s call M the number of integers contained in our sphere.1 ⎛ 4π3 ⎞M = ⎜ R ⎟8 ⎝ 3 ⎠The number of integers in our little shell from r to r + dr (or ε to ε + dε) is:dMW ( ε ) dε= M ( ε + dε) − M ( ε ) = dεdεIf we take the energy equation we had earlier and solve for R we can substitute into the Mequation.2⎛ 8maε ⎞R =⎜2⎟⎝ h ⎠12Mπ ⎛ 8mε⎞= ⎜2⎟6 ⎝ h ⎠32V(if a 3 = V)322Wπ ⎛ 8m⎞⎜2⎟4 ⎝ h ⎠1( ε ) dε= VεdεW is the density of states.Let’s go one step further and take a look at the probability of occupying a certain state−εiβepi= if we rewrite this in terms of levels rather than statesqp( ε )W=( ε )eq−εβW increases rapidly with energy, and the Boltzmann factor decreases rapidly with energy.The result is Figure 12.5 in your book.4