Question

8. (10 pts) (a) (6 pts) Suppose a system consists of N noninteracting, indistin guishable, identical particles and that each particle has available to it only two quantum electronic states, whose energies are 0 and a. Find expressions for z, Z and U. Note: Because we don 't consider translational energy here, so the 1/N! should be om itted when writing Z in terms of z (b) (4 pts) For NO, the ground electronic level and the first exited electronic level are each doubly deg enerate. The separation between these two electronic labels is 0.0149 eV. There are no other low-lying electronic levels. For NO, calculate za at 150 K. Note: leV 16022 * 101 J 9, (12 pts) (a) (3 pts) Show that if the hamonic-oscillaor approximation is used, the fractional population of the diatomic-molecule vibrational level v is (b) (2 pts) show that z, N/

Answer 1

of the too available Since, each panticle quantom electronic states eoith enngies , 0 & e, a can be any 8. one syshm The pontihion funcion for the single pantiele Can be toritten as 2quilibriom Here, Tis the e temperatare of the system e- Boltzmann's Cowat. 1 + T + e

N-panticle system the panttion funchon can be writlen ohile ignoring the tranelational enong) as For ie, intarnl enngy given by for micro canoriaal The avenage enengy of the system Systems InZ U = 1 Ba (-a) e Ba Na e NT 13 Iepa 1+e Ba Na Na Ba + 1 + N

For No, sine the groond state electronic level & frst excited dectronic level are both deobly degennate, the electronic penhtior Can be writen an 20 ground chute enng y Here, O ev & Ist exated stale enngy -19 0.0149 eV : 0.D149x1-6022 x 1o = -21 2.387 xID & Since, the cystem is at T 150k -23 x 150 21 2.07 x 10 J kgT 1.38 x 10

Hence, electronie panhhion funchon betomes after putling the velues of &, e, ka, T -2) 2.383 x10 .OX10 21 2 2, e R.387 2 2 e 2 + 0.631 2 631 is Hence, for NO Z at iso K 2 631 (