Question

2. Consider a closed system with three possible energy values, O, E, and 2€, under constant V and T condition. The third energy level with E = 2€, however, has a degeneracy of y: i.e. There are states that have the identical energy value of 2€. (a) Express the partition function in terms of %, E, and T. (b) Write the probability to sample each energy level (P1, P2, and P3) in terms of 7, €, and T. (c) Write an expression for the average energy of this system. (d) Write an expression for the heat capacity of this system. (e) If € = 2 kcal/mol and y = 1000, find the temperatue at which Pi = P3. Note that Boltzmann constant is, k= 0.002kcal/molK.

Answer 1

Answer to the problem w wo 9-11 Molecular partition funelion, f = ge Elkt + gea where go, I, and q, degeneracy of energy & level Eo, E, and E. 4Partition funetion in terms of V, , am T: f = n. eft + 1. ee/kt ve 2/ KT f = 1 + e Elverd. ZE/KT 6) Probability of ith state is given by - Pia ni gië / KT gi è lijkt N { gië Eilkt F f = gecolat + geelkt & ge scalks... f = & gie Eilkt wordt = 4 ; P = b

c) Average energy E = ? E; (E) E = (x1 + x (KT 25 24/kr) {e{kt + 2€ 0 25/KT Ē = & & & & KT + 20 28/kr) d) Heat Capacity co= ( 2 ) E=E= Average Energy e) { = 2 Kcal/mot, V=1000 if P = P₂ ,T=?, k=0.002 kcallmook : P = P3 + = ve 2E/KT or on or é 25/kt = 5 & 22/KT = 1000 inle 2E/K) = ln (%1000)

ov - 20 = -6.90 or TE LE 6.goxk T = 2x2 kcal moit 6.90 x 0.002 ka moi' k1 or T=289.85k T = 290k