Question

Thermodynamics

5. A system has three energy eigenstates (microstates), with energies 0, E1, and E2 » Ei. It is sitting in a heat bath (reservoir) with temperature T. a. Find the partition function Z(T). b. Find simple approximate expressions for Z when t > E2, E2 »T» Ei, and T < E1. For the high- and medium-temperature regimes, your expressions should be zeroth-order, i.e., should not contain t, but for the low-temperature regime you should include the leading T-dependence. c. Use your approximate expressions to find simple approximate expressions for the entropy S as well. Do any of your expressions for Zor Shave intuitive interpretations?

Answer 1

We will use canonical ensemble formulas. because the syslem is in a heat bath with temp tatt Partition function, z(1) is given by Z(T) - Egue Blu where gh is degeneracy kt I Eu = energies of system for 1-0, 2- :. Ми од Е. = 0, 6 , за да enegin > z= e tette te Z = 1 + e- & vit te te te (b) Simple o when approximate expensione fou z I » >>E, KT 21k as it is required, I should be independent 2 of tent high temp regene I we can neglect a fo term Z = 1 + 1 + 1 - 12=3

1 when 4 << T ELĘ - it e ke & et Let ECT - E - 0 & >> E 0 so that we get I independent of t. (he why we are mading such approximation) 6 - za 1 teo teo - 1 + 1 to T DUDUUUUUUU (11) large when t « Ę L Ę ty very large , et ne e et le 1 e >> 1 z= Ite that te the z = &- Elke & e/k (neglect 1) Kr> 1 e kt » 1 > ЭТ we know , S = - DA & A= - KT enz AKT р, африо хі та охрли ІВт Кочо оле. gwens by ☺ I >> E >> E

A = - KT en z A c - KT lng ( 2=3) sol TJEJE & S=DA = + Kln 32T a Is- tk ln 3 6 <TKE Z= 28 for & «TKE S--DA = +2 (en 2), KT (AKT lnz) S = t klar] 1 << <E__ 2 & tek ЭТ rena -KT enZ = - KT. E - E I te Kt & » ET - 00 O - 448x + A- E S =- at = ISO we can say that at low demit, T0 - Isol temperature