Question

(b) For a system of N independent harmonic oscillators at temperature T, all having a common vibrational unit of energy, the partition function is Z = ZN. For large values of N, the system's internal energy is given by U = Ne %3D eBe For large N, calculate the system's heat capacity C.

Answer 1

The answers are given in the images below. Initially I have just defined a partition function. From there I have written the partition function for a single harmonic oscillator. In the second part, I have used a simple equation for finding the internal energy. I have also put in brackets the equation from where this simple equation is derived from.

the Answer! The partition function for a single particle is given by; D - Ei/KT zu e where Eo is the possible energy levels of particle. a) For a single harmonie oscillator & 4- +%) two. - 1 (1 kw) - (3½ 2w) Z= e te + het B2 1 / 4 -Bhco - 3 Btw 5Btw Z = ete 2 te - Taking e Phot common; -Bho e lite + -Bho 2Bhuo 2 z = e t ... h

Given that no 1-v. In equation () if x="ßh w then terms in brackett becomes Citex + ( 24 .... ) = 2x -ßt w - e equation () becomes I – Bhow/ Z = e 1- - ptico e -BE/ OR za e (where ez hw) as given in question. -B& - eße

6. For A midependent harmonie osillatoxs z = 2 YN Bel BE 11-ė BE The internal energy is the total energy of the system. It is given by the equation Ur KT² & Inz FOR U= 2 i et kta InZ L 2T IN – BE/ cé Uz kt? 2 lol é at I The Be -Be/ KT? 2 N In [e liepos NRT 21, 10(-e43) BE - NKT e-Be - ln (1-e

= NKT NKT* [L*** )- .- L -BE -E k and [ Aboue we have substituted B taken derivative cort T] - BE Dividing second term in brackett throughout with e BE UNE + che, Speutic - The heat capacity a is defined as the Change in internal energy of a system as temperature is raised (changed), « Cv = (30)

- Cv = 2 (NEC) + / BE, )) T = NE 2 ( + ebe) 2 NE. [ 0 + - Bg 2 NE 2 COBE = NE? BE KT² Ceße , ²