Byeong-Joo Lee Byeong-Joo Lee POSTECH - MSE Statistical Thermodynamic s

Byeong-Joo Lee Warming Up – Mathematical Skills 1. Stirling’s approximation 2. Evaluation of the Integral 3. Lagrangian Undetermined Multiplier Method

Byeong-Joo Lee Basic Concept of Statistical Mechanics – Macro vs. Micro View Point Macroscopic vs. Microscopic State Macrostate vs. Microstate

Byeong-Joo Lee Particle in a Box – Microstates of a Particle n = 1, 2, 3, … for 66 : 8,1,1 7,4,1, 5,5,4

Byeong-Joo Lee System with particles – Microstates of a System

Byeong-Joo Lee Macrostate / Energy Levels / Microstates

Byeong-Joo Lee Scope and Fundamental Assumptions of Statistical Mechanics ▷ Microstate: each of the possible states for a macrostate. (n 1, n 2, …, n k ) 로 정의되는 하나의 macrostate 를 만들기 위해, 있을 수 있는 수많은 경우의 수 하나하나를 microstate 라 한다. ▷ Ensemble: mental collection of macrostates 어떠한 시스템에 가능한 (quantum mechanically accessible 한 ) macrostate ( 하나하나가 (n 1, n 2, …, n k ) 로 정의되는 ) 의 mental collection 을 ensemble 이라 한다. ▷ Each microstate is equally probability. 같은 energy level 에서 모든 microstate 의 실현 확률은 동등하다. ▷ Ensemble average = time average

Byeong-Joo Lee Number of ways of distribution : in k cells with g i and E i ▷ Distinguishable without Pauli exclusion principle ▷ Indistinguishable without Pauli exclusion principle for g i with n i ▷ Indistinguishable with Pauli exclusion principle for g i with n i

Byeong-Joo Lee Evaluation of the Most Probable Macrostate – Boltzmann

Byeong-Joo Lee Evaluation of the Most Probable Macrostate – B-E & F-D → Bose-Einstein Distribution Fermi-Dirac Distribution

Byeong-Joo Lee Definition of Entropy and Significance of β → ▷ Consider an Isolated System composed of two part in Thermal equilibrium in Classical Thermodynamics: maximum entropy (S) in Statistical mechanics: maximum probability (Ω) ▷ There exists a monotonic relation between S and Ω

Byeong-Joo Lee Calculation of Macroscopic Properties from Partition Function

Byeong-Joo Lee Ideal Mono-Atomic Gas

Byeong-Joo Lee Ideal Mono-Atomic Gas – Evaluation of k for 1 mol of gas

Byeong-Joo Lee Entropy – S = k ln W

Byeong-Joo Lee Equipartition Theorem translational kinetic energy : rotational kinetic energy : vibrational energy : kinetic energy for each independent component of motion has a form of The average energy of a particle per independent component of motion is

Byeong-Joo Lee Equipartition Theorem The average energy of a particle per independent component of motion is ※ for a monoatomic ideal gas : for diatomic gases : for polyatomic molecules which are soft and vibrate easily with many frequencies, say, q: ※ for liquids and solids, the equipartition principle does not work

Byeong-Joo Lee Einstein & Debye Model for Heat Capacity – Background & Concept 3N independent (weakly interacting) but distinguishable simple harmonic oscillators. for N simple harmonic vibrators average energy per vibrator

Byeong-Joo Lee Einstein & Debye Model for Heat Capacity – number density Let dN v be the number of oscillators whose frequency lies between v and v + dv where g(v), the number of vibrators per unit frequency band, satisfy the condition The energy of N particles of the crystal

Byeong-Joo Lee Einstein & Debye Model for Heat Capacity – Einstein All the 3N equivalent harmonic oscillators have the same frequency v E Defining Einstein characteristic temperature

Byeong-Joo Lee Einstein & Debye Model for Heat Capacity – Debye A crystal is a continuous medium supporting standing longitudinal and transverse waves set

Byeong-Joo Lee Einstein & Debye Model for Heat Capacity – Comparison

Byeong-Joo Lee Einstein & Debye Model for Heat Capacity – More about Debye Behavior of at T → ∞ → x 2 : Debye’s T 3 law at T → ∞ and T → 0 at T → 0

Byeong-Joo Lee Einstein & Debye Model for Heat Capacity – More about Cp for T << T F

Byeong-Joo Lee Statistical Interpretation of Entropy – Numerical Example A rigid container is divided into two compartments of equal volume by a partition. One compartment contains 1 mole of ideal gas A at 1 atm, and the other compartment contains 1 mole of ideal gas B at 1 atm. (a) Calculate the entropy increase in the container if the partition between the two compartments is removed. (b) If the first compartment had contained 2 moles of ideal gas A, what would have been the entropy increase due to gas mixing when the partition was removed? (c) Calculate the corresponding entropy changes in each of the above two situations if both compartments had contained ideal gas A.