Statistical Thermodynamics Byeong-Joo Lee POSTECH - MSE calphad@postech.ac.kr

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

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

Particle in a Box – Microstates of a Particle for 66 : 8,1,1 7,4,1, 5,5,4

System with particles – Microstates of a System

Macrostate / Energy Levels / Microstates –

Scope and Fundamental Assumptions of Statistical Mechanics ▷(n1, n2, …, nk)로 정의되는 하나의 macrostate를 만들기 위해, 있을 수 있는 수많은 경우의 수 하나하나를 microstate라 한다. ▷ 어떠한 시스템에 가능한 (quantum mechanically accessible 한) macrostate (하나하나가 (n1, n2, …, nk)로 정의되는)의 mental collection을 ensemble이라 한다. ▷ 같은 energy level에서 모든 microstate의 실현 확률은 동등하다. ▷ Ensemble average는 time average와 같다.

Number of ways of distribution : in k cells with gi and Ei ▷ Distinguishable without Pauli exclusion principle ▷ Indistinguishable without Pauli exclusion principle for gi with ni ▷ Indistinguishable with Pauli exclusion principle for gi with ni

Evaluation of the Most Probable Macrostate – Boltzman

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

Definition of Entropy and Significance of β ▷ Thermal contact 상태에 있는 두 부분으로 이루어진 Isolated System을 고려. 이에 대한 평형 조건은 Classical Thermodynamics에서는 maximum entropy (S) Statistical mechanics에서는 maximum probability (Ω) ▷ S와 Ω는 monotonic relation을 가지며 →

Calculation of Macroscopic Properties from the Partition Function

Ideal Mono-Atomic Gas

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

Entropy – S = k ln W

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

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

Einstein and 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

Einstein and Debye Model for Heat Capacity – number density Let dNv 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

Einstein and Debye Model for Heat Capacity – Einstein All the 3N equivalent harmonic oscillators have the same frequency vE Defining Einstein characteristic temperature

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

Einstein and Debye Model for Heat Capacity – Comparison

Einstein and Debye Model for Heat Capacity – More about Debye Behavior of at T → ∞ and T → 0 at T → ∞ → x2 at T → 0 : Debye’s T3 law

Einstein and Debye Model for Heat Capacity – More about Cp for T << TF

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.