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Byeong-Joo Lee www.postech.ac.kr/~calphad 이 병 주 포항공과대학교 신소재공학과 calphad@postech.ac.kr

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Byeong-Joo Lee www.postech.ac.kr/~calphad Warming Up – Mathematical Skills 1. Stirling’s approximation 2. Evaluation of the Integral 3. Lagrangian Undetermined Multiplier Method

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Byeong-Joo Lee www.postech.ac.kr/~calphad Basic Concept of Statistical Mechanics – Macro vs. Micro View Point Macroscopic vs. Microscopic State Macrostate vs. Microstate

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Byeong-Joo Lee www.postech.ac.kr/~calphad Particle in a Box – Microstates of a Particle n = 1, 2, 3, … for 66 : 8,1,1 7,4,1, 5,5,4

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Byeong-Joo Lee www.postech.ac.kr/~calphad System with particles – Microstates of a System

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Byeong-Joo Lee www.postech.ac.kr/~calphad Macrostate / Energy Levels / Microstates –

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Byeong-Joo Lee www.postech.ac.kr/~calphad Scope and Fundamental Assumptions of Statistical Mechanics ▷ (n 1, n 2, …, n k ) 로 정의되는 하나의 macrostate 를 만들기 위해, 있을 수 있는 수많은 경우의 수 하나하나를 microstate 라 한다. ▷ 어떠한 시스템에 가능한 (quantum mechanically accessible 한 ) macrostate ( 하나하나가 (n 1, n 2, …, n k ) 로 정의되는 ) 의 mental collection 을 ensemble 이라 한다. ▷ 같은 energy level 에서 모든 microstate 의 실현 확률은 동등하다. ▷ Ensemble average 는 time average 와 같다.

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Byeong-Joo Lee www.postech.ac.kr/~calphad 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

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Byeong-Joo Lee www.postech.ac.kr/~calphad Evaluation of the Most Probable Macrostate – Boltzman

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Byeong-Joo Lee www.postech.ac.kr/~calphad Evaluation of the Most Probable Macrostate – B-E & F-D → Bose-Einstein Distribution Fermi-Dirac Distribution

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Byeong-Joo Lee www.postech.ac.kr/~calphad Definition of Entropy and Significance of β → ▷ Thermal contact 상태에 있는 두 부분으로 이루어진 Isolated System 을 고려. 이에 대한 평형 조건은 Classical Thermodynamics 에서는 maximum entropy (S) Statistical mechanics 에서는 maximum probability (Ω) ▷ S 와 Ω 는 monotonic relation 을 가지며

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Byeong-Joo Lee www.postech.ac.kr/~calphad Calculation of Macroscopic Properties from the Partition Function

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Byeong-Joo Lee www.postech.ac.kr/~calphad Ideal Mono-Atomic Gas

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Byeong-Joo Lee www.postech.ac.kr/~calphad Ideal Mono-Atomic Gas – Evaluation of k for 1 mol of gas

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Byeong-Joo Lee www.postech.ac.kr/~calphad Entropy – S = k ln W

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Byeong-Joo Lee www.postech.ac.kr/~calphad 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

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Byeong-Joo Lee www.postech.ac.kr/~calphad 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

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Byeong-Joo Lee www.postech.ac.kr/~calphad 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

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Byeong-Joo Lee www.postech.ac.kr/~calphad Einstein and 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

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Byeong-Joo Lee www.postech.ac.kr/~calphad Einstein and Debye Model for Heat Capacity – Einstein All the 3N equivalent harmonic oscillators have the same frequency v E Defining Einstein characteristic temperature

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Byeong-Joo Lee www.postech.ac.kr/~calphad Einstein and Debye Model for Heat Capacity – Debye A crystal is a continuous medium supporting standing longitudinal and transverse waves set

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Byeong-Joo Lee www.postech.ac.kr/~calphad Einstein and Debye Model for Heat Capacity – Comparison

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Byeong-Joo Lee www.postech.ac.kr/~calphad Einstein and 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

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Byeong-Joo Lee www.postech.ac.kr/~calphad Einstein and Debye Model for Heat Capacity – More about Cp for T << T F

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Byeong-Joo Lee www.postech.ac.kr/~calphad 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.

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