1. 2002.11.08 N96770 微奈米統計力學 1 上課地點 : 國立成功大學工程科學系越生講堂 (41X01 教室 ) N96770 微奈米統計力學

2. 2002.11.08 N96770 微奈米統計力學 2  Entropy of Ideal Monatomic Gases  Heat Capacity of Monatomic Solids References: D. A. McQuarrie, Statistical Mechanics, Harper & Row Publishers, Inc., 1976. C. L. Tien & J. H. Lienhard, Statistical Thermodynamics, Hemisphere Publishing Corp., 1979. OUTLINES

3. 2002.11.08 N96770 微奈米統計力學 3 Problem 1 How to calculate the entropy (S) of 1 mole of argon (Ar) at 298K and 1 atm? We need to derive the entropy expression in terms of the number of particles (N), volume (V), and temperature (T) for ideal monatomic gas. Ideal Monatomic Gas a gas of single-atom molecules dilute enough that intermolecular interactions can be neglected. Ideal Monatomic Gas

4. 2002.11.08 N96770 微奈米統計力學 4 The Hamiltonian H of a monatomic gas can be divided into translational and internal. Partition Function ( 配分函數 ) a function expressing the partition or distribution of energies over the various energy levels of a system. Ideal Monatomic Gas The partition function Z depends upon the types of energy storage and can be written as translational and internal.

5. 2002.11.08 N96770 微奈米統計力學 5 Ideal Monatomic Gas Translational Partition Function It can be shown Start from solving the wave equation of a particle m translating freely in a 1-D motion between the interval 0 and L. Expand to 3-D and use Maxwell-Boltzmann Distribution, then the translational partition function can be obtained. Maxwell-Boltzmann Distribution where partition function

6. 2002.11.08 N96770 微奈米統計力學 6 Internal Partition Function The only internal mode of energy storage for a single- atom molecule is electronic. Ideal Monatomic Gas g i : degeneracy Degeneracy a number of eigenfunctions or states of the system having the same eigenvalue or energy. It can be shown that the ground energy level  0 is 0 and its degeneracy g 0 is 1, and higher-order terms vanish at ordinary temperatures.

7. 2002.11.08 N96770 微奈米統計力學 7 The partition function of the entire system can be written in terms of the individual atomic partition function. Combine translational and internal partition functions : Ideal Monatomic Gas Recall the Helmhotz free energy A : Using Stirling’s approximation:

8. 2002.11.08 N96770 微奈米統計力學 8 Entropy : Ideal Monatomic Gas where

9. 2002.11.08 N96770 微奈米統計力學 9 Ideal Monatomic Gas Ideal gas : Sackur-Tetrode Equation At practical temperatures,

10. 2002.11.08 N96770 微奈米統計力學 10 How to calculate the entropy (S) of 1 mole of argon (Ar) at 298K and 1 atm? Ideal Monatomic Gas (experimental = 154.72)

11. 2002.11.08 N96770 微奈米統計力學 11 Monatomic Solids Problem 2 How to determine the heat capacity (C v ) of crystalline solids, such as copper? — Need to determine the partition function for monatomic solids. — Start from lattice dynamics. — Einstein theory and Debye theory.

12. 2002.11.08 N96770 微奈米統計力學 12 Monatomic Solids A crystalline solid can be represented by a system of regularly spaced springs and masses, in which atoms vibrate with small amplitude about their equilibrium positions. Minimum total potential energy U for N atoms : U(0;  ) : total potential energy for atoms at rest  = V/N : lattice density  j : displacement at the j-th atom k ij : force constant between the i-th & j-th atoms

13. 2002.11.08 N96770 微奈米統計力學 13 Monatomic Solids The complete partition function is related to lattice vibration only. It can be shown that the vibrational partition function for the j-th atom is given by : v j : vibrational frequency of the j-th oscillator total partition function

14. 2002.11.08 N96770 微奈米統計力學 14 Monatomic Solids where Helmhotz free energy entropy heat capacity if g( ) is known Introducing a function g(v) that gives the number of frequencies between v and v+dv :

15. 2002.11.08 N96770 微奈米統計力學 15 Monatomic Solids Einstein Theory of Solids Assuming there are 3N independent quantum oscillators. Each oscillator having the same vibrating frequency. Applying the Planck theory of quantized oscillators. v E : vibrational frequency assigned to all 3N oscillators (Einstein frequency)

16. 2002.11.08 N96770 微奈米統計力學 16 Monatomic Solids Define Einstein temperature ★ Drawback on the Einstein model : Heat capacity approaches 0 too quickly as T  0

17. 2002.11.08 N96770 微奈米統計力學 17 Monatomic Solids Debye Theory of Solids (to improve Einstein model) Lattice vibrating at low frequencies at low temperatures. Corresponding wavelengths much longer than atomic spacing. Assuming crystal as a continuous elastic body. Applying the concept of phonons. where v : velocity of the elastic wave : wavelength of the elastic wave Treating lattice vibration as elastic vibration.

18. 2002.11.08 N96770 微奈米統計力學 18 Monatomic Solids An elastic wave can be de-coupled into 2 transverse and 1 longitudinal waves : Define average velocity as v trans : velocity of the transverse wave v long : velocity of the longitudinal wave

19. 2002.11.08 N96770 微奈米統計力學 19 Monatomic Solids Define maximum frequency D such that Debye frequency where

20. 2002.11.08 N96770 微奈米統計力學 20 Monatomic Solids T 3 law : meaning at low temperatures C v behaves as T 3 ★ Improvement on the Debye model : Heat capacity is in good agreement with experiment for solids at low temperatures.