Summary: Isolated Systems, Temperature, Free Energy

1. Summary: Isolated Systems, Temperature, Free Energy Zhiyan Wei ES 241: Advanced Elasticity 5/20/2009

2. Isolated Systems • Statistical description of systems • Internal variable of an isolated system • The second law of thermodynamics • Entropy

3. Statistical Description • Specification of the state of the system • Statistical ensemble • The fundamental postulate • Probability calculations

4. Statistical Description • Specification of the state of the system • Microscopic scale • Quantum description: a set of quantum numbers • Classical description: a phase point in the phase space • Macroscopic scale • A subset of quantum states of an isolated system is called a macrostate (conformation, thermodyanamic state, or configuration) • Described by macroscopic parameters

5. Statistical Description • Specification of the state of the system • System Any part of the world • Isolated system A system is said to be isolated if it does not interact with the rest of the world– thermally isolated, mechanically isolated….

6. Statistical Description • Statistical ensemble • A very large number of identical systems prepared under identical macroscopic conditions– same macroscopic state • Ergodic Theorem The average behavior of a system over sufficient amount of time is the same as the average behavior of many identically prepared sytems.

7. Statistical Description • The fundamental postulate ★ An isolated system isolated for an enough long time is equally likely to be found in any of its quantum states!

8. Statistical Description • Probability calculations • The macrostates has ΩA number of quantum states • Ω is the number of quantum states of an isolated system • Probability for the isolated system to be in macrostate A is

9. Statistical Description • Probability calculations– examples • Irreversible change in an isolated system– half glass of wine. Evaporation is spontaneous, but not all the gas molecules will go back to the liquid again, why? • Dispersion of a drop of ink in a glass of wine Ω ~VN V– volume of the glass of wine N– number of ink particles

10. Internal Variable of An Isolated System • A function that maps a quantum state of an isolated system to a number. That is, the domain of the function is the set of the quantum states of the isolated system, and the range of the function is a real number. • Example: half glass of wine!

11. Second Law of Thermodynamics • For a thoroughly isolated system that evolves from one macroscopic state to another, its entropy tend to increase!

12. Entropy • The logarithm of the number of quantum states • Composite of two isolated systems

13. Temperature • Thermal contact • Definition of absolute temperature • Experimental determination of temperature • Experimental determination of the number of quantum states • Heat capacity and latent heat

14. Thermal Contact • Only energy exchange between two systems is allowed • Heat transfer • Empirical observations about hotness: • Two system will reach thermal equilibrium in thermal contact after a long time • Zeroth law of thermodynamics • Levels of hotness are ordered • Levels of hotness are continuous

15. Definition of Absolute Temperature What is the most probable partition of energy? Energy A’ Ω’(U’) A’’ Ω’’(U’’) dU Isolated system

16. Definition of Absolute Temperature • Before energy exchange, the total number of quantum states: • After the energy of the composite is partitioned as U’+dU and U’’-dU, # of quantum states: • The #s of states differ by

17. Definition of Absolute Temperature • Define

18. Experimental Determination of Temperature • Calculate the temperature of a simple system by counting the number of states • Use the simple system to calibrate a thermometer by thermal contact • Use the thermometer to measure temperatures of any other system by thermal contact.

19. Experimental Determination of Temperature • Ideal gas

20. Experimental Determination of The Number of Quantum States • Determine the function Ω(U) of a system up to a multiplicative factor. To fix the multiplication factor, we set Ω=1 as T 0, which is the Third Law of Thermodynamics.

21. Heat Capacity and Latent Heat • Heat Capacity • Latent Heat

22. Free Energy • A system with variable energy • A system with variable energy and an internal variable • Free energy • Co-existent phases of a substance

23. A System with Variable Energy • Open a system: the system can vary its energy U by thermal contact with the rest of the world • When the energy U is fixed at a particular value, the system becomes isolated • Characterized by Ω(U), S(U) and T(U)

24. A System with Variable Energy • Leading characteristics of the curves • The horizontal position: no empirical significance • The vertical position: constricted by the 3rd Law of thermodyanics

25. A System with Variable Energy • The function S(U) is usually convex • Two identical systems, each with energy U • Each part can exchange energy. U-Q and U+Q

26. A System with Variable Energy and An Internal Variable • Entropy S(U,Y) • At a constant U, the most probable Y maximizes S(U,Y)

27. Free Energy • (U,Y) specifies a macrostate of the composite • The entropy of the macrostate of the composite is • The above maximization is equivalent to the minimization below

28. Free Energy • Temperature and entropy is one to one function • Helmholtz free energy • An alternative way to introduce the free energy The free energy of the system is the total energy of the composite of the system and the thermostate in thermal equilibrium

29. Co-existent phases of a substance • N– number of molecules in one phase • The entropy per molecule is • The energy per molecule is • Two phases

30. Co-existent phases of a substance • Graphic representation

31. Co-existent phases of a substance • Examine co-existent phases using the function u(s) • Examine co-existent phases using the free energy

32. Phase Transition of The Second Kind • A crystal has a rectangular symmetry at high temperature

33. Phase Transition of The Second Kind • T>Tc • T<Tc

34. Research Related • Mechanical response of Miura-Ori pattern

35. Research Related

36. Thank you!