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Summary: Isolated Systems, Temperature, Free Energy. Zhiyan Wei ES 241: Advanced Elasticity 5/20/2009. Isolated Systems. Statistical description of systems Internal variable of an isolated system The second law of thermodynamics Entropy. Statistical Description.
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Summary: Isolated Systems, Temperature, Free Energy Zhiyan Wei ES 241: Advanced Elasticity 5/20/2009
Isolated Systems • Statistical description of systems • Internal variable of an isolated system • The second law of thermodynamics • Entropy
Statistical Description • Specification of the state of the system • Statistical ensemble • The fundamental postulate • Probability calculations
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
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….
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.
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!
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
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
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!
Second Law of Thermodynamics • For a thoroughly isolated system that evolves from one macroscopic state to another, its entropy tend to increase!
Entropy • The logarithm of the number of quantum states • Composite of two isolated systems
Temperature • Thermal contact • Definition of absolute temperature • Experimental determination of temperature • Experimental determination of the number of quantum states • Heat capacity and latent heat
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
Definition of Absolute Temperature What is the most probable partition of energy? Energy A’ Ω’(U’) A’’ Ω’’(U’’) dU Isolated system
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
Definition of Absolute Temperature • Define
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.
Experimental Determination of Temperature • Ideal gas
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.
Heat Capacity and Latent Heat • Heat Capacity • Latent Heat
Free Energy • A system with variable energy • A system with variable energy and an internal variable • Free energy • Co-existent phases of a substance
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)
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
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
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)
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
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
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
Co-existent phases of a substance • Graphic representation
Co-existent phases of a substance • Examine co-existent phases using the function u(s) • Examine co-existent phases using the free energy
Phase Transition of The Second Kind • A crystal has a rectangular symmetry at high temperature
Phase Transition of The Second Kind • T>Tc • T<Tc
Research Related • Mechanical response of Miura-Ori pattern