Summary: Isolated Summary: Isolated Systems, Temperature, Free Systems, Temperature, Free EnergyEnergy

Zhiyan WeiES 241: Advanced Elasticity5/20/2009

Isolated SystemsIsolated SystemsStatistical description of systems

Internal variable of an isolated system

The second law of thermodynamics

Entropy

Statistical DescriptionStatistical DescriptionSpecification of the state of the

system

Statistical ensemble

The fundamental postulate

Probability calculations

Statistical DescriptionStatistical DescriptionSpecification 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 scaleA subset of quantum states of an isolated

system is called a macrostate (conformation, thermodyanamic state, or configuration)

Described by macroscopic parameters

Statistical DescriptionStatistical DescriptionSpecification of the state of the

system System

Any part of the world

Isolated systemA system is said to be isolated if it does not interact with the rest of the world– thermally isolated, mechanically isolated….

Statistical DescriptionStatistical DescriptionStatistical ensemble A very large number of identical

systems prepared under identical macroscopic conditions– same macroscopic state

Ergodic TheoremThe average behavior of a system over sufficient amount of time is the same as the average behavior of many identically prepared sytems.

Statistical DescriptionStatistical DescriptionThe fundamental postulate

An isolated system isolated for an enough long time is equally likely to be found in any of its quantum states!

Statistical DescriptionStatistical DescriptionProbability 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 DescriptionStatistical DescriptionProbability 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 wineN– number of ink particles

Internal Variable of An Internal Variable of An Isolated Isolated SystemSystemA 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 Second Law of ThermodynamicsThermodynamicsFor a thoroughly isolated system

that evolves from one macroscopic state to another, its entropy tend to increase!

EntropyEntropyThe logarithm of the number of

quantum states

Composite of two isolated systems

TemperatureTemperatureThermal contactDefinition of absolute temperatureExperimental determination of

temperatureExperimental determination of the

number of quantum statesHeat capacity and latent heat

Thermal ContactThermal ContactOnly energy exchange between

two systems is allowedHeat transferEmpirical observations about

hotness:Two system will reach thermal equilibrium

in thermal contact after a long timeZeroth law of thermodynamicsLevels of hotness are ordered Levels of hotness are continuous

Definition of Absolute Definition of Absolute TemperatureTemperature

What is the most probable partition of energy?

A’ Ω’(U’) A’’ Ω’’(U’’)

Energy

dU

Isolated system

Definition of Absolute Definition of Absolute TemperatureTemperatureBefore 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 Definition of Absolute TemperatureTemperature

Define

Experimental Determination Experimental Determination of Temperatureof 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 Experimental Determination of TemperatureTemperatureIdeal gas

T

P

V

VU

),(log

VNUNfNVU

VUNfNVU N

log),(log),,(log

),(),,(

Experimental Determination Experimental Determination of The Number of Quantum of The Number of Quantum StatesStates

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 Capacity and Latent HeatHeatHeat Capacity

Latent Heat

VVV T

ST

T

UC

PP

P T

ST

T

HC

Free EnergyFree EnergyA 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 A System with Variable EnergyEnergyOpen 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 A System with Variable EnergyEnergyLeading characteristics of the

curves

The horizontal position: no empirical significance

The vertical position: constricted by the 3rd Law of thermodyanics

A System with Variable A System with Variable EnergyEnergyThe 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 A System with Variable Energy and An Internal Energy and An Internal VariableVariableEntropy S(U,Y)

At a constant U, the most probable Y maximizes S(U,Y)

Free EnergyFree 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 EnergyFree 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 Co-existent phases of a substancesubstanceN– number of molecules in one phaseThe entropy per molecule is The energy per molecule is Two phases

Co-existent phases of a Co-existent phases of a substancesubstanceGraphic representation

Co-existent phases of a Co-existent phases of a substancesubstanceExamine co-

existent phases using the function u(s)

Examine co-existent phases using the free energy

Phase Transition of The Phase Transition of The Second Kind Second Kind A crystal has a rectangular symmetry at high

temperature

Phase Transition of The Phase Transition of The Second Kind Second Kind T>Tc

T<Tc

Research RelatedResearch RelatedMechanical response of Miura-Ori

pattern

Research RelatedResearch Related

Thank you!Thank you!