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?
A’
Energy
Ω’(U’)
dU
Isolated system
A’’
Ω’’(U’’)
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
 log (U , V ) P

V
T
(U ,V , N )  f ( N ,U )V N
log (U ,V , N )  log f ( N ,U )  N log V
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
 U 
 S 
CV  
  T 
 T V
 T V

Latent Heat
 H 
 S 
CP  
  T 
 T  P
 T  P
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
Research Related
Thank you!