BEZOUT IDENTITIES WITH INEQUALITY CONSTRAINTS

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THERMODYNAMICS
5 Halliday, David, Resnick, Robert, and Walker,
Jeart, Fundamentals of Physics, 6th edition, John
Wiley, Singapore, 2001, ISBN 0-471-33236-4
5, p. 426-427 thermodynamics is the study of
the thermal energy of systems. The central
concept of thermodynamics is temperature
Discuss intuitive notions of temperature. Why are
they unreliable ? What is a thermometer ?
Zeroth Law of Thermodynamics if bodies A and B
are each in thermal equilibrium with a third body
T, then they are in thermal equilibrium with each
other
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MEASURING TEMPERATURE
5, p. 427-429 We define the triple point of
water to have a temperature value of
We define the temperature of any body by
here is the pressure of the gas in thermal
equilibrium with the body and is the
pressure of the gas at
Gas-filled bulb
Reservoir that can be raised and lowered
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TEMPERATURE AND HEAT
5 9. 433 Heat is energy transferred to a system
from its environment because of a temperature
difference that exists between them
here C equals the heat capacity of an object
One calorie (cal) is the amount of heat required
that would raise the temperature of 1 g of water
from 14.5 degrees Celsius to 15.5 degrees Celsius
(degrees Celsius degrees Kelvin 273.15) (1
cal 4.1860J)
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insulation
lead shot
WORK
W
Pressure
State diagram
Q
thermal reservoir
Volume
Q and W are path dependent, not functions of state
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FIRST LAW OF THERMODYNAMICS
There exists a function of state, called the
internal energy and denoted by ,
such that
or
Special cases of thermodynamic processes include
Adiabatic, Constant-volume, Cyclical, and Free
expansions.
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SECOND LAW REVISITED
Lord Kelvin A transformation whose only final
result is to transform into work heat extracted
from a source which is at the same temperature
throughout is impossible
Rudolph Clausius A transformation whose only
final result is to transfer heat from a body at a
given temperature to a body at a higher
temperature is impossible (this principle implies
the previous one)
Martian Skeptic What temperature ?
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SECOND LAW REVISITED
Definition Body A has higher temperature than
body B ( ) if, when we bring them into
thermal contact, heat flows from A to B. Body A
has the same temperature as B ( ) if,
when we bring them into thermal contact, no heat
flows from A to B and no heat flows from B to A.
( A U U A )
Enrico Fermi (Clausius Reformulated) If heat
flows by conduction from a body A to another body
B, then a transformation whose only final result
is to transfer heat from B to A is impossible.
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SECOND LAW REVISITED
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SECOND LAW REVISITED
Definition Absolute Thermodynamic Temperature
Choose a body D
If
then
then
then
In a reversible process
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SECOND LAW REVISITED
System Cylinder that has a movable piston and
contains a fixed amount of homogeneous fluid
States (Macroscopic) Region in positive
quadrant of the (V volume, T temperature)
plane. Functions (on region) V, T, p pressure
Paths (in region) Oriented curves
Differential Forms can be integrated over paths
WORK
HEAT
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SECOND LAW REVISITED
Definition Entropy Function S
(by thermal equilibrium and by thermal isolation)
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FIRST LAW REVISITED
such that
Therefore
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FIRST SECOND LAWS COMBINED
Therefore, the basic (but powerful) calculus
identity
Yields (after some tedious but straightforward
algebra)
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IDEAL GAS LAW
(Chemists) Boyl, Gay-Lussac, Avogardo
amount of gas in moles
ideal gas constant
ideal gas temperature in Kelvin
(water freezes at 373.16 degrees)
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JOULES GAS EXPANSION EXPERIMENT
We substitute the expression for p (given by the
ideal gas law) to obtain
and observe that the outcome of Joules gas
expansion experiment
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IDEAL GAS LAW
(Physicists)
number of molecules of gas
Boltzmanns constant
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GAS THERMODYNAMICS
Experimental Result (dilute gases)
Therefore
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GAS KINETICS
Monatomic dilute gas, m molecular mass
average kinetic energy / molecule
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GAS KINETICS
Photon gases
Maxwell Equipartition of Energy
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EQUIPARTITION
Number of ways of partitioning N objects into m
bins with relative frequencies (probabilities)
is
Stirlings formula
yields
where
denotes Shannons information-theoretic entropy
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EQUIPARTITION
If the bins correspond to energies, then
and therefore (nearly) C,
is maximized, subject to an energy constraint
by the Gibbs
distribution
Therefore
and free energy
Maxwell dist.
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THIRD LAW
Nernst The entropy of every system at absolute
zero can always be taken equal to zero
inherently quantum mechanical
discrete microstates, a quart bottle of air has
about

Maxwells demon may he rest in peace
Times arrow probably forward ???
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REFERENCES
V. Ambegaokar, Reasoning about Luck
H. Baeyer, Warmth Disperses and Time Passes
F. Faurote, The How and Why of the Automobile
E. Fermi, Thermodynamics
R. Feynman, Lectures on Physics, Volume 1
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REFERENCES
H. S. Green and T. Triffet, Sources of
Consciousness, The Biophysical and Computational
Basis of Thought
K. Huang, Statistical Mechanics
N. Hurt and R. Hermann, Quantum
Statistical Mechanics and Lie Group Harmonic
Analysis
C. Shannon and W. Weaver, The Mathematical Theory
of Communication