Transcript Structure of Thrmodynamics

The Laws of Thermodynamics - Revisited
Arthur Shavit, Professor Emeritus
Department of Mechanical Engineering
Technion – Israel Institue of Technology
Haifa, ISRAEL
Krakow
9/12/2011
The second Law
Usual statement
A PMM2 is impossible
A PMM2 is cyclic device that produces positive work while
interacting with a single reservoir
Questions:
Is that a law of thermodynamics?
What is a law of thermodynamics?
To answer these question some introduction is needed
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The Structure of Thermodynamics
• Definitions
• Experimental facts
• Laws (axioms)
• Theorems and Corollaries
• Applications
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Definitions
• Body
• Environment
• Primitive property
• State
• Identical states
• Change of state
• Allowed states
• System
• Path
• Interaction
• Process
• Cycle
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Definitions – some examples
Body A body is a part of space enclosed by a well defined boundary.
The boundary may be physical or mathematical, fixed or changing
in time, closed or open to passage of matter.
Environment
Everything outside the boundary of the body
Primitive Property
Primitive property of a body is specified by subjecting the body to an
operation or a test, that requires no previous knowledge of the body, the
result of which at a specific time is the value of the primitive property at
that time.
A primitive property may be determined without the need to change the
conditions of the body.
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Definitions – examples cont.
State The condition of the body identified by all its primitive properties
Identical states
States that have the same values of the
corresponding primitive properties.
Change of state
Occurs when the value of at least one primitive
property is changed.
Allowed states
Allowed states of a body are all the states which the
body may inherently attain consistent with the
definition of the body.
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System
An idealization of a body that may includes only part of the allowed
states of the body. It is also required that the system could be isolated
from its environment
The allowed states may be given as an explicit list. or implicitly, by
describing one state and all possible variations of state.
These variations must be consistent with:
1. the laws of matter,
2. the constraints.
3. the passive resistances.
Closed System A system where matter may not cross its boundary.
Open System A system where matter may cross its boundary.
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General Property
An observable characteristic of the system, whose change between
two end states is independent of the path.
Derived property: A property that is not primitive
Examples: – Ampere-hour on a battery.
– Life time of an incandescent lamp.
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Classifications of properties
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• Primitive
– Derived
• Extensive
– Intensive
• Independent
– Dependent
• Conservative
– Non conservative
Equilibrium
An equilibrium state is one that can not be changed without a
corresponding change in the environment.
4 types according to the changes required in the environment
Stable
Unstable
Neutral
Metastable
Mutual equilibrium
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Types of Equilibria
The type of equilibrium is characterized by the required change in the
environment for a finite change in the system
For a
Stable
Unstable
Neutral
Change of state in system
Permanent
Same order
Temporary
Smaller order
Temporary
Smaller order
Metastable Stable up to a limit
Unstable above the limit
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Rate of change of state in system
Permanent
Same order
Permanent
Same order
Permanent
Same order
Stable up to a limit
Unstable above the limit
Neutral vs Themodynamic property
Neutral Property A property of neutral equilibrium that can
change in both directions by only temporary changes in the environment.
Example: The horizontal position of the system in a gravity field.
Substate
A state different from others only by neutral properties.
Thermodynamic Property
Any property that is not neutral.
Note: A thermodynamic property may have several substates.
Thermodynamic State A state that includes only thermodynamic
properties
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Work Interaction
Work is an interaction between two systems such that whatever happened
in each system and its boundary could be repeated exactly while the sole
external effect is a change of level of a weight.
Measure of work The work of a system equals the number of weights,
in the test, that underwent a unit change of level.
Adiabatic process
A process having no interactions other than work.
Modes of quasistatic work Dislacement of a wire pulled by a force
Change of volume under pressure.
Change of magnetization in magnetic field.
Change of surface area with surface tension .
Etc.
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First law / Energy
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First law
The work of a system undergoing an adiabatic
process depends only on the end states.
Energy
A property whose change between two end
states is determined by the adiabatic work.
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The Laws of Thermodynamics
A law is a generalization of all known experimental facts.
Zeroth Law
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Maxwell, 1891
First Law
Clausius, 1850
(Joule, 1848)
Second Law
Clausius, 1850
(Carnot, 1824)
State principle
Kline & Koenig, 1957
Third Law
Nernst, 1906
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Work of a system in a stable state
Theorem:
A system, in a stable equilibrium state, cannot change its state while the
only external effect is the rise of the level of weight.
Proof
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Assume that the theorem is incorrect then there should be at
least one case where a the stable state changes while the only
external effect is a rise in a level of a weight. It possible to
lower the weight and impart a velocity to the system.
In this case the net external effect is zero while the state of the
system changed from a stable state to another.
That violates the definition of a stable state.
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Quasi-stable State
Some non equilibrium states can be made stable by eliminating some of the
allowed stated, while retaining others. This can be achieved by altering
passive resistances and constraints.
A state made stable by altering passive resistances and/or constraints
is called a quasi-stable state.
The stable state so produced is called corresponding stable state.
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Heat Interaction
Heat is an interaction between two systems each in a stable state
with no change in the constraints and the passive resistances.
Heat Interaction between systems not in stable states.
Interaction during which the system vary only through the corresponding
stable states.
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Zeroth Law
If two systems, A and B, are each in mutual equilibrium with a system C
then they are in mutual equilibrium with each other.
Is that trivial???
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Temperature
T
T
T
T

Thermometer
Temperature is a property that is common to all systems in
mutual equilibrium.
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State Principle
The stable state of a system bounded by a fixed boundary and subjected
to prescribed force fields is fully determined by its energy.
The state principle fixes the number of independent properties of a
system in a stable state. These are the parameters of the boundary
and the force fields and the energy.
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Heat Machines
A closed system that undergoes a cycle while having interactions
Heat Machine
 dW
Heat Engine
 dW
Heat Pump - Refrigerator
 dW   dQ  0
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

 dQ  0
 dQ  0
The Second Law
Reservoir A system in a stable state whose temperature stays
constant under finite interactions.
PMM2 A heat engine that communicates with a single reservoir
The Second Law
(Two statements)
• A PMM2 is not possible
• It is not possible to transfer heat from a reservoir at a low
temperature to one at a higher with no other effects
Are these really Laws (axioms)???
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Clausius Inequality

dQ
0
T
(a corollary of the second law)
For a reversible process
Leads to define a property
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 dQ 
  T rev  0
 dQ 
dS  

T

rev
The Law of Stable Equilibrium
A system having specified allowed states can reach, from any given state,
one and only one stable state and leave no effect on the environment.
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Gibbs Principle of General Inertia
A finite rate of change (or a finite rate of a rate of change) cannot be
stopped by means of infinitesimal alteration in the circumstances.
(J.W. Gibbs, Collected Works, Yale University. Press, Vol. 1 p.56,1948)
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The Unified Laws
First law
Second law
Law of stable
equilibrium
Zeroth Law
Gibbs Principle
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Prague 14.04.2003
Structure of Thermodynamics
State principle
Pressure
weight
Theorem
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Pressure is a thermodynamic property.
A weight is an idealized body whose only independent
property is its level in a gravitational field.
A process involving no effects except the lowering
of weights is impossible.
Work Interaction
Work is an interaction between two systems such that whatever
happened in each system and its boundary could be repeated exactly
while the sole external effect is a change of level of a weight.
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Proof of first law
X1 Y1
A1
X1
A2
X2 Y2
B
X2 Y2
B
X0 Y0
X0 Y0
C
weights
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X00 Y00
Proof of the State Principle
According to the Law of Stable States one and only one stable state
is possible for a system of fixed constraints (and passive resistances)
that undergoes no interactions (constant energy)
It follows that the stable state is determined by the constraints
Thus
  ( E, 1, 2 ,...,  1,  2 ,...)
Where  is any property of the system in stable equilibrium
 are the constraints
 are the passive resistances
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Work of a system in combination
with a reservoir
In general
(dW  dWre v ) R
Define
d R  dW rev
rev
12
W
 R  1  2 R
dW ad,rev  d ad
R  dE
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Called Available Work
Entropy
Define entropy
dSR  CR (dE  d R )
dSRad,re v  0
(dE  d R )ad,re v
dSRad  0
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Principle of increase of entropy
Criterion of Equilibrium
It is necessary and sufficient for equilibrium of an isolated system,
not subdivided by adiabatic walls, that all possible variation in
state satisfy: (dS)E ≤ 0
(dS)E  0
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for all p.v.
Stable Equilibrium
(dS)E  0
for all p.v.
For every possible variation for which
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(dS)E  0
(DS)E  0
Unstable Equilibrium
(dS)E  0
for all p.v.
For at least one possible variation for which
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(dS)E  0
(DS)E  0
Metastable Equilibrium
(dS)E  0
for all p.v.
For all possible variation for which
(dS)E  0
And for p.v. smaller than a certain value
(DS)E  0
And for some possible variation, larger
than that value, at least one p.v. is
(DS)E  0
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Neutral Equilibrium
(dS)E  0
for all p.v.
For at least one possible variation for which
and
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(dS)E  0
(DS)E  0
Proof of the Zeroth Law
Any property
  (E, 1 , 2 ,...,1 ,2 ,...)
Thus S  S ( E, 1, 2 ,...,  1,  2 ,...)
S  [S ( E)] ,
Consider two systems A and B in mutual equilibrium
S A  [S A (E A )] ,
S B  [S B (E B )] ,
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A


dS
A
dS   A  dE A
 dE   ,
B


dS
B
dS   B  dE B
 dE   ,
Proof of the Zeroth Law (cont.)
A
B
A
B

dS


dS
dS  dS   A  dE   B 
dE
dE  ,
dE ,
A
B
For equilibrium
A
B
dSA 

dS
 dE A  0

B  
dE A 

dE
 ,
 , 

dS A   dS B 
dE A  , dE B  ,
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B
(dS  dS )(E A E A )  0
Temperature
T
f  dS 
 dE  ,
1
Kelvin scale
 dS 
 dE 
T    
 dE   ,  dS   ,
For a system in equilibrium with the reservoir
  d R 

1  dS 

 CR 1  
  CR


TR  dE   , ,TR
  dE   , ,TR 
CR 
1
273.16
dS 
dE  dR
TR
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For the triple point of water
d R  dE  TRdS
The Laws of Thermodynamics Revisited
41
Alternative Criteria of Equilibrium
For all possible variations
to states of
(dS)E  0
equal E
(dE)S  0
equal S
dE  TdS  0
dA  SdT  0
}
uniform T
dA 0
dE  pdV  0
dH  Vdp  0
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uniform and equal T
}
uniform T and equal T
Alternative Criteria of Equilibrium
For all possible variations
dE  pdV  0
dH  Vdp  0
}
dH  0
From states of
to states of
uniform T
equal S
uniform p
equal p and S
dH  TdS  0
dG  SdT  0
dG  0
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}
uniform p and T
equal p
uniform p and T
equal p and T
Alternative Criteria of Equilibrium
 dS 
Tf

 dE  ,
1
Kelvin scale
 dS 
 dE 
T 
 

dE
dS

  , 
  ,
For a system in equilibrium with the reservoir
  d R 

1  dS 

 CR 1  
  CR


TR  dE   , ,TR
  dE   , ,TR 
1
Select CR  273.16
dE  dR
dS 
TR
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For the triple point of water
d R  dE  TRdS
Many Thanks
‫תודה רבה‬
Dziękuję Bardzo
KrakowKrakow 9/12/2011
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The Laws of Thermodynamics Revisited
45
Questions ???
Discussions ???
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Nomenclaure
A,a
E,e
G,g
f
H,h
m
n
p
Q
S,s
T
U,u
V,v
W


m

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Helmholzenee energy
Energy
Gibbs free energy
Fugacity
Enthalpy
Mass
Number of moles
Pressure
Heat
Entropy
Temperature
Internal energy
Volume
Work
constraint
Passive resistance
Chemical potential
Degree of reaction
Notations
A,a
E,e
G,g
H,h
m
n
p
S,s
T
U,u
V,v
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Helmholz free energy
energy
Gibbs free energy
enthalpy
mass
number of moles
pressure
entropy
temperature
internal energy
volume



m
degree of reaction
passive resistance
constraint
chemical potential
f fugacity
W work
Q heat
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Simple system
A system that has only one boundary quasistatic work parameter.
If the parameter is the volume the system is called
a simple compressible system.
Such a system has exactly 2 independent properties:
the volume and the energy. (V and E)
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The volume, v, and energy, e, of a simple system in a stable state are two
independent properties.
 1   1 (e, v)
Namely, any property:
Solving for e and v yields:
 2   2 (e, v)
v  v ( 1 ,  2 )
e  e( 1 ,  2 )
For example if  1  p and  2  T then
v  v( p, T )
e  u ( 1 ,  2 )
or
v( p, T , v)  0
for a stable state.
In general
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Equation of state
U = Internal energy
Internal energy is the functional relation
of two independent properties.
U E
UE
etc.
U  U (1 ,  2 )
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The Unified Laws
Quantum
mechanics
Thermodynamics
Law of stabke
equilibrium
Gibbs Principle
First law
Zeroth law
Second law
Third law
State principle
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Schrodinger
equation