TEMPERATURE
• I am teaching Engineering Thermodynamics to a class of 75 undergraduate students.
• These slides follow closely my written notes (http://imechanica.org/node/288).
• I went through these slides in four 90-minute lectures.
Zhigang Suo, Harvard University
The play of thermodynamics
ENTROPY
energy
temperature
heat capacity
Helmholtz function
space
pressure
compressibility
enthalpy
matter
chemical potential
charge
electrical potential
capacitance
Gibbs function
thermal expansion
Joule-Thomson coefficient
2
The basic algorithm of thermodynamics
• (entropy) = log (number of sample points).
• Entropy is additive.
• When a constraint internal to an isolated system fixes an internal variable at a
value x, the isolated system flips in a subset of quantum states.
• Let the number of quantum states in the subset be W(x).
• Call S(x) = log W(x) the entropy of the configuration of the isolated system when
the internal variable is fixed at x.
1. Construct an isolated system with an internal variable, x.
2. When the internal variable is constrained at x, the isolated
system has entropy S(x).
3. After the constraint is lifted, x changes to maximize S(x).
3
Classify systems according to how they
interact with the rest of the world
Exchange matter
Exchange energy
by work
Exchange energy
by heat
Open system
yes
yes
yes
Isolated system
no
no
no
Closed system
no
yes
yes
Thermal system
no
no
yes
Adiabatic system
no
yes
no
4
An open system modeled as
a family of isolated systems
• The fire, the weights and the valve make the water an open system.
• Insulate the wall, jam the piston, and shut the valve. Make the water an isolated
system.
• A system isolated for a long time reaches a state of thermodynamic equilibrium.
• Entropy S = log (number of quantum states).
• Isolating water at various (U,V,N), we obtain a family of isolated systems of three
independent variables
• For the family of isolated systems, the entropy is a function, S(U,V,N).
weights
O
vapor
2
open system
vapor
valve
a family of isolated systems
S(U,V,N)
liquid
liquid
fire
5
Basic problem of thermodynamics
adiabatic, stationary, impermeable wall
U’, V’, N’
S’(U’, V’, N’)
open system A’
diathermal, moving, leaky wall
U’’, V’’, N’’
S’’(U’’, V’’, N’’)
isolated system
open system A’’
Isolated system conserves energy, space, and matter
U’ + U’’ = constant. U’ is an internal variable
V’ + V’’ = constant. V’ is an internal variable
N’ + N’’ = constant. N’ is an internal variable
How does the system isolated for a long time choose the values of the three internal
variables?
System isolated for a long time maximizes entropy
Entropy is additive, but not constant.
Choose U’, V’, N’ that maximize S’(U’, V’, N’) + S’’(U’’, V’’, N’’)
6
Calculus
dU ¢ + dU ¢¢ = 0
dV ¢ + dV ¢¢ = 0
dN ¢ + dN ¢¢ = 0
dS ¢ =
dS ¢¢ =
(
) dU ¢ + ¶S¢(U ¢,V ¢, N ¢) dV ¢ + ¶S¢(U ¢,V ¢, N ¢) dN ¢
¶S ¢ U ¢,V ¢, N ¢
¶U ¢
(
¶V ¢
¶N ¢
) dU ¢¢ + ¶S¢¢(U ¢¢,V ¢¢, N ¢¢) dV ¢¢ + ¶S¢¢(U ¢¢,V ¢¢, N ¢¢) dN ¢¢
¶S ¢¢ U ¢¢,V ¢¢, N ¢¢
¶U ¢¢
é ¶S ¢
d S ¢ + S ¢¢ = ê
ê
ë
é ¶S ¢
+ê
ê
ë
é ¶S ¢
+ê
ê
ë
(
)
¶V ¢¢
¶N ¢¢
(U ¢,V ¢, N ¢) - ¶S¢¢(U ¢¢,V ¢¢, N ¢¢) ùúdU ¢
ú
û
U ¢,V ¢, N ¢ ¶S ¢¢ U ¢¢,V ¢¢, N ¢¢ ù
údV ¢
ú
¶V ¢
¶V ¢¢
û
U ¢,V ¢, N ¢ ¶S ¢¢ U ¢¢,V ¢¢, N ¢¢ ù
údN ¢
ú
¶N ¢
¶N ¢¢
û
¶U ¢
¶U ¢¢
(
)
(
)
(
)
(
)
7
Conditions of equilibrium
(
) = ¶S¢¢(U ¢¢,V ¢¢, N ¢¢)
¶S ¢ U ¢,V ¢, N ¢
¶U ¢
¶S ¢ U ¢,V ¢, N ¢
¶U ¢¢
¶S ¢¢ U ¢¢,V ¢¢, N ¢¢
¶V ¢
¶S ¢ U ¢,V ¢, N ¢
¶V ¢¢
¶S ¢¢ U ¢¢,V ¢¢, N ¢¢
¶N ¢
¶N ¢¢
(
)=
(
)=
adiabatic, stationary, sealing wall
(
)
(
)
diathermal, moving, leaky wall
U’, V’, N’
S’(U’, V’, N’)
open system A’
U’’, V’’, N’’
S’’(U’’, V’’, N’’)
open system A’’
isolated system
8
The goal: understand the relation
(
¶S U ,V , N
¶U
)= 1
T
• We understand everything in this equation, except for temperature.
• Temperature is a child of entropy and energy.
9
Count the number of quantum states by
experimental measurement
• For a closed system, entropy is a property, S (U ,V )
• According to calculus,
dS =
(
¶S U ,V
¶U
• In later lectures we will show that
) dU + ¶S (U ,V ) dV
¶V
dS =
1
P
dU + dV
T
T
• Measure entropy incrementally.
weights
No quantum mechanics.
No theory of probability.
vapor
liquid
fire
10
Circular statements
What is temperature?
Answers from teachers in kindergartens:
• Temperature is the quantity measured by a thermometer.
• Thermometer is an instrument that measures temperature.
Answers from textbooks of thermodynamics:
• Temperature is a property shared by two bodies in thermal contact,
when they stop exchanging energy by heat.
• Heat is the transfer of energy caused by difference in temperature.
11
Heat and temperature are distinct quantities,
and can be measured by separate experiments.
• Calorimetry. The art of of measuring heat.
• Thermometry. The art of measuring temperature.
12
What can we do for temperature?
• Temperature as an abstraction from everyday experience of
thermal contact.
• Temperature as a consequence of the two great principles
of nature: an isolated system conserves energy and
maximizes entropy.
And so, my fellow enthusiasts of thermodynamics: ask not what temperature
can do for you—ask what you can do for temperature
13
Plan
• Calorimetry. Find a method to measure heat
without the concept of temperature.
• Empirical observations of thermal contact
• Theory of thermal contact
• Theory of phases
• Alternative independent variables
• Free energy
14
Thermodynamic states of equilibrium
•
•
•
•
A closed system changes under the fire and the weights.
The system isolated for a long time reaches a state of thermodynamic equilibrium.
A fixed amount of matter can be in many thermodynamic states of equilibrium.
For a fixed amount of a pure substance, specify all thermodynamic states of
equilibrium using two thermodynamic properties, P and V.
weights
O
vapor
2
closed
system
vapor
isolated
system
P
state
liquid
liquid
V
fire
15
Experimental determination of internal energy
Internal energy is a thermodynamic property, U(P,V).
•
•
•
•
•
Seal and insulate a system, making it an adiabatic system.
Do work Wadiabatic to the system.
The system changes from state (PA,VA) to state (PB,VB).
The internal energy changes by U(PB,VB) - U(PA,VA) = Wadiabatic
Reach many states to determine the function U(P,V).
P
state B
state A
V
force x displacement
torque x angle
voltage x change
How do we know that
we have sealed and insulated the system well enough?
16
Calorimetry: the art of measuring heat
•
•
•
•
•
We have measured the function U(P,V).
The fire and the weights change the closed system from state
(PA,VA) to state (PB,VB).
We know that the internal energy changes by U(PB,VB) - U(PA,VA).
We also know that the weights do work W = force x displacement.
We determine the heat Q from U(PB,VB) - U(PA,VA) = W + Q.
weights
O
vapor
2
state B
vapor
O
vapor
liquid
liquid
liquid
P
state A
fire
V
Isolated system A
U(PA,VA)
closed system
W, Q
isolated system B
U(PB,VB)
17
Plan
• Calorimetry
• Empirical observations of thermal contact.
Name all places of hotness by a single,
positive, continuous variable.
• Theory of thermal contact
• Theory of phases
• Alternative independent variables
• Free energy
18
Everyday experience of hotness
• (temperature) = (hotness)
• (a value of temperature) = (a place of hotness)
• Hot, warm, cool, cold.
• Words are not enough to name many places of hotness.
• Name all places of hotness by a single, positive, continuous
variable.
• Why is hotness so different from happiness?
19
Ranking Universities
Two-dimensional ranking
average salary of graduates
One-dimensional ranking
• Harvard
• Princeton
• Yale
citations to papers
Why stop at two dimensions?
20
Ranking places of hotness
Imagine you were born 500 years ago.
Galilei (1564-1642)
Middleton, A History of the Thermometer and Its Use in Meteorology
21
Air thermometer
•
•
•
•
Problem: Visualize hotness.
Invention: Air thermometer (Galileo and others, 1612)
Science: Air expands when heated.
Engineering: Map hotness to height.
22
Liquid-in-glass thermometer
•
•
•
•
Problem: Gas thermometer is bulky and sensitive to pressure.
Invention: liquid-in-glass thermometer (Ferdinando 1654)
Science: Liquid expands when heated.
Engineering: Volume of liquid is insensitive to pressure. Glass is transparent.
Mark the glass.
Middleton, A History of the Thermometer and Its Use in Meteorology
Wikipedia page on thermometer
23
Thermal contact
stationary, impermeable, but diathermal wall
heat
•
•
•
•
isolated system
The two systems together form an isolated system.
The two systems do not exchange matter (impermeable wall)
The two systems do not exchange energy by work (stationary wall).
The two systems exchange energy by heat (diathermal wall).
24
Observation 1
Two systems in thermal contact for a long time will stop
transferring energy.
thermal contact
isolated system
The two systems are said to have reached thermal equilibrium.
25
Observation 2 (zeroth law, Fowler 1931)
If two systems are separately in thermal equilibrium
with a third system, the two systems are in thermal
equilibrium with each other.
Use thermal contact to discover places of hotness.
26
Observation 3
For a fixed amount of a pure substance, once pressure
and volume are fixed, the hotness is fixed.
A
weights
P
B
fire
C
V
27
Observation 4
For a pure substance in a state of coexistent solid and
liquid, the hotness remains fixed as the proportion of
liquid and gas changes.
This place of hotness is specific to substance, but is insensitive to pressure.
liquid
For water, this place of hotness has many names
• Melting point
• Freezing point
• 0 Celsius
• 32 Fahrenheit
• 273.15 Kelvin
28
Name places of hotness
The same way as we name streets
•Harvard, Cambridge, Oxford…
•Washington, Lincoln,…
•5th Avenue, 6th Avenue,…
After physical events.
•
Streets are physical objects.
•WATER at the melting point
•
Names of the streets are arbitrary.
•LEAD at the melting point
•ALUMINUM at the melting point
•GOLD at the melting point
•Steam at pressure 0.1 MPa and specific volume 2000 m3/kg
Relative terms
•Cold
•Cool
•Warm
•Hot
29
Thermometry: the art of measuring hotness
Match system X in thermal equilibrium with a system in the library.
X
An isolated system at a
unknown place of hotness
A library of isolated systems preserved at
previously named places of hotness
30
Observation 5 (Fermi’s improved version of the Clausius
statement of the second law of thermodynamics)
When a system of hotness A and a system of hotness B are
brought into thermal contact, if energy goes from B to A,
energy will not go in the opposite direction.
Places of hotness are ordered.
•
When two systems are in thermal contact, a difference in hotness gives heat a direction.
•
By convention, the system losing energy is said to be hotter than the system gaining energy.
Fermi, Thermodynamics
31
Hotness “WATER” is lower than hotness “LEAD”
heat
liquid
liquid
solid
solid
Water at melting point
Lead at melting point
Calorimetry determines the direction of heat and the quantity of heat.
Thermometry uses only the direction of heat, not the quantity of heat.
32
In thermodynamics, the word “hot” is used strictly within the
context of thermal contact.
It makes no thermodynamic sense to say that one movie is
hotter than the other, because the two movies cannot
exchange energy.
33
Observation 6 (A generalization of the zeroth law)
If hotness A is lower than hotness B, and hotness B is
lower than hotness C, then hotness A is lower than
hotness C.
heat
heat
heat
liquid
liquid
liquid
liquid
solid
solid
solid
solid
WATER
LEAD
ALUMINUM
GOLD
hotness
Order all places of hotness in one dimension.
34
Scale of hotness
An ordered array of places of hotness
Scales of other things
• Scale of earthquake
• Scale of hurricane
• Scale of happiness
• Scale of terrorism threat
35
Observation 7
Between any two places of hotness there exists another
place of hotness.
heat
heat
liquid
liquid
liquid
solid
solid
solid
WATER
X
LEAD
hotness
Name all places of hotness by a continuous variable.
36
Numerical scale of hotness
• Problem: Every thermometer is unique. Newton’s thermometer disagreed with
Galileo’s thermometer.
• Invention (Fahrenheit 1720): Name two places of hotness after physical
events. Name other places by thermal expansion of mercury
• Science: Melting point. Boiling point. Name all places of hotness by a single,
continuous variable.
• Engineering: Why mercury?
hotness
32
Freezing point of water
212
boiling point of water
37
Map one numerical scale of hotness to another
Any increasing (linear or nonlinear) function will do.
C = (5/9)(F – 32)
38
Long march toward naming places of hotness
using a single continuous variable
•
•
•
•
No useful way to name all streets by an ordered array.
Cannot name streets with a continuous variable.
We don’t know how to name all places of happiness.
We laugh at rankings of universities.
A library of isolated systems of
previously named places of hotness
39
Non-numerical vs. numerical scales of hotness
• A non-numerical scale of hotness perfectly captures
all we care about hotness.
• Naming places of hotness by using numbers makes it
easier to memorize that hotness 80 is hotter than
hotness 60.
• Our preference to a numerical scale reveals more the
nature of our brains than the nature of hotness.
40
Numerical values of hotness do not obey arithmetic rules
•
Adding two places of hotness has no empirical significance. It is as meaningless as
adding the addresses of two houses. House number 2 and house number 7 do not
add up to become house number 9.
•
Raising the temperature from 0C to 50C is a different process from raising
temperature from 50C to 100C.
41
Observation 8
All places of hotness are hotter than a certain place of
hotness.
•
•
•
•
There exists a coldest place of hotness, but not a hottest place of hotness.
Name the coldest place of hotness zero.
Name all other places of hotness by a single, positive, continuous variable.
Such a scale of hotness is called an absolute scale.
Under rare conditions, however, negative absolute temperature has been attained.
We will not consider these conditions in this course.
42
Observation 9
Thin gases obey the law of ideal gases.
thermal contact
Gas A’
Gas A’’
P’,V’,N’
P’’,V’’,N’’
Experimental discovery: The two gasses reach thermal equilibrium when
P’V’/N’ = P”V”/N”
Ideal-gas scale of hotness: t = PV/N.
This scale of hotness has the same unit as energy, J
43
Observation 10
For a pure substance, its solid phase, liquid phase and gaseous phase coexist at a
specific hotness and a specific pressure.
44
Kelvin scale of hotness
1.
2.
3.
The Kelvin scale of hotness, T, is proportional to the ideal-gas scale of
temperature. Write kT = PV/N.
The unit of the Kelvin scale, K, is defined such that the triple point of pure water
is T = 273.16 K exactly.
Experimental value: k = 1.38x10-23 J/K. k is the conversion factor between the
two scales of hotness, and is known as Boltzmann’s constant.
• Experimental value: melting point of water at 0.1 MPa: 273.15K.
• Modern definition of the Celsius scale: C = T - 273.15
45
Today’s temperature is…
•
•
•
•
•
20 degree Celsius
68 Fahrenheit
293 Kelvin
404.34x10-23 J
0.0253 eV
C = (5/9)(F – 32)
K = C + 273.15
1 K = 1.38x10-23 J
1 eV = 1.6x10-19J
46
Thermometry is a growing art
Temperature affects everything. Everything is a thermometer.
Today’s opportunity: The Internet of things.
Air thermometer
resistance thermometer
liquid-in-glass thermometer
pyrometer
bimetallic thermometer
thermocouple
47
Plan
• Calorimetry
• Empirical observations of thermal contact
• Theory of thermal contact. Identify
temperature as a child of entropy and energy.
• Theory of phases
• Alternative independent variables
• Free energy
48
The play of thermodynamics
ENTROPY
energy
temperature
heat capacity
Helmholtz function
space
pressure
compressibility
enthalpy
matter
chemical potential
charge
electrical potential
capacitance
Gibbs function
thermal expansion
Joule-Thomson coefficient
49
The basic algorithm of thermodynamics
• (entropy) = log (number of sample points).
• Entropy is additive.
• When a constraint internal to an isolated system fixes an internal variable at a
value x, the isolated system flips in a subset of quantum states.
• Denote the number of quantum states in the subset by W(x).
• Call S(x) = log W(x) the entropy of the configuration of the isolated system when
the internal variable is fixed at x.
1. Construct an isolated system with an internal variable, x.
2. When the internal variable is constrained at a value x, the
isolated system has entropy S(x).
3. After the constraint is lifted, x changes to maximize S(x).
50
A thermal system modeled as
a family of isolated systems
•
•
•
•
The fire heats up a thermal system.
Insulate the wall and an isolated system.
A system isolated for a long time reaches a state of thermodynamic equilibrium.
Isolating at various values of U, we obtain a family of isolated systems of one
independent variable.
• Use U as a coordinate.
• Each state of thermodynamic equilibrium corresponds to a point on the coordinate.
state
U
fire
thermal system
Fire changes U
Fix V and N
a family of isolated systems
of a single independent variable, U
51
Entropy is a thermodynamic property
• Entropy S = log (number of quantum states).
• Isolating at various values of U, we obtain a family of isolated systems of one
independent variable.
• For this family of isolated systems, entropy is a function of a single variable, S(U).
S
S(U)
fire
thermal system
U
a family of isolated systems
52
A bit of high-school mathematics
three ways to represent
a function of a single variable, f(x)
• Table
• A curve in a plane
• An equation
53
Construct an isolated system with an internal variable
Diathermal wall. Thermal contact
U’
S’(U’)
thermal system A’
U’’
S’’(U’’)
isolated system
themal system A’’
Isolated system conserves energy
U’ + U’’ = constant. U’ is an internal variable
How does the isolated system partition energy between the two subsystems?
Isolated system maximizes entropy
Entropy is additive, but not constant.
Change U’ to maximize S’(U’) + S’’(U’’)
54
Calculus of changes
U’
S’(U’)
heat
dU’
thermal system A’
U’’
S’’(U’’)
themal system A’’
Isolated system conserves energy. dU ¢ + dU ¢¢ = 0
Isolated system changes entropy
• Entropies of the two subsystems: dS ¢ =
( ) dU ¢,
dS ¢ U ¢
dU ¢
( )
dS ¢¢ =
)
dU ¢¢
( ) ùúdU ¢
é dS ¢ U ¢ dS ¢¢ U ¢¢
• Entropy of the isolated system: d S ¢ + S ¢¢ = êê
dU ¢¢
ë dU ¢
(
( ) dU ¢¢
dS ¢¢ U ¢¢
ú
û
55
Thermodynamic scale of temperature
The internal variable U’ changes to
maximize the entropy of the isolated system:
( )
U’
S’(U’)
( ) ùúdU ¢ ³ 0
é dS ¢ U ¢ dS ¢¢ U ¢¢
ê
ê dU ¢
dU ¢¢
ë
ú
û
heat
dU’
thermal system A’
U’’
S’’(U’’)
thermal system A’’
Thermal equilibrium
•If
( ) = dS¢¢(U,¢¢the
) entropy of the isolated system does not change when energy flows either way.
dS ¢ U ¢
dU ¢
dU ¢¢
Irreversibility and direction of heat:
•If
•If
¢¢)
( ) < dS¢¢(U, the
energy of the isolated system increases when energy flows in the direction dU’ < 0.
dS ¢ U ¢
dU ¢
dU ¢¢
¢¢)
( ) > dS¢¢(U, the
energy of the isolated system increases when energy flows in the direction dU’ > 0.
dS ¢ U ¢
dU ¢
dU ¢¢
Thermodynamic scale of temperature defined
(Clausius-Gibbs equation):
( )
1 dS U
=
T
dU
56
Theory explains empirical observations
Thermal contact
U’, S’(U’)
U’’, S’’(U’’)
isolated system
1/T’ = dS’(U’)/dU’
thermal system A’
•
•
•
•
•
•
Observation 1.
Observation 2.
Observation 5.
Observation 6.
Observation 7.
Observation 8.
1/T’’ = dS’’(U’’)/dU’’
themal system A’’
Thermal contact leads to thermal equilibrium
Zeroth law
Arrow of heat (Second law)
Temperature is ordered
Temperature is continuous
Temperature is positive
57
An open system modeled as
a family of isolated systems
weights
O
open system
vapor
2
vapor
A family of isolated systems
S(U,V,N)
valve
liquid
liquid
fire
Clausius-Gibbs equation
(
1 ¶S U ,V , N
=
T
¶U
)
58
Thermodynamic scale of temperature
coincides with ideal-gas scale of temperature
Thermodynamic scale
Ideal-gas scale
(
1 ¶S U ,V , N
=
T
¶U
)
T = PV / N
• Historically the law of ideal gases was discovered empirically.
• Later we will derive the law of ideal gases theoretically.
59
Unit of entropy
Temperature in the unit of energy
• Definition: Entropy = log (number of sample points), s = log W.
• Entropy is dimensionless.
• Define temperature (Clausius-Gibbs equation): 1/t = ds/dU.
• Temperature has the same unit as energy.
Temperature in the unit of Kelvin
• The committee adopts Kelvin as the unit for temperature T. t = kT.
• To preserve the Clausius-Gibbs equation 1/T = dS/dU, we write S = k logW.
• The unit for entropy is the same as that of k: J/K.
60
Experimental determination of
thermodynamic scale of temperature
Division of labor
1. For a simple system (e.g., a thin gas), create a theory to relates the thermodynamic
scale of temperature to measurable quantities (theoretician).
2. Use the simple system to calibrate a thermometer by thermal contact
(manufacturer).
3. Use the thermometer to measure the temperature of any other system by thermal
contact (patient).
Theoretician
manufacturer
patient
T = PV / N
61
Experimental determination of
Energy-temperature curve
T
U
fire
thermal system
•
•
•
•
a family of isolated systems
Properties: T(U), S(U)
Calorimetry measures internal energy U.
Thermometry measures temperature T.
Both U and T are thermodynamic properties.
A thermal system is a system of a single independent variable. T(U).
62
Practical calorimetry
Increase internal energy by a known amount without changing volume.
liquid
solid
fire
Thermal system
How much heat?
Thermal system
Ice calorimetry
Adiabatic system
Frictional heating
Adiabatic system
Joule heating
63
Count the number of quantum states
• U, T, S are thermodynamic properties.
• A thermal system is a system of a single independent variable. T(U), S(U).
Clausius-Gibbs equation
( )= 1
dS U
dU
T(U)
T
T
U
dS =
dU
T U
( )
S
S(U)
1
U
dU
S U = S U0 +
T U
U0
( ) ( ) ò
( )
T
U
64
Plan
•
•
•
•
Calorimetry
Empirical observations of thermal contact
Theory of thermal contact
Theory of phases. Ask what temperature CAN
do for you.
• Alternative independent variables
• Free energy
65
Graphic derivation of the condition of equilibrium
(
)
Entropy is additive: Scomposite = S (U ) + SR U composite -
U
TR
( )
1 dS U
=
In equilibrium, the isolated system maximizes entropy:
TR
dU
entropy
1
TR
( )
S U
energy U
reservoir
SR(UR) = SR(Ucomposite) – U/TR
small system
S(U)
heat U
66
Convex function
Define temperature (Clausius-Gibbs equation):
Define heat capacity:
( )
1 dS U
=
T
dU
S
S(U)
1
( )
1 dT U
=
C
dU
T
U
T
Equivalent statements
• S(U) is convex
• T(U) is an increasing function
• C(U) > 0
U(T)
1
C
U
67
Non-convex S(U) causes phase transition
(i.e., discontinuous change of state)
S
( )
1 dS U
=
T
dU
U
T
U
68
Co-existent phases of a pure substance
water
water
water
T
mixture of ice and water
ice
ice
ice
Tm
latent heat
U
fire
thermal system
isolated system
T(U), S(U)
dS =
dU
T
Ssolid - Sliquid =
U solid -U liquid
Tm
69
Thermodynamic theory of co-existent phases
water
ice
s'(u’)
s’’(u’’)
( )
s¢¢ u¢¢
s
water
ice
( )
s¢ u¢
u
Each phase has its own s(u) function.
70
Rule of mixture corresponds to a line in the s-u plane
water
ice
isolated system
( )
s¢¢ u¢¢
s
u = (1 - x)u¢ + xu¢¢
( )
s¢ u¢
s¢¢
s¢
u¢
s = (1 - x)s¢ + xs¢¢
s
u
u¢¢
u
71
Isolated system conserves energy and maximizes entropy
roll a tangent line
( )
s¢¢ u¢¢
s
( )
1 dS U
=
T
dU
s¢¢
s
( )
s¢ u¢
s¢
u¢
u
u
u¢¢
s¢¢ - s¢ =
u¢¢ - u¢
Tm
water
T
mixture of ice and water
Tm
ice
latent heat
u¢
u¢¢
u
Coexistent phases correspond to a common tangent.
72
Plan
•
•
•
•
•
Calorimetry
Empirical observations of thermal contact
Theory of thermal contact
Theory of phases
Alternative independent variables. So many
choices!
• Free energy
73
Thermal system
state
U
fire
thermal system
A family of isolated systems
• Many states of thermodynamic equilibrium
• One independent variable: U
• Four thermodynamic properties: USTC
• Three equations of state: S(U), T(U), C(U)
T
T(U)
U
Define temperature (Clausius-Gibbs equation):
( )
1 dS U
=
T
dU
dT (U )
Define heat capacity: 1 =
C
dU
Given a thermal system, we obtain the equations of state as follows.
• Perform calorimetry and thermometry to measure T(U).
• Integrate the Clausius-Gibbs equation to obtain S(U).
• Differentiate T(U) to obtain C(U).
74
Heat capacity
T
T(U)
1
fire
thermal system
C
U
A family of isolated systems
of a single independent variable, U
( )
1 dT U
=
C
dU
75
4 properties: USTC
6 curves: US, UT, UC, ST, SC, TC
Measure one curve.
Calculate the other five.
T
S
T(U)
C
S(U)
U
C(U)
U
intensive-extensive
extensive-extensive
•
•
•
U
extensive-extensive
Cross-plot T(U) and S(U) to obtain T(S).
Cross-plot S(U) and C(U) to obtain C(S).
Cross-plot T(U) and C(U) to obtain C(T).
76
Entropy as independent variable
O
vapor
vapor
2
U
T
liquid
liquid
1
fire
S
thermal system
A family of isolated systems
T(S), U(S)
( )
dU = T S dS
T
( )
S
( ) ò T ( S ) dS
U S -U S0 =
S0
U
S
77
Temperature as independent variable
• Perform thermometry and calorimetry to measure U(T).
• Differentiate U(T) to obtain C(T) = dU(T)/dT.
• Integrate the Clausius-Gibbs equation to obtain S(T)
dS =
( )
dU C T
=
dT
T
T
T
U(T)
S
T
S(U)
1
C
1
S(T)
T
U
U
S
78
Experimental control of temperature
water
ice
Ice-water mixture
thermal bath (heat reservoir)
thermostat
79
Thermodynamic model of reservoir of energy
•
•
•
A reservoir of energy is a thermal system, and has a single independent variable, UR.
Entropy of the reservoir of energy is a thermodynamic property, SR(UR).
The reservoir of energy has a fixed temperature, TR.
Clausius-Gibbs equation: dSR =
Integration:
(
dU R
TR
)
(
)
S R U R2 - SR U R1 =
U R2 -U R1
TR
Reservoir of water
Reservoir of energy
Potential energy PE
Internal energy UR
Height h
Temperature TR
Weight w
Entropy SR
DPE = hDw
DUR = TRDSR
80
A small system in equilibrium with
a reservoir of energy
•
•
(isolated system) = (small system) + (reservoir)
Internal variable: the internal energy of the small system U
•
The isolated system conserves energy: U composite =U +U R = constant
•
The isolated system changes entropy:
(
)
U
Scomposite = S U + SR U composite TR
( )
( )
1 dS U
=
• In equilibrium, the isolated system maximizes entropy:
TR
dU
reservoir
SR(UR) = SR(Ucomposite) – U/TR
small system
S(U)
heat U
81
A small system in thermal contact, but out of
equilibrium, with a reservoir of energy
•
•
(isolated system) = (small system) + (reservoir)
Internal variables: the internal energy of the small system U, and something else Y
• Entropy is additive:
(
)
U
Scomposite = S U ,Y + SR U composite TR
(
)
Q
• Internal variables change to increase entropy (Clausius inequality): DS ³
TR
• Irreversible change in U because T ¹ TR
• Irreversible change in Y.
S(U,Y)
vapor
reservoir
SR(UR) = SR(Ucomposite) – U/TR
heat U = Q
liquid
82
Plan
•
•
•
•
•
•
Calorimetry
Empirical observations of thermal contact
Theory of thermal contact
Theory of phases
Alternative independent variables.
Free energy
83
Calculus: Legendre transform
Turn a derivative into an independent variable
A dance of 4 variables: USTF
S as independent variable:
( )
U S
( )
dU S
Derivative:
T=
Increment:
dU = TdS
Define Legendre transform:
Product rule:
dS
F =U -TS
dF = dU -TdS - SdT
T
-F
U
S
Increment:
dF = -SdT
Derivative:
-S =
T as independent variable
Helmholtz function (free
energy, Helmholtz free energy):
( )
dF T
dT
( )
F T
84
Debye model (1912)
U = aT 4
Solid at low temperature:
Clausius-Gibbs equation :
Heat capacity:
Helmholtz function:
Eliminate T from U(T) and
S(T):
dS =
C=
( )
dU T
T
( )
dU T
dT
F =U -TS
4
S = aT 3
3
C = 4aT 3
1
F = - aT 4
3
4
S U = a1/4U 3/4
3
( )
85
Free energy for a system with internal variable
•
•
(isolated system) = (small system) + (reservoir)
Internal variables: the internal energy of the small system U, and something else Y
•
Entropy is additive:
•
(
)
U
Scomposite = S U ,Y + SR U composite TR
1 ¶S U ,Y
Thermal equilibrium:
TR = T. U(T,Y).
=
TR
¶U
(
(
)
)
•
•
At a fixed T, the internal variable Y changes to maximize S(U,Y) – U/T.
Define the Helmholtz free energy (or Helmholtz function): F = U-TS.
•
At a fixed T, the internal variable Y changes to minimize F(T,Y).
S(U,Y)
vapor
reservoir
SR(UR) = SR(Ucomposite) – U/TR
heat U
liquid
86
Free energy and co-existent phases
•
•
•
•
Each phase has its own free-energy function. f’(T), f’’(T).
Mixture obeys the rule of mixture, f (x,T) = (1-x)f’(T) + xf’’(T)
The quality x is an internal variable.
For a fixed T, x changes to minimize the free energy f(x,T).
f’(Tm) = f’’(Tm)
u’ - Tms’ = u’’ –Tms’’
f
f '(T)
f ’’(T)
Tm
T
87
summary
•
Heat and temperature are distinct concepts, and are measured by separate experiments:
calorimetry and thermometry.
•
Define temperature by the Clausius-Gibbs equation:
•
Define heat capacity by the slope of the energy-temperature curve:
•
Define Helmholtz function (free energy) by the Legendre transform: F = U – ST. F(T)
•
Thermal system: many states of thermodynamic equilibrium, one independent variation, five
thermodynamic properties (USTCF). Alternative choices of independent variables: U, S, T.
Functions of state: S(U), U(S), U(T), C(T), F(T)…
•
Law of ideal gases: PV = NkBT. Debye model for a solid near absolute zero: U(T) = aT4.
•
Theory of phases. Rules of mixture: u = (1-x)u’ + xu’’, s = (1-x)s’ + xs’’. Common tangent line.
Latent heat u’’ - u' = Tm(s’’-s’).
•
For a thermal system held at fixed temperature T and with an internal variable Y, the internal
variable Y changes to minimize the Helmholtz free energy F(T,Y).
( )
1 dS U
=
T
dU
( )
1 dT U
=
C
dU
88