Embed Size (px)
Citation preview
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
2
energy space matter charge
ENTROPY
temperature pressure chemical potential electrical potential
heat capacity compressibility capacitanceHelmholtz function enthalpy
Gibbs free energythermal expansionJoule-Thomson coefficient
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
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. • The number of quantum states in the subset is (x). • Call S(x) = log (x) the entropy of the configuration of the isolated system when
the internal variable is fixed at x.
Classify systems according to how they interact with the rest of the world
4
Exchange matter Exchange energyby 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
An open system modeled asa family of isolated systems
5
liquid
2Ovapor a family of isolated systemsS(U,V,N)
• 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 flips to every quantum state with equal probability.• 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).
liquid
weights
fire
vapor valveopen system
Basic problem of thermodynamics
Isolated system conserves energy, space, and matterU’ + U’’ = constant. U’ is an internal variableV’ + V’’ = constant. V’ is an internal variableN’ + N’’ = constant. N’ is an internal variableHow does the system isolated for a long time choose the values of the three internal variables?
System isolated for a long time maximizes entropyEntropy is additive, but not constant. Choose U’, V’, N’ that maximize S’(U’, V’, N’) + S’’(U’’, V’’, N’’) 6
U’, V’, N’S’(U’, V’, N’)
isolated systemU’’, V’’, N’’S’’(U’’, V’’, N’’)
adiabatic, stationary, impermeable wall diathermal, moving, leaky wall
open system A’ open system A’’
Calculus
7
Conditions of equilibrium
8
U’, V’, N’S’(U’, V’, N’)
isolated system
U’’, V’’, N’’S’’(U’’, V’’, N’’)
adiabatic, stationary, sealing wall diathermal, moving, leaky wall
open system A’ open system A’’
The goal: understand the relation
9
• We understand everything in this equation, except for temperature.• Temperature is a child of entropy and energy.
Count the number of quantum states by experimental measurement
10
liquid
weights
fire
vapor
• For a closed system, entropy is a property,
• According to calculus,
• In later lectures we will show that
• Measure entropy incrementally.
No quantum mechanics.No theory of probability.
Circular statements
11
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.
What is temperature?
Heat and temperature and 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?
13
• 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
Plan
• Calorimetry. Find a method to measure heat without the concept of temperature.
• Empirical observations of thermal contact• Theory of thermal contact• Refinements and applications
14
Thermodynamic states of equilibrium
15
• 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.
liquid
weights
fire
vapor
liquid
2Ovapor isolated systemclosed
system
P
V
state
16
Experimental determination of internal energy
P
V
state A
state B
• 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).
Internal energy is a thermodynamic property, U(P,V).
force x displacement torque x angle voltage x change
How do we know that we have sealed and insulated the system well enough?
17
Calorimetry: the art of measuring heat
Pstate B
state A
• 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.
liquid
weights
fire
vapor
liquid
2Ovapor
Isolated system A closed system isolated system BU(PA,VA) W, Q U(PB,VB)
liquid
Ovapor
V
Plan
• Calorimetry• Empirical observations of thermal contact.
Name all places of hotness by a single, positive, continuous variable.
• Theory of thermal contact• Refinements and applications
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
20
One-dimensional ranking•Harvard•Princeton•Yale
citations to papers
aver
age
sala
ry o
f gr
adua
tes
Two-dimensional ranking
Why stop at two dimensions?
Ranking places of hotness
21
Galilei (1564-1642)
Middleton, A History of the Thermometer and Its Use in Meteorology
Imagine you were born 500 years ago.
Air thermometer
22
• Problem: Visualize hotness. • Invention: Air thermometer (Galileo and others, 1612)• Science: Air expands when heated. • Engineering: Map hotness to height.
Liquid-in-glass thermometer
23
• 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 MeteorologyWikipedia page on thermometer
Thermal contact
• 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
isolated system
stationary, impermeable, but diathermal wall
heat
Observation 1Two systems in thermal contact for a long time will stop transferring energy.
25The two systems are said to have reached thermal equilibrium.
isolated system
thermal contact
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.
26Use thermal contact to discover places of hotness.
Observation 3 For a fixed amount of a pure substance, once pressure and volume are fixed, the hotness is fixed.
27
weights
fire
P
V
A
B C
Observation 4For 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.
28
liquidFor water, this place of hotness has many names•Melting point•Freezing point•0 Celsius•32 Fahrenheit•273.15 Kelvin
Name places of hotnessThe same way as we name streets•Harvard, Cambridge, Oxford…•Washington, Lincoln,…•5th Avenue, 6th Avenue,…
After physical events.•WATER at the melting point•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
• Streets are physical objects.• Names of the streets are arbitrary.
Thermometry: the art of measuring hotness
30
A library of isolated systems preserved at previously named places of hotness
X
An isolated system at a unknown place of hotness
Match system X in thermal equilibrium with a system in the library.
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.
31
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
Hotness “WATER” is lower than hotness “LEAD”
32
liquid
solid
liquid
solid
heat
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.
33
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.
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.
34Order all places of hotness in one dimension.
liquid
WATER LEAD ALUMINUM GOLD
hotness
solid
liquid
solid
heat
liquid
solid
heat
liquid
solid
heat
Scale of hotnessAn ordered array of places of hotness
Scales of other things•Scale of earthquake•Scale of hurricane•Scale of happiness•Scale of terrorism threat
35
liquid
Observation 7Between any two places of hotness there exists another place of hotness.
36Name all places of hotness by a continuous variable.
WATER X LEAD
hotness
solid
liquid
solid
heat
liquid
solid
heat
Numerical scale of hotness
37
hotness
32 212Freezing point of water boiling point of water
• 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?
Map one numerical scale of hotness to another
38
Any increasing (linear or nonlinear) function will do.
C = (5/9)(F – 32)
Long march toward naming places of hotness using a single continuous variable
39
A library of isolated systems of previously named places of hotness
• 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.
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 8All places of hotness are hotter than a certain place of hotness.
42
• 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.
Observation 9Thin gases obey the law of ideal gases.
43
Experimental discovery: The two gasses reach thermal equilibrium when
P’V’/N’ = P”V”/N”
Gas A’
P’,V’,N’
Gas A’’
P’’,V’’,N’’
thermal contact
Ideal-gas scale of hotness: = PV/N.This scale of hotness has the same unit as energy, J
Observation 10For 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. The Kelvin scale of hotness, T, is proportional to the ideal-gas scale of
temperature. Write kT = PV/N. 2. The unit of the Kelvin scale, K, is defined such that the triple point of pure water
is T = 273.16 K exactly.3. 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.
45
• Experimental value: melting point of water at 0.1 MPa: 273.15K.• Modern definition of the Celsius scale: C = T - 273.15
Today’s temperature is…
• 20 degree Celsius• 68 Fahrenheit• 293 Kelvin• 404.34x10-23 J• 0.0253 eV
46
C = (5/9)(F – 32)K = C + 273.151 K = 1.38x10-23 J1 eV = 1.6x10-19J
Thermometry is a growing art Temperature affects everything. Everything is a thermometer.
Today’s opportunity: The Internet of things.
47
Air thermometer liquid-in-glass thermometer bimetallic thermometer
resistance thermometer pyrometer thermocouple
Plan
• Calorimetry• Empirical observations of thermal contact• Theory of thermal contact. Identify
temperature as a child of entropy and energy.• Refinements and applications
48
The play of thermodynamics
49
energy space matter charge
ENTROPY
temperature pressure chemical potential electrical potential
heat capacity compressibility capacitanceHelmholtz function enthalpy
Gibbs free energythermal expansionJoule-Thomson coefficient
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
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 (x). • Call S(x) = log (x) the entropy of the configuration of the isolated system when
the internal variable is fixed at x.
A thermal system modeled asa family of isolated systems
51
a family of isolated systemsof a single independent variable, U
• 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.
fire
thermal systemFire changes UFix V and N
U
state
Entropy is a thermodynamic property
52
• 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).
U
S(U)
S
a family of isolated systems
fire
thermal system
A bit of high-school mathematicsthree 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
Isolated system conserves energyU’ + U’’ = constant. U’ is an internal variableHow does the isolated system partition energy between the two subsystems?
Isolated system maximizes entropyEntropy is additive, but not constant. Change U’ to maximize S’(U’) + S’’(U’’)
54
U’S’(U’)
isolated systemU’’S’’(U’’)
Diathermal wall. Thermal contact
thermal system A’ themal system A’’
Isolated system changes entropy
•Entropies of the two subsystems:
•Entropy of the isolated system:
Calculus of changes
55
U’S’(U’)
U’’S’’(U’’)
thermal system A’ themal system A’’
heat dU’
Isolated system conserves energy.
Thermal equilibrium
•If , the entropy of the isolated system does not change when energy flows either way.
Irreversibility and direction of heat: •If , the energy of the isolated system increases when energy flows in the direction dU’ < 0.
•If , the energy of the isolated system increases when energy flows in the direction dU’ > 0.
Thermodynamic scale of temperature
56
U’S’(U’)
U’’S’’(U’’)
thermal system A’ thermal system A’’
heat dU’
Thermodynamic scale of temperature defined (Clausius-Gibbs equation):
The internal variable U’ changes to maximize the entropy of the isolated system:
Theory explains empirical observations
• Observation 1. Thermal contact leads to thermal equilibrium• Observation 2. Zeroth law• Observation 5. Arrow of heat (Second law)• Observation 6. Temperature is ordered• Observation 7. Temperature is continuous• Observation 8. Temperature is positive
57
U’, S’(U’)
1/T’ = dS’(U’)/dU’isolated system
U’’, S’’(U’’)
1/T’’ = dS’’(U’’)/dU’’
Thermal contact
thermal system A’ themal system A’’
58
liquid
weights
fire
vapor valveopen system
An open system modeled asa family of isolated systems
2OA family of isolated systemsS(U,V,N)liquid
vapor
Clausius-Gibbs equation
Thermodynamic scale of temperature coincides with ideal-gas scale of temperature
59
Thermodynamic scale
Ideal-gas scale
• Historically the law of ideal gases was discovered empirically.• Later we will derive the law of ideal gases theoretically.
Unit of entropy
Temperature in the unit of energy•Definition: Entropy = log (number of sample points), = log .•Entropy is dimensionless.•Define temperature (Clausius-Gibbs equation): 1/ = d/dU.•Temperature has the same unit as energy.
Temperature in the unit of Kelvin•The committee adopts Kelvin as the unit for temperature T. kT.•To preserve the Clausius-Gibbs equation 1/T = dS/dU, we write S = k log.•The unit for entropy is the same as that of k: J/K.
60
Experimental determination of thermodynamic scale of temperature
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).
61
Theoretician manufacturer patient
Division of labor
Experimental determination of Energy-temperature curve
• 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
a family of isolated systemsProperties: T(U), S(U)
thermal system
T
Ufire
Practical calorimetryIncrease internal energy by a known amount without changing volume.
63
fire
Thermal system Thermal system Adiabatic system Adiabatic systemHow much heat? Ice calorimetry Frictional heating Joule heating
liquid
solid
Count the number of quantum states
64
T
U
S
U
T
1
T(U)
S(U)
• U, T, S are thermodynamic properties.• A thermal system is a system of a single independent variable. T(U), S(U).
Clausius-Gibbs equation
Heat capacity
65
A family of isolated systemsof a single independent variable, U
fire
thermal system
T
U
C
1
T(U)
Plan
• Calorimetry• Empirical observations of thermal contact• Theory of thermal contact• Refinements and applications. Ask what
temperature can do for you.
66
Thermal system
67
Define temperature (Clausius-Gibbs equation):
Define heat capacity:
U
state
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)
fire
thermal system
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).
T
T(U)
U
4 properties: USTC6 curves: US, UT, UC, ST, SC, TC
Measure one curve. Calculate the other five.
68
U
CT
U
T(U) C(U)
U
S(U)
S
intensive-extensive extensive-extensive 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).
Entropy as independent variable
69
S
U
1
T
S
T
U
2O
isolated systemT(S), U(S)
liquid
fire
vapor
thermal system
liquid
vapor
• 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)
Temperature as independent variable
70S
TT
U
C
1
U(T)S(T)
S
U
T
1
S(U)
Experimental control of temperature
71
water
ice
Ice-water mixture thermal bath (heat reservoir) thermostat
• 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.
Thermodynamic model of reservoir of energy
72
Reservoir of water Reservoir of energy
Potential energy PE Internal energy UR
Height h Temperature TR
Weight w Entropy SR
PE = hw UR = TRSR
Clausius-Gibbs equation:
Integration:
• (isolated system) = (small system) + (reservoir)• Internal variable: the internal energy of the small system U
• The isolated system conserves energy:
• The isolated system changes entropy:
• In equilibrium, the isolated system maximizes entropy:
A small system in equilibrium with a reservoir of energy
73
reservoirSR(UR) = SR(Ucomposite) – U/TR
small systemS(U)
heat U
• (isolated system) = (small system) + (reservoir)• Internal variables: the internal energy of the small system U, and something else Y
• Entropy is additive:
• Internal variables change to increase entropy (Clausius inequality):• Irreversible change in U because • Irreversible change in Y.
A small system in thermal contact, but out of equilibrium, with a reservoir of energy
74
reservoirSR(UR) = SR(Ucomposite) – U/TR
heat U = Q
liquid
vapor
S(U,Y)
• (isolated system) = (small system) + (reservoir)• Internal variables: the internal energy of the small system U, and something else Y
• Entropy is additive:
• Thermal equilibrium: TR = T. U(T,Y).
• At a fixed T, the internal variable Y changes to maximize S(U,Y) – U/T. Function (T,Y)
• Define the Helmholtz free energy (or Helmholtz function): F = U-TS. • Helmholtz function is an extensive property.• At a fixed T, the internal variable Y changes to minimizes F(T,Y).
Free energy
75
reservoirSR(UR) = SR(Ucomposite) – U/TR
heat U
liquid
vapor
S(U,Y)
Free energy, entropy, and temperature
76
A thermal system modeled by a function S(U)
Define temperature by the Clausius-Gibbs equation: dU = TdS
Define the Helmholtz free energy: F = U - TS
Calculus: dF = dU - TdS - SdT
dF = -SdT
T
F(T)U
Graphic derivation of the condition of equilibrium
77
energy U
entropy
1
Entropy is additive:
In equilibrium, the isolated system maximizes entropy:
reservoirSR(UR) = SR(Ucomposite) – U/TR
small systemS(U)
heat U
Convex function
78
T
UC
1
U(T)
S
T
1
S(U)Define temperature (Clausius-Gibbs equation):
Define heat capacity:
Equivalent statements•S(U) is convex•T(U) is an increasing function•C(U) > 0
U
Non-convex S(U) causes phase transition (i.e., discontinuous change of state)
79
U
S
U
T
Co-existent phases of a pure substance
80
water
ice
mixture of ice and water
latent heat
water
ice
fire
water
ice
isolated systemT(U), S(U)
thermal system
T
U
Tm
Thermodynamic theory of co-existent phases
81
ice
s'(u’) s’’(u’’)
ice
water
water
Each phase has its own s(u) function.
Rule of mixture corresponds to a line in the s-u plane
82
water
ice
isolated system
Isolated system conserves energy and maximizes entropy
83
water
ice
mixture of ice and water
latent heat
T
u
Tm
Coexistent phases correspond to a common tangent.
Free energy and co-existent phases
84
• Each phase has its own free-energy function. f’(T), f’’(T).• Mixture obeys the rule of mixture, f = (1-x)f’(T) + xf’’(T)• x changes to minimize the free energy. f’ = f’’
u’ - Tms’ = u’’ –Tms’’
T
f '(T)f
f ’’(T)
Tm
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 capacitance:
• Define free energy: F = U – ST.
• Thermal system: many states of thermodynamic equilibrium, one independent variation, five properties (USTCF)
• Thermodynamics provides a theory of co-existent phases.
85
