Thermodynamics (Lecture Notes) -...
Transcript of Thermodynamics (Lecture Notes) -...
Thermodynamics (Lecture Notes)
Heat and Thermodynamics (7th Edition)
by
Mark W. Zemansky & Richard H. Dittman
2
Chapter 1
Temperature and the Zeroth Lawof Thermodynamics
1.1 Macroscopic Point of View
If no matter crosses the boundary, =⇒ a closed system.If an exchange of matter, =⇒ an open system.
Two points of view:
• Macroscopic: ∼ the human scale or larger.
• Microscopic: ∼ the molecular scale or smaller.
Macroscopic Coordinates: (provide a macroscopic description of a system)
1. No special assumptions (e.g., the structure of matter)
2. Few in number (to describe the system)
3. Fundamental (as suggested by our sensory perceptions)
4. Directly measurable
Including the macroscopic coordinate of temp. =⇒ Thermodynamics.
1.2 Microscopic Point of View
A microscopic description of a system:
1. Assumptions
2. Many quantities
3. Mathematical models
4. Theoretical calculation
=⇒ Statistical mechanics (Ch. 12)
3
4 CHAPTER 1. TEMPERATURE AND THE ZEROTH LAW OF THERMODYNAMICS
1.3 Macroscopic VS. Microscopic Points of View
=⇒ (Macroscopic description) = 〈Microscopic description〉ave
1.4 Scope of Thermodynamics
Thermaodynamic systems: a gas, a vapor, a mixture (e.g., vapor and air), surface films, electriccells, wire resistors, electric capacitors, and magnetic substances.
1.5 Thermal Equilibrium and The Zeroth Law
The 0th law of thermodynamics:
If A ≡ C and B ≡ C, ⇒ A ≡ B. (≡: thermal equilibrium)
1.6 Concept of Temperature
Isotherm: The locus of all points representing states in which a system is in thermal equilibriumwith one state of another system. (See Fig. 1-3)
=⇒ If (X1, Y1) ≡ (X ′1, Y
′1), (X2, Y2) ≡ (X ′
1, Y′1), (X3, Y3) ≡ (X ′
1, Y′1), · · · , =⇒ Isotherm I .
=⇒ If (X ′1, Y
′1) ≡ (X1, Y1), (X ′
2, Y′2) ≡ (X1, Y1), (X ′
3, Y′3) ≡ (X1, Y1), · · · , =⇒ Isotherm I ′.
The temperature of a system is a property that determines whether or not a system is in thermalequilibrium with other systems.
1.7 Thermometers and Measurement of Temperature
(See Fig. 1-4)θ(X) = a X (const. Y ) · · · · · · (1)
θTP = 273.16 K · · · · · · (2)
(1) =⇒ θ(XTP ) = a XTP(∗)=⇒ a = 273.16 K
XT P
=⇒ θ(X) = (273.16 K) XXT P #
(const. Y )
See Fig. 1-5 (Triple-pt. cell)
1.8 Comparison of Thermometers
In Table 1.1, =⇒ six thermometers.
For a gas, =⇒ θ(X) = (273.16 K) PPT P
(const. V ).
1.9. GAS THERMOMETER 5
For a wire resistor, =⇒ θ(R′) = (273.16 K) R′
R′
T P
(const. tension).
For a thermocouple, =⇒ θ(E) = (273.16 K) E
ETP(const. tension).
In Table 1.2, =⇒ choose a gas thermometers as the standard thermometer.
1.9 Gas Thermometer
See Fig. 1-6. (The volume of the gas is kept constant.)
1.10 Ideal-Gas Temperature
Ideal-gas law:
PV = n R T, · · · · · · (1)
where
n: the number of moles,R: the molar gas const.,T : theoretical thermodynamic temp. (Sec. 7.7).
Experiment: (T → θ)
(1) =⇒ PV = n R θ
PTP V = n R θTP , θTP = 273.16 K
=⇒ θ = (273.16 K) PPT P #
(const. V ) · · · · · · (2)
Measuring T : (at the normal boiling pt. of water)
1. With the triple-pt. cell =⇒ PTP = 120 kPa (suppose).With steam, measure PNBP =⇒ θ(PNBP ) = (273.16 K) PNBP
120 .
2. Remove some of the gas =⇒ PTP ↓, say, 60 kPa.Measure the new PNBP =⇒ θ(PNBP ) = (273.16 K) PNBP
60 .
3. Repeat the procedures 1 and 2.
4. Plot θ(PNBP ) − PTP ,
=⇒ T = limPT P →0
θ(PNBP ) = 273.16 K limPT P →0
P
PTP #
(const. V )
(See Fig. 1-7.)
6 CHAPTER 1. TEMPERATURE AND THE ZEROTH LAW OF THERMODYNAMICS
1.11 Celsius Temp. Scale
θ(0C) = T (K) − 273.15.
e.g.,θNBP (0C) = TNBP (K)
︸ ︷︷ ︸
373.124
−273.15 = 99.974 0C#
1.12 Platinum Resistance Thermometry
Range: 13.8033 ∼ 1234.93 K (−259.3467 ∼ 961.78 0C)
=⇒ R′(T ) = R′
TP (1 + at + bT 2), a, b : consts. (empirical formula)
1.13 Radiation Thermometry
Blackbody radiation (> 11000C)
1.14 Vapor Pressure Thermometry
Use 3He or 4He (isotopes of He). (0.3 ∼ 5.2 K)
1.15 Thermocouple
E = c0 + c1θ + c2θ2 + c3θ
3, c′is : consts. (−270 ∼ 1372 0C)
1.16 International Temperature Scale of 1990 (ITS-90)
(See Table 1.3)
ITS-90 = a set of defining fixed points measured with the primary gas thermometer+a set of procedures for interpolation between the fixed points using secondary thermometers.
1.17 Rankine and Fahrenheit Temp. Scales
T (R) = 95 T (K)
θ(0F ) = T (R) − 459.67
θ(0F ) = 95 θ(0C) + 32
Chapter 2
Simple Thermodynamic Systems
2.1 Thermodynamic Equilibrium
mechanical equil. + chemical equil. + thermal equil. =⇒ thermodynamic equil. state
2.2 Equation of State
For a closed system, the eq. of state relates the temperature to two other thermodynamic variables.
e.g., (a gas)
=⇒ PV = nRT (very low pressure),
or
Pv = RT, v(= V/n): molar volume (or volume per mole),
van der Waals eq.:
(P +a
v2(v − b) = RT (higher pressure).
XYZ systems =⇒ Simple systems (e.g., a gas, 1-dim stretched wire, 2-dim. surface,...)
2.3 Hydrostatic Systems
E.g., a solid, a liquid, a gas, or a mixture of any two.
=⇒ a PV T system
The eq. of state,
=⇒ V = func. of (T, P ) or V = V (T, P ) · · · · · · (1)
or
=⇒ P = func. of (T, V ) or P = P (T, V ) · · · · · · (2)
7
8 CHAPTER 2. SIMPLE THERMODYNAMIC SYSTEMS
or=⇒ T = func. of (P, V ) or T = T (P, V ) · · · · · · (3)
Exact differentials:
If dz is an exact differential of a func. of x and y, then
dz =
(∂z
∂x
)
y
dx +
(∂z
∂y
)
x
dy,
(1) =⇒ dV =
(∂V
∂T
)
P
dT +
(∂V
∂P
)
T
dP, · · · · · · (4)
If P =const,
=⇒ β = (∆V/V )∆T (coeff. of volume expansion)
∆ → ∂,
=⇒ β = 1V
(∂V∂T
)
P(coeff. of volume expansion)
If T =const,
=⇒ B = − ∆P(∆V/V ) (isothermal bulk modulus)
∆ → ∂,
=⇒ B = −V(
∂P∂V
)
T(isothermal bulk modulus)
=⇒ κ = − 1V
(∂V∂P
)
T(isothermal compressibility)
(2) =⇒ dP =(
∂P∂T
)
VdT +
(∂P∂V
)
TdV, · · · · · · (5)
(3) =⇒ dT =(
∂T∂P
)
VdP +
(∂T∂V
)
PdV, · · · · · · (6)
(4)−(6) =⇒ The dV , dP , and dT are differentials of actual functions. =⇒ exact differentials#.
2.4 Mathematical Theorems
(∂x
∂y
)
z
=1
(∂y/∂x)z
;
(∂x
∂y
)
z
(∂y
∂z
)
x
(∂z
∂x
)
y
= −1.
e.g., a PVT system,
(∗) =⇒(
∂P∂V
)
T
(∂V∂T
)
P
(∂T∂P
)
V= −1, β = 1
V
(∂V∂T
)
P& κ = − 1
V
(∂V∂P
)
T,
=⇒
(∂P
∂V
)
T
=
(∂V∂T
)
P
−(
∂T∂P
)
V
=β
κ#
2.5. STRETCHED WIRE 9
Therefore, (5) =⇒ dP =(
∂P∂T
)
VdT +
(∂P∂V
)
TdV = β
κ dT − 1κV dV.
If V = const, =⇒ dP = βκ dT,
∫=⇒
∫ Pf
PidP =
∫ Tf
Ti
βκ , dT, =⇒ Pf − Pi = β
κ (Tf − Ti)#
(V = const)
2.5 Stretched Wire
=⇒ a FLT system, F : tension (in N), L: length (in m), T : temp. (in K).
If T= const, (within the limit of elasticity)
=⇒ F = −k(L − L0), L0: the length at zero tension. (Hooke’s law)
Since L = L(T,F),
=⇒ dL =(
∂L∂T
)
FdT +
(∂T∂F
)
TdF ,
If F = const,
=⇒ α = 1∆T
(∆LL
)(linear coeff. of exansion)
∆ → ∂,
=⇒ α = 1L
(∂L∂T
)
F(linaer coeff. of expansion)
If T =const,
=⇒ Y = (∆F/A)(∆L/L) (Young’s modulus)
∆ → ∂,
=⇒ Y = LA
(∂F∂L
)
T(isothermal Young’s mdulus)
Since(
∂F∂L
)
T
(∂L∂T
)
F
(∂T∂F
)
L= −1
=⇒(
∂F∂T
)
L= −
(∂F∂L
)
T
(∂L∂T
)
F= −α A Y #
2.6 Surfaces
=⇒ a γAT system, γ: surface tension (in N/m), A: area (in m2), T : temp. (in K).
e.g.,
(1) For most pure liquids in equil. with their vapor phase,
10 CHAPTER 2. SIMPLE THERMODYNAMIC SYSTEMS
=⇒ γ = γ0(1 − T/Tc)n, γ0: the surface tension at 20 0C, Tc: critical temp., n: betw. 1-2.
(2) A thin filem of oil on water,
=⇒ (γ − γw) A = a T , γw: the surface tension of a clean water surface, a: a const.
2.7 Electrochemical Cell
=⇒ a EZT system, E : emf (in Volts), Z: charge (in coulombs C), T : temp. (in K).
Eq. of state, (by Exp.)
=⇒ E = E20 + α(θ − 200) + β(θ − 200)2 + γ(θ − 200)3,
where E20: the emf at 20 0C, θ: temp. in Celsius, α, β, γ: consts.
2.8 Dielectric Slab
=⇒ an E PT system,
where E: electric field (in V/m), P : tot. polarization emf (in C·m), T : temp. (in K).
Eq. of state,
=⇒ P
V = (a + b/T ), a, b: consts. (for T > 10 K)
2.9 Paramagnetic Rod
=⇒ a HMT system,
where H: magnetic field (in A/m), M: tot. magnetization (in A·m2), T : temp. (in K).
Eq. of state,
=⇒ M = CcH
T , Cc: Curie const.
2.10 Intensive and Extensive Coordinates
Intensive coords. (indept of the mass):
2.10. INTENSIVE AND EXTENSIVE COORDINATES 11
e.g., P , F , γ, E , E, H, T , density(ρ),...
Extensive coords. (propotional to the mass):
e.g., V , L, A, Z, E, P , M, mass(m), U , S, ...
=⇒ extensive × intensive = extensive
12 CHAPTER 2. SIMPLE THERMODYNAMIC SYSTEMS
Chapter 3
Work
3.1 Work
If work is done on the system, =⇒ W > 0.
If work is done by the system, =⇒ W < 0.
3.2 Quasi-Static Process
quasi-static process (thermodynamics)::
⇐⇒ massless springs (mechanics) or wires with no resistance (circuit)
3.3 Work in Changing the Volume of a Hydrostatic System
See Fig. 3-1 (quasi-static compression)
dW = F dx = PA dx, −dV = A dx
⇐⇒ dW = −PdV
=⇒ Wif = −∫ Vf
ViP dV (a quasi-static path i → f)
i ↔ f ,
=⇒ Wfi = −∫ Vi
VfP dV (a quasi-static path f → i)
= −Wif#
13
14 CHAPTER 3. WORK
3.4 PV Diagram
See Fig. 3-2 (a), (b), and (c).
3.5 Hydrastatic Work Depends on the Path
See Fig. 3-3,
i → a → f : W = −2P0 V0
i → b → f : W = −P0 V0
i → f : W = − 32P0 V0
=⇒ W is path-dependent.
=⇒ W is not a state function.
=⇒ W is an exact differential.
3.6 Calculation of∫
P dV for Quasi-Static Processes
Quasi-static isothermal expansion or compression of an ideal gas:
=⇒ W = −∫ Vf
ViP dV, PV = nRT
= −∫ Vf
Vi
nRTV dV
= −nRT∫ Vf
Vi
dVV = −nRT ln
Vf
Vi #
Quasi-static isothermal increase of pressure on a solid:
=⇒ W = −∫
P dV, dV =(
∂V∂P
)
TdP +
(∂V∂T
)
PdT = −κV dP
= −∫ Pf
PiκV P dP
= κV∫ Pf
PiP dP = κV
2 (P 2f − P 2
i )#
3.7. WORK IN CHANGING THE LENGTH OF A WIRE 15
3.7 Work in Changing the Length of a wire
dW = F dL, F = F(L, T ),
=⇒ W =∫ Lf
LiF dL
#
3.8 Work in Changing the Area of a Surface Film
dW = γ dA,
=⇒ W =∫ Af
Aiγ dA
#
3.9 Work in Moving Charge with an Electrochemical Cell
dW = E dZ,
=⇒ W =∫ Zf
ZiE dZ
#
3.10 Work in Changing the Total Polarization of a DielectricSolid
dW = E dP ,
=⇒ W =∫ Pf
PiE dP
#
3.11 Work in Changing the Total Magnetization of a Para-
magnetic Solid
dW = µ0H dM,
=⇒ W = µ0
∫ Mf
MiH dM
#
16 CHAPTER 3. WORK
3.12 Generalized Work
See Table 3.1 (Work of simple systems)
3.13 Composite Systems
See Figs. 3-8 & 3-9.
In general, a five-coords. system (Y , X , Y ′, X ′, and T ),
=⇒ dW = Y dX + Y ′ dX ′
=⇒ Choose T , X , and X ′ as indept coords.
17
18 CHAPTER 4. HEAT AND THE FIRST LAW OF THERMODYNAMICS
Chapter 4
Heat and the First Law ofThermodynamics
4.1 Work and Heat
4.2 Adiabatic Work
4.3 Internal-Energy Function
4.4 Mathematical Formulation of the First Law
4.5 Concept of Heat
4.6 Difference Form of the Firat Law
4.7 Heat Capacity and its Measurement
4.8 Specific Heat of Water; the Calorie
4.9 Equation for a Hydrostatic System
4.10 Quasi-Static Flow of Heat; Heat Reservoir
4.11 Heat Conduction
4.12 Thermal Conductivity and its Measurement
4.13 Heat Convection
4.14 Thermal Radiation; Blackbody
4.15 Kirchhoff’s Law; Radiation Heat
4.16 Stefan-Boltzmann Law
Chapter 5
Ideal Gas
5.1 Equation of State of a Gas
5.2 Internal Energy of a Real Gas
5.3 Ideal Gas
5.4 Experimental Determination of Heat Capacities
5.5 Quasi-Static Adiabatic Process
5.6 Ruchhardt’s Method of Measuring γ
5.7 Velocity of a Logitudinal Wave
5.8 The Microscopic Point of View
5.9 Kinetic Theory of the Ideal Gas
19
20 CHAPTER 5. IDEAL GAS
21
22 CHAPTER 6. THE SECOND LAW OF THE THERMODYNAMICS
Chapter 6
The Second Law of theThermodynamics
6.1 Conversion of Work into Heat and Vice Versa
6.2 The Gasoline Engine
6.3 The Diesel Engine
6.4 The Steam Engine
6.5 The Stirling Engine
6.6 Heat Engine; Kelvin-Planck Statement of the Second
Law
6.7 Refrigerator; Clausius’ Statement of the Second Law
6.8 Equivalence of the Kelvin-Planck and Clausius State-ments
6.9 Reversibility and Irreversibility
6.10 External Mechanical Irreversibility
6.11 Internal Mechanical Irreversibility
6.12 External and Internal Thermal Irreversibility
6.13 Chemical Irreversibility
6.14 Conditions for Reversibility
Chapter 7
The Carnot Cycle and theThermodynamic TemperatureScale
7.1 Carnot Cycle
7.2 Examples of Carnot Cycles
7.3 Carnot Refrigerator
7.4 Carnot’s Theorem and Corollary
7.5 The Thermodynamic Temperature Scale
7.6 Absolute Zero and Carnot Efficiency
7.7 Equality of Ideal-Gas and Thermodynamic Temperatures
23
24CHAPTER 7. THE CARNOT CYCLE AND THE THERMODYNAMIC TEMPERATURE SCALE
Chapter 8
Entropy
8.1 Reversible Part of the Second Law
8.2 Entropy
8.3 Principle of Caratheodory
8.4 Entropy of the Ideal Gas
8.5 TS Diagram
8.6 Entropy and Reversibility
8.7 Entropy and Irreversibility
8.8 Irreversible Part of the Second Law
8.9 Heat and Entropy in Irreversible Processes
8.10 Entropy and Nonequilibrium States
8.11 Principle of Increase of Entropy
8.12 Application of the Entropy Principle
8.13 Entropy and Disorder
8.14 Exact Differentials
25
26 CHAPTER 8. ENTROPY
Chapter 9
Pure Substances
9.1 PV Diagram for a Pure Substance
9.2 PT Diagram for a Pure Substance; Phase Diagram
9.3 PV T Surface
9.4 Equation of State
9.5 Molar Heat Capacity at Constant Pressure
9.6 Volume Expansivity; Cubic Expansion Coefficient
9.7 Compressibility
9.8 Molar Heat Capacity at Constant Volume
9.9 TS Diagram for a Pure Substance
27
28 CHAPTER 9. PURE SUBSTANCES
Chapter 10
Mathematical Methods
10.1 Characteristic Functions
10.2 Enthalpy
10.3 Helmholtz and Gibbs Functions
10.4 Two Mathematical Theorems
10.5 Maxwell’s Relations
10.6 T dS Equations
10.7 Internal-Energy Equations
10.8 Heat-Capacity Equations
29
30 CHAPTER 10. MATHEMATICAL METHODS
Chapter 11
Open Systems
11.1 Joule-Thomson Expansion
11.2 Liquefaction of Gases by the Joule-Thomson Expansion
11.3 First-Order Phase Transitions: Clausius-Clapeyron Equa-
tion
11.4 Clausius-Clapeyron Equation and Phase Diagrams
11.5 Clausius-Clapeyron Equation and the Carnot Engine
11.6 Chemical Potential
11.7 Open Hydrostatic Systems in Thermodynamic Equilib-
rium
31
32 CHAPTER 11. OPEN SYSTEMS
Chapter 12
Statistical Mechanics
12.1 Fundamental Principles
12.2 Equilibrium Distribution
12.3 Significance of Lagrangian Multipliers λ and β
12.4 Partition Function for Canonical Ensemble
12.5 Partition Function of an Ideal Monatomic Gas
12.6 Equipartition of Energy
12.7 Distribution of Speeds in an Ideal Monatomic Gas
12.8 Statistical Interpretation of Work and Heat
12.9 Entropy and Information
33
34 CHAPTER 12. STATISTICAL MECHANICS
Chapter 13
Thermal Properties of Solids
13.1 Statistical Mechanics of a Nonmetallic Crystal
13.2 Frequency Spectrum Crystals
13.3 Thermal Properties of Nonmetals
13.4 Thermal Properties of Metals
35