thermodynamics (lecture notes) -...

Thermodynamics (Lecture Notes)

Heat and Thermodynamics (7th Edition)

by

Mark W. Zemansky & Richard H. Dittman

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,


=⇒ 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

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.

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

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

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