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14 THERMODYNAMICS

103

14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 14.2 Macroscopic Description of Matter . . . . . . . . . . . . . . . . . . . . . . . . . 104 14.2.1. State variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 14.2.2. Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 14.2.3. Phase changes, phase diagrams . . . . . . . . . . . . . . . . . . . . . . . 106 14.2.4. Ideal Gas Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

15 Heat, the First Law of Thermodynamics

113

15.1 Work and Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 15.1.1Work done on/by ideal-gas processes . . . . . . . . . . . . . . . . . . . . . 114 15.1.2 Heat . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . 115

15.2 The First-Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 115 15.3 Thermal Properties of Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 15.3.1 Heat of transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 15.3.2 The specific heat of gases . . . . . . . . . . . . . . . . . . . . . . . . . . 117 15.3.3 More on adiabatic process . . . . . . . . . . . . . . . . . . . . . . . . . . 118 16 From Micro to Macro, Entropy of the 2nd Law of Thermodynamics

119

16.1 The Kinetic Theory of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 16.1.1 Maxwell speed Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 119 16.1.2 Mean Free Path (MFP): the average distance between collision . . . . . . 121 16.1.3 Microscopic origin of PRESSURE . . . . . . . . . . . . . . . . . . . . . . 121 16.1.4 Microscopic View of TEMPERATURE . . . . . . . . . . . . . . . . . . . 121 16.2 Thermal energy and specific heat . . . . . . . . . . . . . . . . . . . . . . . . . . 121 16.3 Thermal interaction & Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 16.4 Irreversible Processes, Entropy and the 2nd Law of Thermodynamics . . . . . . . 124

17 Heat Engines & Refrigerators

125

17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 17.2 Heat to work and work to heat . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

iv


17.3 Heat engines and refrigerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 17.4 Ideal-gas engines and refrigerator . . . . . . . . . . . . . . . . . . . . . . . . . . 127 17.5 The Carnot Cycle and the limit of efficiency . . . . . . . . . . . . . . . . . . . . 129

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Chapter 14 THERMODYNAMICS 14.1

Introduction

Thermodynamics is the science of energy & energy-conversion. •

Macroscopic description of matter State variables, temperature and the zeroth law of thermodynamics, phase change, ideal gas processes

Heat, and the First law of thermodynamics Heat as energy transfer, heat & work for ideal gas processes, 1st law of thermodynamics, thermal property of matter

From MICRO to MACRO, entropy, and the 2nd law of thermodynamics Molecular properties of gases, thermal energy and specific heat, the concept of entropy, 2nd law of thermodynamics

Heat engine & refrigerator Heat to work and vice versa, ideal gas engines, the Carnot cycle, limit of efficiency (perfect vs. real engine)


Chapter 14 Thermodynamics

14.2

104

Macroscopic Description of Matter

Thermodynamics deals with MACROSCOPIC systems, rather than the “particles”. It is all about energy and energy conversion, especially that of converting HEAT ENERGY into MECHANICAL WORK. (So, the word thermo-dynamics)

14.2.1.

State variables

The set of parameters used to characterize or describe the “state” of a macro-system. e.g., mass, volume, pressure, temperature, thermal energy, entropy… (are not all independent, however) Change of state variable: ∆x = x f − xi •

A system is in THERMAL EQUILIBRIUM if the state variables stay constant with time. Two or more systems are in thermal equilibrium with each other when their respective variables are unchanged upon making thermal constant.

If system A and B are each in thermal equilibrium with a third system, then A and B are in thermal equilibrium with each other. (zeroth law)

Mass density: ρ = M

V

,

Number density: N

V

m(12 C ) = 12 µ Atomic/Molecular mass: m(1 H ) = 1.0078µ ≈ 1µ , µ --atomic mass unit m(O2 ) ≈ 32µ

Moles and Molar Mass: 1 mol ≈ 6.02 × 1023 basic particles

N A = 6.02 ×1023 mol −1 --Avogadro’s number The number of moles in a substance containing N basic particles is n = N

NA

.


Chapter 14 Thermodynamics

105

The number of atoms in a system of mass M (in kg) is found by N =

M , m is the m

atomic mass.

The molar mass is the mass in grams of 1 mol. of substance. M mol (12 C ) = 12 g M mol (O2 ) ≈ 12 g

mol

mol

,

.

For a system of mass M consisting of atoms/molecules with molar mass M mol , the number of moles of the atoms/molecules in system is n =

M . M mol

Example The 12C atoms weigh 12 g by definition, so the mass of one 12C atom is m(12 C ) = 12 g

NA

= 1.993 × 10−26 kg .

On the other hand, we also defined that m(12 C ) = 12 µ , so 1µ =

m(12 C ) = 1.661× 10−27 kg . 12

For any other substance, one way to find its atomic mass is, e.g.

m(O2 ) = 32µ = 32 × 1.661× 10−27 kg = 5.315 × 10−26 kg

14.2.2. Temperature

A measure of system’s THERMAL ENERGY, the kinetic and potential energy of atoms/molecules in a system as they vibrate and/or move around. Two systems that are in thermal equilibrium have the same temperature. In other words, the temperature of a system is a property that determines whether or not a system is in thermal equilibrium with other systems! •

Temperatures scales 1. Kelvin scale ( K): Ttr = 273.16 K , T ( K ) ≥ 0


Chapter 14 Thermodynamics

106

9 2. Celsius and Fahrenheit: Tc = T − 273.15 , TF = Tc + 32 5 The temperature in Kelvin scale is adopted as fundamental in physics! It is sometimes called the absolute temperature scale. At the absolute zero temperature ( T = 0 K ), Eth = 0 . •

Measuring the temperature – thermometers Use the properties of a substance that vary with temperature. For example the pressure of a gas at constant volume, the electrical resistance of a wire, the length of a metal strip, the color of a lamp filament, etc.

Let X be a parameter property that is linearly dependent on T , T * = αX . At the triple point of water, 273.16 K = α X tr , from which, α = 273.16 at any other temperature, T * = (273.16 K )

X tr

is found. Then

X . X tr

The pressure in a constant-volume gas thermometer extrapolates to zero at

T0 = −273o C . This is the basis for the concept of absolute zero.

14.2.3. Phase changes, phase diagrams

A substance may change phase, e.g. from solid to liquid by heating. For example, water solidify (freeze) at the freezing point, but vaporize (boil) at the boiling point. At the freezing (melting) point, the solid phase (ice) and the liquid phase (water) are in Phase


Chapter 14 Thermodynamics

107

equilibrium, meaning that any amount of solid can coexist with any amount of liquid. Similarly, at the boiling (condensation) point, the liquid and vapor phases of water are in phase equilibrium. Note that only at those boiling and melting points that phase equilibrium can be maintained!

â&#x20AC;˘

A phase diagram is a diagram showing how the phases & phase changes of a substance vary with both temperature and pressure.

Examples

The following figures show the phase diagrams of water & CO2 . Three phases of matter are the solid, liquid and gas.


Chapter 14 Thermodynamics

108

The right figure below shows the temperature as a function of time as water is transformed from solid to liquid to gas.

14.2.4. Ideal Gas Processes

â&#x20AC;˘

Ideal Gas

The potential-energy diagram for the interaction of two atoms is shown in figure. Solid and liquid are systems where the atomic separation is close to req . A gas is a system where the average spacing of atoms is much greater than req , so atoms are usually not interacting.


Chapter 14 Thermodynamics •

109

An idealized hard-sphere model of the interaction potential energy of two atoms

A gas of atoms obeying such an interacting potential is called Ideal Gas. The ideal gas model can be good approximation of a real gas when its density is low and its temperature is well above the condensation point.

Molecular speed Atoms in a gas are in random motion at T > 0 K . The distribution of speed is outlined as follows. (a) The most probable speed vP = (b) The average speed vav =

2kT m

8kT πm

(c) The root-mean-square speed vrms =

3kT m


Chapter 14 Thermodynamics

110

A histogram showing the distribution of speeds in a beam of N 2 molecules at T = 20o C .

The Ideal Gas Law and Ideal Gas Processes The ideal gas law (thermal equilibrium): pV = nRT = Nk BT Universal gas constant: R = 8.31 J Boltzmann's constant: k B =

In a sealed container, we have gas is given by

mol ⋅ K

R = 1.38 × 10−23 J K NA

pV = nR = constant, and the number density of atoms in a T

N p . = V k BT

An ideal gas process is the means by which the gas changes from one state to another. The p-V diagram is a graph of PRESSURE against VOLUME. A point on the p-V diagram represents a unique state of a (sealed) gas.


Chapter 14 Thermodynamics

111

A quasi-static process is one that when the system changes state from, say 1 to 2, it is done so slowly that the system remains (approximately) at thermal equilibrium. Thus, a quasi-static process is reversible.

V f = Vi

(a)

Constant-Volume (ISOCHORIC) process:

(b)

Constant-Pressure (ISOBARIC) process: Pf = Pi

(c)

Constant-Temperature (ISOTHERMAL) process: T f = Ti

(d)

ADIABATIC (no heat transfer) process: Q = 0

Example

A gas at 2.0 atm pressure and a temperature of 200o C is first expanded isothermally until its volume has doubled. It then undergoes an isobaric compression until its original volume is restored. Find the final temperature and pressure.


Chapter 14 Thermodynamics

112

Solutions:

For process 1 → 2 , T2 = T1 = cons tan t . Hence, we have

P2V2 = P1V1 , or. P2 =

V1 P P1 = 1 = 1.0atm V2 2

For process 2 → 3 , P3 = P2 = 1.0atm . Since

V3 V2 = , we have T3 T2 T3 =

V3 V 1 1 T2 = 1 T1 = T2 = × (200 + 273.16) = 236.5 K = −36.5o C . V2 2V1 2 2


Chapter 15 Heat, the First Law of Thermodynamics 15.1 •

Work and Heat Work is the energy transferred to or from a system due to force acting on it over a distance.

Heat is the energy that flows between a system and its environment due to a temperature difference between them.

Energy conservation says: ∆Esys = ∆Emech + ∆Eth = Wext + Q , (Note: not ∆W &

∆Q !!) where E sys = E mech + Eth is the total energy of the system E mech = K + U is the mechanical energy associated with the motion of the system as a whole

(macroscopic E ), K is kinetic energy and U is potential energy.

Eth = K micro + U micro is the energy associated with the motion of atoms/molecules within the system (microscopic E ). It is one form of the “internal” energy.

Wext is the work done by external forces (environment). Q is the heat transferred to the system from its environment. Work and heat are the energies transferred between systems and the environment. They are NOT the state variables or state functions! Heat is transferred by


Chapter 15 Heat, the First Law of Thermodynamics

114

one of the following three mechanisms: (a)

Thermal conduction

(b)

Convection

(c)

Radiation

H = kA

∆T ∆x

I = σT 4

15.1.1 Work done on/by ideal-gas processes

Vf

The work done on a gas is defined by W = − ∫ pdV . It is the negative of the area under the curve Vi

between Vi and V f !

(a) Isochoric process ( V = const ): W = 0 (b) Isobaric process ( p = const ): W = − p∆V , ∆V = V f − Vi (c) Isothermal process ( T = const , pV = const ): Vf

W = −∫ Vi

Vf Vf Vf nRT dV = −nRT ln( ) = − piVi ln( ) = − p f V f ln( ) V Vi Vi Vi

(d) Adibatic process ( Q = 0 , pV γ = const , γ : ratio of specific heats) Vf

W = −∫ Vi

piVi γ dV = − piVi γ Vγ

Vf

Vi

piVi γ 1−γ pV V 1 dV (Vi − V f1−γ ) = i i [( i )γ −1 − 1] = ( P V − PV = − i i) γ γ −1 γ −1 Vf γ −1 f f V


Chapter 15 Heat, the First Law of Thermodynamics

115

The above expression equals to nCv ∆T . The work done during an ideal gas process depends on the path followed through the p-V diagram! The work done during these two ideal-gas processes is not the same.

15.1.2 Heat

Heat is the energy transfer, it is process-specific. One needs to distinguish heat from thermal energy and temperature.

THERMAL ENERGY is a form of energy of the system.

TEMPERATURE is a measure of “hotness” of the system. It is related to the thermal

energy per molecule. It is also a state variable.

HEAT is the energy transferred between the system and its environment as they

interact. It is NOT a particular form of energy, nor a state variable.

15.2 The First-Law of Thermodynamics It is about the conservation of energy of a thermodynamic system. A thermodynamic system is one where the internal energy is the only type of energy the system may have. So, we have ( ∆E mech = 0 )

∆Eint = W + Q If the change of internal energy is solely in the form of thermal energy, then the above


Chapter 15 Heat, the First Law of Thermodynamics

116

statement becomes ∆Eth = W + Q .

15.3

Thermal Properties of Matter

Here, we look at the consequences of ∆E th to a system, be the thermal energy change due to work W or heat Q.

Temperature change: ∆Eth = Mc∆T , M is the mass, and c is specific heat. Specific heat is the amount of energy that raises the temperature of 1 kg of a substance by 1 K. It is material specific. If ∆Eth = Q (i.e., W = 0 ), then Q = Mc∆T . Molar specific heat is the amount of energy that raises the temperature of 1 mol. of a substance by 1 K. Q = nC∆T , n is the number of moles of the substance and C is molar specific heat. For most elemental solids, C ~ 25 J

mol ⋅ K

.

Phase change, as characterized by a thermal energy change without changing the temperature. (Solid ↔ Liquid ↔ Gas)

Q = ML, where M is the mass and L is the heat of transformation.

15.3.1 Heat of transformation Heat of transformation is the amount of heat energy that causes 1 kg of a substance to


Chapter 15 Heat, the First Law of Thermodynamics

117

undergo a phase change. The heat of transformation for a phase change between a solid and a liquid is called Heat of Fusion ( L f ).The heat of transformation for a phase change between a liquid and a gas is called Heat of Vaporization ( Lv ). ⎧± ML f . Q=⎨ ⎩± MLv

15.3.2 The specific heat of gases For a gas, one needs to distinguish between the molar specific heat at constant volume Cv and the molar specific heat at constant pressure C p , where

Q = nCv ∆T (temperature change at constant volume “A”) Q = nC p ∆T (temperature change at constant pressure “B”) The thermal energy of a gas is associated with temperature, so the change of thermal energy

∆Eth will be the same for any two processes that have the same ∆T . Similarly, any two processes that change the thermal energy of the gas by ∆Eth will cause the same temperature change ∆T . Process A and B have the same ∆T and the same ∆E th , but they require different amounts of heat.

For process “A”, ( ∆Eth ) A = W + Q = Q = nC v ∆T


Chapter 15 Heat, the First Law of Thermodynamics

118

For process “B”, ( ∆Eth ) B = − p∆V + nC p ∆T So, nCv ∆T = − p∆V + nC p ∆T Since pV = nRT , ∆( pV ) = p∆V = ∆( nRT ) = nR∆T Thus, nCv ∆T = −nR∆T + nC p ∆T and C p = Cv + R , ∆E th = nCv ∆T . Remarks:

1. The change in thermal energy when temperature changes by ∆T is the same for any processes, i.e., ∆E th = nCv ∆T . 2. The heat required to bring about the temperature change depends on the process itself. It is different for different processes. (Heat depends on path, just like the work does!)

15.3.3 More on adiabatic process

As ∆E th = Q + W = nCv ∆T , so for an adiabatic process ( Q = 0 ), W = nCv ∆T .

As that dEth = dW nCv dT = − pdV = −nRT

Note that

dT R dV dV or =− V T CV V

Cp R C P − Cv = = γ − 1 , where γ = , the specific heat ratio (>1). Cv Cv Cv

Tf

Vf

dT dV = −(γ − 1) ∫ So ∫ T V Ti Vi

ln(

Tf Ti

) = ln(

For ideal gas, T =

Tf Vi γ −1 V ) = ( i )γ −1 or Vf Ti Vf

pV , so p f V fγ = piVi γ = const. . nR


Chapter 16 From Micro to Macro, Entropy of the 2nd Law of Thermodynamics 16.1 The Kinetic Theory of Gases A gas consists of a vast number of atoms/molecules ceaselessly colliding with each other and the walls of their container. For an ideal gas, (a) these atoms/molecules (referred to as “particles”) are in RANDOM motion and obey Newton’s laws of motion; (b) the total number of the particles is “large” and yet the volume occupied by these particles is negligibly small comparing to the volume the gas occupies; (c) no force acts on a molecule except during collision; (d) all collisions are elastic and of negligible duration.

16.1.1

Maxwell speed Distribution

Particles in a gas move randomly with different speeds. The distribution of speeds is described by the so-called Maxwell speed distribution: 3

m 2 2 − mv 2 2 kT N ( v ) = 4πN ( ) v e 2πkT ∞

N = ∫ N (v)dv 0


Chapter 16 From Micro to Macro, Entropy of the 2nd Law of Thermodynamics

120

1 2 mv , one may derive the Distribution molecule energies, the so-called 2

Note as E =

Maxwell-Boltzman energy distribution: N( E ) =

2N

1

π ( kT )

1

3

2

E e

E kT

2

N = ∫ N ( E )dE 0

2 kT m

The most probable speed ( dN dv = 0 ): v p =

The average speed v avg =

The root-mean-square (RMS) speed v rms = ( v 2 )avg = ( v 2 )avg =

1∞ ∫ vN ( v )dv = N0

1∞ 2 3 kT v N ( v )dv = ∫ N0 m

8 kT πm

3 kT m


Chapter 16 From Micro to Macro, Entropy of the 2nd Law of Thermodynamics

16.1.2

Mean Free Path (MFP): the average distance between collision 1

λ=

16.1.3

121

2πd ( N v ) 2

=

1

kT 2πd p 2

Microscopic origin of PRESSURE

The pressure of a gas exerted on the walls of a container in due to the steady rain of a vast number of atoms/molecules striking the walls. •

The force (averaged) exerted to the wall by a single atom upon collision is

Favg = •

2 mv avg

∆t coll

The total force due to collision of all atoms/molecules

N N m( v 2 )avg A m( v x2 )avg A = 3V V N 2 mvrms The pressure then is p = = NkT V 3V Fnet = N coll Favg =

16.1.4

Microscopic View of TEMPERATURE

The average translational kinetic energy per molecule is

ε avg =

1 1 2 m( v 2 )avg = mv rms 2 2

∵ v rms =

3 kT 3 ∴ ε avg = kT 2 m

So, temperature is simply a measure of translational kinetic energy per molecule!

16.2 Thermal energy and specific heat Thermal energy Eth = K micro + U micro

Monatomic Gases, U micro = 0

Eth = K micro = N ε avg = So, CV =

3 3 NkT = nRT 2 2

3 R = 12.5 J mol ⋅ K 2

For a monatomic gas, the energy is exclusively translational. As the translational motions are


Chapter 16 From Micro to Macro, Entropy of the 2nd Law of Thermodynamics

122

INDEPENDENT on the three space-coordinate axes, the energy is stored in 3 independent modes (degree of freedom). The thermal energy of a system of particles is equally divided among all the possible energy modes. For a system of N particles at temperature T , the energy stored in each mode (or 1 1 NkT or, in terms of moles, nRT . 2 2 1 A monatomic gas has 3 degree of freedoms, so Eth = 3 × nRT . 2

in each degree of freedom) is

Remarks: Solids: an atom in solid has 3 degrees of freedom associated with the vibrational kinetic energy, another 3 modes associated with the stretch/compress of the bonds (potential 1 energy), so Eth = 6 × nRT = 3nRT , C = 3R = 25.0 J . mol ⋅ K 2

Diatomic molecules (a) 3 modes for translational kinetic energy (b) 3 modes for rotational degrees of freedom (c) 2 modes for stretching/compressing bonds 1 So, it would suggest Eth = 8 × nRT , and C = 4 R which is inconsistent with 2 experiment. The reason is due to the “quantum effect”, which prevent 3 modes from being active. So, for diatomic molecules Eth =

5 5 nRT , C = R = 20.8 J mol ⋅ K 2 2


Chapter 16 From Micro to Macro, Entropy of the 2nd Law of Thermodynamics

123

16.3 Thermal interaction & Heat Atoms in system 1 have higher kinetic energy than those in system 2 ( T1 > T2 ), upon collision between a “fast” atom in 1 and a slow atom in 2, energy is transferred. So, heat is the energy transferred via collisions between the more energetic (warmer) atoms on one side and the less energetic (cooler) atoms on the other. At thermal equilibrium, there is no net energy transfer, or atoms at both sides have the same average translational kinetic energy.

(ε 1 ) avg = (ε 2 ) avg

∵ ε avg =

3 kT 2

∴ T1 = T2 = T f (at thermal equilibrium) Two thermally interacting systems reach a common final temperature by exchanging energy via collisions, until atoms on each side have, on average, equal translational kinetic energies!

∵ Eth = Nε avg Therefore, at equilibrium,

then E1 f =

E1 f N1

=

E2 f N2

=

Etot , Etot = E1i + E 2i N1 + N 2

N1 N2 Etot . Etot , E2 f = N1 + N 2 N1 + N 2

Hence, the heat flowed from “1” to “2” is Q = ∆E2 = E2 f − E2i .


Chapter 16 From Micro to Macro, Entropy of the 2nd Law of Thermodynamics

124

16.4 Irreversible Processes, Entropy and the 2nd Law of Thermodynamics An irreversible process is one that happens only in one direction. e.g., heat is transferred from the warmer side to the cooler side, but not the other way. Why? After all, heat transfer is by collision, a micro-process that is irreversible! It lies with the “probability”. Reversible microscopic events lend to irreversible macroscopic behavior because some macroscopic states are vastly more probable than others. The equilibrium is actually the MOST probable state in which to be! •

Entropy is a variable that measures the “orderedness” of a system. It is a measure of the probability that a macroscopic state will occur. ∆S = Q (reversible, isothermal) T

The 2nd Law of Thermodynamics The entropy of an isolated system never decreases. It either increases until the system reaches equilibrium, or if the system is in equilibrium, stays the same. An isolated system never spontaneously generates order out of randomness.

∆S ≥ 0


Chapter 17 Heat Engines & Refrigerators 17.1 Introduction •

Heat engine is a device that uses a cyclical process to transform heat energy into work.

Refrigerator is a device that uses work to move heat from a cold object to a hot object.

17.2 Heat to work and work to heat •

Energy-transfer-diagram

(1) Energy reservoir is an object or part of environment so large that its temperature and thermal energy do not change when heat is transferred between the system


Chapter 17 Heat Engines & Refrigerators

126

and the reservoir. (2) A reservoir at a higher temperature than the system is called Hot reservoir. (3) A reservoir at a lower temperature than the system is called Cold reservoir.

The First Law of Thermodynamics: Q = QH − QC = W + ∆Eth •

Work to heat is easy and straight forward, which can have 100% efficiency.

Heat to work is difficult, the 2nd law makes the transfer efficiency <100%.

The reason lies on the fact that for a practical device that transform heat into work must return to its initial state at the end of the process and be ready for continued use!

17.3

Heat engines and refrigerators

For any heat engine, the closed-cycle device periodically return to its initial conditions, so ( ∆Eth ) net = 0 for a full cycle. Therefore, W = Qnet = Q H − QC . The engine’s THERMAL EFFICIENCY η =

Q W = 1− C . QH QH

For a refrigerator, the close-cycle device uses external work to extract heat from a cold reservoir and exhaust heat to a hot reservoir. Again, ∆Eth = 0 . So, QH = QC + W .


Chapter 17 Heat Engines & Refrigerators

The refrigerator Coefficient of Performance K =

127 QC . W

The 2nd law suggests that there is no perfect refrigerator with K = ∞ ! It also suggests there is no perfect heat engine with η = 1 . For the 1st statement, K = ∞ ⇒ W = 0 , then it implies the refrigerator spontaneously draw heat from cold reservoir to hot reservoir! For the 2nd statement, if η = 1 , then QH 1 = W , using this work, we draw heat from a cold reservoir, so QH 2 = QC 2 + W , which equivalently drawing heat from “cold” to “hot” spontaneously.

17.4

Ideal-gas engines and refrigerator

We use a gas as the working substance, and the close-cycle is represented by a closed-cycle trajectory in the p − V diagram. The net work done for such a closed-cycle is simply the area inside the closed curve.


Chapter 17 Heat Engines & Refrigerators

128

Summary of ideal gas processes Process

Gas Law

Isochoric

pi

Isobaric

Vi

Ti Ti

=

pf

=

Vf

Work W

Heat Q

Thermal Energy

0

nCV ∆T

∆E th = Q

− p ∆V

nC p ∆T

∆Eth = Q + W

Q = −W

∆Eth = 0

0

∆Eth = W

Tf Tf

V − nRT ln( f piVi = p f V f

Isothermal

Adiabatic

Vi

piViγ = p f V fγ

( p f V f − piVi )

TiVi γ −1 = T f V fγ −1

nCV ∆T

piVi

Any

V − pV ln( f

Vi

Ti

=

p fVf Tf

)

)

γ −1

∆Eth = nCV ∆T

−(area under PV curve)

Properties of monatomic and diatomic gases Monatomic

Diatomic

Eth

3 nRT 2

5 nRT 2

CV

3 R 2

5 R 2

Cp

5 R 2

7 R 2

γ

5 = 1.67 3

7 = 1 .4 5

Note: For refrigerator, as the heat always transfers from hotter object to a colder object, it has to use the ADIBATIC process to lower the temperature of the gas to below TC and to increase it to TH .


Chapter 17 Heat Engines & Refrigerators

17.5

129

The Carnot Cycle and the limit of efficiency

A perfectly reversible engine is one that can be operated as either a heat engine or a refrigerator between the same two energy reservoirs and with the same energy transfer, with only their direction changed.

A perfectly reversible engine has MAXIMUM efficiency. Otherwise, it again violate the 2nd law. Similarly, no refrigerator can have a coefficient of performance larger than that of a perfectly reversible refrigerator.

A perfectly reversible engine must use only two types of processes: (1) Frictionless mechanical interactions with no heat transfer, Q = 0 (2) Thermal interactions in which heat is transferred in an isothermal process, â&#x2C6;&#x2020;Eth = 0 The engine that uses only there two types of processes is called CARNOT engine.


Chapter 17 Heat Engines & Refrigerators

130

Carnot engine has maximum thermal efficiency η max , and it operated as a refrigerator has the maximum coefficient of performance K max . •

The Carnot cycle We now analyze the Carnot cycle and find the η max .

The Carnot cycle is an ideal gas cycle that consists of two adiabatic ( Q = 0 ) and two isothermal ( ∆Eth = 0 ) processes.

η =1−

QC QH

QC = Q12 = nRTC ln(

V1 ) V2

Q H = Q34 = nRTH ln(

V4 ) V3

For adiabatic process, TC V2γ −1 = TH V3γ −1 , TC V1γ −1 = TH V4γ −1 . So,

V1 V4 T = and η max = 1 − C . V2 V3 TH

For refrigerator, KCarnot =

TC . TH − TC

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