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Chapter 20

The Second Law of Thermodynamics 20.1 Directions of Thermodynamic Processes Thermodynamic processes that occur in nature are all irreversible processes. These are processes that occur spontaneously in one direction but not the other. Irreversible processes are all nonequilibrium processes, in that the system is not in thermodynamic equilibrium at any point until the end of the process. The second law of thermodynamics determines the preferred direction for such processes. A reversible process is an idealized process that a system is taken through such that the system is always very close to thermodynamic equilibrium within itself and with its surroundings. Reversible processes are equilibrium processes, with the system always in thermodynamic equilibrium. A cyclic process is a sequence of processes that eventually leaves the working substance in the same state in which it started. In an internal-combustion engine, such as that used in an automobile, the working substance is a mixture of air and fuel.

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20.2 Heat Engines A heat engine is a device that transforms heat energy partly into mechanical energy (work). A heat engine is a mechanical device by whose agency a system is caused to undergo a cycle where QH is larger than Qc and work

outengW is done by the system.

QH = heat energy absorbed by the system from the hot (at high temperature) reservoir during one cycle. Qc = heat energy rejected by the system into the cold (at low temperature) reservoir during one cycle. outengW = work done by the system during one cycle.

Conservation of energy yields

!

QH =Wengout + Qc

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The thermal efficiency e of a heat engine is defined as:

H

outeng

QW

e !

Combining the last two equations gives,

!

Wengout = QH " Qc so that the thermal efficiency becomes

!

e =QH " Qc

QH

!

!

e =1" Qc

QH

A heat reservoir is a body of such a large mass that it may absorb or reject an unlimited quantity of heat without suffering an appreciable change in temperature or any other thermodynamic coordinate. 20.6 Carnot Engine (1824) The Carnot engine is an idealized reversible engine operating between two reservoirs at different temperatures. The hot reservoir is maintained at a temperature TH and the cold reservoir is maintained at a temperature Tc . The maximum work that you can get out of an engine operating between two reservoirs is the work

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out of a Carnot engine operating between these same two reservoirs. For a Carnot engine, it turns out that the heat energies are related to the temperatures of the reservoirs by:

Qc

QH

=TcTH

(Carnot engine)

Thus, the thermal efficiency of a Carnot engine also equals

!

eCarnotmax =1"

TcTH

The PV diagram for a Carnot engine consists of two isotherms (one at TH and the other at Tc) and two adiabats.

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20.4 Refrigerators Heat engines are used to do useful work as energy flows from a hot place to a cold place. That is, a heat engine has a net output of mechanical work. A refrigerator is used to transfer energy from a cold place to a hot place. Thus energy must be spent for a refrigerator to operate. A refrigerator requires a net input of mechanical work. The working substance is a refrigerant fluid. QH = heat energy rejected by the system into the hot (at high temperature) reservoir during one cycle. Qc = heat energy absorbed by the system from the cold reservoir (at low temperature) during one cycle. inW = work done on the system during one cycle.

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Conservation of energy yields:

!

Qc +Win = QH The definition of Coefficient Of Performance, K, for a refrigerator (and air conditioner) is

!

Krefrigerator "Qc

Win

!

!

Krefrigerator =Qc

QH " Qc=

1QH

Qc"1

which for a Carnot refrigerator becomes

!

!

Kcarnotmax (refrigerator )= Tc

TH "Tc

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The compressor takes in fluid, compresses it adiabatically, and delivers it to the condenser coil at high pressure. The fluid temperature is then higher than that of the air surrounding the condenser, so the refrigerant fluid gives off heat energy |QH| and partially condenses to liquid. The fluid then expands adiabatically into the evaporator at a rate controlled by the expansion valve. As the fluid expands, it cools considerably, enough that the fluid in the evaporator coil is colder than its surroundings. It absorbs heat energy |Qc| from its surroundings, cooling them and partially vaporizing. The fluid then enters the compressor and begins another cycle.

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20.5 The Second Law of Thermodynamics In all of nature there is a one-way drive toward thermodynamic equilibrium. There is no way to undo these one-way drives. (a) mechanical one-way drive: frictional dissipation of kinetic energy into internal energy as heat. (b) thermal one-way drive: thermal energy flows from hot to cold. (c) chemical one-way drive: chemicals flow from high concentration to low concentration. This is diffusion. This is the general irreversibility statement of the second law of thermodynamics. A. Kelvin – Planck statement of the second law: the dissipation of work into heat cannot be undone. Therefore, it is impossible to build a perfect heat engine. A perfect engine would undo the dissipation of work into heat, violating the second law of thermodynamics.

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B. Clausius Statement of the second law: Heat flow from hot to cold cannot be undone. Hence, it is impossible to build a perfect refrigerator. A perfect refrigerator would undo heat flow from hot to cold, violating the second law of thermodynamics. 20.7 Entropy S Isolated systems tend toward disorder and entropy is a measure of that disorder. The second law of thermodynamics can be stated in terms of the concept of entropy, a quantitative measurement of the degree of randomness (or disorder) of a system. The one-way irreversibility of macroscopic processes in nature can be described by an inequality law. There is a

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quantity S called the entropy which has the following properties: (a) additive: S = S1 + S2 +… (b) each Si is a function of the thermodynamic state of the system. It is a function of temperature and volume. (c) The entropy is not conserved. It can only be created, but never destroyed. Entropy is created whenever a one-way process occurs. Hence,

0=!universetotalS (for a reversible process, i.e., not a one-

way process)

0>!universetotalS (when a one-way process occurs).

Here, Suniverse is the total entropy of the objects involved. Entropy is a state variable (like pressure, volume, and temperature). Thus S! depends only on the properties of the initial and final equilibrium states of the system. An isentropic process is one for which S! = 0.

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Entropy Changes in Irreversible Processes The change in entropy S! of a system as the system undergoes a quasistatic process between two equilibrium states is given by

1. This is so because the differential form of the entropy function is

!

dS =dQT

integrate both sides of the equation to obtain

!

"S =dQT

i

f

# .

2. More generally, one can show that the differential

form of the entropy function may also be written as

!

dS = n cvdTT

+dPdT" # $

% & ' VdV

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A. For an isochoric (isovolumetric) process of an ideal gas, dQ = n cv dT so that

!!"

#$$%

&===' ((

i

fv

T

T

v

f

i

v

TT

ncTdTnc

TdTcnS

f

i

ln

B. For an isobaric process of an ideal gas, dQ = n cp dT

!!"

#$$%

&===' ((

i

fp

T

T

p

f

i

p

TT

ncTdTnc

TdTcn

Sf

i

ln

20.8 Microscopic Interpretation of Entropy A microstate (or microscopic state) is a particular combination of numbers in an output. For example, in a throw of four coins on the floor, a description of a microstate of the system includes information about each coin: coin 1 was heads, coin 2 was heads, coin 3 was tails, coin 4 was heads. So in a toss of 4 coins there are six possible microstates in which half the coins are heads and half the coins are tails. The macrostate for the example above is 3 heads and 1 tail. Thus there are four microstates corresponding to the macrostate of 3 heads and 1 tail. See figure.

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Essential idea: The number of microstates associated with a given macrostate is not the same for all macrostates, and the most probable macrostate is that with the largest number of possible microstates (most disordered !). Macrostate of 2 heads and 2 tails with 6 microstates more disordered. Macrostate of 4 heads with 1 microstate more ordered.

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All physical processes tend toward more probable macrostates for the system and its surroundings. The mathematical relationship between entropy S and the number of microstates W associated with a given macrostate is

( )WkS B ln=