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THERMODYNAMICS
Lecture 5: Second Law ofThermodynamics
Thermodynamics
Second Law of Thermodynamics
In the course of discussions on the First Law of Thermodynamicswe concluded that all kinds of energy are equally useful, none ofthem is a preferred one, but in the case of isolated system it hasto be conserved.
We can however discern different kinds of energy such as mechanical work and internal energy, which can be changed throughapplied work. We discern also the heat, which is defined by internalenergy and work.
From the above we can conclude that both kinds of energy areequivalent. Second Law of Thermodynamics assumes the fact thatheat and work are not equivalent and provides a series of relations, supplementary to the First Law of Thermodynamics ininvestigations of thermodynamical systems.
∫ ∫ =+ 0WQ δδ
Thermodynamics
Thermodynamics
The first statement is reduced to a point that a cup of water in a fridge will not boil, despite it would be possible from the point ofview of energy conservation. The flow of heat is only in one direction, which is not a result of energy conservation.
3. A battery will discharge through the resistor releasing duringsuch process some heat, but the reverse process will not be possible.
4. It is not possible to construct a continuously operatingmachine which would cool one reservoir and perfom anequivalent amount of work.
2. Two gases in an isolated compartment will mixe homogeneouslyin the tank and will not spontaneously separate
1. Heat flows from higher temperature to lower temperature andnot the other way around!. A hot body will cool in a contact with a colder body and also not the other way around!.
Let’s remind ourselves four formulations of the second law andscrutinise first and fourth.
Thermodynamics
Statement number 4 says that construction of perpetuum mobileof the second kind is not possible. Perpetuum mobile of the first kind would be a device, which producesenergy without consideration of the I Law of Thermodynamics.
Summarising we can say that II Law of Thermodynamics assumes a one directional flow of heat and some pre-determined types ofenergy conversion.
We will attempt to formulate the II Law by providing analyticalrelations, basing on macroscopic arguments and assuming thefourth formulation as an experimental axiomat.
Thermodynamics
Second Law of Thermodynamics
1850 Clausius statement:It is impossible for any system to operate in such a way thatthe sole result would be an energy transfer by heat from a cooler to a hotter body.
T2
T1
Q2
Q1
T2 > T1
Q1 = Q2
Thermodynamics
Second Law of Thermodynamics
1851 Kelvin-Planck statement:It is impossible for any system to operate in a thermodynamicscycle and deliver a net amount of energy by work to itssurroundings while receiving energy by heat transfer from a single thermal reservoir.
Entropy statement:It is impossible for any system to operate in a way that entropy is destroyed.
T
W
Q = W
Thermodynamics
Reversible processes and cycles
Let’s define first the quasistatic process. That is a process whichtakes place so slowly that all the time the system is arbitrarily closeto the equilibrium.
When the process can be named a reversible?
If we consider a portion of gas in a cylinder then we could notice thea quasistatic expansion of that gas is related to performing work andremoval of heat. If we can return to initial conditions by adding thesame amount of heat and performing same amount of work then thegas in a cylinder can be compressed to the initial conditions. Hence itcan be said that a quasistatic process is a reversible process.
In other words the reversible process it is such a process after theoccurence of which the initial conditions can be recovered only by imposing a constraint condition removed at the beginning of theprocess.
Thermodynamics
And yet another formulation:
If we are capable of carrying out the process in such a way thatthe Second Law is not infringed then we can say that theprocess is reversible..
An irreversible process is such a process which is not reversible.
A reversible cycle is a sequence of constituting reversibleprocesses such that a system returns to the initial conditions in a periodical way.
Thermodynamics
Conversion of work into heatis a reversible process.
An inverse situation infringesthe formulation of SecondLaw due to Kelvin-Planck
Heat transport caused by a finite temperature differenceis an irreversible process. An inverse situation infringesthe formulation of Second Law due to Kelvin-Planck
T
W
Q = W
T2
T1
Q T2 > T1
Thermodynamics
We ought also define the power cycle, i.e. the cycle delivering workas a result of supplied heat, and refrigeration cycle, which isaccomplished as a result of supplied work.
A reversible power cycle can be converted to a reversiblerefrigeration cycle by inversion of the rate of heat and work.
The identity of Clausius statement and a Kelvin-Planck formulationof the Second Law can be shown in a following way.
Let’s assume that possible is a transfer of heat contrary to theClausius statement, i.e. directly from a lower temperaturereservoir to a higher temperature reservoir. Such picture can be supplemented by a reversible engine, as in the middle of picture. A result of adding of such processes is development of an enginewhich operates only using one heat source. Such process isexcluded by the Kelvin-Planck formulation of the Second Law
Thermodynamics
T2
T1
Q2
T2
T1
Q2
Q1
T2
T1
Q2-Q1
+W=Q2-Q1
=W=Q2-Q1
We have shown that violation of the Clausius formulationcauses violation of the Kelvin-Planck formulation of theSecond Law of Thermodynamics.
Thermodynamics
T2
T1
Q1
W=Q2-Q1
Q2
For the sake of completnesslet’s formulate also therefrigeration/heat pump cycle.
If we want to sort out thesign convention of heat andwork then:
Energy added to the system=Energy accumulated in thesystem+Energy removed from thesystem.
Thermodynamics
T2
T1
Q2A
Q1A
T2
T1
Q2B
Q1B
Let’s consider that we have at our disposal two reversible thermalmachines operating in a cyclic manner:
WA WB
Thermal efficiency ηt of a cyclic machine is defined in a followingway:
Thermodynamics
ηt = = useful energyenergy input
obtained workused heat
(6.2)
In a considered case that would be:
.
2QW
t =η(6.3)
Cycles A and B can be constructed in a differentmanner. Let’s assume that the efficiency of cycle A isgreater then that of cycle B, and
Q2A = Q2B
In such case WA > WB and Q1A < Q1B.Due to the fact that both machines are reversible the machine B can be inversed and be connected with machine A. In such way we obtain a situation depicted in the next slide.
Thermodynamics
T2
T1
Q2A
Q1A
WA WB
T2
T1
Q2B
Q1B
+ WA - WB
T2
T1
Q1B – Q1A
=
We can see that we would obtain a cycle whereWA - WB = Q1B – Q1A, violates the Kelvin-Planck formulation ofsecond Law. Therefore our assumption thatηA > ηB was incorrect.
Thermodynamics
Therefore we can conclude, that: „all reversiblethermal machines operating in the same temperaturerange have the same efficiency”.
2
1
2
12
2
1QQ
QQQ
QW
t −=−
==η (6.4)
We can also conclude that Q1/Q2 is a function of thesetemperatures. Therefore we would obtain a relation:
),( 212
1 TTfQQ
= (6.5)
We can show that,
)()(
2
1
2
1
TFTF
= (6.6)
where F is a new function.
Thermodynamics
Relation (6.6) can be obeyed by several functions F.Kelvin suggested to assume the simplest form of thatfunction, i.e.
2
1
2
1
TT
= (6.7)
And at the same time regard that equation as a definition of absolute thermodynamical temperature.
Efficiency of reversible thermal machine operatingbetween two heat reservoirs with temperatures TL –lower and TH – higher, is given by an expression;
H
L
t TT
−= 1η(6.8)
Thermodynamics
Clausius inequality
TH
d’QH
d’WZ
d’Q
d’WCCTL
Reversiblemachine
Cyclicmachine
Let’s consider a device whichreceives the heat d’QH from a tank with constant temperature TH andtransports that heat to thereversible machine Z producing workin the amount d’WZ.
The heat rejected by themachine Z feeds the cyclicmachine C producing workin the amount d’WC.Considering these twomachines as a single system, the total workaccomplished is equal to:
d’W =d’WZ + d’WC
Basing on the efficiency of reversible engine Z, we can write:
Z
Thermodynamics
QdWdTTQd
TTQdWd
C
H
H
HZ
''
)1(')1(''
=
−=−=
i.e.
TTQd
TTQdWd HH ')11('' =+−= (6.9)
In the case of a full cycle equation (6.9) assumes a form
∫∫ =TQdTWd
H
'' (6.10).
The machine presented in the schematic in the diagram cannotperform work as that is a contradictory process to the Kelvin-Planck statement.The device can only work with a cyclic input of work and cyclictransfer of heat to the tank.
Thermodynamics
Mathematically that means
∫ ≤ 0'Wd (6.11)
where d’W is a resulting work. We can also write that
0'≤∫ T
Qd (6.12)
The latter is named the Clausius inequality.
Up till now we did not considered the fact that the engine C canbe reversible. Let’s assume that it is so and that
∫ < 0'Wd
If C is a reversible engine then we get,
Thermodynamics
∫ > 0'Wd
That is not possible as we would develop a perpetuum mobile of thesecond kind.It results from here that in case of reversible processes inequation (6.12) the following relation must hold
0)'( =∫ odwrTQd
(6.13)
Macroscopic definition of entropy and the principle ofentropy increaseIn equation (6.13) the expression inside the integral must be an absolute differential of some function of state. Hence we can write
odwrTQddS )'(= (6.14)
That function S is entropy
Thermodynamics
That equation presents the macroscopic definition of entropy. Entropy is defined only for reversible processes and the changeof entropy can be calculated from the relation;
odwrTQdSSS )'(
2
112 ∫=−=Δ (6.15)
Let’s consider two arbitrary points of state of our system.
1
2
Irreversible process
Reversible process
Cycle=N+O
According to (6.1)
0''' 1
2
2
1
<+= ∫∫∫ TQd
TQd
TQd ON
The inequality sign has been used as the whole cycle is irreversible.
Thermodynamics
Recalling that
21
2
1
' SSTQd O −=∫
The last equation can be written as:
∫
∫
>−
<−+
2
1
12
21
2
1
'
0'
TQdSS
orSSTQd
N
N
In a general case we can write:
∫≥−2
112
'TQdSS (6.16)
The inequality sign is important in the case of irreversible processesand the equal sign in case of reversible equations
Thermodynamics
In case of adiabatic process d’Q = 0, hence S2 – S1 = 0. If that isgoing to be an adiabatic process then the change of entropy will be equal zerp. Such process is an isentropic process.
We can say that none of the real processes is reversible. If theprocess is irreversoble and adiabatic then entropy must increase.In case of isolated system,
0≥izolSΔ (6.17)
Basing on equation (6.14) we can find that for an arbitraryreversible isothermal process
STQ izotermodwr Δ=−(6.18)
In T - S system of coordinates the adiabatic process can be presented as reversible and irreversible.
.
.
Thermodynamics
1
22’
T
S
Adiabaticreversible process
Adiabaticirreversibleprocess
ΔSirreversible adiabatic.
The greater the increase ofentropy the more irreversibleprocess is. The reason for a smaller or greater irreversibilityof processes are various kinds offriction (stirring of the soup).
Entropy of a pure substanceWe have shown that entropy is a property of thermodynamicalsystem, an extensive property. It is the same property as totalenergy, internal energy and enthalpy. It can be calculated fromthe specific entropy.
Thermodynamics
smS = (6.19)
In case of pure substances the entropy can be tabulated in a same way as enthalpy, specific volume or other thermodynamicproperty. A twofold type of diagrams is usually presented, relation of temperature and entropy or enthalpy and entropy. Thelatter is named the Molier diagram (h-s).
Entropy of ideal gas
Basing on already derived relations,
dTcdhdTcdu
p
V
==
As well as information that in case of reversible process d’Q=Tdsand assuming that the ideal gas is a compressible liquid we canwrite:
Thermodynamics
dsTdvpduQd =+='i.e.
dvTp
Tduds += .
Using the ideal gas equation we obtain
vR
Tp
=
i.e.
vdvR
TdTcds V += .
In case of cV = const we obtain the expression describing thechange of entropy between two states of ideal gas
1
2
1
212 lnln
vvR
TTcss V +=− (6.20)
Thermodynamics
Such equation can also be written basing on the relations
pdpR
TdTcds
TdsvdpdhQd
p −=
=−='
as
,
2
1
2
112 lnln
ppR
TTcss p −=−
(6.21)
Both in equations (6.20) and (6.21) the change of entropy iscalculated between two states of thermodynamical system (p1,v1,T1) and (p2,v2,T2). As the entropy is a function of state , its change isnot dependent upon the course of the process.
