1 2 the second law of thermodynamics it will arouse changes while the heat transfers from low...
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1
2 The second law of thermodynamics
It will arouse changes while the heat transfers from low temp.substance to high temp. one.
Content2.1 Common characters of spontaneous process 2.2 The second law of thermodynamics2.3 Carnot cycle and Carnot law2.4 Entropy2.5 Clausius inequality2.6 Calculation of entropy change2.7 The nature of the second thermodynamics law2.8 Helmholz and Gibbs free energy2.9 Equilibrium condition2.10 G calculation2.11 Relation among thermodynamics functions2.12 Clausius-Clapeyron equation2.13 The 3rd thermodynamics law
3
2.1 Common character of spontaneous process
2.1.1 Some examples• Work changes into heat automatically;• Gas inflates toward vacuum;• Heat transfers from the high temp. object to the
low temp. one;• The solution of different concentration can mi
xed evenly;• The contrary process of these process can not proceed automati
cally. When it gets back to the original state by external force, the effect which can not be wore away will leave.
4
2.1.2 Spontaneous process & its common character
Spontaneous process (change) Some changes have the trend of sp
ontaneity, once happens, it can proceed automatically without the help of outside force.
Common character– irreversible, the contrary process of every spo
ntaneous process is non-spontaneous.
5
2.2 The 2nd law of thermodynamics
Clausius saying,
“It will arouse other changes while the heat is transferred from the low temp. object to the high temp. one”
Kelvin saying,
“It will arouse other changes while the heat from the single thermal source is taken out and is totally changed into work.”
• Ostward expression: “The 2nd kind of perpetual motion machine can not be made” 。
6
2.3 Carnot cycle and Carnot law
• In 1824, French engineer N.L.S. Carnot designed a Carnot cycle.
• Ideal gas absorbs heat Qh from Th thermal source, exports work W through ideal thermal machine, other heat Qc discharged to Tc thermal source.
2.3.1 Processes in Carnot cycle
7
2.3.1.1 Process 1
• The work is showed as the following area under the curve AB.
1
22
1
ln21 VVV
VRTpdVW hTT 2
12 WQQ h 01 U
Reversible expansion at Th from P1,V1 to P2 ,V2 , (A→B)
8
2.3.1.2 Process 2
Adiabatic reversible expansion from P2 ,V2 ,Th to P3,V3,Tc, (B→C)
Q=0
1
2,22
T
T mV dTCUW
The work is showed area under the curve BC.
9
2.3.1.3 Process 3
Reversible compress at Tc from P3,V3 to P4 ,V4 ,
(C→D)
03 U
4
33
4ln13
V
V VVRTpdVWcQQ 1
cTT 1
The work is showed as the area under the curve DC.
10
2.3.1.4 Process 4
Adiabatic reversible compress from P4 ,V4 ,Tc to P,V,T, (D→A)
0Q
2
1,44
T
T mV dTCUW
The work is showed as the area under the curve DA.
11
2.3.1.5 General heat & work
The whole cycle:
0UQ Q Q
ch
Qh >0
Qc <0.
W=
(W2 and W4 can be eliminated)
3
4
1
2 lnln 1231 VV
VV RTRTWW
12
2.3.1.6
According to the formula of the adiabatic reversible process
1
3c
1
2h
VTVT
4
3
1
2
V
V
V
V
1
4c
1
1h
VTVT
3
4
1
2 lnln 1231 VV
VV RTRTWW
1
2ln)( 12 VVTTR
Process 2:
Process 4:
2 divide 4:
13
2.3.2 Efficiency of the engine
Thermal machine absorbs heat Qh from Th source, part of heat is changed into work, other Qc go back to Tc source.
2h c
1
2h
1
( ) ln( )
ln( )
VnR T T
VV
nRTV
or
h c
h
c
h
1TT
T
T
T
1
)0(
ch
ch
hQ
Q
Q
W
14
2.3.3 Carnot lawCarnot law: The efficiency of the
reversible machine is the highest.Deduction of Carnot law: The efficiency of all r
eversible machines is the same.The importance of Carnot law:
(1) In principle inequality (ηI <ηR) has solved the direction problem of the chemical reaction;
(2) it solves the limitation problem of the thermal machine efficiency.
15
2.4 Entropy
Conclusion from Carnot cycle
h
c
h
c 11T
T
Q
Q
h
h
c
c
T
Q
T
Q
c h
c h
0QQ
T T
Or:
That is, in the Carnot cycle, the summation of heat effect and temperature is zero.
h
ch
h
ch
h T
TT
Q
Q
W
16
2.4.1 Thermal temperature quotient of any reversible cycle
The summation of every reversible cycle thermal temperature quotient is zero.
iR
i i
( ) 0Q
T
R( ) 0Q
T
or
•Adiabatic lines: RS &TU•ΔPVO= Δ OWQ •Same work, Δ U, Q
17
2.4.1.2 Thermal temperature quotient
• In many little Carnet cycles, adiabatic inflation line of the previous cycle is the adiabatic reversible compression line of the next, the work of these two process counteract.
• The total effect corresponds the closed curve, so the thermal temperature quotient addition is zero. Or the integral of cycle process is zero.
18
2.4.2 Derivation of entropy
R( ) 0Q
T
1 2
B A
R RA B( ) ( ) 0
Q Q
T T
Choose two points A, B from the reversible cycle curve at random, the cycle is divided into two reversible processes A→B and B → A.
19
2.4.2.2 Derivation of entropy Transposition
1 2
B B
R RA A( ) ( )
Q QT T
every reversible process
The thermal temperature quotient addition of every reversible process depends on the initial and final state, but it has nothing to do with the reversible approach, this thermal temperature quotient addition has the properties of state function.
20
2.4.3 Definition of entropy
For a little change
B
B A RA( )
QS S S
T
R( )i
i i
QS
T
or R( ) 0i
i i
QS
T
Rd ( )Q
ST
These entropy change formulas are called the definition of entropy, that is, the entropy change can be scaled by the thermal temperature quotient addition of the reversible process.
Clausius defined this state function as “entropy”.
Mark: S ; Unit: J.K-1 1
21
2.5 Clausius inequality
Suppose there is a reversible machine between the high and low thermal source.
h
c
h
chIR 1
Q
Q
Q
so
：
h
c
h
chR 1
T
T
T
TT
The same, there is an irreversible machine which have the same temperature with the previous one.
22
2.5.1 Thermal temp quotient addition of irreversible process
According to the Carnot law: IR R
0h
h
c
c T
Q
T
Qso
iIR
i i
( ) 0QT
If an irreversible process contacts with many thermal source:
23
2.5.1.2 Thermal temp quotient addition
of irreversible process An irreversible cycle has two parts.
A
IR,A B RBi
( ) ( ) 0Q Q
T T
A
R A BB( )
QS S
T
B A IR,A B
i
( )Q
S ST
A B IR,A Bi
( ) 0Q
ST
If A→B is a reversible process:
A B R,A Bi
( ) 0Q
ST
24
2.5.2 Clausius inequality
or
A B IR,A Bi
( ) 0Q
ST
A B R,A Bi
( ) 0Q
ST
A B A Bi
( ) 0Q
ST
Combine the two formulas together:
25
2.5.2.2 Clausius inequality
For a little change:
d 0Q
ST
dQ
ST
or
All these are called Clausius inequality, the math expression of the second law of thermodynamics.
26
2.5.3 Importance of Clausius inequality
The sign of inequality can be used as the criterion of changing direction and limitation in thermodynamics.
dQ
ST
“>” means irreversible process
“=” means reversible process
0d iso S “>” means spontaneous process“=” means equilibrium state
27
2.5.3.2 Importance of Clausius inequality
Sometimes we conclude the surrounding which has the close relationship of the system to judge the spontaneity of the process, that is:
“>” means spontaneous process“=” means reversible process
ΔSiso= ΔS(system)+ΔS(surrounding)≥0
28
2.5.4 Principle of entropy increasing
d 0S • Under the adiabatic condition, the process which approaches equilibrium makes the entropy increased. • Under the adiabatic condition, the process in which its entropy decreases can not happen.• As for an isolated system the principle can be stated as: the entropy of the isolated system never decrease.
For adiabatic system,
29
2.6 Calculation of entropy change
Entropy change of isothermal process
Entropy change of non-isothermal process
Entropy change of chemical process Entropy change of surrounding
Work out the entropy change by the relationship of thermodynamics
30
2.6.1 Isothermal process
(3) the mixed process of the ideal gas.
B BmixB
lnS R n x Example A
)(
)()(
gephase chanT
gephase chanHgephase chanS
(1) Ideal gas
)ln(1
2
V
VnRS )ln(
2
1
p
pnR
(2) the isothermal isobaric reversible change
T
QS R
31
Example 1
Under the same temperature, 1 mol ideal gas pass through:
(1) reversible inflation,
(2) vacuum inflation, the volume increases by 10 times, work out the change of entropy separately.
32
Answer
(1) Reversible inflation
1
2lnV
VnR
1ln10 19.14 J KnR
(1) It is a reversible process.
0)()()( gsurroundinSsystemSisolatedS
T
W
T
QsystemS
max
)()( R
33
Answer-2
Entropy is state function, the initial and the final state are the same, the change of system entropy are the same too,
So:
(2) It is an irreversible process
But the surrounding entropy do not change, so:
114.19)( KJsystemS
014.19)()( 1 KJsystemSisolatedS
34
2.6.2 Non-isothermal process
The isometric non-isothermal process of the fixed substantial quantity
2
1
dm,T
T
V
T
TnCS
2
1
dm,T
T
p
T
TnCS
The isobaric non-isothermal process
of the fixed substantial quantity
35
2.6.2.2 Non-isothermal process
(3) the fixed physical quantity change from p1,V1,T1 to p2,V2,T2.
Isothermal first and then isometric2
1
,m2
1
dln( )
T V
T
nC TVS nR
V T
Isothermal first and then isobaric2
1
,m1
2
dln( )
T p
T
nC TpS nR
p T
2 2,m ,m
1 1
ln( ) ln( )p V
V pS nC nC
V p
Isobaric first and then isometric
36
2.6.3 Chemical reaction
(1) under the standard pressure and 298.15K, every substance standard molar entropy can be found out in the table, we can work out the entropy of the reaction.
r m B mB
(B)S S $ $
37
2.6.3.2 Chemical reaction
(2) Under the standard pressure, work out the entropy of the reaction temperature T.
The entropy’s change at 298.15K can be found out from the table:
B ,mB
r m r m 298.15K
(B)d( ) (298.15K)
pT
C TS T S
T
$ $
38
2.6.3.3 Electrochemical reaction
Work out the entropy from the thermal effect of the reversible battery or the temperature change rate of the electromotive force.
T
QS R
mr
r m ( ) p
ES zF
T
39
2.6.4 Surrounding
(1) the changing surrounding entropy of every reversible changes
If the system heat effect is irreversible, however, because the surrounding is large enough, as to the surrounding, it can be considered as reversible heat effect
)(/)()( gsurroundinTsystemQgsurroundindS R
)(/)()( gsurroundinTsystemQgsurroundindS
40
2.6.5 The other calculation
According to the definition of the Gibbs free energy G H TS
TGHS
STHG
/)(
Apply this method to every thermo- dynamics equilibrium system.
For every isothermal changing process
41
2.7 The nature of the 2nd thermodynamics law
The irreversible properties of heat and work
• Heat is the confusion movement exhibition of molecules, while work is the regular movement result of molecules.
• Work transfers to heat, that is the regular movement changes to the irregular one, confusion increases, it is a spontaneous process.
42
2.7.2 The irreversible properties of the gas mixed process
• Put N2 and O2 separately into both sides of
the box with clapboard inside, then take out t
he clapboard, N2 and O2 mixed automatically t
ill it gets equilibrium.
• This is an confusion increasing process,also an
increasing process of entropy, it is a sponta
neous process, its contrary process will neve
r happen automatically.
43
2.7.3 The nature of the 2nd thermodynamics
• Every spontaneous process is irreversible. • Every irreversible process proceeds t
oward the confusion increasing, while entropy function can be considered as a measurement of the system confusion, this is the essence of the irreversible process.
44
2.8 Helmholz free energy and Gibbs free energy
Why new function is necessary? The state function of thermodynamics energy
is deduced by the 1st thermodynamics law. In dealing with the problem of the thermochemistry, the enthalpy is also defined. The state function of entropy is deduced by the 2nd thermodynamics law. However, when the entropy is used as criterion, the system must be isolated , that is, the entropy changes of both system and surrounding must be considered at the same time, but it is not so convenient.
45
2.8.1 Helmholz free energy
So it is necessary to bring in the new thermodynamics function of System itself to judge the direction and limitation of the spontaneous change.
F is called Helmholz free energy .
TS U F def
Helmholz define a state function:
46
2.8.1.2 Helmholz free energy
(Isothermal, reversible
SdTTdS dU dF
max)( δWdF T,R or
)( δWδQ dU SdTTdSδWδQ
maxδW
TdS)δQ
47
2.8.1.2 Helmholz free energy
That is: during the isothermal, reversible process, the most work which export to the surrounding by the system is equal to the decreasing value of the system Helmholz free energy, so F is called work function. If it is an irreversible process, the work which is done by the system is less than the decreasing value of F.
48
2.8.1.3 Helmholz free energy
If the system is under the isothermal, isobaric condition and do not export other works
“=” : reversible process
“>” : irreversible spontaneous process
The spontaneous change always proceeds toward the direction that Helmholz free energy decreases.
0)( 0 T,V,WfdF or 0)( 0 T,V,Wf dF
49
2.8.2 Gibbs free energy
Gibbs J.W., 1839-1903 define a state function
def G H TS
G is called Gibbs free energy, it is a state function, it has capacity properties.
50
2.8.2.2 Gibbs free energy
d d d dG H T S S T
)( pdVWVdp pdVδWδWδQ efe
VdpδWδQ f
SdTTdSVdpδWδQdG f maxf,δW
)( pVddU dH because
))( maxf,T,p,R δWdGor
reversibledpdT ,0,0(
51
2.8.2.2 Gibbs free energy
That is: in the isothermal, isobaric and reversible process, the most non-inflation work which exports to the surrounding by the system equals to the decrease value of system Gibbs free energy. If it is the irreversible process, the work which is done by the system less than the decrease value of system Gibbs free energy .
52
2.8.2.3 Gibbs free energy
If the system is in the condition of isothermal, isobaric and exports non-inflation work,
0)d( 0,, f WpTG
0)d( 0,, fWpTGor
“=”: reversible process “<”: irreversible spontaneous process. The spontaneous change always happens toward decreasing of Gibbs free energy.
dG is also called isothermal, isobaric potential.
53
2.8.2.4 Gibbs free energy
In the isothermal, isobaric and reversible battery reaction
n is the substantial quantity of electron in the battery reaction,E is the electromotive force of the reversible battery, F is the Faraday constant.
nEF maxf,r WG
54
2.9 Equilibrium condition
R( ) 0i
i i
QS
T
“=” means reversible, equilibrium
“<” means irreversible, spontaneity
“=” means reversible, equilibrium
“<” means irreversible, spontaneity
ΔSiso= ΔS(system)+ΔS(surrounding)≥0
0)( 0 T,V,WfdF
0)( 0 T,p,WfdG
55
2.10 ∆G calculation
2.10.1 G of isothermal physical change
(1) the ∆G of the change phase of the isothermal, isobaric and reversible reaction.
Because it do not export non-inflation work
0
pVFG
eδWdF VdppdVdFdG
VdppdVWe )0,( dppdVWe
56
2.10.1.2 ∆G of isothermal physical change
(2) Under the same temperature, the system changes from P1,V1 to P2,V2,
suppose Wf=0
pVd2
1
dp
pG V p
For the ideal gas:2 1
1 2
ln lnp V
G nRT nRTp V
VdppdVWdG e )( pdVWe
57
2.10.2 ∆G of isothermal reaction change
D E F Gd e f g
Gr
Fm
D E
ln lnf g
p d e
p pRT K RT
p pG ln lnp pRT K RT Q
D E F Gd e f g
ΔG1 ΔG3
ΔrG2=0
ΔrG
E
E
D
D
p
peRT
p
pdRT
''
1 lnlnG ''3 lnlnGG
G
F
F
p
pgRT
p
pfRT
Δ rG =ΔG1+ ΔG3
58
2.10.3 van’t Hoff isotherm
This is called as van’t Hoff isotherm, or chemical reaction isotherm. Kp is the eq
uilibrium constant Qp is the pressure ratio.
r m ln lnp pG RT K RT Q
59
2.10.3.2
• When QP<KP, ∆rGm<0, Reaction proceed towards the positive
way.
• When QP=KP, ∆rGm=0, Reaction is in the equilibrium state.
• When QP > KP, ∆rGm>0, Reaction proceed towards the opposite wa
y.
r m ln lnp pG RT K RT Q
60
2.11 The relation among thermodynamics functions
2.11.1 definition of thermodynamics function
pVUH TSUF TSHG pVFSo G
61
2.11.2 Figure relationship among thermodynamics function
函数间关系的图示式
Example C
HpVU
TS G
TS pVF
pVUH
TSHG pVF
TSUF
62
2.11.3 Four basic formulas
• This is the combine formula of the first and second law of thermodynamics, it can be used in the closed system which has invariable component and do not expert non-inflation work.
• Formula (1) is the most basic formulas among four.
d d dU T S p V
d dU Q p V because
2.11.3.1
63
2.11.3.2 2nd basic formula
d d dH T S V p (2)
pVUHbecause
d d d dH U p V V p
VpSTU ddd
pVSTHtherefore ddd
64
2.11.3.3 Four basic formulas
pdVSdT dF (3)
TSU Fbecause SdTTdSdUdF
pdVSdT dFtherefore
pdVTdSdU
65
2.11.3.4 Four basic formulas
(4) d d dG S T V p
TSSTHG dddd pVSTH ddd
TSHGbecause
VdpSdT dGtherefore
66
2.11.4 Derivative formulas
Deduce from formulas of (1), (2)
VpSTU ddd (1)pVSTH ddd (2)
( ) ( )V p
U H
ST
S
Deduce from formulas of (1), (3)
VdpSdTdG (4)
pdVSdTdF (3)
)()(V
F
V
U
S T
p
67
2.11.4.2 Derivative formulas
• Deduce from formulas of (2), (4)
( ) ( )S T
H G
pV
p
•Deduce from formulas of (3), (4)
VpSTU ddd (1)pVSTH ddd (2) VdpSdTdG
pdVSdTdF
(4)
(3)
)()(T
G
T
F
V pS
68
2.11.5 Character functions
Character function and their character variables.
VpSTU ddd (1)
pVSTH ddd (2)
( , ) G T p ( , )S H p ( , )H S p ( , ) U S V
),( VTF
VdpSdTdG
pdVSdTdF
(4)
(3)
69
2.11.6 Maxwell formulas
2.11.6.1 Review of full differential coefficient
( , )z z x y
d ( ) d ( ) dy x
z zz x y
x y
d dM x N y
dxdy
ydxxdydxdyydxxdy
xydyydxxdZ
))((
xdyZ x )( ydxZ y )(
70
2.11.6.2 Maxwell formulas
M and N are the x, y functions2 2
( ) , ( )x y
M z N z
y x y x x y
( ) ( )x y
M N
y x
so
( ) ( )VSpT
V S
VpSTU ddd (1)
( ) ( ) pST Vp S
pVSTH ddd (2)
71
2.11.6.2.2 Maxwell formulas
So, we can use these partial differential coefficient which can be measured in the experiments instead of those which can not be measured directly.
( ) ( ) pTS Vp T
pVTSG ddd (4)
pdVSdT dF )3( )()(T
p
V
S
T V
72
2.11.7 Application of Maxwell formulas
2.11.7.1 Relationship between U and V
According to 1st basic formula,
VpSTU ddd
( ) ( )T T
U ST p
V V
Work out the partial differential coefficient of V at the same temperature
( ) ( )T V
S p
V T
so( ) ( )T V
U pT p
V T
73
2.11.7.1.2 Why U=f (T) for ideal gas?
( )Vp nRT V
As we know, /pV nRT p nRT V
So, Ideal gas’s U is only the function of temperature.
( ) ( )VTpT pT
UV
0nRT pV
74
2.11.7.2 Relationship between H and p
According to 2nd basic formula,
d d dH T S V p
( ) ( )T T
H ST V
p p
Work out the partial differential coefficient of p at the same temperature
( ) ( )T p
S V
p T
( ) ( )T p
H VV T
p T
(for ideal gas)=V-T(nR/p)=V-V=0
75
2.12 Gibbs-Helmholtz equation
Relationship of temperature and ∆rG, ∆rF.
( )(1) [ ] p
G G H
T T
2
( )(2) [ ]p
GHT
T T
T
UF
T
F
])(
[V
)3(T
U
TTF
]
)([
V)4(
2
76
2.12.2 Gibbs-Helmholtz equation
The deducing of formula (1)
( )(1) [ ] p
G G H
T T
d d dG S T V p
According to the basis formula:
( ) p
GS
T
( )
[ ]p
GS
T
77
2.12.2 Gibbs-Helmholtz equation
According to the definition:
G H T S
G HS
T
S
o
therefore( )
[ ]p
G G H
T T
Under certain T:
G H TS
78
2.12.2.2 Gibbs-Helmholtz equation
2
( )[ ]p
GHT
T T
(1)two sides 1
T 2
1 ( )[ ]p
G G H
T T T
2 2
1 ( )[ ]p
G G H
T T T T
2
( )(2) [ ]p
GHT
T T
2d( ) dp
G HT
T T
79
2.13 Clausius-Clapeyron equation
2.13.1 Clapeyron equation
Under certain T and p, when two phases of
pure substances are in equilibrium, the change
rate of vapor pressure which changes with
temp. can be figure out by the following eq
uation:
dp / dT= ∆H / T∆V
80
2.13.2 Deducing of Clapeyron equation
Phase A and Phase B are balanceable in T and P. Ga=Gb. Changing dT and dP, and the system is balanceable again. △G=0.
Phase A Phase B T P Ga GbT+dT P+dP Ga +dGa Gb +dGb
Ga +dGa =Gb +dGb; dGa =dGb
dPVdTSdPVdTS 2211
12
12
VV
SS
dT
dP
VT
H
81
2.13.3 Clausius-Clapeyron equation
As to the gas-liquid equilibrium of two phases, 1 mol ideal gas, neglecting the liquid volume,
)/(g)(d
d mvap
m
mvap
pRTT
H
TV
H
T
p
vap m
2
d ln
d
Hp
T RT
If ∆vapHm has nothing to do with temp,
vap m2
1 1 2
1 1ln ( )
Hp
p R T T
82
2.14 Trouton’s Rule
Trouton concluded a approximate rule.
Most non-polarity liquids vapor at normal
boiling point Tb, their ∆vapHms are almost
constant.
vap m -1 1
b
85 J K molH
T
83
2.15 The 3rd thermodynamics law
2.15.1 Thermometric scale of thermodynamics
In the thermodynamics temp scale, temperature of three-phase-point of water is thought as 273.16 K. Choose 1/273.16 as the unit of thermo- dynamics temperature, which is called one Kelvin degree. Every system thermodynamics temperature is the value compared to it.
c
h
273.16KQ
TQ
84
2.15.2 The relationship among
∆H, ∆G and T in gathering system In 1902, T.W. Richard researched the
relationship of T with ∆G or ∆H, which were measured by some battery reaction. ∆G and H are almost the same size at low temperature.
0lim( ) 0T
G H
, H G 0KT
85
2.15.3 Nernst heat theorem
In 1906, Nernst put forward a supposition when he studied systematically the reaction of the coacervate system at quite low temperature
0 0lim( ) lim( ) 0p TT T
GS
T
In the isothermal process, when the temperature approaching 0 K, entropy of the system is unchangeable.
86
2.15.4 The 3rd thermodynamics law
• The temperature of the substance can not drop to 0 K by finite ways
• When the temp is approaching 0 K, the entropy of system is unchangeable.
• At 0 K, entropy of every integrity crystals is zero.
87
2.15.5 Conventional entropy
TTCS p d)/(d
0 0( / )d
TpTS S C T T
T
p TC0
lnd
89
Exercises
• P178-4, 6
• P179-10
• P180-21, 25
