IDEAL-GAS MIXTURE
• I am teaching Engineering Thermodynamics to a class of 75 undergraduate students.
• I went through these slides in one 90-minute lecture.
Zhigang Suo, Harvard University
Plan
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Ideal gas, a review
PVT relation of ideal-gas mixture
Mixing (TV-representation)
Mixing (TP-representation)
Adiabatic mixing
Steady-flow, adiabatic mixing
Isentropic mixing. Stop generating entropy!
Do work.
2
Law of ideal gases
Oscar Wilde: We are all in the gutter, but some of us are looking at stars.
We all generate entropy, but some of us are doing work.
mechanics
chemistry
PV = Nt
geometry
•
•
•
•
P = pressure
V = volume
N = number of molecules
t = temperature in the unit of energy
thermometry
Boyle (1662)-Mariotte (1679) law. PV = constant for a fixed amount of gas and fixed temperature.
Charles’s law (1780). V/t = constant for a fixed amount of gas and fixed pressure.
Avogadro’s law (1811). V/N = constant for all gases at a fixed temperature and fixed pressure.
Clapeyron (1834) combined the above laws into the law of ideal gases.
3
Human folly
To every beautiful discovery, we add many ugly ideas.
Generating entropy is natural.
The discovery
Number of molecules
mole
PV = NkBT
PV = nRT
PV = Nt
Ugly idea 1
Kelvin temperature kBT = t
Boltzmann constant
-23
kB = 1.38´10
J/K
Mass
PV = mRT
Ugly idea 2
Avogadro constant
NAvogadro = 6.022 x 1023
Mole n = N/NAvogadro
Ugly idea 3
Specific gas constant
R = R / M = kB / mmolecule
Universal gas constant
(
R = kB N Avogadro = 8.314J/ K ×mole
Gas
Formula
Molar mass, M
kg/kmol
Air
)
R
kJ/kgK
0.2870
Steam
H2O
18
0.4615
Hydrogen
H2
2
4.124
4
Model a closed system as a family of isolated systems
weight
closed system
vapor
2O
vapor
liquid
Isolated system
liquid
fire
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•
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Each member in the family is a system isolated for a long time, and is in a state of thermodynamic
equilibrium. The system can have many species of molecules. A state can have coexistent phases.
Change state by fire (heat) and weights (work).
2 independent variables name all members of the family (i.e., all states of thermodynamic equilibrium).
6 functions of state: TVPUSH
4 equations of state.
The basic task: Obtain S(U,V) from experiment or theory.
•
Definition of temperature (Gibbs equation 1)
•
Definition of pressure (Gibbs equation 2)
•
Definition of enthalpy
(
)
(
)
1 ¶S U ,V
=
T
¶U
P ¶S U ,V
=
T
¶V
H = U + PV
5
Law of ideal gases derived from
molecular picture and fundamental postulate
When molecules are far apart, the probability of finding
a molecule is independent of the location in the
container, and of the presence of other molecules.
Number of quantum states of the gas scales with VN
Definition of entropy S = kBlog W
(
)
(
W U ,V , N = V N x U, N
)
(
S = kB N logV + kB log x U, N
(
)
Gibbs equation 1:
1 ¶S U ,V , N
=
,
T
¶U
Gibbs equation 2:
P ¶S U ,V , N
=
,
T
¶V
(
)
)
( )
U = Nu T
PV = NkBT
6
4 equations of state
Change state
T0,V0
T,V
2 independent variables (T,V) name all states of thermodynamic equilibrium.
4 equations of state: PUSH
PV = nRT , R = 8.314J/K × mole
U T ,V ,n = n éëu T0 + cv T -T0 ùû
é
æT ö
æ V öù
S T ,V ,n = n ês T0 ,V0 + cv log çç ÷÷ + R log çç ÷÷ú
êë
è T0 ø
è V0 øúû
H T ,V ,n = n éëh T0 + cP T -T0 ùû, cP = cv + R
(
)
( )
(
)
(
(
)
(
)
(
)
)
( )
7
Plan
•
•
•
•
•
•
•
Ideal gas, a review
PVT relation of ideal-gas mixture
Mixing (TV-representation)
Mixing (TP-representation)
Adiabatic mixing
Steady-flow, adiabatic mixing
Isentropic mixing. Stop generating entropy!
Do work.
8
Two species of molecules
B
A
Number of moles of species A
nA
Number of moles of species B
nB
Number (mole) fraction of species A:
yA =
Number (mole) fraction of species B:
yB =
Algebra:
nA
n A + nB
nB
n A + nB
y A + yB = 1
9
Dalton’s law (1801)
Dalton’s law:
(
)
PV = n A + nB RT
Partial pressures: PA = y A P, PB = yB P
Total pressure: P = PA + PB
T,V,nA
PA
T,V,nB
PB
T,V,,nA,nB
PA + PB
Boxes of the same volume and temperature
10
Dry air (no water)
name
formula
Molar mass M
kg/kmol
number fraction y
nitrogen
N2
28
0.78
oxygen
O2
32
0.21
(
)
PV = (m A + mB ) RmixT
PV = n A + nB RT
Rmix =
n A + nB
m A + mB
Rair »
R=
n A + nB
n A M A + nB M B
R=
R
y A M A + yB M B
R
» 0.27kJ/kgK
yN M N + yO MO
2
2
2
2
11
Molecular picture of an ideal-gas mixture
U,V,S,P,T,NA,NB,W
When molecules are far apart, the probability of
finding a molecule is independent of the location in
the container, and of the presence of other
molecules.
B
A
(
)
(
Number of quantum states of the gas scales with volume as: W U ,V , N , N = V N A V N B x U , N , N
A
B
A
B
Definition of entropy S = kBlog W S = kB ( N A + N B ) logV + kB log x (U, N A , N B )
(
)
Gibbs equation 2:
P ¶S U ,V , N A , N B
=
T
¶V
Delton’s law:
PV = N A + N B kBT
(
)
PV = (n A + nB ) RT
12
)
Plan
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•
•
•
Ideal gas, a review
PVT relation of ideal-gas mixture
Mixing (TV-representation)
Mixing (TP-representation)
Adiabatic mixing
Steady-flow, adiabatic mixing
Isentropic mixing. Stop generating entropy!
Do work.
13
Energy and entropy of mixing
Ta,Va,nA
T,V,nA,nB
mix
Tb,Vb,nB
(
)
( )
(
)
(
( )
(
)
( )
( )
U T ,V ,n A ,nB ¹ n A u A Ta + nB uB Tb , U mix = U T ,V ,n A ,nB - éën A u A Ta + nB uB Tb ùû
)
(
)
(
)
(
)
(
)
S T ,V ,n A ,nB ¹ n A s A Ta ,Va + nB sB Tb ,Vb , Smix = S T ,V ,n A ,nB - éën A s A Ta ,Va + nB sB Tb ,Vb ùû
14
Internal energy of an ideal-gas mixture
At a fixed temperature, mixing two ideal gases do not change internal energy.
T,Va,nA
T,V,nA,nB
Mix at a
constant temperature
T,Vb,nB
(
)
( )
( )
U T,V ,n A ,nB = n Au A T + nBuB T
15
Internal energy of mixing
Change state of pure A
T,Va,nA
Ta,Va,nA
Mix at
constant temperature
T,V,nA,nB
Change state of pure B
Tb,Vb,nB
T,Vb,nB
(
)
( ) (
)
U mix = n AcvA (T -Ta ) + nBcvB (T -Tb )
( )
(
)
U T ,V ,n A ,nB = n A éëu A Ta + cvA T -Ta ùû + nB éëuB Tb + cvB T -Tb ùû
16
Entropy of an ideal-gas mixture
At a fixed volume and a fixed temperature, mixing two ideal gases do not change entropy
Mix at
constant temperature
constant volume
T,V,nA
Isentropic mixing
T,V,nA,nB
T,V,nB
W = W AW B
S = S A + SB
(
)
(
)
(
)
(
)
S T ,V ,n A ,nB = n A s A T ,V + nB sB T ,V
é
é
æT ö
æ V öù
æT ö
æ V öù
S T ,V ,n A ,nB = n A ê s A Ta ,Va + cvA log çç ÷÷ + R log çç ÷÷ú + nB êsB Tb ,Vb + cvB log çç ÷÷ + R log çç ÷÷ú
êë
êë
è Ta ø
è Va øúû
è Tb ø
è Vb øúû
(
)
(
)
17
Entropy of mixing
Change state of pure A
Ta,Va,nA
Mix at
constant temperature
constant volume
T,V,nA
T,V,nA,nB
Change state of pure B
Tb,Vb,nB
T,V,nB
é
é
æT ö
æ V öù
æT ö
æ V öù
ê
ú
ê
S T ,V ,n A ,nB = n A s A Ta ,Va + cvA log çç ÷÷ + R log çç ÷÷ + nB sB Tb ,Vb + cvB log çç ÷÷ + R log çç ÷÷ú
êë
êë
è Ta ø
è Va øúû
è Tb ø
è Vb øúû
é
é
æT ö
æ V öù
æT ö
æ V öù
Smix = n A êcvA log çç ÷÷ + R log çç ÷÷ú + nB êcvB log çç ÷÷ + R log çç ÷÷ú
êë
êë
è Ta ø
è Va øúû
è Tb ø
è Vb øúû
(
)
(
)
(
)
18
Enthalpy of an ideal-gas mixture
H = U + PV
(
)
PV = n A + nB RT
( )
( )
U = n A u A T + nB u B T
( )
( )
( )
( ) + RT , hB (T ) = uB (T ) + RT
(
)ùû + nB éëhB (Tb ) + cPA (T - Tb )ùû,
H = n Ah A T + nBhB T , hA T = u A T
H T ,V ,n A ,nB = n A éëhA Ta + cPA T -Ta
(
)
( )
cPA = cvA + R, cPB = cvB + R
19
Ideal-gas mixture (using mole)
T,V,nA,nB
4 independent variables (T,V, nA, nB) name all states of thermodynamic equilibrium.
4 equations of state: PUSH
(
)
PV = n A + nB RT
( )
( )
S = n A s A (T ,V ) + nB sB (T ,V )
H = n Ah A (T ) + nBhB (T )
U = n A u A T + nB u B T
20
Ideal-gas mixture (using mass)
m
, nB = B
MA
MB
R
R
RA =
, RB =
MA
MB
nA =
(
)
PV = n A + nB RT
(
mA
)
U = m AcvA + mBcvB T
é
é
æT ö
æ V öù
æT ö
æ V öù
ê
ú
ê
ç
÷
ç
÷
ç
÷
S = m A s A Ta ,Va + cvA log ç ÷ + RA log ç ÷ + mB sB Tb ,Vb + cvB log ç ÷ + RB log çç ÷÷ú
êë
êë
è Ta ø
è Va øúû
è Tb ø
è Vb øúû
(
(
)
(
)
)
H = m AcPA + mBcPB T , cPA = cvA + RA , cPB = cvB + RB
21
Plan
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•
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•
Ideal gas, a review
PVT relation of ideal-gas mixture
Mixing (TV-representation)
Mixing (TP-representation)
Adiabatic mixing
Steady-flow, adiabatic mixing
Isentropic mixing. Stop generating entropy!
Do work.
22
Entropy of an ideal-gas mixture
Change state of pure A
Ta,Va,nA
Pa
T,V,nA
Pressure = yAP
Mix at constant entropy
T,V,nA,nB
P
Change state of pure B
Tb,Vb,nB
Pb
T,V,nB
Pressure = yBP
é
é
æT ö
æ V öù
æT ö
æ V öù
S T ,V ,n A ,nB = n A ês A Ta ,Va + cvA log çç ÷÷ + R log çç ÷÷ú + nB êsB Tb ,Vb + cvB log çç ÷÷ + R log çç ÷÷ú
êë
êë
è Ta ø
è Va øúû
è Tb ø
è Vb øúû
é
é
æT ö
æ y P öù
æT ö
æ y P öù
A
÷÷ú + nB êsB Tb ,Pb + cPB log çç ÷÷ - R log çç B ÷÷ú
S T ,P,n A ,nB = n A ês A Ta ,Pa + cPA log çç ÷÷ - R log çç
êë
êë
è Ta ø
è Pa øúû
è Tb ø
è Pb23øúû
(
)
(
)
(
)
(
)
(
)
(
)
Ideal-gas mixture (TP-representation)
(
)
PV = n A + nB RT
(
)
U = m AcvA + mBcvB T
é
é
æT ö
æ y P öù
æT ö
æ y P öù
S = m A ês A Ta ,Pa + cPA log çç ÷÷ - RA log çç A ÷÷ú + mB ê sB Tb ,Pb + cPB log çç ÷÷ - RB log çç B ÷÷ú
êë
êë
è Ta ø
è Pa øúû
è Tb ø
è Pb øúû
(
(
)
(
)
)
H = m AcPA + mBcPB T , cPA = cvA + RA , cPB = cvB + RB
24
Entropy of mixing
at constant temperature and pressure
P,T,VA,nA
P,T,VB,nB
P,T,V,nA,nB
Thermostat, T
é
é
æT ö
æ y P öù
æT ö
æ y P öù
A
÷÷ú + nB êcPB log çç ÷÷ - Rlog çç B ÷÷ú
Smix = n A êcPA log çç ÷÷ - R log çç
êë
êë
è Ta ø
è Pa øúû
è Tb ø
è Pb øúû
(
Smix = -R n A log y A + nB log yB
(
Smix
R n A + nB
)
)
= -y A log y A - yB log yB
25
Plan
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Ideal gas, a review
PVT relation of ideal-gas mixture
Mixing (TV-representation)
Mixing (TP-representation)
Adiabatic mixing
Steady-flow, adiabatic mixing
Isentropic mixing. Stop generating entropy!
Do work.
26
Adiabatic mixing
Ta,Pa,nA
T,P,nA,nB
Adiabatic mixing
Tb,Pb,nB
• Know the initial states in the two boxes (Ta,Pa,nA) and (Tb,Pb,nB)
• Also know the pressure of the mixture, P.
• Assume the mixing is adiabatic.
• Determine the temperature of the mixture, T.
• Determine the entropy of mixing, Smix.
27
Conservation of energy
Ta,Pa,nA
T,P,nA,nB
Adiabatic mixing
Tb,Pb,nB
( )
( )
( )
( )
n Au A Ta + nB uB Tb = n Au A T + nB uB T
n AcvATa + nBcvBTb = n AcvAT + nBcvBT
n c T +n c T
T = A vA a B vB b
n AcvA + nBcvB
28
Entropy of mixing
Ta,Pa,nA
T,P,nA,nB
Adiabatic mixing
Tb,Pb,nB
é
é
æT ö
æ y P öù
æT ö
æ y P öù
A
÷÷ú + nB êsB Tb ,Pb + cPB log çç ÷÷ - R log çç B ÷÷ú
S T ,P,n A ,nB = n A ês A Ta ,Pa + cPA log çç ÷÷ - R log çç
êë
êë
è Ta ø
è Pa øúû
è Tb ø
è Pb øúû
(
)
(
)
(
(
)
(
)
)
(
Smix º S T,P,n A ,nB - n A s A Ta ,Pa - nB sB Tb ,Pb
)
é
é
æT ö
æ y P öù
æT ö
æ y P öù
A
÷÷ú + nB êcPB log çç ÷÷ - Rlog çç B ÷÷ú
Smix = n A êcPA log çç ÷÷ - R log çç
êë
êë
è Ta ø
è Pa øúû
è Tb ø
è Pb øúû
29
Plan
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•
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•
•
•
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•
Ideal gas, a review
PVT relation of ideal-gas mixture
Mixing (TV-representation)
Mixing (TP-representation)
Mixing at constant temperature and pressure
Adiabatic mixing
Steady-flow, adiabatic mixing
Isentropic mixing. Stop generating entropy! Do
work.
30
Steady-flow, adiabatic mixing
Adiabatic chamber
TA ,P,m A
T ,P,m A ,mB
TB ,P,mB
• Know the inlet conditions (TA ,P,m A ) (TB ,P,mB )
• The pressure at the outlet is the same as that at the inlets, P.
• The mixing chamber is adiabatic.
• Determine the temperature at the outlet, T.
• Determine the entropy generation.
31
Conservation of energy
Adiabatic chamber
TA ,P,m A
T ,P,m A ,mB
TB ,P,mB
hA,inm A + hB,inmB = hA,out m A + hB,out mB
cPATAm A + cPBTBmB = cPATm A + cPBTmB
c T m +c T m
T = PA A A PB B B
cPAm A + cPBmB
32
Generation of entropy
Adiabatic chamber
TA ,P,m A
T ,P,m A ,mB
TB ,P,mB
(
) (
)
Sgen = m A s A,out + mB sB,out - m A s A,in + mB sB,in
é
ù
é
ù
æT ö
æT ö
= m A êcPA log çç ÷÷ - RA log y A ú + mB êcPB log çç ÷÷ - RB log yB ú
êë
úû
êë
úû
è TA ø
è TB ø
33
Plan
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•
•
•
•
•
•
Ideal gas, a review
PVT relation of ideal-gas mixture
Mixing (TV-representation)
Mixing (TP-representation)
Adiabatic mixing
Steady-flow, adiabatic mixing
Isentropic mixing. Stop generating entropy!
Do work.
34
Isolated system
When confused, isolate.
Isolated system conserves mass over time:
dmIS
dt
Isolated system conserves energy over time:
dEIS
Isolated system generates entropy over time:
dSIS
Define words:
dt
dt
Isolated
system
IS
=0
=0
³0
ì > 0, irreversible process
dSIS ïï
í =0, reversible process
dt ï
ïî <0, impossible process
35
Carnot: “The steam is here only a means
of transporting the caloric (entropy).”
High-temperature source, TH
Q
DSin = H
TH
High-temperature source, TH
Engine
Q
Low-temperature sink, TL
QH
Q
DSout = L
TL
W
QL
Generator
W = QH - QL
Low-temperature sink, TL
Isolated system = source + sink
Thermal contact transports and generates entropy
dSIS
dt
=
Q Q
>0
TL TH
Isolated system = source + sink + engine + generator
Reversible engine transports but does not generate entropy
dSIS
Q
Q
= L - H =0
dt
TL TH
36
The world according to entropy
ì > 0, irreversible process
dSIS ïï
í =0, reversible process
dt ï
ïî <0, impossible process
• Irreversible process transports and generates entropy. Natural process.
Non-equilibrium process. e.g., Friction, mixing, conduction.
• Reversible process transports but does not generate entropy. Idealized
process. Quasi-equilibrium process. Isentropic process. e.g. Carnot cycle,
Stirling cycle, a frictionless pendulum.
• Impossible process. Entropy of an isolated system can never decrease over
time.
• Equilibrium. A system isolated for a long time reaches a state of
thermodynamic equilibrium, and maximizes entropy.
• Every reversible process (i.e., natural process) is an opportunity to do work.
37
Isentropic mixing and separation
Balance osmosis with external force.
Air
Pressure = P
Temperature = T
Number fraction = yN2,yO2
Equilibrium
Weight = A (P –yN2P)
P
Pure nitrogen
Pure nitrogen
Pressure = yN2P
Temperature = T
Weight
P = yN2P + yO2P
Direct mixing generates entropy
Semipermeable membrane
Permeable to nitrogen
Impermeable to oxygen
Isentropic mixing transports entropy
38
39
Summary
(
)
PV = n A + nB RT
(
)
H = ( m AcPA + mBcPB ) T ,
U = m AcvA + mBcvB T
cPA = cvA + RA , cPB = cvB + RB
é
é
æT ö
æ y P öù
æT ö
æ y P öù
A
ê
ú
ê
ç
÷
ç
÷
ç
÷
S = m A s A Ta ,Pa + cPA log ç ÷ - RA log ç
+ mB sB Tb ,Pb + cPB log ç ÷ - RB log çç B ÷÷ú
÷
êë
êë
è Ta ø
è Pa øúû
è Tb ø
è Pb øúû
(
)
(
)
40