1. 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

2. 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.

3. 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
P = pressure
V = volume
N = number of molecules
t = temperature in the unit of energy
geometry
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.

4. Human folly To every beautiful discovery, we add many ugly ideas.
Generating entropy is natural.
The discovery
Number of molecules
mole
Mass
Ugly idea 1
Kelvin temperature kBT = t
Boltzmann constant
Ugly idea 2
Avogadro constant
NAvogadro = x 1023
Mole n = N/NAvogadro
Universal gas constant
Ugly idea 3
Specific gas constant
Gas
Formula
Molar mass, M kg/kmol R
kJ/kgK
Air
0.2870
Steam
H2O 18
0.4615
Hydrogen H2 2
4.124

5. Model a closed system as a family of isolated systems
weight 2O
vapor
Isolated system
closed system
vapor
liquid
liquid
fire
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

6. 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 Gibbs equation 1: Gibbs equation 2:

7. 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

8. 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.

9. Two species of moleculesB A
Number of moles of species A
Number of moles of species B
Number (mole) fraction of species A:
Number (mole) fraction of species B:
Algebra:

10. Dalton’s law (1801) Dalton’s law: Partial pressures: Total pressure:
T,V,nA
T,V,nB
T,V,,nA,nB PA PB
PA + PB
Boxes of the same volume and temperature

11. Dry air (no water) name formula Molar mass M kg/kmol number fraction y
nitrogen N2 28
0.78
oxygen O2 32
0.21

12. 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.
Number of quantum states of the gas scales with volume as:
Definition of entropy S = kBlog W
Gibbs equation 2:
Delton’s law: B A

13. 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.

14. Energy and entropy of mixing
Ta,Va,nA
T,V,nA,nB
mix
Tb,Vb,nB

15. 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

16. 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

17. 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

18. Entropy of mixing Change state of pure A Mix at constant temperature
constant volume
Ta,Va,nA
T,V,nA
T,V,nA,nB
Change state of pure B
Tb,Vb,nB
T,V,nB

19. Enthalpy of an ideal-gas mixture

20. 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

21. Ideal-gas mixture (using mass)

22. 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.

23. 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

24. Ideal-gas mixture (TP-representation)

25. Entropy of mixing at constant temperature and pressure
P,T,VA,nA
P,T,VB,nB
P,T,V,nA,nB
Thermostat, T

26. 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.

27. 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.

28. Conservation of energy
Ta,Pa,nA
T,P,nA,nB
Adiabatic mixing
Tb,Pb,nB

29. Entropy of mixing
Ta,Pa,nA
T,P,nA,nB
Adiabatic mixing
Tb,Pb,nB

30. Plan 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.

31. Steady-flow, adiabatic mixing
Adiabatic chamber
Know the inlet conditions
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.

32. Conservation of energy
Adiabatic chamber

33. Generation of entropy
Adiabatic chamber

34. 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.

35. Isolated system When confused, isolate.
Isolated system conserves mass over time:
Isolated system conserves energy over time:
Isolated system generates entropy over time:
Define words:

36. Carnot: “The steam is here only a means of transporting the caloric (entropy).”
High-temperature source, TH QH
High-temperature source, TH
Engine
Generator Q W
W = QH - QL
Low-temperature sink, TL QL
Low-temperature sink, TL
Isolated system = source + sink Isolated system = source + sink + engine + generator
Thermal contact transports and generates entropy Reversible engine transports but does not generate entropy

37. The world according to entropy
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.

38. Isentropic mixing and separation Balance osmosis with external force.
Air
Pressure = P
Temperature = T
Number fraction = yN2,yO2
Equilibrium
Weight = A (P –yN2P)
Pure nitrogen
Pressure = yN2P
Temperature = T
P = yN2P + yO2P P
Pure nitrogen
Weight
Semipermeable membrane
Permeable to nitrogen
Impermeable to oxygen
Direct mixing generates entropy Isentropic mixing transports entropy

40. Summary