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 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.

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.

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 = 6.022 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

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

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:

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

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.

Two species of molecules B 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:

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

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

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

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.

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

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

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

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

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

Enthalpy of an ideal-gas mixture

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

Ideal-gas mixture (using mass)

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.

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

Ideal-gas mixture (TP-representation)

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

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.

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.

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

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

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.

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.

Conservation of energy Adiabatic chamber

Generation of entropy Adiabatic chamber

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.

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:

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

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.

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

Summary