1Review from Ch. 1• Thermodynamic quantities• Composition• Pressure• Density• Temperature• Kinetic Theory of GasesCurry and Webster, pp. 1-17Feynman, Book I, ch. 39Thermodynamic Quantities• Classical vs. Statistical thermodynamics• Open/closed systems• Equation of state f(P,V,T)=0• Extensive/intensive properties• Thermal, engine, heat/work cyclesIntensive quantities: P, T, v, nExtensive quantities: V, NConcentration: n=N/VVolume: v=V/NSystemEnvironment<-- ClosedOpen -->Composition• Structure– Comparison to other planets• N2, O2, Ar, CO2, H2O: 110 km constitute 99%• Water, hydrometeors, aerosolPressure• Force per unit area• 1 bar = 105 Pa; 1 mb = 1 hPa; 1 atm = 1.013 bar• Atmosphere vs. Ocean2AtmosphereOceanDensity• Specific volume: v=V/m• Density: ρ=m/V– 1.29 kg/m3• Mean free path– frequency of intermolecular collisionsTemperature• “Zeroeth” Law of Thermodynamics– Equilibrium of two bodies with third– Allows universal temperature scale• Temperature scale– Two fixed points: Kelvin, Celsius– Thermometer• Lapse Rate Γ = -∂T/∂z– Change in temperature with altitude– Typically Γ=6.5 K/km• Temperature inversion Γ<0– Boundary layer “cap”– Tropopause between troposphere and stratosphereKinetic Theory of Gases• Pressure of a gas• Kinetic energy• Internal energy• Temperature of a gas• Pressure-volume-temperature relationship• The “fine print”Initial Momentum: mvxFinal Momentum: -mvxIf all atoms had same x-velocity vx:Momentum Change for one Atom-Collision: [Initial]-[Final] = mvx-(-mvx) = 2mvxNumber of Atom-Collisions-Per-Time: [Concentration]*[Volume] = [n]*[vxA]Force = [Number]*[Momentum Change] = [nvxA]*[2mvx] = 2nmAvx2Pressure = [Force]/[Area] = 2nmvx2For atoms with average velocity-squared of <vx2>:Pressure = [Force]/[Area] = nm<vx2>Force: FArea: ACollision Distance-Per-Time: vxt/t=vxIndividual collisionsPerfect reflectionIdeal gas Monatomic gas3Population-averaged Velocity: <v2>=[vi2 + vii2 + viii2 +…+ vn2]/nScalar multipliers: <mv2/2>=[mvi2 + mvii2 + mviii2 +…+ mvn2]/2nHow many will hit “right” wall? n/2viiviiivi3D velocity: <v2>=<vx2>+<vy2>+<vz2>Random motion (no preferred direction): <vx2>=<vy2>=<vz2><vx2>= <v2>/3vvxvzvyP = nm<vx2>=[2/2]*[nm]*[<v2>/3]=[2/3]n*<mv2/2>=[2/3]n*[kinetic energy of molecule]PV =[2/3]*[N*<mv2/2>]=[2/3]*U=[2/3]*EkConcentration: n=N/VTotal “internal” energy: UKinetic energy of gasPV =[2/3]*EkEk =[3/2]* PVDefine T = f(Ek)For scale choose T=(2/3Nk)*Ek Ek =(3/2)*NkT Then PV = NkT = nR*TKinetic energy of gasRHS is independent of gas--> so scale can be universalMean k.e.: Ek/N=(3/2)kTk=1.38x10-23 J/KR*=N0k=8.314 J/mole/KTemperatureis defined to beproportional to the averagekinetic energy of the molecules.Lecture Ch. 2a• Energy and its properties– State functions or exact differentials– Internal energy vs. enthalpy• First law of thermodynamics• Heat/work cycles– Energy vs. heat/work?– Adiabatic processes– Reversible “P-V” work• Homework problem Ch. 2, Prob. 2Curry and Webster, Ch. 2 pp. 35-47Van Ness, Ch. 24Internal Energy vs. Enthalpy• Difference b/w U and H– U depends on v– H depends on p• Specific heats [a.k.a. heat capacity]– cv is constant v– cp is constant pHeat CapacityFor an ideal gas• Simplify to• [Types of processes]– Constant pressure– Constant volumeLord Kelvin(a.k.a William Thomson)James P. Joule• The First Law of Thermodynamics• ConsequencesΔE = mc2Q = 0, ΔE = 0 ⇒ W = 0(Relativity)Impossibility of perpetual motion machineConservation of energyDefinition of energyUniqueness of work valuesQ = 0,W = 0 ⇒ ΔE = 0 ⇒ E2= E1Q = 0 ⇒ ΔE = WWrev= − pdv∫ReversibleAdiabaticState functionSee also 2nd law!Proof follows..Other Kinds of Energy• In addition to changes in internal energy, asystem may change– Potential energy for height change Δz– Kinetic energy for velocity change Δv– Nuclear energy for mass change ΔmVan Ness, p. 13€ ΔE = ΔU p,V,T( )+ mgΔz +12mΔv2− c2Δm = Q + Wif ΔE ≈ ΔU p,V,T( ), then ΔU p,V,T( )= Q + WWork• Expansion work W=-pdV or w=-pdv– Lifting/rising– Mixing– Convergence• Other kinds of work?– Electrochemical (e.g. batteries)5• Work and heat are path-dependenttransfers– W work– Q heat• State functions are unique “states”– U internal energy– H enthalpy– η (also S) entropy– A Helmholtz free energyCyclesExact Differentials• State functions are exact differentialsHeat/Work Cycles• The efficiency with which work is accomplished in a reversible cyclic processdepends only on the temperature of the reservoirs to which heat istransferredQ1Q2WFLUIDT1T2STEP 1: Expand isothermally and reversibly at T1STEP 2: Expand adiabatically and reversiblySTEP 3: Compress isothermally and reversibly at T2STEP 4: Compress adiabatically and reversiblyW1= Q1= RT1lnPAPBW2= Q2= RT2lnPCPDW = CvT2− T1( )W = CvT1− T2( )THE CARNOT CYCLECarnot was an engineer inNapoleon’s defeated armywith an interest in engines.Efficiency:HotColdTT−= 1η6P-V diagrams of work• Work is determined by pathwayOther Work CyclesNikolausOttodevelopedthe Ottocycle in1876.RudolfDieseldevelopedthe Dieselcycle in1892.The Diesel Cycle worksby compressing air andthen adding fuel directlyto the piston. Thecompressed air thencombusts the mixture.The Otto Cycle worksby compressing amixture of air and fuelin a piston and thenigniting the mixturewith a spark.The compression ratio of the Diesel Cycle rangesfrom 14:1 to 25:1, while the Otto Cycle range issignificantly lower, from 8:1 to 12:1.Efficiency: Efficiency:DACBTTTT−−−= 1η−−−=DACBTTTTγη1135=γfor monatomic ideal gasIdeal GasesReversible-Adiabatic-WorkReversible-Adiabatic-WorkAdiabaticFirst LawReversibleInternal EnergyIdeal Gasp1v1T1= R =p2v2T2Δu = cvdTW = − pdvΔu = Q + WQ = 0T2T1=P2P1 γ−1γ€ γ=Rcvthick wallsLow P, Low TFrictionlessReversible, Adiabaticmass is conservedReversible ProcessesReversible ProcessesReversible€ Wrev= − pdvFrictionlessmass is conserved• Always at or infinitesimally close to equilibrium• Infinitesimally small steps• Infinite number of steps• Each step can be reversed with infinitesimal forcegrain of sand7Homework Ch. 2 Problem
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