Adiabatic Expansion (DQ = 0)Occurs if:• change is made sufficiently quickly• and/or with good thermal isolation.Governing formula:PVg = constantwhere g = CP/CVBecause PV/T is constant (ideal gas): Vg-1 T = constant (for adiabatic)PVAdiabatIsothermsProof of PVg=constant (for adiabatic process)1) Adiabatic: dQ = 0 = dU + dW = dU + PdV2) U only depends on T:dU = n CV dT (derived for constant volume, but true in general)3) Ideal gas: T = PV/(nR) dT = [(dP)V + P(dV)]/(nR)Plug into 2): dU = (CV/R)[VdP + PdV]Plug into 1): 0 = (CV/R)[VdP + PdV] + PdVPVIsotherms(constant T)Same DURearrange: (dP/P) = - (CV+R)/CV (dV/V) = - g (dV/V)where g = (CV+R)/CV = CP/CVIntegrate both sides:ln(P) = - g ln(V) + constantorln(PVg) = constantorPVg = constantQEDIrreversible ProcessesExamples:• Block sliding on table comes to restdue to friction: KE converted to heat.• Heat flows from hot object to coldobject.• Air flows into an evacuated chamber.Reverse process allowed by energyconservation, yet it does not occur. arrow of timeWhy?2nd Law of Thermodynamics (entropy)Heat EnginesHeat engine: a cyclic device designed toconvert heat into work.2nd Law of TD (Kelvin form):It is impossible for a cyclic process toremove thermal energy from a system at asingle temperature and convert it tomechanical work without changing thesystem or surroundings in some other way.Hot Reservoir, THCold Reservoir, TCQHQCWork, WFor a cyclic engine DU = 0,So work done is equal toheat in minus heat out:W = QH - QCDefine the Efficiency of the engine: e = W/QH = (QH-QC)/QH = 1 - QC/QHCorollary of the 2nd Law of TD:It is impossible to make a heat enginewhose efficiency is 100%.RefrigeratorsRefrigerator: a cyclic device which useswork to transfer heat from a coldreservoir to hot reservoir.2nd Law of TD (Clausius form):It is impossible for a cyclic process tohave no other effect than to transferthermal energy from a cold object to a hotobject.Hot Reservoir, THCold Reservoir, TCQHQCWork, WA measure of refrigerator performance isthe ratio:K = QC / W“Coefficient of performance”(The larger the better.)Corollary of the 2nd Law of TD:It is impossible for the coefficient ofperformance to be infinite.Equivalence of Kelvin andClausius StatementsFor example:You could combine an ordinaryrefrigerator with a perfect engine(impossible)...to obtain a perfect refrigerator (alsoimpossible).QHQCW WQQH-QQCThe Carnot Engine2nd Law of TD says: 100% efficient Heat Engine is impossible.What is the maximum possible efficiency?No engine working between 2 heatreservoirs can be more efficient than anideal engine acting in a Carnot cycle.(Sadi Carnot, 1824)Properties of the Carnot cycle:1. It is reversible: no friction or otherdissipative forces.2. Heat conduction only occursisothermally at the temperatures ofthe two reservoirs.Derivation of Carnot Efficiencye = 1 - TC/THPVQinQoutTHTC12341-2: Isothermal (Qin at TH)2-3: Adiabatic expansion3-4: Isothermal (Qout at TC)4-1: Adiabatic compressionThe Stirling EngineInvented by Robert Stirling in 1816.Its operating cycle is:The two temperature-changing steps areperformed at constant volume; A heattransfer occurs at these steps also. eStirling < eCarnotPVQinQoutTHTC1234QQEntropyConsider a reversible process for an idealgas:dQ = dU + dW = n CV dT + P dV = n CV dT + n R T (dV/V)We cannot write a general integral of this,because dW (and therefore dQ) dependson the functional form of T(V) (i.e. thepath). However, if we divide by T:dQ/T = n CV (dT/T) + n R (dV/V)is integrable independent of path.This suggests a new state function,Entropy, defined by: dQ DS = Sf - Si = ∫ T(Valid for any system)ifIn general, the process may be toocomplicated to do the integral(particularly if irreversible process):However, because entropy is a statefunction, we can choose any convenientpath between i and f to integrate.For an ideal gas: DS = n CV ln(Tf/Ti) + n R ln(Vf/Vi)This only depends on the initial state(Vi,Ti) and final state (Vf,Tf), but not thepath.PVif12Isothermal Expansion: Tf=Ti, Vf>ViThe amount of heat which leaves thereservoir and enters the gas isQ = n R T ln(Vf/Vi).The entropy change of the gas isDSgas = + Q/T = n R ln(Vf/Vi).The entropy change of the reservoir isDSreservoir = - Q/T.The net entropy change isDSuniverse = DSgas + DSreservoir = 0.This illustrates a general result:In a reversible process, the entropychange of the universe (system +surroundings) is zero.Adiabatic Free Expansionof an Ideal GasTwo containers connected by stopcock.They are thermally insulated so no heatcan flow in or out.Initial: One container is evacuated. Gas isin volume Vi at temperature Ti.Final: Stopcock opened, gas rushes intosecond chamber. Gas does no work(nothing to push against) and there isno heat transfer. So internal energydoes not change. Final volume Vf>Vi attemperature Tf=Ti.Because there is no heat transfer, youmight think DS = 0. WRONG! This is anirreversible process. We can’t integratedQ ∫ . TBut entropy is a state function, and we doknow the initial and final conditions forthe Free Expansion. They are exactly thesame as for an Isothermal Expansion. SoDSgas = n R ln(Vf/Vi).just as for an isothermal expansion.However, since it is thermally isolatedfrom its surroundings,DSsurround = 0andDSuniverse = DSgas + DSsurround = n R ln(Vf/Vi) > 0.In an irreversible process, the entropy ofthe universe increases.Entropy and Heat EnginesFor a reversible cycle: dQ DS = ∫ TThis implies that dQ cannot be strictlypositive. There must also be heat releasedin the cycle.Carnot cycle: (Qin/TH) + (-Qout/TC) = 0.2nd Law of TD (Entropy form):DSuniverse ≥ 0.(greater-than sign for irreversibleprocesses, and equals sign for reversibleprocesses)PVQinQoutTHTCEntropy and Probability(A statistical view)Entropy ~ a measure of the disorder of a system.A state of high order = low probabilityA state of low order = high probabilityIn an irreversible process, theuniverse moves from a state of lowprobability to a state of higherprobability.We will illustrate the concepts byconsidering the free expansion of a gasfrom volume Vi to volume Vf.The gas always expands to fill theavailable space. It never spontaneouslycompresses itself back into the originalvolume.First, two definitions:Microstate: a description of a system thatspecifies the properties (positionand/or momentum, etc.) of eachindividual