Rutgers University Department Of Physics & Astronomy

Rutgers UniversityDepartment of Physics & AstronomyHome PageTitle Page01:750:271 Honors Physics IFall 2015JJIIJIPage 1 of 33Go BackFull ScreenCloseQuit

The adiabatic expansion of an ideal gasHome PageTitle PageJJIIJIPage 2 of 33Go BackFull ScreenAdiabatic process: Q 0 (no heat exchange withthe environement)CloseQuit

dEint Q pdV ndT pCVHome PageTitle Page dV(a)pV nRTJJIIJIPage 3 of 33 Go BackpdV (dp)V nRdT (b)Adiabatic process(a) and (b): dpCp dV 0pCVVFull ScreenCloseQuit

dp p CpCV dV 0V Home PageTitle Pageln(p) γln(V ) constantγ Cp/CV JJIIJIPage 4 of 33Go BackAdiabatic processγpV constantFull ScreenCloseQuit

piViγ pf VfγHome Page Title PagenRTi γnRTf γVi Vf ViVf TiViγ 1 Tf Vfγ 1JJIIJIPage 5 of 33Go BackFull ScreenAdiabatic processCloseQuit

Free expansion of an ideal gasHome PageAn adiabatic process with no work or change in internal energy.W 0 Eint 0Ti TfJJIIJIQ 0piVi pf VfnotTitle PagePage 6 of 33Go BackpiViγ pf VfγFull ScreenCloseQuit

Ideal gas processesHome PageTitle Page1. Isobaric2. IsothermalJJIIJIPage 7 of 33Go Back3. AdiabaticFull Screen4. IsochoricCloseQuit

Example: a bubble of n 5.00 mol of helium issubmerged at a certain depth in liquid water when thewater (and thus the helium) undergoes a temperatureincrease T 20.0 C at constant pressure. As aresult, the bubble expands. The helium is monatomicand ideal.(a) How much energy is added to the helium as heatduring the increase and expansion?p constant Q nCp T n(CV R) TFor monatomic gases: CV 3R/2. Hence5nR TQ .2Home PageTitle PageJJIIJIPage 8 of 33Go BackFull ScreenCloseQuit

(b) What is the change Eint in the internal energyof the helium during the temperature increase? Eint3nR T nCV T .2(c) How much work W is done by the helium as itexpands against the pressure of the surrounding waterduring the temperature increase?Home PageTitle PageJJIIJIPage 9 of 33Go Back Eint Q W W Q Eint nR TFull ScreenCloseQuit

Example: adiabatic expansionHome Pagen 1 mol of oxygen (assumed to be an ideal gas)has temperature Ti 310 K and volume Vi 12 L.What would be the final temperature if the gas expands adiabatically to a volume Vf 2Vi? Oxygen(O2) is diatomic.TiViγ 1 Tf Vfγ 1Title PageJJIIJIPage 10 of 33Go BackCp1 f /27/27 CVf /25/25 2/5ViTf Ti 2 2/5TiVfγ Full ScreenCloseQuit

20. Entropy and the second law of thermodynamicsHome Page Reversible and irreversible processesTitle PageReversible process: consists of a sequence of welldefined equilibrium states during the intermediate stagesof the change from initial state i to final state f .Each such intermediate state is characterized by someintermediate values (p, V, t)JJIIJIPage 11 of 33Go BackReversible processes can be represented by a graphin the (p, v) plane.Such processes can be reversed, evolving from thefinal to the initial state through the inverse sequenceof equilibrium states.Full ScreenCloseQuit

Home PageTitle PageJJIIJIPage 12 of 33Go BackFull ScreenThe lead shot is slowly removed such that the expansion takes place through a sequence of intermediateequilibrium states at constant T .CloseQuit

Such a slow process corresponds to an isothermal inthe (p, V ) planeHome PageTitle PageJJIIJIPage 13 of 33Go BackFull ScreenIn principle, such a process can be reversed by slowlyplacing the lead weight back on the piston.CloseQuit

Irreversible process: the evolution from the initialto the final state does not consist of a sequence ofequilibrium states.At any given time between the initial and the finalstate the system is not in thermal equilbrium, henceit cannot be characterized by well defined values of(p, V, T ).For example the temperature and the pressure maynot be uniform throughout a sample of ideal gas.Home PageTitle PageJJIIJIPage 14 of 33Go BackFull ScreenSuch processes cannot be reversed. There is no inverse process from the final to the initial state.CloseQuit

Home PageIrreversibleTitle PageprocessJJIIJI Page 15 of 33The stopcock is suddenly opened. The gas expandsirreversibly into the second container.An inverse process is impossible in this case becauseone cannot confine all the gas molecules back in thefirst container by closing the stopcock.Go BackFull ScreenCloseQuit

Such a process cannot be represented by a graph inthe (p, V ) plane.Home PageTitle PageJJIIJIPage 16 of 33Go BackFull ScreenCloseQuit

If we leave the stopcock open,why don’tall molecules spontaneouslymove to the first container,recovering the initial stateof the system?For suchprocessahypotheticalHome PageTitle PageJJIIJIPage 17 of 33Q 0 (insulated container)Go BackThe hypothetical inverseW 0 (rigid container)process would not contradict energy conservation Thereforeor the 1st law of thermo Eint 0dynamics.Full ScreenCloseQuit

Need a second principle, or law forbidding such processes. Entropy: a new physical quantity which measuresthe degree of disorder in a physical system. Entropyis denoted by S and it is a function of the state of thesystem.Home PageTitle PageJJIIJIPage 18 of 33Entropy postulateIf an irreversible process occurs in a closed system, the entropy S of the system always increases;it never decreases.Go BackFull ScreenCloseQuit

Closed system: no heat or matter exchange withthe environement:Q 0and the number of particles is constant.Home PageTitle PageJJIIJIFor all irreversible processes in closed systems: S 0Page 19 of 33This reflects the natural tendency of physical systems to evolve from order to disorder.Go BackFull ScreenHow do we define and compute the entropy of anideal gas?CloseQuit

Key point: for a system in equilibrium entropy isonly a function of the state of the system Hence Sdepends only on the initial and final states (pi, Vi, Ti),(pf , Vf , Tf ). To find the entropy change for an irreversibleprocess occurring in a closed system, replace thatprocess with any reversible process that connectsthe same initial and final states.Home PageTitle PageJJIIJIPage 20 of 33Go BackFull ScreenCloseQuit

Change in entropy for a reversible processHome Page Isothermal reversible processTitle PageJJIIJIT S Q Eint WPage 21 of 33 General reversible processGo BackdQdS S TfZidQTFull ScreenCloseQuit

Reversible process for n moles of ideal gasHome PagedQ dEint dWTitle Page pdV nCV dTfZ S idV nRT nCV dTV dQVfTf nRln nCV lnTViTiJJIIJIPage 22 of 33Go BackFull ScreenNote: as expected, S depends only on the initialand final states, not on the process connecting thetwo states. S Sf SiCloseQuit

Example: free expansionHome PageTitle PageIrreversibleJJIIJIprocess Page 23 of 33Go BackSuppose n 1.0 mol of nitrogen gas is confined tothe left side of the container. You open the stopcock, and the volume of the gas doubles. What is theentropy change of the gas for this irreversible process?Full ScreenCloseQuit

The containers are insulated Q 0 (closed system).The total volume of the system does not change W 0. Eint Q W 0 Ti TfChange in entropy for gas: Vf S nRln nRln2ViHome PageTitle PageJJIIJIPage 24 of 33Go BackFull ScreenCloseQuit

The second law of thermodynamicsHome PageIf a process occurs in a closed system, the entropy of the system increases for irreversible processes and remains constant for reversible processes. It never decreases.Title PageJJIIJINote: by definitionPage 25 of 33dQdS Tfor reversible processes. Closed system:dQ 0 dS 0Go BackFull ScreenCloseQuit

Home PageTitle PageJJIIJIPage 26 of 33Go BackFull ScreenThe gas itself is not a closed system, but the gasand the thermal reservoir consitute a closed system.CloseQuit

Home PageTitle PageJJIIJIPage 27 of 33Go BackFull Screen( S)gas ( S)reservoir 0( S)gas 6 0,( S)reservoir 6 0CloseQuit

Example: two identical copper blocks L, R ofmass m 1.5 kg ina thermally insulatedbox, separated by aninsulating shutter. TiL 60 C andTiR 20 C . S ? Remove the shutter irreversibleprocess.At theend – thermal equilibriumHome PageTitle PageJJIIJIPage 28 of 33Go BackFull ScreenCloseTf 40 C Quit

The final state of equilibrium can be reached by asequence of reversible processes.Step 1. Slowly cool downblock L from TL to TfdTdQ mcdT dS mcTZ fdT( S)L mc mcln(Tf /TL)L THome PageTitle PageJJIIJIPage 29 of 33Go BackStep 2. Slowly heat up block R from TR to TfZ fdT( S)R mc mcln(Tf /TR)R TFull ScreenCloseQuit

Total entropy variation:Home Page S ( S)L ( S)RTitle Page mc[ln(Tf /TL) ln(Tf /TR)]JJIIJIPage 30 of 33Go BackFull ScreenCloseQuit

Entropy in the real world: EnginesHome Page Heat engine: a device that takes in energy by heatand, operating in a cyclic process, expels a fraction ofthat energy by means of work A perfect heat engine is adevice which converts heat QHfrom a high temperature reservoir to work W with 100% efficiency i.e. W QH .Title PageJJIIJIPage 31 of 33Go BackFull ScreenCloseQuit

The 2nd law of thermodynamics a perfect heat engine cannot exist. The system can only absorb heat QH 0 from thereservoir. Therefore its entropy can only increaseHome PageTitle PageJJIIJI S QH /TH 0Page 32 of 33 The cycle cannotclose. The system can neverspontaneously return to astate with smaller volume atconstant TH since such a process would have S 0Go BackFull ScreenCloseQuit

Kelvin-Planck formulation of the 2nd lawHome PageTitle PageJJIIJIPage 33 of 33Go BackNo series of processes is possible whose soleresult is the transfer of energy as heat from athermal reservoir and the complete conversionof this energy to work.Full ScreenCloseQuit

Reversible and irreversible processes Reversible process: consists of a sequence of well-de ned equilibrium states during the intermediate stages of the change from initial state ito nal state f. Each such intermediate state is characterized by some intermediate values (p;V;t) Reversible processes can be represented by a graph in the (p;v) plane.