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Chapter 17: The First Law of Thermodynamics Thermodynamic Systems Interact with surroundings Heat exchange Q = heat added to the system(watch sign!) Some other form of energy transfer Mechanical Work, e.g. W = done by the system (watch sign!) protosystem: ideal gas p203/4c17:1 Work done during volume changes dW Fdx pAdx Adx dV dW pdV W V2 pdV V1 Depends up on process! Work = Area under curve! p203/4c17:2 Isobaric Expansion expansion at constant pressure reversed => compression W V2 pdV V1 V2 p dV V1 W p(V2 V1 ) can be rewritten, with pV = nRT W nR(T2 T1 ) p203/4c17:3 Isothermal Expansion (example 17-1) expansion at constant temperature reversed => compression W V2 pdV V1 with pV = nRT W V2 V2 nRT 1 dV nRT V V V V dV 1 1 V2 W nRT ln( ) V1 can be rewritten, with p1V1 p2V2 p1 W nRT ln( ) V2 p203/4c17:4 Example: 1 m3 of an ideal gas starting at 1 atm of pressure expands to twice its original volume by one of two processes: isobaric expansion or isothermal expansion. How much work is done in each case? p203/4c17:5 Work Done Depends upon path!!!! A B p203/4c17:6 Heat Transfer and “Heat Content” Two constant Temperature processes, same initial state slow expansion rapid expansion Final States (T, V and P) are the same Heat added in first process, not in second “Heat Content” not a valid concept p203/4c17:7 Internal Energy U : Sum of microscopic kinetic and potential energies Changes in response to heat addition (Q) to the system Changes in response to Work done (W) by the system The First Law of Thermodynamics: DU = Q W or Q = DU + W = Conservation of energy => DU is independent of path! U is a function of the state of the system (function of the state variables). U = U(p,V,T) for an ideal gas. Infinitesimal processes dU = dQ - dW dU = dQ - dW p203/4c17:8 For an isolated system W=Q=0 DU = 0 Uf=Ui the internal energy of an isolated system is constant For a cyclic process system returns to its initial state state variables return to their initial values Uf = Ui => Wnet = Qnet p203/4c17:9 p Example: Thermodynamic processes not an ideal gas a-b 150 J of heat added b-d 600 J of heat added Step ab bd abd Q W DU Step ac cd acd Q W DU b d 8.0 x 104 Pa 3.0 x 104 Pa a c 2.0 x 10-3 m3 5.0 x 10-3 m3 p203/4c17:10 Thermodynamic Processes Isothermal: constant temperature generally dV 0 ; dW 0 ; dQ 0 Adiabatic: no heat transfer Q = 0 ; dQ = 0 p Isochoric: constant volume (isovolumetric) dV = 0 => dW = 0 => Q = DU V Isobaric: constant pressure dW = pdV => W = pDV Polytropic processes: one generalization, not (necessarily) isoanything. pVr = const p203/4c17:11 Internal Energy of an ideal gas. adiabatic free expansion Q = 0; W = 0 => DU = 0 Dp 0 ; DV 0 ; DT = 0 => U depends upon T only Kinetic Theory U = sum of microscopic kinetic and potential energies each microscopic DOF averages 1/2 kT => U depends upon T only p203/4c17:12 Molar Heat Capacities of Ideal Gases 1 dQ CV n dT V const 1 dQ Cp n dT p const First Law dQ dU dW Constant V olume Process V const dW 0 dQ nCV dT dU dU nCV dT for all processes! !! p203/4c17:13 Constant pressure process: dQ = nCp dT = dU + dW vs dQ = nCp dT = dU for constant volume => Cp > CV for any material which expands upon heating For an ideal gas: dQ nC p dT dU nCV dT (still!!) p const , dW pdV and pV nRT dW pdV = d ( pV ) = d (nRT ) = nRdT dQ dU dW nC p dT nCV dT nRdT C p CV R p203/4c17:14 With the Equipartition Theorem DOF CV R 2 DOF dU CV dT RdT 2 3 5 5 7 7 9 CV R R , R Cp R R, R 2 2 2 2 2 2 , another effective way of characterizing an ideal gas Cp CV 5 7 9 , 3 5 7 p203/4c17:15 Adiabatic Processes dQ 0 dW pdV dU nCV dT nRT dV nCV dT V R dV dT 0 CV V T R C p CV 1 CV CV dV dT ( 1) 0 V T ( 1) ln V ln T const lnV 1 T const V 1 T const p203/4c17:16 Adiabatic Processes (cont’d) V 1 T const ideal gas law 1 pV V const pV const nR CV W DU nCV ( T1 T2 ) ( p1V1 p2V2 ) R p203/4c17:17 Summary of basic thermodynamic processeses for an ideal gas special W Q Isobaric Isochoric p const V const nC p DT nCV DT Isothermal T const p(V2 V1 ) 0 V2 nRT ln V1 Adiabatic Q0 TV 1 , pV const p1 nRT ln p2 DU CV ( p1V1 p2V2 ) R W 0 p203/4c17:18 Polytropic process for an ideal gas because not all processes are the basic types (iso*) or can be approximated by one of the basic process types polytropic exponent r pVr = const r = 0 > isobaric lim r> infinity > constant volume r = > adiabatic r =1 > isothermal p203/4c17:19