Lecture 2 The First Law of Thermodynamics (Ch.1)

Transcription

1 Lecture he Frst Law o hermodynamcs (Ch.) Outlne:. Internal Energy, Work, Heatng. Energy Conservaton the Frst Law 3. Quas-statc processes 4. Enthalpy 5. Heat Capacty

2 Internal Energy he nternal energy o a system o partcles, U, s the sum o the knetc energy n the reerence rame n whch the center o mass s at rest and the potental energy arsng rom the orces o the partcles on each other. system boundary envronment system Derence between the total energy and the nternal energy? U knetc + potental A he nternal energy s a state uncton t depends only on the values o macroparameters (the state o a system), not on the method o preparaton o ths state (the path n the macroparameter space s rrelevant). In equlbrum [ (,,)0 ] : U U (, ) U depends on the knetc energy o partcles n a system and an average nter-partcle dstance (~ -/3 ) nteractons. For an deal gas (no nteractons) : U U () - pure knetc

3 Internal Energy o an Ideal Gas he nternal energy o an deal gas wth degrees o reedom: U Nk 3 (monatomc), 5 (datomc), 6 (polyatomc) (here we consder only trans.+rotat. degrees o reedom, and neglect the vbratonal ones that can be excted at very hgh temperatures) How does the nternal energy o ar n ths (not-ar-tght) room change wth the external const? U N n room k N n room k - does not change at all, an ncrease o the knetc energy o ndvdual molecules wth s compensated by a decrease o ther number.

4 Work and Heatng ( Heat ) WORK We are oten nterested n ΔU, not U. ΔU s due to: Q - energy low between a system and ts envronment due to Δ across a boundary and a nte thermal conductvty o the boundary heatng (Q >0)/coolng (Q <0) (there s no such physcal quantty as heat ; to emphasze ths act, t s better to use the term heatng rather than heat ) W - any other knd o energy transer across boundary - work HEAING Work and Heatng are both dened to descrbe energy transer across a system boundary. Heatng/coolng processes: conducton: the energy transer by molecular contact ast-movng molecules transer energy to slow-movng molecules by collsons; convecton: by macroscopc moton o gas or lqud radaton: by emsson/absorpton o electromagnetc radaton.

5 he Frst Law he rst law o thermodynamcs: the nternal energy o a system can be changed by dong work on t or by heatng/coolng t. ΔU Q + W conservaton o energy. Sgn conventon: we consder Q and W to be postve energy lows nto the system. For a cyclc process (U U ) Q - W. I, n addton, Q 0 then W 0 An equvalent ormulaton: erpetual moton machnes o the rst type do not exst.

6 Quas-Statc rocesses Quas-statc (quas-equlbrum) processes sucently slow processes, any ntermedate state can be consdered as an equlbrum state (the macroparamers are welldened or all ntermedate states). Advantage: the state o a system that partcpates n a quas-equlbrum process can be descrbed wth the same (small) number o macro parameters as or a system n equlbrum (e.g., or an deal gas n quasequlbrum processes, ths could be and ). y contrast, or nonequlbrum processes (e.g. turbulent low o gas), we need a huge number o macro parameters. Examples o quasequlbrum processes: sochorc: const sobarc: const sothermal: const adabatc: Q 0 For quas-equlbrum processes,,, are well-dened the path between two states s a contnuous lnes n the,, space.

7 Work A the pston area he work done by an external orce on a gas enclosed wthn a cylnder tted wth a pston: W (A) dx (Adx) - d orce Δx he sgn: thevolumesdecreased, W s postve (by compressng gas, we ncrease ts nternal energy); the volume s ncreased, W s negatve (the gas decreases ts nternal energy by dong some work on the envronment). W - d - apples to any shape o system boundary W (, ) d du Q d he work s not necessarly assocated wth the volume changes e.g., n the Joule s experments on determnng the mechancal equvalent o heat, the system (water) was heated by strrng.

8 W D A W and Q are not State Functons (, ) d dagram ΔU Q + W C - we can brng the system rom state to state along nnte # o paths, and or each path (,) wll be derent. Snce the work done on a system depends not only on the ntal and nal states, but also on the ntermedate states, t s not a state uncton. W U s a state uncton, W - s not thus, Q s not a state uncton ether. net W A + WCD ( ) ( ) ( )( ) < 0 - the work s negatve or the clockwse cycle; the cyclc process were carred out n the reverse order (counterclockwse), the net work done on the gas would be postve.

9 Comment on State Functons U,,, and are the state unctons, Q and W are not. Specyng an ntal and nal states o a system does not x the values o Q and W, we need to know the whole process (the ntermedate states). Analogy: n classcal mechancs, a orce s not conservatve (e.g., rcton), the ntal and nal postons do not determne the work, the entre path must be speced. In math terms, Q and W are not exact derentals o some unctons o macroparameters. o emphasze that W and Q are NO the state unctons, we wll use sometmes the curled symbols δ (nstead o d) or ther ncrements (δq and δw). d U d S d - an exact derental U S ( ) ( ) y z(x,y ) dz Ax x, y dx + Ay x, y dy - t s an exact derental t s the derence between the values o some (state) uncton z(x,y ) z(x,y) at these ponts: dz z ( x + dx, y + dy) z( x, y) x A ( ) A ( x y) x x, y y, A necessary and sucent condton or ths: y x I ths condton z z z ( ) ( x, y) z ( ) ( x, y) Ax x, y Ay x, y d z dx + dy holds: x y x y y x e.g., or an deal gas: δq du + d Nk d + d - cross dervatves are not equal

10 roblem Imagne that an deal monatomc gas s taken rom ts ntal state A to state by an sothermal process, rom to C by an sobarc process, and rom C back to ts ntal state A by an sochorc process. Fll n the sgns o Q, W, and ΔU or each step., 0 5 a A Step Q W ΔU const A C C C A, m U Nk Nk

11 Quasstatc rocesses n an Ideal Gas Nk Nk, Nk Nk sochorc ( const ) W 0 3 Q Nk ( ) > ( C Δ ) 0 du Q sobarc ( const ) ( ) 0 W (, ) d < 5 Q Nk 0 du W Q ( ) > ( C Δ ) + (see the last slde)

12 Isothermal rocess n an Ideal Gas sothermal ( const ) : W Nk W Nk ln W d (, ) d Nk Nk ln Q du 0 W W - > 0 > (compresson) W - < 0 < (expanson)

13 Adabatc rocess n an Ideal Gas adabatc (thermally solated system) he amount o work needed to change the state o a thermally solated system depends only on the ntal and nal states and not on the ntermedate states. Nk Nk U Nk W Q 0 du W (, ) d to calculate W -, we need to know (,) or an adabatc process du Nk d d ( the # o unrozen degrees o reedom ) Nk d + d Nkd d + d d d Adabatc d d + + 0, exponent d ln ln const

14 Adabatc rocess n an Ideal Gas (cont.) const Nk Nk An adabata s steeper than an sotherma: n an adabatc process, the work lowng out o the gas comes at the expense o ts thermal energy ts temperature wll decrease. + (, ) + W d d +/3.67 (monatomc), +/5.4 (datomc), +/6.33 (polyatomc) (agan, neglectng the vbratonal degrees o reedom) W Δ NkΔ ΔU rove ( )

15 Summary o quas-statc processes o deal gas ΔU U U Quas-Statc process sobarc (Δ0) sochorc (Δ0) sothermal (Δ0) adabatc (Q0) ΔU Q W Δ U NkΔ Δ 0 + Δ Δ U Nk Δ ( Δ ) ( ) W Δ Δ 0 Nk ln Δ U NkΔ Δ( ) 0 ΔU Ideal gas law

16 roblem Imagne that we rapdly compress a sample o ar whose ntal pressure s 0 5 a and temperature s 0 C ( 95 K) to a volume that s a quarter o ts orgnal volume (e.g., pumpng bke s tre). What s ts nal temperature? Rapd compresson approx. adabatc, no tme or the energy exchange wth the envronment due to thermal conductvty Nk Nk Nk For adabatc processes: also / const K 4 95 K K - poor approx. or a bke pump, works better or desel engnes const

17 Free expanson Non-equlbrum Adabatc rocesses. const ncreases decreases (coolng). On the other hand, ΔU Q + W 0 U ~ unchanged (agrees wth expermental ndng) Contradcton because approach # cannot be justed volent expanson o gas s not a quasstatc process. must reman the same. const - apples only to quas-equlbrum processes!!!

18 Isobarc processes ( const): he Enthalpy du Q - Δ Q -Δ() Q Δ U + Δ() H U + - the enthalpy he enthalpy s a state uncton, because U,, and are state unctons. In sobarc processes, the energy receved by a system by heatng equals to the change n enthalpy. sochorc: sobarc: Q Δ U Q Δ H n both cases, Q does not depend on the path rom to. Consequence: the energy released (absorbed) n chemcal reactons at constant volume (pressure) depends only on the ntal and nal states o a system. he enthalpy o an deal gas: (depends on only) H U + Nk + Nk + Nk

19 Heat Capacty he heat capacty o a system - the amount o energy transer due to heatng requred to produce a unt temperature rse n that system C s NO a state uncton (snce Q s not a state uncton) t depends on the path between two states o a system ( sothermc C, adabatc C 0 ) +d Q C δ Δ 3 he specc heat capacty c C m

20 C and C C U C Q du + d δ d d H C the heat capacty at constant volume the heat capacty at constant pressure o nd C and C, we need (,,) 0 and U U (,) For an deal gas C Nk U nr Nk H + Nk C + nr # o moles For one mole o a monatomc deal gas: C 3 R C 5 R

21 Another roblem Durng the ascent o a meteorologcal helum-gas lled balloon, ts volume ncreases rom m 3 to.8 m 3, and the pressure nsde the balloon decreases rom bar (0 5 N/m ) to 0.5 bar. Assume that the pressure changes lnearly wth volume between and. (a) I the ntal s 300K, what s the nal? (b) How much work s done by the gas n the balloon? (c) How much heat does the gas absorb, any? (a) (b) (c) Nk δw ( ) d δw ON ΔU δ ON W Y δw Y δ Q + δw ON Nk 0.5bar.8m 300K 3 bar m 3 ( ) 0.65 bar/m bar - work done on a system δw Y ( ) d 70K ( bar m bar m ) 0.6bar m 6 0 J ( ) d 3 δq ΔU δwon Nk ON Y - work done by a system ( ) W + δw.5 0 J ( 0.) J J