Physics 207 Lecture 27

Transcription

1 hyscs 07 Lecture 7 hyscs 07, Lecture 7, Dec. 6 Agenda: h. 0, st Law o Thermodynamcs, h. st Law o thermodynamcs ( U Q + W du dq + dw ) Work done by an deal gas n a ston Introducton to thermodynamc cycles (hater ) Ideal gas at the molecular level, Internal Energy Molar Secc Heat (Q m c T n T) Ideal Molar Heat aacty (and U nt Q + W) onstant : v / R, onstant : / R + R 5/R Degree o Freedom and Equartton theorem Adabatc rocesses (no heat transer) Assgnments: roblem Set 0 (h. 0 & ) due Tuesday, Dec., :59 M h. 0:,,8,,50,68 h.:,6,9,6,70 Monday, hater ( nd Law o Thermdynamcs) hyscs 07: Lecture 7, g st Law: Work & Heat Two tyes o varables State varables: descrbe the system (e.g. T,,, U). Transer varables: descrbe the rocess (e.g. Q, W). 0 unless a rocess occurs nvolve change n state varables. Work done on gas (mnus sgn because system volume s reerenced) W F d cosθ -F y - A y - ald only or sobarc rocesses ( ( constant) I not, use average orce or calculus: W area under curve NkB T) / dagram hyscs 07: Lecture 7, g Work: st Law: Work & Heat Deends on the ath taken n the -dagram (It s not just the destnaton but the ath ) Same or Q (heat), deends on ath st Law: Work (Area under the curve) Work deends on the ath taken n the -dagram : (a) W a W to + W to (here ether or constant) W a - ( - ) + 0 > 0 (work done on system) (b) W b W to + W to (here ether or constant) W b 0 - ( - ) > W a > 0 (work done on system) (c) Need exlct orm o versus but W c > 0 hyscs 07: Lecture 7, g hyscs 07: Lecture 7, g Reversng the ath ( ) Work deends on the ath taken n the -dagram : st Law: Work (gong ull cycle) Work deends on the ath taken n the -dagram : 5 (a) W a W to + W to (here ether or constant) W a 0 - ( - ) < 0 (work done on system) (b) W b W to + W to (here ether or constant) W b - ( - ) + 0 < W a < 0 (work done on system) (c) Need exlct orm o versus but W c < 0 hyscs 07: Lecture 7, g 5 (a) W a W to + W to (here ether or constant) W a - ( - ) > 0 (work done on system) (b) W b W to + W to 5 (here ether or constant) W b - ( - ) < 0 (work done by system > 0) (a) & (b) W a + W b - ( - ) - ( - ) ( - ) x ( - ) < 0 but net work done by system (what I get to use) s ostve. hyscs 07: Lecture 7, g 6 age

2 hyscs 07 Lecture 7 Lecture 7: Exercse (relude) Work done by system onsder the ath connectng onts and on the dagram. Lecture 7: Exercse Work done by system onsder the two aths, and, connectng onts and on the dagram. The magntude o the work, W, done by the system n gong rom to along ath s. W on system > 0, by system < 0 ( deal gas, Nk B T). W on system < 0, by system > 0. W +W on system < 0, by system > 0 (area o trangle) (A) W > W (B) W W (A) W < W Work (W) and heat (Q) both deend on the ath taken n the -dagram! hyscs 07: Lecture 7, g 7 hyscs 07: Lecture 7, g 8 Frst Law o Thermodynamcs wth heat (Q) and/or work (W) Frst Law o Thermodynamcs U Q + W work done on the system heat low n (+) or out (-) varaton o nternal energy U s ndeendent o ath n -dagram Deends only on state o the system (,,T, ) Isolated system s dened as one wth No nteracton wth surroundngs Q W 0 U 0. U U : nternal energy remans constant. hyscs 07: Lecture 7, g 9 Other Alcatons yclc rocess: rocess that starts and ends at the same state (, T T and ) Must have U 0 Q -W. Adabatc rocess: No energy transerred through heat Q 0. So, U W. Imortant or Exanson o gas n combuston engnes Lquacton o gases n coolng systems, etc. Isobarc rocess: ( s constant) Work (on system) s: W d ( ) hyscs 07: Lecture 7, g 0 Other Alcatons (contnued) Isovolumetrc rocess: onstant volume W 0. So U Q all heat s transerred nto nternal energy e.g. heatng a can (and no work done). Isothermal rocess: T s constant I deal gas: nrt, we nd nrt/. Work (on system) becomes : W d d nrt nrt ln s constant. -dagram: sotherm hyscs 07: Lecture 7, g Lecture 7: Exercse rocesses Identy the nature o aths A, B,, and D Isobarc, sothermal, sovolumetrc, and adabatc A D B T T T T hyscs 07: Lecture 7, g age

3 hyscs 07 Lecture 7 Heat Engnes We now try to do more than just rase the temerature o an object by addng heat. We want to add heat and get some work done! Heat engnes: urose: onvert heat nto work usng a cyclc rocess Examle: ycle a ston o gas between hot and cold reservors * (Strlng cycle) ) hold volume xed, rase temerature by addng heat ) hold temerature xed, do work by exanson ) hold volume xed, lower temerature by dranng heat ) hold temerature xed, comress back to orgnal hyscs 07: Lecture 7, g Heat Engnes Examle: The Strlng cycle TT TT H TT TT H We can reresent ths cycle on a - dagram: x start a b T T H * reservor: large body whose temerature does not change when t absorbs or gves u heat hyscs 07: Lecture 7, g Heat Engnes Identy whether Heat s or REMOED rom the gas ostve work s done or the gas or each ste o the Strlng cycle: U Q + W (reerences system) ste HEAT REMOED ostve WORK W 0 U Q > 0 U REMOED REMOED a b T T H U 0 W Q < 0 0 W Q > 0 W 0 U Q < 0 hyscs 07: Lecture 7, g 5 REMOED Lecture 7: Exercse yclc rocesses Identy A gas s taken through the comlete cycle shown. The net work done on the system was (A) ostve (B) negatve () zero hyscs 07: Lecture 7, g 6 Lecture 7: Exercse yclc rocesses (gong n crcles) Identy A gas s taken through the comlete cycle shown. The net work done on the system (by the world) was (A) ostve (B) negatve () zero Work s done only on the horzontal aths, and the area under the thrd ath segment s ostve and larger than the area under the rst ath segment, whch s negatve. Hence the net work (on the system) s ostve. (We, the world, are not ganng ostve work.) hyscs 07: Lecture 7, g 7 h. : Knetc Theory o an Ideal Mcroscoc model or a gas Goal: relate T and to moton o the molecules Assumtons or deal gas: Number o molecules N s large They obey Newton s laws (but move randomly as a whole) Short-range nteractons durng elastc collsons Elastc collsons wth walls (an mulse) ure substance: dentcal molecules Ths mles that temerature, or an deal gas, s a drect measure o average knetc energy o a molecule N k B T N mv mv k B T hyscs 07: Lecture 7, g 8 age

4 hyscs 07 Lecture 7 onsder a xed volume o deal gas. When N or T s doubled the ressure ncreases by a actor o. mv k B T. I T s doubled, what haens to the rate at whch a sngle molecule n the gas has a wall bounce? (A) x. (B) x () x.. I N s doubled, what haens to the rate at whch a sngle molecule n the gas has a wall bounce? (A) x Lecture 7, Exercse N k B T (B) x. () x hyscs 07: Lecture 7, g 9 Knetc Theory o an Ideal Theorem o equartton o energy (A key result o classcal hyscs) v x v y v z v mv k T B Each degree o reedom contrbutes k B T/ to the energy o a system (e.g., translaton, rotaton, or vbraton) Total translatonal knetc energy o a system o N deal gas molecules K N mv tot trans NkBT Internal energy o monoatomc gas: U K deal gas K tot trans Root-mean-square seed: v rms v k BT m nrt hyscs 07: Lecture 7, g 0 Lecture 7, Exercse & 5 A gas at temerature T s mxture o hydrogen and helum gas. Whch atoms have more KE (on average)? (A) H (B) He () Both have same KE Lecture 7, Exercse 6 An atom n a classcal sold can be characterzed by three ndeendent harmonc oscllators, one or the x, y and z- drectons? How many degrees o reedom are there? How many degrees o reedom n a D smle harmonc oscllator? (A) (B) () (D) (E) Some other number (A) (B) () (D) (E) Some other number hyscs 07: Lecture 7, g hyscs 07: Lecture 7, g Ideal Molar Heat aactes Denton o molar heat caactes (relates change n the nternal energy to the temerature) Ideal Internal Energy n lmq / T n δ Q / δt 0 K tot trans U NkBT nrt There s only mcroscoc knetc energy (.e., no srngs) n a monoatomc deal gas (He, Ne, etc.) At constant, work W s 0 so U Q R At constant : U Q + W Q - nrt R + R hyscs 07: Lecture 7, g Lecture 7, Exercse 6 An atom n a classcal sold can be characterzed by three ndeendent harmonc oscllators, one or the x, y and z- drectons ( U er atom RT)? What s the classcal molar heat caacty ( 0)? (A) nr (B) nr () nr (D) nr (E) Some other number hyscs 07: Lecture 7, g age

5 hyscs 07 Lecture 7 Adabatc rocesses By denton a rocess n whch no heat traner (Q) occurs W For an Ideal : const d const d Adabatc rocess: I deal gas then s constant nrt but not sothermal Work (on system) becomes : const ( ) hyscs 07: Lecture 7, g 5 Reca, Lecture 7 Agenda: h. 0, st Law o Thermodynamcs, h. st Law o thermodynamcs ( U Q + W du dq + dw ) Work done by an deal gas n a ston Introducton to thermodynamc cycles (hater ) Ideal gas at the molecular level, Internal Energy Degree o Freedom and Equartton theorem Adabatc rocesses (no heat transer) Assgnments: roblem Set 0 (h. 0 & ) due Tuesday, Dec., :59 M h. 0:,,8,,50,68 h.:,6,9,6,70 Fnsh h., Monday, Read hater ( nd Law o Thermdynamcs) hyscs 07: Lecture 7, g 6 age 5