PHYSICS 212 MIDTERM II 19 February 2003

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1 PHYSICS 1 MIDERM II 19 Feruary 003 Exam s losed ook, losed notes. Use only your formula sheet. Wrte all work and answers n exam ooklets. he aks of pages wll not e graded unless you so request on the front of the page. Show all your work and explan your reasonng (exept on #1). Partal redt wll e gven (not on #1). No redt wll e gven f no work s shown (not on #1). If you have a queston, rase your hand or ome to the front. 1. ( ponts) For eah of these multple hoe questons, ndate the orret response (A, B, C, or D (where needed)) on the page for prolem 1 n your exam ooklet. ) An deal gas s taken through two losed yle proesses. Proess A s the yle aa, and proess B s the yle ada n the fgure at rght. For whh proess s the amount of heat transferred to the gas the largest? p A) Proess A B) Proess B C) Both the same. a d V ) A metall rng wth a small gap s shown at rght. he metal has a postve lnear expanson oeffent. As the temperature of the rng nreases, does the sze of the gap nrease, derease, or reman the same? A) Inreases. B) Dereases. C) Remans the same. ) An deal gas undergoes an adaat ompresson. Does the temperature of the gas nrease, derease, or reman the same? A) Inreases. B) Dereases. C) Remans the same. v) he volume of an deal gas s dereased at onstant pressure. Does the entropy of the gas nrease, derease, or reman the same? A) Inreases. B) Dereases. C) Remans the same. v) In an deal refrgerator, s the heat Q h transferred to the hot reservor per yle less than, greater than, or equal to the work W done per yle? A) Less than. B) Greater than. C) he same as.

2 . (0 ponts) A sold steel sphere has a dameter of 8.0 m at C. A thn alumnum plate has a rular hole n t wth a dameter of 8.00 m at C. Assume that steel has a lnear expanson oeffent α S 10 x 10-6 / C and that alumnum has a lnear expanson oeffent α Al x 10-6 / C. a) o what temperature must the alumnum plate e heated n order for the steel sphere (not heated) to just ft through the plate's hole? ) If plate and sphere are oth heated, what must the ommon temperature e for the sphere to ft through the hole? 3. (0 ponts) A 3.0 kg alumnum uket ontans.0 kg of e, and oth have an ntal temperature of 0 C. A hot 6.0 kg lok of alumnum s added to the uket and the whole system omes to thermal equlrum at a fnal temperature of f 30 C. Assume Al 900 J/kgK, 4000 J/kgK, L F, 300 kj/kg. a) What was the ntal temperature of the alumnum lok? ) After the equlrum aove s reahed, more e s added to the system and a new equlrum temperature of 0 C s reahed wth no e remanng. How muh extra e was added? 4. (3 ponts) Fve moles of an deal monatom gas are taken through the yle ada shown elow. Proesses a and d are adaat. Assume P 8.0 x 10 Pa, V 0.10 m 3, V V, V a 10 V and R 8.3 J/molK. a) How muh work s done y the gas durng the omplete yle? ) What s the effeny of ths heat engne? ) Is the effeny found n part () larger, smaller, or the same as the effeny of an deal engne operatng etween the hghest and lowest temperatures that our n the yle? d) Fnd the entropy hange of the gas durng the proess. P P d a V a V

3 PHYSICS 1 MIDERM II SOLUIONS 19 Feruary ) A For a yl proess E nt 0, sne the nternal energy s a state funton (path ndependent). hus the frst law of thermodynams mples that Q W for the yle. he work done y the gas s just the area of the enlosed path. Both paths are lokwse, so result n postve work done y the gas. he yle aa has a larger area, so more work s done, and hene more heat s transferred to the gas. ) A For a materal wth a postve lnear expanson oeffent, eah lnear dmenson nreases as the temperature nreases. hus the rumferene of the rng nreases, as do all parts of the rumferene nludng the part that s there and the part that s not. he gap nreases n the same manner that the mssng pee tself would expand. ) A Durng an adaat proess Q 0. hs mples that E nt -W. In any ompresson, the gas does negatve work, so n ths ase the nternal energy must nrease. Sne E nt nc V, the temperature must nrease. hat s, the work done on the gas nreases the nternal energy of the gas. v) B he dfferental hange n entropy s gven y ds dq/, so the sgn of any hange n entropy s the same as the sgn of the heat transferred. For a onstant pressure proess, the heat s gven y dq nc P d and the deal gas law (pv nr) mples that s proportonal to V. hus a derease n V mples a derease n and hene a negatve heat transfer (.e., heat leaves the gas). hus the hange n entropy s negatve. v) B In a refrgerator, the heat n plus the work done on the system equals the heat out: Q + W Q, or Q + W Q. hus the heat transferred to the hot reservor s larger n out than the work done. h

4 PHYSICS 1 MIDERM II SOLUIONS 19 Feruary 003. a) In the frst ase only the plate expands. he hole also expands n the same proportons. Equate the expanded hole to the sphere to see when they wll just ft: dal dal, + α AldAl, ds, ds α AldAl, ds α AldAl, ds + α AldAl, 80. m 800. m C ( 8. 00m) 1 C ) In the seond ase oth sphere and hole expand, ut the hole expands more sne t has a larger expanson oeffent. d d + α d d + α d Al Al, Al Al, S, S S, ( α ) AldAl, αsds, ds, dal, ds αaldal, αsds, ds + αaldal, αsds, 80. m 800. m C m m 19 C ( ) ( )

5 PHYSICS 1 MIDERM II SOLUIONS 19 Feruary a) o fnd the ntal temperature of the lok, equate the heat transferred to the uket and e & to the heat transferred from the lok, takng aount of the mnus sgn. Quket + Qe+ Qlok Quket muketal( ) Qe+ melf, + m( ) Qlok mlokal(, lok ) muketal( ) + melf, + m( ) mlokal(, lok ) ( muket Al + m )( ) + melf, +, lok m, lok lok +, 30 C+, lok 434 C f lok Al ( ) + m + m ( ) m L m uket Al f e F, lok Al [ ] + ( ) 3kg( 900J kgk) + kg( 4000J kgk) 30 C kg 300kJ kg 6kg( 900 J kgk) ) Now equate the heat that s requred to melt the new e wth the amount of heat that s transferred from the uket, lok, and as they are ooled from f1 30 C to f 0 C. Q m L Q m + m Q Q m L m + m m m m e, melt e F, Al, uket+ lok Al f f1 e, melt Al, e F, uket+ lok Al f f1 e e ( ) m + + m L uket lok Al f1 f F, ( ) ( ) ( ) ( ) ( ) ( ) 9kg( 09. kj kgk) + kg( 4kJ kgk) 30C 300kJ kg. kg e 81 3

6 PHYSICS 1 MIDERM II SOLUIONS 19 Feruary a) he work done n a yle s the area enlosed y the yle n the p-v dagram, ut the adaat s hard to ntegrate, so we use the 1 st law of thermodynams. Around the full yle, the heat must equal the work, sne there s no hange n the nternal energy. he adaats have no heat transfer and the other two steps are smply at onstant volume and onstant pressure. o fnd the temperatures, relate the ponts usng the deal gas law and the adaat defnton. W Qnet Q + Qda ncp + ncv da nr W nr 3 ( )+ nr ( a d ) ( ( )+ ( a d )) 3 PV Pa m PV nr ( 01. ) : K nr mol( 8 3J molk) 198. PV P : V P P, V V 38K nr nr d: P V P V P P V γ γ V PV P V d d d P γ γ d d 10 d 1 3 ( ) 10 1 V 10V nr nr ( ) K γ γ a P V P V P P V V PV : P a a a a Va 10V nr mol( 83. J molk) W ( ( 38K 198K) + 3( 41K 1319K) ) d γ γ P10V nr a a K ( ) ( ) W 144kJ ) Heat s transferred nto the gas durng the proess, whh s a onstant pressure proess. Q ncp Q n R( ) mol ( 8. 3J molk) ( 38K 198K) 00kJ W W 144J ε Q Q 00J n ε 7% ) he hghest and lowest temperatures n the yle our at and a, respetvely. hus the effeny of an deal engne would e: C a 41K ε deal H 38K ε 89% ε < ε deal deal d) Proess s a onstant pressure proess, so the heat transferred s Q nc P. he entropy hange s thus: dq nc d P d S ncp ncp nc p ln ln S n Rln mol ( 83. J molk)( ) S J K 4

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