SOLUTIONS TO TKP 2016

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Collin McDowell
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1 SOUIONS O KP 6 BSc/MSci Exmintion by course unit. uesdy 4th My 6 :m :pm SPA54 Durtion: hours minutes herml & Kinetic Physics YOU ARE NO PERMIED O READ HE CONENS O HIS QUESION PAPER UNI INSRUCED O DO SO BY AN INIGIAOR. Instructions: Answer A questions in section A. Answer ONY WO questions rom section B. Section A crries 5 mrks, ech question in section B crries 5 mrks. I you nswer more questions thn speciied, only the irst nswers (up to the speciied number) will be mrked. Cross out ny nswers tht you do not wish to be mrked. Only non-progrmmble clcultors re permitted in this exmintion. Plese stte on your nswer book the nme nd type o mchine used. Complete ll rough workings in the nswer book nd cross through ny work tht is not to be ssessed. Importnt note: he cdemic regultions stte tht possession o unuthorised mteril t ny time when student is under exmintion conditions is n ssessment oence nd cn led to expulsion rom QMU. Plese check now to ensure you do not hve ny notes, mobile phones or unuthorised electronic devices on your person. I you hve ny, then plese rise your hnd nd give them to n invigiltor immeditely. It is lso n oence to hve ny writing o ny kind on your person, including on your body. I you re ound to hve hidden unuthorised mteril elsewhere, including toilets nd clokrooms it will be treted s being ound in your possession. Unuthorised mteril ound on your mobile phone or other electronic device will be considered the sme s being in possession o pper notes. A mobile phone tht cuses disruption is lso n ssessment oence. EXAM PAPERS MUS NO BE REMOED ROM HE EXAM ROOM Exminers: K.J.Donovn.J.S.Dennis Queen Mry University o ondon, 6

2 Pge SPA54 (6) Question A SECION A. Answer ll questions in Section A. During n dibtic process on n idel gs wht is the reltionship between initil temperture nd volume i nd i nd inl temperture nd volume nd? Explin ny terms used. i i [4] C P C Where Question A C P nd C re the het cpcities t constnt pressure nd volume. [] Write down n expression or the Gibbs potentil, G, or P-- system. Question A G U S P [4] [6 mrks] [4 mrks] A system consisting o gs contined in cylinder with rictionless piston is tken round the pth b c s shown in igure. In tking the system rom c the internl energy chnges by +75 J. In one cycle the system does J o work. P b c b P P b

3 SPA54 (6) Pge i) he work in going rom b c is W b 5 J. Wht is the het evolved in going rom c. Be creul to include the sign nd mke cler whether the het is bsorbed or expelled by the system. c W c Wbc J W c J Wbc 5 J Q c Uc Wc J [] Het is expelled by the system. [5 mrks] ii) During the trnsition rom b c the internl energy chnges by - J Wht is the het evolved during the process b? Be creul to include the sign nd mke cler whether the het is bsorbed or expelled by the system. Qb Ub Wb Ub Uc Ubc Q b Uc Ubc 75 5 J [] Question A4 Het is bsorbed by the system. [5 mrks] Wht is the root men squre velocity, bsolute temperture? v rms, o molecule o mss m in gs t urn Over

4 Pge 4 SPA54 (6) Question A5 k v B rms [6] m [6 mrks] Write down deinition o the isotherml bulk modulus,, o P- system in terms o thermodynmic vribles nd prtil dierentil. Mke it cler which thermodynmic vrible is being held constnt. P [6] [6 mrks] Question A6 A Crnot het pump opertes between hot nd cold reservoirs t 45 K nd 5 K respectively..6 kw o work is done on the het pump. i) How much het is delivered to the hot reservoir? ii) How much het will be extrcted rom the cold reservoir to llow this? Q HP W 45.5 Q.5.6 kw.4 kw [] Q Q W.4 kw,.6 kw 8 W [] [6 mrks]

5 SPA54 (6) Pge 5 Question A7 Write down n expression or the idel gs temperture mesured by constnt volume gs thermometer. G imit 7.6 PP P PP [6] [6 mrks] Question A8 In one cycle het engine operting between hot reservoir nd cold reservoir receives 5 J o het rom the hot reservoir nd does J o work; i) Wht is the eiciency, E, o the engine? ii) Wht is the het given up to the cold reservoir? W i) E. 57 Q 5 [] Q Q W 5 J 5 [] ii) J [6 mrks] urn Over

6 Pge 6 SPA54 (6) Question B SECION B. Attempt two o the our questions in this section. ) i) In n isochoric process one mole o vibrting ditomic idel gs o volume,, hs its pressure decresed rom initil pressure, pressure, P i. Wht is the het trnser during this process in terms o Pi, to inl P i nd? Be creul to include the sign nd stte whether the het ws bsorbed or expelled by the gs. ii) 7 7 Q U W U i Pi Pi Pii he het is expelled by the gs. In n isotherml process one mole o montomic idel gs t temperture,, is expnded rom n initil volume, i, to inl volume, het trnserred in this process in terms o nd i. Wht is the i? Be creul to include the sign nd stte whether the het ws bsorbed or expelled by the gs. i d Q U W W R R ln i R ln i i he het is bsorbed by the gs. iii) In n isobric process one mole o rigid ditomic idel gs t pressure, P, is compressed rom initil volume, i, to inl volume, the work done during this process in terms o P nd i. Write down i. Be creul to include the sign nd stte whether the work ws done by or on the gs. i i W P d Pi Pi i Work ws done on the gs. [6 mrks]

7 SPA54 (6) Pge 7 b) A useul description o rel gs is the vn der Wls gs with n eqution o stte or one mole o the gs given by; nd b re positive constnts. P b R. i) By considering the vn der Wls eqution o stte t = clculte the potentil energy contribution to the internl energy nd show tht the internl energy o one mole o vn der Wls gs is given by; U vdw R Hint: Clculte the work done in tking the gs rom n ininite dilution to some inite volume. At zero temperture the kinetic energy is zero nd ll o the internl energy is potentil energy. hereore I we tke n ininitely dilute gs ( i P b P ) with zero potentil energy (the molecules re too r prt to hve ny interction) nd compress it to some volume the work done is turned into potentil energy; d W P d At inite temperture there is the kinetic energy to include thus or one mole o gs U vdw R urn Over

8 Pge 8 SPA54 (6) ii) Show tht, when ignoring terms o second order in smllness, the enthlpy, H dw U P o one mole o vn der Wls gs my be pproximted by; he enthlpy H is deined s or vn der Wls gs. H dw 5 R H dw U P R Expnding the eqution o stte to ind P P Ignoring terms o nd order smllness bp H U P P b b P Pb R P R Using this in the enthlpy eqution H dw R 5 H dw R b Pb P R Pb R R Pb Pb QED [7 mrks]

9 c) SPA54 (6) Pge 9 In throttling process gs on the let hnd side o cylinder with dibtic wlls is orced by rictionless piston through porous plug into the right hnd side o the cylinder where second rictionless piston consequently withdrws. he initil pressure, volume nd temperture on the let hnd side re P i, i nd i nd inl pressure, volume nd temperture P, nd re obtined on the right hnd side s shown i igure. Initil P i + dp i P i, i, i Adibtic wlls inl P,, P - dp i) By inding the work done in tking the system rom its initil to its inl stte nd the internl energy chnge show tht the throttling process is isoenthlpic, H. igure ind the work crried out on the HS nd on the RHS seprtely; WHS Pi d Pii i Porous plug WRHS P d P urn Over

10 Pge SPA54 (6) here is no het low so W represents the internl energy chnge. U U Ui P Pii U P Ui Pii H H i H H Hi QED ii) By considering the gs to obey the vn der Wls eqution o stte o b) nd using i) nd nswers rom b) demonstrte using (nd clerly stting) pproprite pproximtions tht the Joule homson coeicient, J P H is given by; J b R 5 P R R rom b we hve H dw 5 R Pb Eliminte rom this expression by replcing with P P P R An expression or HdW involving only P nd is obtined; H dw 5 R P R Pb Becuse the throttling is n isoenthlpic process dierentiting wrt to P gives zero thus; H dw P 5 R P P R 5 Pb R P R P R b P 5 P R R P R b [] P H b R 5 P R R J QED

11 SPA54 (6) Pge iii) Give n expression or the inversion temperture, inv. ooking t the expression or J it is cler tht the denomintor is lwys positive while the numertor chnges sign going rom negtive to positive s the temperture is reduced the chnge occurring where the numertor becomes zero, the inversion temperture. b Rinv inv Rb iv) Explin wht hppens to the temperture o gs initilly t temperture i inv when it undergoes throttling process with decresed inl pressure. I the gs is t temperture then i inv J is positive nd i we decrese P P then P is negtive nd must be negtive too. ie. there will be decrese in temperture when the pressure is reduced i the gs is below the inversion temperture. [] [ mrks] urn Over

12 Pge SPA54 (6) Question B ) i) Write down the Clusius inequlity. ii) dq S R i iii) dq. kg o wter is heted rom o C to 5 o C. Clculte the entropy chnge o the wter including sign s result o this process. mc Pd i mc ln P i 4..5 ln JK. kg o ice t o C melts nd then wrms to 5 o C. Clculte the entropy chnge o the ice/wter including sign s result o this process. he entropy chnge must be clculted in two prts; b) melt 5 ml. Smelt mc wrm Pd S mc ln P i i or one mole o n idel montomic gs;,9 JK 4. ln JK S net, JK JK [] [6 mrks] i) Use the thermodynmic identity or P-- system nd n eqution o stte to show tht the entropy chnge, s volume nd temperture re chnged, is given by: S S(, ) S( i, i ) Rln Rln. i i

13 SPA54 (6) Pge he thermodynmic identity is; du ds Pd Re-rrnging du P ds d Using the equtions o stte; U R du Rd And P R P R ds R d R d Integrting this wrt t constnt S Rln i const Integrting this wrt t constnt S Rln const i Putting these two together; S R ln i R ln i QED ii) he gs is compressed in n isobric process rom i to i entropy chnge tht occurs s result.. Evlute the In n isobric process when the volume chnges rom P temperture chnge i i to R i to i. here is Pi i. R he entropy chnge is; urn Over

14 Pge 4 SPA54 (6) S R ln i R ln i i i 5 R ln R ln R ln i i [7 mrks] c) he equtions o the sublimtion nd the vporistion curves o prticulr mteril re given by: 8 lnp.75 sublimtion nd 5 lnp.65 vporistion where P is in tmospheres. i) ind the temperture nd pressure o the triple point. At the triple point the curves intersect nd thereore lnpp lnpp 8.75 P 5.65 P Subtrct the second rom the irst P P P P. Now put this vlue bck into either originl eqution lnp P 8.75 P P P.5 Atm ii) Use the Clusius Clpeyron eqution;

15 SPA54 (6) Pge 5 Re-rrnge where Integrte either side s nd dp d s, v v re the chnges in speciic entropy nd speciic volume respectively ollowing phse chnge, to demonstrte tht the molr ltent hets o sublimtion nd vporistion re 8R nd 5R respectively. You my ssume tht the speciic volume in the vpour phse is much lrger thn those in the liquid nd solid phses. dp d vp vp vp s l l l P v vp liq vp v v v R dp P vp l d R l lnp vp const R Compring with the given eqution or the vporistion curve 5 lnp.65 l vp R 5 So l vp 5 R Similr rguments my be pplied to the sublimtion curve to show tht l sub 8R iii) Remembering tht S is stte unction, consider smll closed loop round the triple point in the P - digrm to show tht l sublim e l vpour melt l P P P urn Over

P P [] ooking t the digrm to identiy the vrious ltent hets l melt l vpour l S sublim e P P P And thereore l melt 5R 8R l

16 Pge 6 SPA54 (6) where the l s re the ltent hets o the phse chnges nd hence ind the ltent het o usion or this mteril. On crossing phse boundry the entropy chnge per unit mss is l nd thereore going round the phse digrm clockwise l 4 l l S 65 P P P [] ooking t the digrm to identiy the vrious ltent hets l melt l vpour l S sublim e P P P And thereore l melt 5R 8R l melt R [ mrks] HIN: Consider wht hppens to unction o stte when system goes round cycle nd wht hppens to the entropy s system chnges phse?

17 SPA54 (6) Pge 7 Question B ) he incrementl work done on n elstic bnd extended by n ininitesiml length d when under tension is given by; dw d. he eqution o stte o the elstic bnd t temperture,, is given by; where is its length while nd re positive constnts. he Gibbs ree energy is deined s; G U S i) Using the bove deinition o G nd the thermodynmic identity or n elstic bnd ind the nturl vribles o G. Begin by writing the ininitesiml o G using the deinition; dg du dssd d d Simpliy using the hermodynmic Identity or rubber bnd du ds d dg Sd d rom this the nturl vribles cn be identiied s S nd G G, ii) ind the reltionships between the prtil dierentils o G with respect to ech o the two nturl vribles nd two other thermodynmic vribles. urn Over

18 Pge 8 SPA54 (6) Given the nturl vribles we my write the ininitesiml o G in nother wy; G G dg d d Compring with the nswer rom i) dg Sd d G S G iii) rom i) nd ii) ind Mxwell reltion between prtil dierentils o S nd with respect to the nturl vribles o G. We use the ct tht dg is perect dierentil nd tht thereore; G G Or otherwise written; S [] his is the required Mxwell reltion. [8 mrks] b) Demonstrte tht the therml expnsion coeicient,, is given by

19 SPA54 (6) Pge 9 urn Over We need to ind the prtil dierentil rom the eqution o stte; here re three wys (t lest?) o doing this dierentition;. Dierentiting unction o unction. Re-writing the eqution o stte with s the subject Now ind nd use the reciprocl reltion to obtin Simpliy using the eqution o stte [] Now use the reciprocl reltion nd thereore

20 Pge SPA54 (6) QED [6 mrks]. Implicit Dierentition. he result cn lso be rrived t by implicit dierentition by dierentiting both sides o the eqution o stte with respect to whilst holding constnt; [] Solving or [] nd. Use the Cyclic nd reciprocl rules. A third wy o chieving the result is to use the cyclicl reltionship;

21 SPA54 (6) Pge c) [6 mrks] he eect o surce tension mens tht work hs to be done in creting new re in, or exmple, sop ilm. he ininitesiml work done in creting tht new re is given by; where is the surce tension o the ilm. dw A da Along with this re work there is lso the volume work given by; dw Pd to be ccounted or when considering sop bubble. i) Write n expression or the totl incrementl work in terms o the rdius R o or the re bubble. A da 8 RdR 4R dw A da 8 RdR [] or the volume 4 R d R 4 dr dw Pd 4PR dr [] urn Over

22 Pge SPA54 (6) dwotl dwa dw 8RdR 4PR dr ii) A sop bubble, t temperture,, is expnded isothermlly rom n initil rdius, R i, to inl rdius, R,. Wht is the totl work done in this process? You my ssume tht the gs tht inltes the bubble my be treted s n idel gs. Be creul to include the sign. R Wotl dwotl 8 RdR 4 i Ri R Ri PR dr We need to replce P in the second integrl on the RHS by treting the gs s n idel gs; P nr nr P nr 4 R iii) R R dr Wotl 8 RdR nr R R R Wotl 4 i R R R R nr lnr i Ri R R R n ln Wotl 4 i R Ri Wht is the het low in the process described in i)?. Be creul to include sign. his is n isotherml process nd thereore U By the irst lw Q U W W Q 4 R R R n ln i i R Ri [ mrks]

23 SPA54 (6) Pge Question B4 ) i) Demonstrte the truth o the cyclic reltion or prtil dierentils by using P-- system nd the eqution o stte o n idel gs. he cyclic reltion or P-- system is; P P P he eqution o stte or n idel gs is; P R R P R P P R P R P R P P R P P P R R P R R P QED ii) P he isotherml bulk modulus or gs is deined s nd the isobric therml expnsion coeicient is deined s. Using the P cyclicl nd reciprocl reltions ind prtil dierentil reltionship or the product. P P P P Using the cyclic reltions; urn Over

24 Pge 4 SPA54 (6) P Using the reciprocl reltions; P iii) One mole o vn der Wls gs is described by the eqution o stte; P b R. Ignoring terms o second order in smllness show tht the isobric therml expnsion coeicient o vn der Wls gs,, s deined in ii) is given by; R P o ind the isobric therml expnsion coeicient we need to ind the eqution o stte; P using It is esier to re-write the eqution o stte with s the subject nd then ind beore using the reciprocl reltion to ind P P P R b P R b Pb P P R b

25 SPA54 (6) Pge 5 Getting everything in brckets on the RHS over common denomintor; P P b R R P P b And inlly R P P b Discrding terms o nd order smllness R P QED [ mrks] b) or gs o molecules o mss m t temperture the Mxwell velocity distribution unction is given by; m mv ( v ) exp. kb kb i) Write down n expression or the Mxwell speed distribution unction P(v). P v m mv 4v ( v) 4 v exp kb kb ii) he mode speed, v m expression or, o molecule is the most probble speed. ind n v m using the Mxwell speed distribution. urn Over

26 Pge 6 SPA54 (6) o ind the mode speed dierentite P(v) wrt v nd set the dierentil equl to zero. d m mv 4 v exp dv kb kb m d mv 4 v exp kb dv kb Setting this to zero; m mv m mv 4 v exp v v exp kb kb kb kb m v kb v kb m iii) he number o prticles striking surce per unit re per unit time or lux,, is given by; nv. 4 Where n is the prticle density nd v is the men velocity where or prticles o mss m t temperture irst we ind v 8k B. m Write n expression or in terms o P nd or n idel gs. N n where N is the totl number o prticles occupying the volume P Nk B N P n kb nv 4 4 P k B 8k B m P k B m [6 mrks]

27 SPA54 (6) Pge 7 c) A irst chmber contining mixture o two gses is connected to second chmber, under vcuum, by hole whose dimeter is smll compred to the men ree pth. he number o molecules escping rom the irst to the second chmber vi the hole will depend on the species. he process is known s eusion. i) Using some o the inormtion given in prt b) show tht the rtio o the number o species molecule to species tht hve escped in given time, e N e N, is given by e N e N n n m m he lux o prticles incident upon the hole cn ll be ssumed to escpe into the second chmber. In time t i we tke the hole to be o re A then the number o prticles o species i tht escpe will be; e N i i At ni 4 8k B At mi he rtio is then e N e N At At n n m m QED ii) Suppose we hve mixture o % He nd 9% 4 He in the irst chmber. Wht is the rtio o these isotopes tht collects in the second chmber? N N He 4 He n n He 4 He m m 4 He He [] urn Over

28 Pge 8 SPA54 (6) iii) o produce mixture with % He we would need to repet this procedure on the newly produced mixture nd continue this recycling pproximtely how mny times? Strting with rction.) we hve ter round o the process 4 Ater two rounds 4 4 Ater m rounds m 4 m m.. 7 m log. log.857 log m 9.7 log..5 m must be integer so m = [] his will give rtio o % [9 mrks] End o Pper An ppendix o pge with vlues o physicl constnts etc. ollows

29 SPA54 (6) Pge 9 DAA SHEE You my wish to use some o the ollowing dt. k B = Boltzmnn s constnt =.8 - J K - = Sten s constnt = J s - m -.K -4 c = elocity o light in vcuo =. 8 m s - N A = Avgdro s number = 6. mol - R = Gs constnt = 8. J mol - K - P tm = Atmospheric pressure = tm =. 5 P S = Ice point o wter = 7.5 K H O c P = Speciic het o wter t constnt pressure = 4. J K - kg - Ice c = Speciic het o ice t constnt pressure =. J K - kg - P S l = tent het o melting ice =. 5 J kg - mu = One tomic mss unit =.66-7 kg urn Over

Part I: Basic Concepts of Thermodynamics

Part I: Basic Concepts of Thermodynamics Prt I: Bsic Concepts o Thermodynmics Lecture 4: Kinetic Theory o Gses Kinetic Theory or rel gses 4-1 Kinetic Theory or rel gses Recll tht or rel gses: (i The volume occupied by the molecules under ordinry