Thermodynamic Relations

Transcription

1 7 hermodynamic elation 7.1. General aect. 7.. undamental of artial differentiation Some general thermodynamic relation Entroy equation (d equation) Equation f internal energy and enthaly Meaurable quantitie : Equation of tate, co-efficient of exanion and comreibility, ecific heat, Joule-homon co-efficient 7.7. Clauiu-Claeryon equation ighlight bjectie ye Quetion Exercie GENEA ASPECS n thi chater, ome imtant thermodynamic relation are deduced ; rincially thoe which are ueful when table of roertie are to be comiled from limited exerimental data, thoe which may be ued when calculating the wk and heat tranfer aociated with rocee undergone by a liquid olid. t hould be noted that the relation only aly to a ubtance in the olid hae when the tre, i.e. the reure, i unifm in all direction ; if it i not, a ingle alue f the reure cannot be alloted to the ytem a a whole. Eight roertie of a ytem, namely reure (), olume (), temerature (), internal energy (u), enthaly (h), entroy (), elmholtz function (f) and Gibb function (g) hae been introduced in the reiou chater. h, f and g are ometime referred to a thermodynamic otential. Both f and g are ueful when conidering chemical reaction, and the fmer i of fundamental imtance in tatitical thermodynamic. he Gibb function i alo ueful when conidering rocee inoling a change of hae. f the aboe eight roertie only the firt three, i.e.,, and are directly meaurable. We hall find it conenient to introduce other combination of roertie which are relatiely eaily meaurable and which, together with meaurement of, and, enable the alue of the remaining roertie to be determined. hee combination of roertie might be called thermodynamic gradient ; they are all defined a the rate of change of one roerty with another while a third i ket contant. 7.. UNDAMENAS PAA DEENAN et three ariable are rereented by x, y and z. heir functional relationhi may be exreed in the following fm : f(x, y, z) 0...(i) x x(y, z)...(ii) y y(x, z)...(iii) z z(x, y)...(i) et x i a function of two indeendent ariable y and z x x(y, z)...(7.1) 341

2 34 ENGNEENG EMDYNAMCS hen the differential of the deendent ariable x i gien by dx where dx i called an exact differential. f x J M y z and x x dy yj G z J dz...(7.) z z y y x J N hen dx Mdy Ndz...(7.3) Partial differentiation of M and N with reect to z and y, reectiely, gie M x z yz M z and N x y zy N y...(7.4) dx i a erfect differential when eqn. (7.4) i atified f any function x. Similarly if y y(x, z) and z z(x, y)...(7.5) then from thee two relation, we hae y y dy x J dx z z J dz...(7.6) x z dz x J z dx y yj dy...(7.7) x y dy x J dx y z z z z J dx dy x x J y G y J x QP y y z x J G z z J x G x J yqp dx y z z J x yj dy x y y z x J G z z J x G x J dx dy yqp y y z x J G z z J x G x J 0 y y z z J x G x J y y x J z x z y yj x J G z J 1...(7.8) z y x n term of, and, the following relation hold good J J J 1...(7.9) \M-therm\h7-1.m5

3 EMDYNAMC EANS SME GENEA EMDYNAMC EANS he firt law alied to a cloed ytem undergoing a reerible roce tate that dq du d Accding to econd law, dq d J re. Combining thee equation, we get d du d du d d...(7.10) he roertie h, f and g may alo be ut in term of,, and a follow : dh du d d d d elmholtz free energy function, df du d d...(7.11) d d...(7.1) Gibb free energy function, dg dh d d d d...(7.13) Each of thee equation i a reult of the two law of thermodynamic. Since du, dh, df and dg are the exact differential, we can exre them a du dh df dg u J u d J d, h J h d J d, f J f d J g g J d J d, d. Comaring thee equation with (7.10) to (7.13) we may equate the creonding co-efficient. examle, from the two equation f du, we hae u J u and J he comlete grou of uch relation may be ummaried a follow : u J h J...(7.14) u J J J h f f J...(7.15) J g...(7.16) J g...(7.17) \M-therm\h7-1.m5

4 344 ENGNEENG EMDYNAMCS Alo, J J...(7.18) J J...(7.19) J J...(7.0) J...(7.1) J he equation (7.18) to (7.1) are known a Maxwell relation. t mut be emhaied that eqn. (7.14) to (7.1) do not refer to a roce, but imly exre relation between roertie which mut be atified when any ytem i in a tate of equilibrium. Each artial differential co-efficient can itelf be regarded a a roerty of tate. he tate may be defined by a oint on a three dimenional urface, the urface rereenting all oible tate of table equilibrium ENPY EQUANS (d Equation) Since entroy may be exreed a a function of any other two roertie, e.g. temerature and ecific olume, f(, ) i.e., d d d J d J J d J But f a reerible contant olume change dq c (d) (d) But, c J ence, ubtituting in eqn. (7.), we get d...(7.) J...(7.3) J [Maxwell eqn. (7.0)] d c d J d...(7.4) hi i known a the firt fm of entroy equation the firt d equation. Similarly, writing f(, ) d J d J d...(7.5) \M-therm\h7-1.m5

5 EMDYNAMC EANS 345 where c Alo J J whence, ubtituting in eqn. (7.5) d c d...(7.6) J [Maxwell eqn. (7.1)] J d...(7.7) hi i known a the econd fm of entroy equation the econd d equation EQUANS NENA ENEGY AND ENAPY (i) et u f(, ) o ealuate du u J d u J u d c d J d...(7.8) u J let u f (, ) u J u d J d u J u u J G J J u J, J u J, J hen du But ence u J J...(7.9) hi i ometime called the energy equation. rom equation (7.8), we get du c d S J U V W d...(7.30) (ii) o ealuate dh we can follow imilar te a under h f(, ) dh h J d h c d h d J d...(7.31) J \M-therm\h7-1.m5

6 346 ENGNEENG EMDYNAMCS o find h J ; let h f(, ) hen, dh But ence h J h d J d h h J J h J J h J,, J h J J J h J...(7.3) rom eqn. (7.31), we get S dh c d G J U V W d...(7.33) 7.6. MEASUABE QUANES ut of eight thermodynamic roertie, a earlier tated, only, and are directly meaurable. et u now examine the infmation that can be obtained from meaurement of thee rimary roertie, and then ee what other eaily meaurable quantitie can be introduced. he following will be dicued : (i) Equation of tate (ii) Co-efficient of exanion and comreibility (iii) Secific heat (i) Joule-homon co-efficient Equation of State et u imagine a erie of exeriment in which the olume of a ubtance i meaured oer a range of temerature while the reure i maintained contant, thi being reeated f ariou reure. he reult might be rereented grahically by a three-dimenional urface, by a family of contant reure line on a - diagram. t i ueful if an equation can be found to exre the relation between, and, and thi can alway be done oer a limited range of tate. No ingle equation will hold f all hae of a ubtance, and uually me than one equation i required een in one hae if the accuracy of the equation i to match that of the exerimental reult. Equation relating, and are called equation of tate characteritic equation. Accurate equation of tate are uually comlicated, a tyical fm being A B C... where A, B, C,... are function of temerature which differ f different ubtance. An equation of tate of a articular ubtance i an emirical reult, and it cannot be deduced from the law of thermodynamic. Neerthele the general fm of the equation may be \M-therm\h7-1.m5

7 EMDYNAMC EANS 347 redicted from hyothee about the microcoic tructure of matter. hi tye of rediction ha been deeloed to a high degree of reciion f gae, and to a leer extent f liquid and olid. he imlet otulate about the molecular tructure of gae lead to the concet of the erfect ga which ha the equation of tate. Exeriment hae hown that the behaiour of real gae at low reure with high temerature agree well with thi equation Co-efficient of Exanion and Comreibility rom -- meaurement, we find that an equation of tate i not the only ueful infmation which can be obtained. When the exerimental reult are lotted a a erie of contant reure line on a - diagram, a in ig. 7.1 (a), the loe of a contant reure line at any gien tate i J. f the gradient i diided by the olume at that tate, we hae a alue of a roerty of the ubtance called it co-efficient of cubical exanion β. hat i, ig Determination of co-efficient of exanion from -- data. β 1 J...(7.34) Value of β can be tabulated f a range of reure and temerature, lotted grahically a in ig. 7. (b). olid and liquid oer the nmal wking range of reure and temerature, the ariation of β i mall and can often be neglected. n table of hyical roertie β i uually quoted a an aerage alue oer a mall range of temerature, the reure being atmoheric. hi aerage co-efficient may be ymbolied by β and it i defined by 1 β 1...(7.35) ( 1) ig. 7. (a) can be relotted to how the ariation of olume with reure f ariou contant alue of temerature. n thi cae, the gradient of a cure at any tate i J. When thi gradient i diided by the olume at that tate, we hae a roerty known a the comreibility of the ubtance. Since thi gradient i alway negatie, i.e., the olume of a ubtance alway decreae with increae of reure when the temerature i contant, the comreibility i uually made a oitie quantity by defining it a \M-therm\h7-1.m5

8 348 ENGNEENG EMDYNAMCS ig. 7.. Determination of comreibility from - data. 1...(7.36) J can be regarded a a contant f many uroe f olid and liquid. n table of roertie it i often quoted a an aerage a alue oer a mall range of reure at atmoheric temerature, i.e., 1 1 ( 1) When β and are known, we hae Since J J J J β and J 1 J, β...(7.37) When the equation of tate i known, the co-efficient of cubical exanion and comreibility can be found by differentiation. a erfect ga, f examle, we hae and ence J β 1 1 and J 1, J J Secific eat ollowing are the three differential co-efficient which can be relatiely eaily determined exerimentally. \M-therm\h7-1.m5

9 EMDYNAMC EANS 349 J u Conider the firt quantity. During a roce at contant olume, the firt law infm u that an increae of internal energy i equal to heat ulied. f a calimetric exeriment i conducted with a known ma of ubtance at contant olume, the quantity of heat Q required to raie the temerature of unit ma by may be meaured. We can then write : u J Q J. he quantity obtained thi way i known a the mean ecific heat at contant olume oer the temerature range. t i found to ary with the condition of the exeriment, i.e., with the temerature range and the ecific olume of the ubtance. A the temerature u range i reduced the alue aroache that of J, and the true ecific heat at contant u olume i defined by c J. hi i a roerty of the ubtance and in general it alue arie with the tate of the ubtance, e.g., with temerature and reure. Accding to firt law of thermodynamic the heat ulied i equal to the increae of enthaly during a reerible contant reure roce. herefe, a calimetric exeriment carried out h with a ubtance at contant reure gie u, J Q J which i the mean ecific heat at contant reure. A the range of temerature i made infiniteimally mall, thi become the rate of change of enthaly with temerature at a articular tate defined by and, and thi i h true ecific heat at contant reure defined by c J. c alo arie with the tate, e.g., with reure and temerature. he decrition of exerimental method of determining c and c can be found in text on hyic. When olid and liquid are conidered, it i not eay to meaure c owing to the tree et u when uch a ubtance i reented from exanding. oweer, a relation between c, c, β and can be found a follow, from which c may be obtained if the remaining three roertie hae been meaured. he irt aw of hermodynamic, f a reerible roce tate that dq du d Since we may write u φ(, ), we hae du J d u J d J S J du u u dq d W d c d u W d hi i true f any reerible roce, and o, f a reerible contant reure roce, S S J U V W J J 1 u S J J J dq c (d) c (d) u ence c c Alo J c c U V J U V u W (d) U V W, and therefe S J U V \M-therm\h7-1.m5

10 350 ENGNEENG EMDYNAMCS Now, from eqn. (7.34) and (7.37), we hae c c β...(7.38) hu at any tate defined by and, c can be found if c, β and are known f the ubtance at that tate. he alue of, and are alway oitie and, although β may ometime be negatie (e.g., between 0 and 4 C water contract on heating at contant reure), β i alway oitie. t follow that c i alway greater than c. he other exreion f c and c can be obtained by uing the equation (7.14) a follow : Since c u J u J We hae c Similarly, c ence, c J J J...(7.39) h J h J J...(7.40) Alternatie Exreion f nternal Energy and Enthaly (i) Alternatie exreion f equation (7.9) and (7.3) can be obtained a follow : u J J...(7.9) But J J Subtituting in eqn. (7.9), we get hu, du c d Similarly, But by definition, ence J J J J 1 β β u J β...(7.41) h J J u J β β J d...[7.8 (a)] J...(7.3) h (1 β)...(7.4) \M-therm\h7-1.m5

11 EMDYNAMC EANS 351 hu dh c d (1 β) d...[7.31 (a)] (ii) Since ence J u J u h J h J β Joule-homon Co-efficient u β...(7.43) et u conider the artial differential co-efficient. We know that if a fluid i flowing h through a ie, and the reure i reduced by a throttling roce, the enthalie on either ide of the retriction may be equal. he throttling roce i illutrated in ig. 7.3 (a). he elocity increae at the retriction, with a conequent decreae of enthaly, but thi increae of kinetic energy i diiated by friction, a the eddie die down after retriction. he teady-flow energy equation imlie that the enthaly of the fluid i reted to it initial alue if the flow i adiabatic and if the elocity befe retriction i equal to that downtream of it. hee condition are ery nearly atified in the following exeriment which i uually referred to a the Joule-homon exeriment. J Contant h line, 1 1, luid,, 1 1 Sloe µ (a) ig Determination of Joule-homon co-efficient. hrough a ou lug (inerted in a ie) a fluid i allowed to flow teadily from a high reure to a low reure. he ie i well lagged o that any heat flow to from the fluid i negligible when teady condition hae been reached. urtherme, the elocity of the flow i ket low, and any difference between the kinetic energy utream and downtream of the lug i negligible. A ou lug i ued becaue the local increae of directional kinetic energy, caued by the retriction, i raidly conerted to random molecular energy by icou friction in fine aage of the lug. rregularitie in the flow die out in a ery ht ditance downtream of the lug, and (b) \M-therm\h7-1.m5

12 35 ENGNEENG EMDYNAMCS temerature and reure meaurement taken there will be alue f the fluid in a tate of thermodynamic equilibrium. By keeing the utream reure and temerature contant at 1 and 1, the downtream reure i reduced in te and the creonding temerature i meaured. he fluid in the ucceie tate defined by the alue of and mut alway hae the ame alue of the enthaly, namely the alue of the enthaly creonding to the tate defined by 1 and 1. rom thee reult, oint rereenting equilibrium tate of the ame enthaly can be lotted on a - diagram, and joined u to fm a cure of contant enthaly. he cure doe not rereent the throttling roce itelf, which i irreerible. During the actual roce, the fluid undergoe firt a decreae and then an increae of enthaly, and no ingle alue of the ecific enthaly can be acribed to all element of the fluid. f the exeriment i reeated with different alue of 1 and 1, a family of cure may be obtained (coering a range of alue of enthaly) a hown in ig. 7.3 (b). he loe of a cure [ig. 7.3 (b)] at any oint in the field i a function only of the tate of the fluid, it i the Joule-homon co-efficient µ, defined by µ. he change of temerature due h to a throttling roce i mall and, if the fluid i a ga, it may be an increae decreae. At any articular reure there i a temerature, the temerature of inerion, aboe which a ga can neer be cooled by a throttling roce. Both c and µ, a it may be een, are defined in term of, and h. he third artial differential co-efficient baed on thee three roertie i gien a follow : ence h J J J h h J h 1 µ may be exreed in term of c,, and a follow : he roerty relation f dh i dh d d rom econd d equation, we hae d c d dh c d J µc...(7.44) J d J a contant enthaly roce dh 0. herefe, 0 (c d) h S (c d) h µ J h an ideal ga, ; QP U S J V W U J V d W QP h 1 c J QP d...(7.45) d QP h...(7.46) \M-therm\h7-1.m5

13 EMDYNAMC EANS 353 J µ 1 c J 0. herefe, if an ideal ga i throttled, there will not be any change in temerature. et h f(, ) hen dh But h dh throttling roce, dh 0 J c 0 J h d h J h c 1 µ h J J J d c d h J i known a the contant temerature co-efficient CAUSUS-CAPEYN EQUAN J h h d...(7.47) c...(7.48)...(7.49) Clauiu-Claeryon equation i a relationhi between the aturation reure, temerature, the enthaly of eaation, and the ecific olume of the two hae inoled. hi equation roide a bai f calculation of roertie in a two-hae region. t gie the loe of a cure earating the two hae in the - diagram. uion Critical oint cure iquid Vaouriation cure Vaour Solid rile oint Sublimation cure ig diagram. \M-therm\h7-1.m5

14 354 ENGNEENG EMDYNAMCS he Clauiu-Claeryon equation can be deried in different way. he method gien below inole the ue of the Maxwell relation [eqn. (7.0)] J J et u conider the change of tate from aturated liquid to aturated aour of a ure ubtance which take lace at contant temerature. During the eaation, the reure and temerature are indeendent of olume. d d J g f g f where, g Secific entroy of aturated aour, f Secific entroy of aturated liquid, g Secific olume of aturated aour, and f Secific olume of aturated liquid. Alo, g f fg h fg and g f fg where fg ncreae in ecific entroy, fg ncreae in ecific olume, and h fg atent heat added during eaation at aturation temerature. d g f fg hfg...(7.50) d g f fg. fg hi i known a Clauiu-Claeryon Claeryon equation f eaation of liquid. he deriatie d i the loe of aour reure eru temerature cure. nowing thi loe d and the ecific olume g and f from exerimental data, we can determine the enthaly of eaation, (h g h f ) which i relatiely difficult to meaure accurately. Eqn. (7.50) i alo alid f the change from a olid to liquid, and from olid to a aour. At ery low reure, if we aume g ~ fg and the equation of the aour i taken a, then eqn. (7.50) become d d h fg h fg...(7.51) g h fg d...(7.5) d Eqn. (7.5) may be ued to obtain the enthaly of aouriation. hi equation can be rearranged a follow : d hfg d. ntegrating the aboe equation, we get z d h z fg d ln h fg 1 1 M P...(7.53) N 1 1 Q \M-therm\h7-1.m5

15 EMDYNAMC EANS 355 nowing the aour reure 1 at temerature 1 we can find the aour reure creonding to temerature from eqn. (7.53). rom eqn. (7.50), we ee that the loe of the aour reure cure i alway e, ince g > f and h fg i alway e. Conequently, the aour reure of any imle comreible ubtance increae with temerature. t can be hown that the loe of the ublimation cure i alo e f any ure ubtance. oweer, the loe of the melting cure could be e e. a ubtance that contract on freezing, uch a water, the loe of the melting cure will be negatie. Examle 7.1. a erfect ga, how that N M J u c c Q P J β β G J u where β i the co-efficient of cubical/olume exanion. Solution. he firt law of thermodynamic alied to a cloed ytem undergoing a reerible roce tate a follow : dq du d...(i) A er econd law of thermodynamic, d dq J re. Combining thee equation (i) and (ii), we hae d du d Alo, ince h u dh du d d d d hu, d du d dh d Now, writing relation f u taking and a indeendent, we hae du J u d c d u J u J Similarly, writing relation f h taking and a indeendent, we hae dh J h d c d n the equation f d, ubtituting the alue of du and dh, we hae c u d J d d c h d d d u c d h J d h J d d d d c QP d h d QP...(ii) \M-therm\h7-1.m5

16 356 ENGNEENG EMDYNAMCS Since the aboe equation i true f any roce, therefe, it will alo be true f the cae when d 0 and hence By definition, u (c c ) (d) (d) QP u (c c ) β 1 he aboe equation become, c c QP u β QP β β u Proed. Examle 7.. ind the alue of co-efficient of olume exanion β and iothermal comreibility f a Van der Waal ga obeying a b ( ). Solution. Van der Waal equation i a b ( ) earranging thi equation, we can write Now f β we require ence b u a. hi can be found by writing the cyclic relation, 1 rom the Van der Waal equation, Alo ence β 1 b ( b) a 3 1 J J QP \M-therm\h7-1.m5

17 EMDYNAMC EANS 357 β 1 b a M 3 ( b) Alo, M N 1 QP. ( b) 3 a( b) 1 1 a ( b) 3 QP. (An.) ( b) 3 a( b). (An.) Examle 7.3. Proe that the internal energy of an ideal ga i a function of temerature alone. Solution. he equation of tate f an ideal ga i gien by But u 0. [Eqn. (7.9)] hu, if the temerature remain contant, there i no change in internal energy with olume (and therefe alo with reure). ence internal energy (u) i a function of temerature () alone....proed. Examle 7.4. Proe that ecific heat at contant olume (c ) of a Van der Waal ga i a function of temerature alone. Solution. he Van der Waal equation of tate i gien by, a b Now ence J 0 dc d J c J 0 b J hu c of a Van der Waal ga i indeendent of olume (and therefe of reure alo). ence it i a function of temerature alone. Examle 7.5. Determine the following when a ga obey Van der Waal equation, a ( b) (i) Change in internal energy ; (ii) Change in enthaly ; (iii) Change in entroy. Solution. (i) Change in internal energy : he change in internal energy i gien by du c d d QP \M-therm\h7-1.m5

18 358 ENGNEENG EMDYNAMCS But, z z S a b UVW QP N M N M b du c d d bj QP z z c d a S 1 1 b b z P J z z a c d 1 1 b b z z a c d. d 1 1 u u 1 c ( 1 ) a (ii) Change in enthaly : he change in enthaly i gien by h dh c d et u conider f(, ) d (d) rom equation (1), 0 d P Q P d J. QP d (An.) U V WQ P...(1) d d 0 a d 0...() P Q (dh) P (d). Subtituting the alue of (d) from eqn. (), we get (dh) QP Uing the cyclic relation f,, which i d QP J J J J 1 d d...(3) \M-therm\h7-1.m5

19 EMDYNAMC EANS 359 Subtituting thi alue in eqn. (3), we get (dh) Van der Waal equation b a ( b) a 3 a b QP J QP QP d...(4) b Subtituting the alue of eqn. (5) and (6) in equation (1), we get M S a (dh) ( b) 3 W d b QP z ( dh) z z d d a 1 1 ( b) 1 (h h 1 ) log b e b b (iii) Change in entroy : he change in entroy i gien by d c d Van der Waal equation, b U V J J P 1 1 b b b ( b) ( b) 1. d S QP a 1 N d z1 ( b) UVW QP J 1 1 a 1 1 M P. 1 Q loge (An.) 1...(5)...(6) b b J...a er eqn. (6) d c d d b z z d d d c 1 QP z 1 1 ( b) 1 c log e N M 1 Q P log b e N M 1 bq P. (An.) Examle 7.6. he equation of tate in the gien range of reure and temerature i gien by C 3 where C i contant. Derie an exreion f change of enthaly and entroy f thi ubtance during an iothermal roce. \M-therm\h7-1.m5

20 360 ENGNEENG EMDYNAMCS Solution. he general equation f finding dh i gien by S P Q U J V W QP dh c d P d z dh 0 1 z1 a d 0 f iothermal change. rom the gien equation of tate, we hae C 3 4 z 3 M S C 3C h h 1 d 3 1 J W QP N Mz 4C 4C d J Q P [( )] he general equation f finding d i gien by d c d d z d z 1 J d 1 QP a d 0 f iothermal change. Subtituting the alue from eqn. (i), we get ( 1 ) Mz 3C d J P 1 4 N log e 3C 1 J 4 ( 1 ) (An.) Examle 7.7. a erfect ga obeying, how that c and c are indeendent of reure. Solution. et f(, ) hen d Alo u f(, ) hen du Alo, d u d du d d d d c d d c d 1 u Q d U V P u d c d d u d QP u d...(i) \M-therm\h7-1.m5

21 EMDYNAMC EANS 361 and Equating the co-efficient of d in the two equation of d, we hae c J rom eqn. (7.0), Alo c c J J J c J J 0 c J hi how that c i a function of alone, c i indeendent of reure. Alo, rom eqn. (7.1), Again, c c J J c J J J 0 ; hi how that c i a function of alone c i indeendent of reure. J c J (Gien)...(Gien) \M-therm\h7-.m5

22 36 ENGNEENG EMDYNAMCS \M-therm\h7-.m5 Examle 7.8. Uing the firt Maxwell equation, derie the remaining three. Solution. he firt Maxwell relation i a follow :...(i) (Eqn. 7.18) (1) Uing the cyclic relation (ii) Subtituting the alue from eqn. (i) in eqn. (ii), we get....(iii) Uing the chain rule, (i) Subtituting the alue of eqn. (i) in eqn. (iii), we get hi i Maxwell hird relation. () Again uing the cyclic relation () Subtituting the alue from eqn. (i) into eqn. ()....(i) Again uing the chain rule,.. 1 Subtituting the alue of (i) into (), we get hi i Maxwell econd relation. (3).. 1

23 EMDYNAMC EANS 363. Subtituting the alue from eqn. (i), we get J S J J U V J W.. hi i Maxwell fourth relation. Examle 7.9. Derie the following relation : (i) u a (iii) c a J a where a elmholtz function (er unit ma), and g Gibb function (er unit ma). Solution. (i) et a f(, ) hen da a d Alo da d d Comaring the co-efficient of d, we get Alo ence a a u u a a u a (ii) et g f(, ) hen dg a. g d Alo dg d d Comaring the co-efficient of d, we get g (ii) h g (i) c a a d (An.) g d g g J \M-therm\h7-.m5

24 364 ENGNEENG EMDYNAMCS Alo ence h g g h g (iii) rom eqn. (7.3), we hae Alo a c rom eqn. (i) and (ii), we get c (i) rom eqn. (7.6), we hae Alo c g. J g (An.)...(i) a...(ii) g rom eqn. (i) and (ii), we get c J a. (An.) J...(i) g...(ii) J g. (An.) Examle ind the exreion f d in term of d and d. Solution. et f(, ) hen d A er Maxwell relation (7.1). d Subtituting thi in the aboe equation, we get d d he enthaly i gien by dh c d d d Diiding by d at contant reure h c 0 d. d...(i) (a d 0 when reure i contant) \M-therm\h7-.m5

25 EMDYNAMC EANS 365 Now ubtituting thi in eqn. (i), we get But d c d β 1 Subtituting thi in eqn. (ii), we get G J. d...(ii) d c d βd (An.) Examle Derie the following relation : (i) β (ii) c where β Co-efficient of cubical exanion, and othermal comreibility. Solution. (i) Uing the Maxwell relation (7.19), we hae i.e., i.e., Alo rom eqn. (7.34), c β 1 β c β. (An.) c (ii) Uing the Maxwell relation (7.18) Alo hen Alo c 1 c β. c (Eqn. 7.3) (Eqn. 7.36) 1 1 β β \M-therm\h7-.m5

26 366 ENGNEENG EMDYNAMCS β. c (An.) Examle 7.1. Derie the third d equation d c and alo how that thi may be written a : and where d c d c β d c β d. Solution. et f(, ) hen d But ence Alo d d c and d c 1 β d d 1 J d d J d c c d d...proed. β Subtituting thee alue in the aboe d equation, we get d c d c β β d...proed. Examle Uing Maxwell relation derie the following d equation d c d Solution. f (, ) c Alo, J d Subtituting thee in eqn. (i), we get d c d d. (U.P.S.C. 1988) d d...(i) d. (An.)...Maxwell relation \M-therm\h7-.m5

27 EMDYNAMC EANS 367 Examle Derie the following relation u c J. Solution. can be exreed a follow : u u u u u u Alo d du d du d d u u u...(i) u...(ii) Diiding eqn. (i) by eqn. (ii), we get...(iii) u Alo c and J... Maxwell relation Subtituting thee alue in eqn. (iii), we get u c J Examle Proe that f any fluid (i) h...proed. (ii) h Show that f a fluid obeying an der Waal equation a b where, a and b are contant h (enthaly) b a b f() where f() i arbitrary. \M-therm\h7-.m5

28 368 ENGNEENG EMDYNAMCS i.e., i.e., and i.e., Solution. We know that d c d d [Eqn. (7.4)] Alo Putting d 0, we get (ii) Alo Eqn. (i) become h J h h J J h dh d d dh c d c d d d J J h J N M J J Now b h h a ( b) b a 3 a ( b) 3...Proed. J d d QP Q P J...(i)...Proed. QP b a a ( b) b ( b) b ( b) a ( b) b a ( b) b a ( b) h b a f()...proed. b hi how h deend on and. Examle Derie the following relation : (i) h c h (ii) u \M-therm\h7-.m5

29 EMDYNAMC EANS 369 With the aid of eqn. (ii) how that he quantity c u h effect f a ga obeying the equation of tate ( b) Solution. We know that Alo i known a Joule-homon cooling effect. Show that thi cooling C i equal to 3C b J. h µc...[eqn. (7.44)] h µ 1 c Alo µ h h. h c (ii) et u f(, ) du QP P Q u d c d P u Alo du d d Subtituting the alue of d [from eqn. 7.4], we get rom (i) and (ii), we get Alo u u J u J u J du c d d... Proed....[Eqn. (7.46)] u d...(i) N M J c d u J J d d Q P d J J J N M Q P J J J...Proed....Proed....(ii) \M-therm\h7-.m5

30 370 ENGNEENG EMDYNAMCS We know that Alo J u J J 1 and µ 1 c Now b J u C J QP C 3 Subtituting thi alue in the exreion of µ aboe, we get N M µ 1 C c 3 µc C 3 J J J Q P...Already roed....[eqn. (7.46)]...[Gien] C b 3 C b c 3 C h b...proed. Examle he reure on the block of coer of 1 kg i increaed from 0 bar to 800 bar in a reerible roce maintaining the temerature contant at 15 C. Determine the following : (i) Wk done on the coer during the roce, (ii) Change in entroy, (iii) he heat tranfer, (i) Change in internal energy, and () (c c ) f thi change of tate. Gien : β (Volume exanitiity /, (thermal comreibility) m /N and (ecific olume) m 3 /kg. Solution. (i) Wk done on the coer, W : Wk done during iothermal comreion i gien by z W d 1 he iothermal comreibility i gien by 1 d (.d) z z W. d d 1 1 Since and remain eentially contant W ( 1 ) [( ) ( ) ] \M-therm\h7-.m5

31 EMDYNAMC EANS [(800) (0) ] ( ) J/kg. (An.) he negatie ign indicate that the wk i done on the coer block. (ii) Change in entroy : he change in entroy can be found by uing the following Maxwell relation : J β a β (d) β (d) ntegrating the aboe equation, auming and β remaining contant, we get 1 β ( 1 ) [ ] (800 0) J/kg. (An.) (iii) he heat tranfer, Q : a reerible iothermal roce, the heat tranfer i gien by : Q ( 1 ) (15 73)( ) 18 J/kg. (i) Change in internal energy, du : he change in internal energy i gien by : du Q W 18 ( 3.135) 14.8 J/kg. (An.) () c c : he difference between the ecific heat i gien by : c c β... [Eqn. (7.38)] (An.) ( 5 ) 5 10 ( ) J/kg. (An.) Examle Uing Clauiu-Claeryon equation, etimate the enthaly of aouriation. he following data i gien : where, At 00 C : g m 3 /kg ; f m 3 /kg ; Solution. Uing the equation d d J hfg ( ) g f h fg Enthaly of aouriation. Subtituting the ariou alue, we get d d J 3 kpa/ h fg ( 00 73)( ) h fg (00 73)( ) J J/kg kj/kg. (An.) \M-therm\h7-.m5

32 37 ENGNEENG EMDYNAMCS Examle An ice kate i able to glide oer the ice becaue the kate blade exert ufficient reure on the ice that a thin layer of ice i melted. he kate blade then glide oer thi thin melted water layer. Determine the reure an ice kate blade mut exert to allow mooth ice kate at 10 C. he following data i gien f the range of temerature and reure inoled : h fg(ice) 334 kj/kg ; liq m 3 /kg ; ice m 3 /kg. Solution. Since it i a roblem of hae change from olid to liquid, therefe, we can ue Clauiu-Claeryon equation gien below : d d h fg. 1 fg Multilying both the ide by d and integrating, we get z d h 1 ( 1 ) h fg fg z fg fg 1 log e d J 1 But at 1 1 atm., t 1 0 C hu, bar, ?, Subtituting thee alue in eqn. (i), we get ( ) b g log 63 e log e N/m N/m bar. (An.) hi reure i coniderably high. t can be achieed with ice kate blade by haing only a mall tion of the blade urface in contact with the ice at any gien time. f the temerature dro lower than 10 C, ay 15 C, then it i not oible to generate ufficient reure to melt the ice and conentional ice kating will not be oible. Examle 7.0. mercury, the following relation exit between aturation reure (bar) and aturation temerature () : log / 0.65 log 10 Calculate the ecific olume g of aturation mercury aour at 0.1 bar. Gien that the latent heat of aouriation at 0.1 bar i kj/kg. Neglect the ecific olume of aturated mercury liquid. Solution. atent heat of aouriation, h fg kj/kg (at 0.1 bar)...(gien) Uing Clauiu-Claeryon equation d d h fg hfg...(i) fg ( g f ) Since f i neglected, therefe eqn. (i) become d d h fg g Now, log log 10...(i) \M-therm\h7-.m5

33 EMDYNAMC EANS 373 Differentiating both ide, we get d d d d rom (i) and (ii), we hae hfg g (ii)...(iii) We know that log log (gien) At 0.1 bar, log 10 (0.1) log log log log Soling by hit and trial method, we get 53 Subtituting thi alue in eqn. (iii), we get g g i.e., g.139 m 3 /kg. (An.) ( 53) GS 1. Maxwell relation are gien by J J J J ; ;. he ecific heat relation are c c β ; c 3. Joule-homon co-efficient i exreed a µ J. h J J J J. J ; c J. \M-therm\h7-.m5

34 374 ENGNEENG EMDYNAMCS 4. Entroy equation (d equation) : d c d d c d J d...(1) J d...() 5. Equation f internal energy and enthaly : u J J...(1) S J S du c d h J J dh c d U V W d...[1 (a)]...() J U V W BJECVE YPE QUESNS Chooe the Crect Anwer : 1. he ecific heat at contant reure (c ) i gien by (a) c (c) c J (b) c J (d) c. he ecific heat relation i (a) (c c ) β (c) (c c ) β 3. he relation of internal energy i (a) du (c) du 4. d equation i c d β J c β J d c β d (b) du J c β d (d) du J d...[ (a)] J J. (b) (c c ) β (d) (c c ) β. c β J c d c β J d β J d c β J d. (a) d c d β d (b) d c d β d (c) d c d β d (d) d c d β d. Anwer 1. (a). (a) 3. (a) 4. (a). \M-therm\h7-.m5

35 EMDYNAMC EANS 375 EXECSES 1. Define the co-efficient of : (i) Volume exanion (ii) othermal comreibility (iii) Adiabatic comreibility.. Derie the Maxwell relation and exlain their imtance in thermodynamic. 3. Show that the equation of tate of a ubtance may be written in the fm d d βd. 4. A ubtance ha the olume exaniity and iothermal comreibility : β 1 ; 1 ind the equation of tate. 5. a erfect ga, how that the difference in ecific heat i c c. An. contant Q P 6. the following gien differential equation, du d d and dh d d roe that f erfect ga equation, u J h 0 and J Uing the cyclic equation, roe that J β. 8. Proe that the change in entroy i gien by d c 9. Deduce the following thermodynamic relation : h (i) J J c J h 10. Show that f a Van der Waal ga N M Q c d. β βp d. u (ii) J J. c c 1 a ( b) / A ga obey ( b), where b i oitie contant. ind the exreion f the Joule-homon coefficient of thi ga. Could thi ga be cooled effectiely by throttling? 1. he reure on the block of coer of 1 kg i increaed from 10 bar to 1000 bar in a reerible roce maintaining the temerature contant at 15 C. Determine : (i) Wk done on the coer during the roce (ii) Change in entroy (iii) he heat tranfer (i) Change in internal energy () (c c ) f thi change of tate. he following data may be aumed : Volume exaniity (β) / othermal comreibility () m /N Secific olume () m 3 /kg [An. (i) 4.9 J/kg ; (ii) 0.57 J/kg ; (iii) 164 J/kg ; (i) J/kg ; 9.5 J/kg ] \M-therm\h7-.m5