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B)' the Same Authorsl. Optics for B.Sc. (Twenty-Third Edition)2. Properties of Matter (Fifth Edition)3. Atomic and NuclearPhysics (Sixth Edition)4. Numerical Problems in Physics for B'Sc'5. PrinciPles of PhYsics6. I.l.T. Physics for Engineering Entrance Examination'

t* Ii

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HEATTHERMODYNAMICS

lFor B.Sc. (Pass, Gendral Suhsidiary), B.Sc. (Hons. and Engineering) andMedical and Engineering colleges Entrance, IAS Examinationsl

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,.(c, ( t ),) l,:5".i'tl-l',1{' q"or

lU Heat and Thermoilynamictt.',rrce l,er unit area on the walil BCIO or ADEE is equal to the

1rr errure Py

Px : #r(urr .| zrrfurr+...... +z,r)Similarly the pressurc Py on the watls CDEF and lBO.& isgiven by

,, : # (ur21arr;......+r"2)and the prcssure P7 on the walls ABCD and EFGE is given by

Pz - fr {rrr+rrz+......+wnzl

a" $qpresrurc of a gas is the same in all directions, the meanpressure P is given byp _ Px*Pv*Pz

3tt, f:

5F L (urz { x rz 4 ut') + (ur' +u"'. + r'r"l1- (aaz + o s2 + wzz) + ......

* (u,2+r,"r*.,r) ]: #[ cf +c22+cJ+......c", ] ...(d),

But volume, Y _: lr. Let C be the root.mean-square velocity olthe molecules (R.M.S. velocity).

C, : C r' *Cr' *Cr'*.-....CJ7L

Then

or nCz -

Cr2+C22+Ct21.......Cn2Substituting this value in equation (i), v".e get

D _ m.n0z'- 3v- ...(ii)

Brrt .4f : rntl rvhere .0f is the mass of the gas of volume V, m isthe mass o[ each molecule and z is the number of molecules in avolume [/.

L;From

Noture ol deot

or e: 13tr- Vp

155

...lirr'tNotc. R.M.S. volocity C ia tho Equaro root of tbo moan of tbe squ-aree

"f rh"!;;;'id;;i"if"iifr-.*ind it,ienot'oqual to tbe meenvolooily of tJre

moloculea.]TABLE

Molecular Velociticr et OoC

Molccvlar Wciiht Root mcan cqtare wlocitY

in cm/s

EydrogenIleliumNitrogenOrygonArSOn

Carbon dioxide

Chlorine

2.016

1

?a

20

.10

41

7L

18.4 x 10r

13.l x l0'4.95 x lO'a.ut * ro.

.1.14 x l0r3'95 x lo'3.11 x l0r

t5(;157\ r! rt ,

T.,)1.2

lf,tFrom equation (iii1

?2

T2

: 173 K:?:16

-

mrOrtT,mtCt'

: 16xl73: 2768 K:2768-273: 2{950c

H eat ond T her modynamic,, Nature oJ Heat

t'- t )I nt3vIPy:;MC2

Consider one gram molecule of-a gas at anrure ?. The meanlnersy of the molecules

-lMc..2: )- N'nc'

rv : \' Nmc': I tt.-f,'.*c'

Mean kinetic energy of a molecule: LmCz /

2

:3kT2

PY:-|rv.*r,PY

-

NkTBut /Vx& :.8

absolute tempera'

T2

E*.r-plq.s,&dculcs al 27"C.

Culculate rle RMS velocity o! the oxygen rnole-

First calculate the n.dfS velocity of oxygcn at N.T.p.n

-

l-gP"- V p

Here P : 76X 13.6 X980 dynesTcmrP : 16x0'000089

,:JmrC : 4'6x 10. cm/s

Let the 8.tr48 velocity of the molecules at 27oC be C1

+:'[Tc1 :'cy J THere C : 4.6X l0r cm/sT :273 K?r :27oC

: 27 *273:300K

Ct : 4'6t,* Js-. C, : {.8*Xl0t cm/t

5,17 Derivatioo of Gas Equation' F.o* kinetic theory,

PV:frT ...(d,Note on the Gas Equation.In the gas cquation

PV:RTP is in dynes/sq cmI : B'31X I0? ergs/g mol'K?isinK/ is the volume in cc per gram molecule.

Eraople 5'7 Calculale the volume occupied by 3'2 gtams ofoaygetu at ?6 cm o! Hg and 27"C,

Here P : 76x 13'6 x9B0 dYnes/sq cmr :27,+273

:300KB : 8'31 x l0? ergs/g mol-K

PV:Bf8'1lxl0?x300l : % xT36x98o cc Per g-mor

I/ -

2{610 cc per g molP: + pc2www.

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,ao

Heat and Thennoily namiu

Volume for 3'2 * "t "at.rX;rr* r.,

32: 2461 cc

Noture of fleof IOZ

(iii1 Begnault'a LauFrom equation (dd)

PY -

NKTWhen 7 is constant,

Pc., lIIt means that for a given mass of gas, the presurc is dircc4y

proportional to its absolute temPerature provided the volume remainrconstant. /

r'.'ample 1.8 Shoo tha,t n, the nuwber of nolcoula pr u;nitooluma of an iileal gaa ie gfuen by

PNO: -ffi

whera N io Aaogadro's number.For an ideal gas, for one gram molecule of a gas,

?V:RTR: NK

PY : NKT.Let z be the number of moleculcs per cc. fn that case,

constant, rn.r"ro.. "r(']

akolute temperature

...(d)

But

But

P:nkTPo:

-Tr-n*:-fr

PNfl, : nT

where lV is theconstant.

If

Erample 5'9. Calatbte the nunfrer of molcculca in one ctliamelre of an iileol gaa ot N.T.P.

Let the number of molecules per cc be to.PAIO: 'ffi

Number of molecules in one cubic mcue volumc,: nXlS

PIV x l6cu: -m-

ilere P :76x l3'6x980 dynes/cmr.l[ * 6.023x lOaI : B'31 x l0? crgig rnol'K

Ir means that for a given mass of gas,prg;rt ional to the absolutc temperaturercrnains c()nstant. This i: Charles' Law. 9*JI

.)

the volume is directlyprovided the pressure

fi'l{tDerivation of Gas Laws(t*) (i) ggP\Iw-v According to the kinetic theory,':igP* *#*.

Pr- **o

At a constant tempcratureT, O isconstant tempcrature

tJ MC': constant

Hence PV -

const. -at constant tetnperature.

Li;l -gMk*I&;D-Accoidirrg io the kinetic theory

G9 ": *.,P- *#*PY:**o

Consider oue gram molccule of a gas atT. M:mN

pv: Iy,lr.r.JThe mean Linetic energy of a molecule

I - 3 --Z *C': ;AImCn :3bT

Substituting this value in equation (i),PY : NHI

...(rr)Avogadro's uumber and & is the Boltzmann's

P is constant,Yc.T

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762 Eeat anil Thermdynaniu

T :273K76x 13'6x980i6'023x lOax 1064-t:

- : r.Utt*t,,u

Erample 5'10 Calculate the number of moleaules in one likeoJ an iileal gae al 136'd Q temperoture onil 3 armoepheres prellure.

Let the number of molecules Per cc : 7tPlvn-w

Number of molecules in one litre,c:7,X103

P][x lOEt: --w-

Hcre P:3x76x13'6x980 dYnes/cmr-l[:6'023x104E

-

8'31xl07 ergr/g mol-KI' : (273+ 136'5)

: 409.5 K3 x 76 x l3'6x980 x6'023 x lOax l0r

B'31 x l0?x409'5/, --t','U*'0"

,Zfzlcrote 5'11 The number of moleculea per cc ol o gae ie'/., *r*, at-N.t.P. Calcul,ate the number of rnoleculee per ce of lhegu.

' () of 0C anil 10'6 mm pressu?e of mercury and(id) al 39oC ond f0-6 mm pressure

'nercuty.For a unit volume of a gas

P:LmnCN5

(i) At 0"C, R.M.S. velocity is equal to CAt N.T.P.

IP, : -g- mn1 Cr

At OoC and l0-'mm PressurePt : i mnror

From (d) "td (dd)p,

: n,Pr n,nrxPl

or ,rt : -FI

... (i)

.. . (ii)

Naturc of Ecot

Hcrc n1 : 2'7 X l0rePt : 76x l3'6 x 980 dYnes/curP1 : lQ-e mm of Hg

-

l0-? cm of Hg& : l0-z x l3'6x980 dYnes/cmt

2'7x l01e x l0-?x 13'6x980fi1 : ------7il13i-;580--

z1 :3'553 ll0ro(ddt Let thc R.M.S. velocity at OoC be C1 and at 39oC bc Ot'

Numb'cr of molecules Per cc at OaC and l0-0 mm of Hg pressuree tl,tr

and at 39oC and l0-c mm of Hg pressure: fL,

P:!mnC'JHere presure is the same in both the cases

ImnP;:Imn,cg,n1?1t : tus0gs

C,'7lg : tl2 W

But C e' t/-TC,, T,rl2xTrtt':

-Tl-' Hcre r\: 3'553 x l01o

f1 :0o C:273K

fs:39oC:273+39:312K

3'553x1010x273ms: _*____m_

ro ,, \: 3'109 x loloI 5't9 Avoga&o's HYPothesistJ-L, " konrider two gases A and B at a pressure P and each having a1 vo)ume I/'

r C Z Mass of each molecule of the first gas : m1\V Number of molecules of the first gas : rLLwww.

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Mean square velocitl of the molecules of the fint gas _ qtFor thc 6nt gar.

p: I e,e.: + '# ...(,)Similarly for the second gas

P: $ r,4,. _+ryL

...(,,)where zn. ,spres@l the ,nass of each molecule,. z, the :rumber ofmolecules atia crr tue meu,, rq""i"".r.1iry ot.the morecules.

From (d) and (dr),l **C!__ I .mrnr1szT ---7-: 3or n*t7f

_ ,irnrCrr',.,_^^,lf the twogasesn g at- the same temperaturc ?, the;:::f,rnettc energy of the morecures oi [.iiiirr" gases is the same

+m,c,,-|wc,"From (r:di) and (iu),- " h \ 6i\-8 i ...(du)

.--- r-:'""' "qpd ":lfn :,: :.,#= ::r.;-; conditions or;:ffifi :lX"U:J,;ff f$;91.";;;l,u*u..;;.i;il.il'tni,5.20 Grahan's La,w of Difueion of GeeegAccording to the trinetic tfr"orf

P:*r*or

'- /FCn l

^c - It means *, *:311,,:J[r"re vcrocity of the morecures3:1,ffr .,"#,x,mr:::lt{;t,:1T..d,",?::Lb,}"Tu

C,_ T P,c'-Vp,

Nalarc of EealHere, r, and r, represent the ratesg?ret.

,*-,;ff;?l;,u;;;::1"#;ti::'tototranitomhineticenersvo!oncTotal random kinetic energ.y per gram-molecule of oxygen

I' ,

m@xNe

= ; kT.N: +. #,,

c- J-R?2

. : f *e sx r0?x300: 3'735 X I0ro ergs-

S7Sijoules _/

ffi ill"tlgdH'"".H#:o,:r1ffi ff :.?"i:'#:t-'H:ffi ;1'"'""",*",#Icvle

@,ffff':;:',!a!,;];*:;'!;ov.'*'kineticenersvo!omote-Average K.E. of a molecule.---=---r

=j-m@:Here & is Boltzuann,s constant.

& : I.38x l0-rc erg/molecule-degAveragc K.E. of a molecule : f *l.3Bx l0_rcx300Noto. The aven

: 6'2r x ro-r' ergs

m:*ild&,'H?fl ,YirlT*""i,""9,"k.:;';ffi '"#"TI.iT"#;EaTplc 5.14. Colculnte the mean bandationol bhetic cnery!gw mol,eculc of a gaa at |Zt"e, i;;r:; n":';.JZ joulcs/motc_KAoogailro'e nurfier, (Dcki' 1974)

N -

6.06x10u

rrerc i::?*:ii-il"1*"iI:6.06xIOD

L_!:l 8.32 \,T - (606;10-"/ jouler/molecule'KMean tranrlational tinctic cnergy per molecul

" - |Ul

165

of diffusion of the two

*o* -

+-/f ,

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!ii:-l166 Eeat anil llhermollynamice3x8'32x1000: T;TT6;TT

: 2.059 a l0-ro jouleErampte 5'15- Colculate the lotal ranilom kinetic energy oJ one

gram of nitrogen at 300 K.Total random energy for one gram molecule of nitrogen

q: ;RT

Total random energy for I gram ofnitrogen3Rr: -2Af

whcre the molecular weight of nitrogen y : 28 g3RTn:-

-

2M3x8'3x107x300

- --ETE-

: 133.4 x 107 crgs: 133.4 Joulc

Notc. Tbc total resdom kinotio energy for I gram of a gar h difierun!for diferent gaaes at the gamo temporature.

-Eranple 1.16. Colculote tlu I'otal random hinctic cnugg o! 2 gof helium at 200 K.Energy for I g of helium

3RT: -zfr-

Energy for 2 g of helium2x3xRT: ----mi-3RT:__fr*3x8.3x107x200:____l-

: 1245x 107 ergs:1245 joulee

_ 1'.n- ple 5.17. Calailate the root mcan aquo?e oelocjty o! amolecule of mercury aqpoltr at 300 K.

Mean Li.ctic encrgy of one molecule of.mercurytc:+mO-;W

Nalure oJ HuAVLet il bc the Avogadro's number.

IeThen j nNCs : ;- kNfI --- 3--v MCz : ;Rr

c:r{ryHere the molecular rveight of mercu\ M

-

221WC:.T@v

-ezi-/ : 1.93 a 10r crn/e/Eramply'5 18. Wilh what speed woulil one grom molecul,e otorygen at 300 K be moaing in oril,er that the translational, kinetiaenegry of its cenlre of mass is equol to the totol ranilom binetic enetggoJ all, ila molecul,es 2. Molecular wei,ght of ozygen (M) : 32.

Total random kinetic energy tf I gru--*olecule of olygen- f,i'nr : +xB'3xlo?x3oo-

3.735 x l01o ergsKinetic energy of .ilf grams of oxygen moving with a velocity o

- iMas|*r,: g'Jggy lQro

Ij xSZxu2 : 3'735 x l01o. 3'735 x l01o

__--]6-r : 4.8 X 10r crn/s

. Exarople 5'19. Calculale the temperature at whiah the r.m.e.

tselocity of o hyilrogen mol,ecule will be equal to the opeeil oJ the eorth'efirst aatellite (i.e. o : 8 km/s).

Energy for I gram molecule of hydrogen:!uor:1ar- 2"-'- 2

m Moz,:-572x(8x105)r: T;3:3;-i3r*

./. : 5'14x1$ KExaople.520. At what temperalure, ptessure remaining e;a.tu,

tant, wil,l the r.m.s, oelocitg oJ a gas be halJ its talue oA OoC IlDel,hi lEone.) 1975J

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?t: ?Tr:273 Kr _ r-r;T: 1, E5'

,:o,,\ + 273

"r:+:68'25K*

o' .

?z: -2(/r'75'c '/

R, 5.21 Degreee of Freedom and r![arwett's Law of Fquiparti-AJ tion of Eaergy --- - --:"6i -'

ffiffi#Hfl,.m#;d =;iffi

"re nas ?ffi ;e -a.ffi;*s*m **[,].T:l*

mol ec uI e" h as thr ee d-egr ees -"i ir;l";;r td* I. t i *

"na;;fl;Hof freedom of rotationl It has in au nve e;;;;;;';'i;;;d:B -Z;$ ;Y^*:*:x r,:ffi:;":H??jffi:];he mean,,^,oi d(n

)*o': * o,But C,

-

*qoe{td*",#"?*Xli#f all equivalent, mean square velocities arong

Here

ur-oE-rdlm(ur):Imgf):lm(wr)I mC' : 3 [l m(ur)i: 3 B rn(u!)]

: 3 [] m(w')7q

:; KTlmut:lbTI moa : lkT*^rd: *lil

. Thgreforc, the average-

-linctic encrgy associateddegree of freedom J yf

Nature of Eeat rcgTn,rs(he energy associated with- each dqgree of frcedom(whether trinsratory 3i

-t"i.ryf i, I iA:) Ew u?trEE .,r rrcec ,,This represents the theorem of equipartition of energy .WE \^e

l'?2 Atom.icity of Gasec / -fe\(l Mono'-otomic- ga.E. A mono-atohic gar molecule has oneAtom'-tach molecure has thrcc d.gr;;r;i?..do- a* io-i."rif,tory

Encrgy associated with each degree of freedom _ | WEnergy associated with three degrees offrcedom

:; bTConsider one gram molecule of a gas.,"r:eryyg:

-rYx |r*:{ g*4r

JI7x,t : .BJ

a:$arI:##*ii::"fl'' " :*; nil

* $:: *l:g.H, il;:*f i1$*

c, lTC, E.V Tncr"r: T

Heal anil Thermoilynamia

..-(')

...(i0...(ddi)...(iu)

with each

(# - the increase in internar energy per,"-R".",r"" )

But . Cp-Cv:BCp : Cv*n

: t**n: f,nFor a mono-atomic gas

Coa-9vq

TR \: -a-- : 167 )]a -/

[But,

c,-#:3raunit dqpec rise of

or

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170 Eeat anil ThermoilynamiaThe value of T is found to be uuc experimentally for mono-

atomic gases like argon and helium.\fi 6 Oiot*r;" g*. A diatomic gas molecule has two atoms.

SuchY molecule has three degrees of freedom of translation anC twodegrees of freedom of rotation.

Energy associated with each degree of freedom: l*r

2

Energy associated with 5 degrees of freedom : $ VtConsider one gram molecule of gas.Energy associated with I gram molecule of a diatomic gas(5: il*i kT :; RT

.(a:;Rr^dUwv: _EF

q:;R

But Cp-Cy : RCy: CvlR

: i.o** - tp nu-C'r- Cv

1+BA

: +-: 1.40 \rp )2'- -/The value of 1 : l'40 has been found to be true experi'

mentally for diatomic gases like hydrogen, oxygen, nitrogen etc.1ii$ Triatomia gae. (a) A triatornic gas h-aving 6 degrees of

freedom has an cnergy associated with I gram molecule-

.Irx $*r : *ra :3RTc":-!fi:sn

, But Cp-Cy: RCp

-

Cy+R: 3E*8

-

4.B

v- cPr- Cv

: #:r.33(b) A triatomic gas having 7 degrees of freedom has an energy

asociated with I gram molecule : N x -2 kT : ; B

7u:-;-Rra.,: !! : 1- ndT2

But C?-Cv : RCp: CylR

7(l: 'R+B:

; Rv- cPr_ Cv

: ffi:1.28Thus the value of 1, Ce and Cv can be calculated depending

upon the degrees offreedom of a gas molecule'

5.2t Marwell's Law of Dirtribution of VelocityAt a particular temperature, a gas molecule has a fixed mean

kinetic eneiry. It does not mean that the molecule is moving with,n. r.*" spJid th.oughout its movement. After each encounter thc."."d of ihc molecule changes and due to a large number oli=oitirio*, the speed is different at different instants. But the root*""r, rouur" vilocity (r.m.s.) C remains the same at a fixed tempera-i"r.. ei iny instant, all the molecules are no-t.moving wit! the same.r"io"iw. So*".t" moving with a vclocity highcr than C and the;tL*'*ittr

"

velocity lowei than C- But the mean kinetic energy ofall thc molecules remains constant at a given temperaturc'

I)erivation of Maxwell'e law of Distribution of Molecularvelocitiee

mean slluare velocity of molecules is defined by the

c,: |J] * arHere dIV is the number of molecules having velocities between

c and oadc. If the total number of rnolecules is iv, then a fracticn

$wiff have the comPonents of velocities in r direction in the range

Nalurc of Ecal lft

\o\ v$The

equation

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r76 EeaC and Thermodynomico - n.',cLi.:,-.f-o 'lr>

yE Eul otil TlurmodYnnmiat

(l.t ,t bc the numbcr of molecules Per cc.ihco, the number of molecules Present is a volume ndlu

: ril,tvXnThis value also represents the number of collisions made by the molc.cule in one second.

The distance moved iir one second : u and the number ofcbllisionr in one second : rd\t v.n. .

llolurc o! Ecal 179

Detcrmiaation of Ereea free 1nth. As dlacussed in article5.27, the relation berween coefficient of tiscosity and the mean freepattr of a molecule is given bY

.l

r : i mnC)t

But 4* crilc : du ihs dwdlv : .u/s

"-bo' du ilv ilwAlso 6t * .y'+-ts2+.toz...(;i)

dN : NAs e-J (rrr+rs+otl du & dw ...(ddd)Equation (irid) has to be modified in case the gas as a wholc

POSSeSses ma$ mouon.Let uo, uo and rao be the components of the mass vclocity'

Therefore, -the

actual vilocity of a molicule consists of two- Parts :(i) the mass velocity comPonentS oe, us and raoldi) the random thermal velocity conPonente

u-uo, u-us and w-ll)scorresponding to thermal motion without mass motion, simi-

lar quantitiis with lnass motion arcA

-

u-uo| -

a-asand W

-

$-woFrom equation (ddi;, Maxwell's law of distribution of velocitl

can be wiitten as

d,N : IiAsc-6(u8+vt+'')d(l dY dw...qixyEquation 1io) holds good only if ao, zo, tto, T and 'Y are cons-

taDt throughout the gas.If the gas is not in an equilibrium state, there are three possi'

bitiiies occurring singly or jointly.

For unit volume,mt: Pr: Ieclr- 3't

pC..'0L//

The root mean square velocity C of a molecule can becalculat'.ed knowing pre$ure, d=ensity and iemperature. The coefficient of,ir".ti,, .itire gas is determined expeiimentally' Hcncc the value ofthe mein free pith ol a molecule can be calculated lrom equation(d).5'26 TranePort Phenomear

According to Maxwell's law of distribution of velocityilN

-

+*NAa "-4ct

,26, ...(i)

,-\ .'.1 u. rfree path ^ : #r*:#)

This equation was deduced by Clausius'

...(i)

...(i0

..{o)of thc law of diltri-

Then' 'llllll : Pm

l--7.o"P

...(did)

The mean frce path is inversely proportional to the denrity ofthe gas.--

'Th" expression for the mean path according to Boltzmann ir

^:h ...(iu)He essumed that all molecules have the samc average rpecd.

Maxwell derived the exPression,

Thc mean &cc path is inversely proportional to the sqrrare ofthc diameter of the molecules.

Let m bc the mass of each molecule'

,*#

^-T.r* )He calculated the value of I on the basis

bution of velocitiesTAsLE

Meaa Srcc Petb (tr)

?'l1x lf cna&6Ox l0-t om2.18 x l0-8 om

3.ag X10-{ sDo

l.t!x lO-. om0.S{lxlH oro2.85xlflom0.009 x l0-! ors

I n",l- Hrurogon ww

w.Job

sCare

.info

/,\J Eeal anil Thumodyramict: /VxgkT:gR?

U :3RI,duaH: -m

: 3.8 : 5'96 cals/g.mole KThus, the atomic heat of solids (.4s) is 38 and this value agrcct

with the Dulong and Petit's Lav.,.E=arople 5'21. In an experiment, lhc aiacceilg o! the gc,a wa,

Jou,,d to l,e 1'04x70-t d1'nssi(:r-rt per unit oelocity gradieat. The.B.ff.S. rclocity of the moleculea ie 4'5 xJ0trcmis. The denaity ollhe gaa ie 1'25 grams ,?e/ litre.

Calcalate 1i1 the nean lree path oJ thc molecu,lcs o! the gac, (ifit,equeneg of collisiott anil (iii) molecular diameter of the gd* malc-cules.

Ifere

Jd : 3;10-8 caEraaple 5.22. Im an experiment the oiacooitg of tie got woa

tound to be 2.25 x 10 -t CGS uzrts. The BMB aelocity of the moleculeai.s 4'5x l0r ernlt. The density of the guis .l gram per litre, Colcu-lcte the mean Jrce path oJ the moleculee .

Nature of Eeot

d:

Here

(r)

! : l'56x l0-'unitsC : 4'5 x 10. cm/sp : 1.25x l0.r g/cc.

grln- @. 3xl'66x10-.^ : rU5;TOTIZE TO.

i : 9110-c c-n(ii) Frequen",

"t.",I'iAi."IlS.;f coilisions per second= Ti6-freem

X:9,t

Ir : gllg'I x lO-cl{

-

5xlffliid; Avogadro's numlru.o23

x loBNurnber of moieculeg p;;"r;r;"

22+00According to Maxwell's relatioa

-l^*Jm,l--l- r'|4iffix'fiI

C : 4.5 x lS cm/rp : I g/titre

. 3ztn- ec

, _ 3_x! 4>< lr'"

- l0-rx4'5x l0'< ,/, l-15 1[0-ccm'> *4"-etc 5'2t. \alaulale the meon lree ytath of a gaa moleouk,'giuen lllo;t ibe molcaubr diameter ie 2 x l0-s cm ond the tumbcr of

moleautre pet cc ie 3x10t.IA:-m

I.:@l:3yl(}rcrn

Noto. Thc moco froc peth ir lor than iho wlvclcngth of lighf in tbcviaiblc rpootrum

Eramptc 5'24. Calculote the meon lree path of gaa molecvleein o c,tnnrbei of 10'o mm o.f rbercury 1)reEsure, uauming thc molcculorilianckr to be 2L. Ane ryam molecule. of the gos occupica 22'4litres olN.T;P. Toke thc lemperolure of thc chamber to bc 27 3 K. (Agro 19? 5l

At 760 mm Hg pre$ure and 273 K temperalure, the numberof molecules in 22'4 litres of a gas

-

6.023 x lOsTherefore, the number of molecules per cmt in the chamber at

r0-! mm pressure and 273 "J:rTiXi;:ll,o_,

22400 x 760r

-

3'538 x l0ro molecules/cmld

-

2L: 2x l0-8 cmMean frce Path,

-l^-r,dro

I

I :2'25 x l0-' CGS units

3'l,t x (2 x l0-r1t* 3'538 x I01s == 2'25 X l0 cmwww.

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194

and

Eeat ord ThanoilynamiaBelow the Boyle temperature, the gases are highly compressible

this suggests the eristence of inter-rnolecular attrattions. Beyond

I

Natlrre o! Ecat'-'\'i '1 ' '-' o,uh(

c-i>fttoz t'ltia (

llere c is a constant and I/ is the vol '*c of the garHence correct pressure

: (P+il: (r*# )wherc P I ihe observed pressure.

Q)f Corr""tion for Volume. The fact that the molcc.rleshave 6hite size shows that the acrual space for the movenrent of thcmolecules is less than the volume of tb6 vessel. The morecules have

the gas : (f-b). .11 a--Dtk, the radius6tdne'motecule be r.

(J- ,'.,u ,-t-

LT "''

196

BoYLEr-l-- / TEMPERATURS

the_1p!9le_-ol-rl_9q9l"q aro-und the'.n- and duC-o t6,is-Ectoi rtx, thei6rrection -foi-vilirEre ls 6 rr&ere 6li-a imateiy four times thesorrcL'rton ror volume -tS

o lr'nere O rS approamateiy lOUr ttmes theactual volume or the molecuies. Thercfoie the corrected volumc of

Thc volume of the rnolecule

the Bgyle temperature, Boyle's law is obeyed and intermolecular(". a,.t7&,tions are less significant.l\ 5/36 Van der \flaals Equation of StateA 5t36 Van der \flaals Equation of Statey

^- (lVntt. deriving the prefect gas equation PV : RT on the basis(S'\ oi kjnetic theory, it was assumed that (r) the size olthe molecule of

/AY-/ the cas is nesliEible and (ii) the forces ol inter-molecular attraction/)\'-/ the qas is neeligible and (ii1 the forces ol inter-molecular attractiont are absent. But in actual practice, at high pre$ture, the size of themolecules of the gas becomes significant and cannot be neglected incomparison with the volume of the gas. Also, at high pressure, thernolecules come closer and the forces of intermolecular attracti'lc areappreciable. Therefore, correct;";* hculd be applied to the ga!e(luatlon;l

1i) Correction for Presgure. A molecule in the intcrior of aqas expericnces forces ofattraction in all directions and the resultantIohesive force is zero, A molecule near the walls of the containerexperi enc es a

.

resu I ran t fo@helaattF=D ue to.thi!'...'9nttr.@gasisIessthan-theactualp..r*r.dThe :orrecliotr foi prestute p depJnds upon (d) the numberbf molectrTes striking unit aiea of the walls of the container Persecond and (id) the n"umber of molecules Present in a giveri volume.Both these far:tors depend on the density of tha gas.

. .'. Correction fcrr pressure p * pt -+

The centre of auy two moleculo ".o.pp$,.ch cach othcr onlyby a

'rinim"m distance of 2r ri.c., the dia-meter of cach moleculi.The volume of the sphere of influence of each molecule,AB-in(2r)!:Br

coarider a container of volume 7. If thc molecules are a[ouiedto enter one by one,

The volume available for first molecule:Y

Volume available for second noolecule: Z-B

Volume available for third molecule: F_25

Volume a,railable .rjt}:#:il iAverage spacc available for each molecule

:n

: v- $l+z*3......(n-rIti: ,'-g (n-'l)n-i' ---T-

OT NB B- 2fn

Fig. 6.18.

op:-7,

P-------+

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t- 'l- ,O* the Vau dcr Waals equation of state for a gas is)- t-' (r**)(r-6) : Rr ...(i) ) ,.wherc c and b are Van der Waals constants.

From the Van der Waals equation of statq(;;+F;;:;)?;"^ $rt, :

,+u_;, I-/-

196 Eeal anil Thermoilyrnmict

Ar thc number of molcculca ir vcry tu.g" $ ca. be neglected-.'. Average rpace available for cach molecule

: f-g (ButS-Bc)'2r--__ Y_n(ul'2: Y-4(ns): y-b

b : 4{tt*l -

tH Jl?iL,*. actual volume ol

Nalurc of Eeat Dldecrease in oressure. It is not possible in actual practicc. The rtater18 and I'D, though ,r*t^ti.,'".r, be ,"at;'.d i_n practice by carcfulexpcrimentalion.

-

At higher tempcraturg, the theoretiial andcxperimenLal isothermals ire similai.

until now "s

qany as 56 differcnt equations of state bave beeosuggested. But no single equation satisfie;aII the observed fact!.

Dicterice (1901) has suggested an equation

P(y -b) - Rr "-#

Berthelot has suggeEted an equation

(r+t:*)v-b) : Rr5.37 GriticelConstante

-f6\" (rn. ";ti.ul temperature and the corresponding ,ulo". . Jf''/Jprerrure.and volume at the critical point are catied tajcritical.cons.rants. At the crrtrcal point, the rate of chnngc of pressurc withvolume (#)is zero. This point is calledthc point of inflexion.

r-/-Accorriing to Van der Waals equation

( ,*#) rr-,r : Rr,:l#)-+

Differentiating P with respect to 7iIP _RT7V : (7:[]r *

At the critical ooint * : ndyv-ToY

-Y"-RT, 2o.

-

I - -

tt(7r-D)8T Vr' - v2o R?oor W : O;D;-.),

Differentiating equarion lrrir;dzP 2Rr@ : ip=rf _

...(rt)

Ug:l{lUlE

Ir.r t:r.q ol

2a-v-

...(0

...(j0

...(di0

V0LUME ------+

Fig. 5.19Graphs betwcen Pressure and volume at various temperaturcs are

,os

zRro(7"-D)t

Dividing (tu) by (u)

Substituting the value of

ry_rvV:Y"

6a7] : t-t6o 2RT"

w: v;6)eY, V

"-b32zVo

-

3l'r-3bYo:3b

Vo : 3b in equation (ic)2a RT"fis: @z

mBat":i_Rb

Eeat onil Thermollynamice

...(u)

.. .(r'i)

...(uir)

At thc critical noint -jS- : o,Noture of Hedl l

TASLECritical Temperature and Pressure of Comraon Gasee

id..

IIelium[[ydrogonNitrogenAitArgonOxygen

Carbon dioxide

Ammonie

Chlorino

Wator

Stfrata,rnoe Criticollfcnpnraturc t0

-2Bgoc--24t0"c

-146.C_140.c

-118.C3I'C

130'c

I4b-U

Critical Preacr.rr.c(atn?p{phcrro)

2.26

t2.80

33.60

39.00

48.00

60'00

73.00

15.00

76.00

218.00

Substituting these values of 7" and ?" in equation (ii)D ExBa a'o: TEW - 56r-

^ \

P": h'. 'lt

t$ ')"'o""/V5.38 Corresponding stater fTwo gases are said-to be in corresponding rtates if the rarios ol

their actual pressurc, vqlprg and temperatura and cridcal pressure,critical volume and critical temperature have the same vilue. IiIDCatrg

Pr Pz::-Ed Po,Yr Y,W: N;

cnd +: #lgl lal

...(i)...(rd)

,.,1,1i1)

...(iu)

5.94 C

?10 Eeal and Thcrmodynomica

...(ui)

Nahrc ol Ecat

(;i)

Notc, Eoro V " -

moloouler w0 x apecif,c volumo.

The experimental value of the critical coefficient of all gases isgreater than thc theoretical value of2'67.

- -E,-gple 5'2{. Calculate the Yan iler Waols constanta lor dryoir, gioen that?6 : 732 K, P"

-

37'2 atmosplwreo,B per mole : 82'07 cml atmos K-1.Here Pr:37'2 atmospheres

llo : 132 KR : 82'07 cm! atmos K-r(0 ":#ryo: /11\(t3'ozl'032)'

- \3rl-r7E-a : lt.3l x l(F etuog clqr

@elhi 1975)

^- RT,

" - -8lE

, 82'07 x I32O:-- 8 x37'2

or D:36'41 crnr5.+0 Reduced Equation of State- -Taking the pressure, volume and tcmpcrature of a gas in tcrmr

.of reduced prcssure, volume and temperature,PVT

-F:-: A, *O- : F, 7,- : Tt c f 2 14P:a.P",T-PTo,T:\To

According to Van der Waals eguation(r*#) rr-rr : Rr( -r"*&)rF r,"-u)

- Ry r"

But P, :;i,Y":3b

and T": h. ly.:,+)re.3a_6t:y#

127 b'rPz t

[.* p', ] [3B-r] : syThis is the reduced equation of state ior a gas. If two gases

have the same val rres o[ a, B and 1 they are said to ie in corresfrnd-ing states.5 4l Properties of Matter Near Critical poiat

Based on the experiments of Andreu,s, Amagat and others, thestate of matrer near the critical pcint can bi srrmmarised zrsfollows :--

(l) The-dersities of the vapour and, the liquid gradualyapproach each other and their ciensities

"r" .qrril at t$'c criticalpoint.(21 At the. critical point or just aear the critical point, the

line of demarcation between the liquid and the vapcur d-lsappears.Consequentiy, there must exist riutual

2tl Eeat anil Thermoilgnamict

Write short notes on :Mean free pathjoule-Thomson EffectCo;rtinuity of stateRowland's experimcnt.for finding JVan der W"als equation of statePressure exerted bY an ideal gasCritical constantsDegrees of freedomAtomicity of gasesMaxwell's iaw of distribution of velocity. (Delhi 1975'SAndrelr's' experimentsAmagat's expcrimentsHalborn's experimentsBehaviour ofgases at high PrcssureCritical Pointftrresponding statesIntermolecular attractionTcopcraturc of invcrsionReduced equation of state for a gas lDelii lfiorta.) 19761Porous plug cxPerimcnt"

Thermodynamics

6'l Therraodynam,ic SysternA thermodynamic sysi em is one which can be described in

terms of the thermodynam,c co-ordinates. The co.ordinates of athermodynamic system'can be specified by any pair of quantities ufz.,pressure (P), volume (Z), temperature (f) and entropy (B). Thethermodynamic systems in engineering are gas, vapour, steam, mix-ture of gasoline vapour and air, ammonia vapo'-:rs and its liquid. InPhysics, thermodynamics includes besides the abovc, systems likcstretched rvires, thermocouples, magnetic materials, e]ectrical con-denser, electrical cells, solids and surface films.

Examples : 1. Stretched wire. In a stretched rvire, to findthe Young's modulus of a wire by stretching, the complete thermo-dynamic co-ordinates are

(o) thc stretching force /(D) the length of the stretching wire and(o) lhe temperature of the wire.The pressure and volume are considered to be constant.2. Surface Fihas. For liquid films, in the study of surface

tension, the thermodynamic co-ordinates are(@) the surface tension(b) the area of the film and(c) the temperature.3. Revcreible Cells. The thermodynamic coordinates to

completely describe a reversible cell are(a) thc E.M.F. of the cell(b) the charge that flows and(c) thc temPeraturc.

its centre of mass is equal to the total- random kinetic energy of allits nrolecules ? (N'lolicular weight of hydrogen : 2l- ^^-[Ao". l'93 x 106 cm/s]

61. Calculate the temPerature at which the r.m.s' velocitv ofa helium molecule will be equal to the speed of the earth's 6rstsatellite i.e., 7, : B km/s. [Ans. 10'28 x lOp K]

62. Calculate the mean kinetic energy of a molecuie of a gasat 1,000'C. Given,

E : B'31 x l0? ergs/gram mol-KN : 6.02x lOa(Delhi 1969) lltta. 2'07x l0-rr ergpl

63. If the density of nitrogen is l'25 g/litre at N'T'P', calcu-late the R.M.S. velocity of its molecules.

lDelhi 1972 ;'Delhi (Eons.\ 19731 lLne. 4'95xlS cm/sl6{. At what temperature is the R.M.S. speed of oxygen mole'cules twice their R.M.S. speed at27"C?

' (Delhi 1973) [Ane. 927'C]65. Calculate the R.M.S. velocity of the molecules of hydro'

gen at 0oC. It4olecular weight of hydrogen : 2'016 and- R: 8'31 x I0? ergs/gram mole oC(Dith; 197 1\ lLtr. I 8'4 x I S cm/rl

66. Calculate the R.M.S. velocity of the hydrogen moleculesat room tmperature, given that one^ Iitre of the gas at room temPc'rature and normal pre-ssure weiglrs-0'086 g-'(Oetni /9761 1Aoe. l'BB x lS cm/s)1

67.(i)

(ii)(dii)(iu)(r)

(oi)(oii1luiii\(ir)

(r)(ril(rii1(riii\{ria)(lcu)\rl,il(roiil

(x',tiii)lair\@r\

lAgro 1962 ; Delhi (826.) 1966)(Agro 1962 ; Delhi 1974, 7 5\,

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216 lleat and Thermoirlnamica62 Therrnal equitibriurn and Conceot of Temperature

.\ thernrodl'narnic s)'stem is said to lle in rherrhal equilibrirtnif arry trvo c,l its independent tlrermrclvnamic co.ordinatei ,t and I:remain c()nstant as long as the e\lernal conditions remain unaltered.Consider a 3as enclosed in a cylinder fittcd rtirh a piston. If thepressure and volume of the enclosed mass of gas are P and 11 at thetempcrature o[ the surroundings, rhese values o[ P and I/ u,illremain constant as long as the external conditions t'iz. temperatureand pressure remain unaltered. The gas is said to be in thermalequilibrium with tlie surroundings.

The zeroth law of thermodynamics u,as formulated alter thefirst and the second laws of thermodynamics have been enunciated.This lau, helps to define the term temperuture of a system.

This law states that if , of three syslems, A, B ar;d, C, A and Bore separately in thernol, equilibrium utith C, then A anil B are olsoin thermal equilibrium with one a.nother:

Conversely the law can be stated as follows :4f ,hrt.y' or more systems are in thermal contact, each to each,

by roeans of diathermal walls and are all in thermal "equilibriumtogether, then any two systems taken separately are in thermalequilibrium with one another.

Consider three fluids .4, B and C. Let Pa, Pa represent thepressure and volume of -4, Ps, /s, the l)ressure and volume of B,ind Pc, Ysare the pressure and volume of C.

If .4 and B are in thermal equilibrium, thenSr(P*Vn) : 6r(Ps,Ysl

or Ir[Pl, Y7., Ps,7a] : 0Expression (d) can be solved, and

Ps:/r[Pa, V*,Ys)If B and C are in thermal equilibrium

flz(Ps, Ps) : #slPc,Yclor frtPr, Ya, Pc,Zcl : 0

AIso Ps: f2l?s, Pc, I'c]From equations (zi; and (diz) for .r{ and C to be

equilibrium separately,.fr(Pe, Ye,Vs) : lzlra, Pc,Yc)

...(d)

...(ir)

. . . (iir)in thermal

...(iu)If d and C are in thermal equilibrium with I separately. then

according to the zeroth law, A and C are also in thermil equiiibriumwith one another.

.'. .F'r[Pl, Ve,, Pc,7c] : 0.Equation (it') conta;ns a variable 7s,

does not contain the variable 7s. It means$r(P* V s) : ps(Pc, I'c)

. ...(t )

whereas eguation (r)...(ui)

rntrrne-tc4 ecl -_ it.z$2ruzr,-Thirmoilynam;cs 217

In general,d,(Ie, I'a)

- cr(,I'o, I'n) : dr(Pc, I'c) ....r,ii)

These three functions have the same numerical valrre thougi,the parameters (P, [/1 of each are diffcrent. This nurnerical valireis termed as tentperatur': (T) af the body.

...(riii;This is called the equation o[ stare o[ the nuid.

-,Q'Therefore, the temperature of a system can be defined as

the property that de termines ,rvhether or not the body is in thermalequilibrium with the neighborrring sy!rems. If a nuntl,rer. oIs,vstemsare in therm"] tquiiibrium, this cor,:rnon property of the system canbe represented by a single numerical value called the temperature,It means that if two systems are rrot in thermal equilibrium, t{reyare at different temperatur r.

_

Example. fn a nr"rcury in glass thermometer, the pressureabove the mercury column is zero and volume of mercury measuresthe temperature. If a thermometer shorvs a constant reading.in twosystems. A and_B separately,. it will show the same reading evenwhen .r{ and B are brought in contact.

6'3 Concept of HeatHeat is defined as energy in transit. As it is not possible to

speak of work in a body, it is also not possible to speak oi heat in abody. Work is either done on a body or by a body. Similarly, heatcan flow from a body or to a body. If a body is at a constant t"m-perature, it has both mechanical and therrnal energies dqe to themolecular agitations and it is not possible to separaie them. So, inthis case, we cannot talk of heat energy. It mtans, if the flovr ofheat stops, the word heat cannot be used.' It is only used whenthere is transfer of energy between two or more systems.

Consider two systems.r{ and B in thermal contact $,ith oneanother and surrounded by adialatic walls.

For the sJstem /, fl : Ar-arlW...(r)

where II is the heat energy transferred, U1 is the initial internalnergy, U2 is the frnal internal energy and W is the work done.

Similarly for the sPtem B, 'E',

-

U2',-Ur',*W',Adding (i) and (ii)

E + E' : (Ur-Ur)+ril + (U 2' -At') +W'

fl +E' - llU t*a r')-(Ur*Ur'))+ (f +W')

...{r4

... (idr)Thc total change in the internal energy of the composite system

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The work Lrne by the cor^rposite system : W +W'It means that the heat transferred by the composite s1'stem

: E +H'. But the composite system is surrounded by adiabaticwalls and the net heat ransferred is zero.

u*'; jo-u, ...(iu)

Thus, for two systems A arrd B in thermal contact with eachother, and the composite system sur-rounded b-y ad.iaba.tic walls, theheat gained by one-system' is equal to the heat lost by the othersystem,

6'4 Quasietatic ProceeeA system in thermodynamical equilibrium must satisfy the

following requirements strictly :-(i) Mechanical Equilibriuro. For a system to be in mecha-

nical equilibrium, there should be no unbalanced forces acting on".ry p.it of the system or the

system as a whole'(di) Thermal Equilibrium. For a system to be in thermal

equiliLrium, rhere should be no-temperature difference between thepirts of the system or between the system and the surroundings.

(iid; Cheaical Equilibrium. {or- a system to be in chemicalequiliLrium, there should be no chemical reaction r.r'ithin the systero.rid ulro no movemerlt of any chemical constituent from one part ofthe system to the other

When a system is in thermodynamic equiiibrium and thesurroundings are kept unchanged, there will be no motion and alsooo *c.k wi'il be dorie. On theother hand, if the sum of the exter'nal forces is altered, resulting in a finite unbalanced force actir:3 onthe system, the condition for mechanical eguilibrium wiil not besatisfied any longer. This results in thc following :-

(i) Due to unbalanced forces within the -system, turbulence,

waves'etc. may be set uP. Thesystem as a whole may possess anaccelerated motion-

(dt) Due to turbulence, acceleration etc' the temperature dis-uibution within the system may not be uniform. There may alsoexist a finite temperature difference between the systelo and thesurrouncl'ngs.

(iii.y Due to the presence of unbalanced fprces and differencein temperature, chemical reaction may take place or there may bemovement of a chemical constituent.

From this discussion, it is clear that a finite unbalanced forcemay cause the system to pass through non'eguilibrdurn states. Ifduring a thermodynamic process, it is desired-to describe every stateof a system by thermodynamic co_ordinates_referred to-the system rua whole, the process should not be brought about by a finite un-balanced force.

Thermodynamicc 21'1A quasistatic process is definei as the Pr99c: in which

-thedeviation from theimodynamic equilibrium is infinitesimal and aUthe states through which the system passes during a quasistaticprocess can be considered as equilibrium states.

In actual practice' many proc$s.e-s :losely approach a quasista'tic orocess and mav be trcated as such with no signihcant eror.Consider the exparsion ofa gas in-a closed-cylinder fitted withapiston. Initially'weights are on the piston and the pressure of the gasinside the cylinder ii trigtrer than the atmospheric pressure. If thewershts are'small and aie taken off slowly one by one, the process"^rii" considered quasistatic. If, however,

all the -weights are re'

'moved at once, expinsion takes place-suddenly and it-will be a non'equilibrium p.oceis. The system will not be in equilibrium at anytiiee during this Process.

A quasistatic process is an idcal concept that is applicable toall thermodynamic iystems includin_g electric and magnetic systems.It should bi noted that conditions for such a process can never bcsatisfied rigorouslY in Practice.6'5 IIeat-A Path Function

Heat is a path function. When a systm chalSes from a stateI to state 2, thiquantity of heat transferred will depend upon thcintermediate stagls through which the- systern passes i;e., its path.Hence heat is an ineract differential and is written as 8II.

On integrating, we get[*m:E'olla laHere, 1EI, represents the heat transferred

-during thc givenprocss bctwien the states I and 2 along a pardcular Path A.6'6 \f,orlr-A Path Function

Suppose that a systen is taken from an- initial equilibriumstate I to'a final equilibrium state 2 by- two different paths A and Btfig. O't). The prbcesses are quasistatic.

H eat anil ThcrmoilYnamics

tI

P

V _---+a[. 0.1ww

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250 Heat and Thermodynomica

l'lrc;.r'els rrndcr these curves are different and hence thelrrrntitics of rvor k clorte .rle :rlso different.

For the p;ulr .\,r2AI pdt'J r.{

r28I PdVJIB

It is customary to represent, \.vork done by the st'stem as {"r'.,work done on the syste:-o as

-ve, heat flowing into tl,e s,'stem ras

^1-ve, and heat flowing out of the system as -ve.

frr)rt First Law of Thermodynarnicsx' vJheat produced. It is true rvhen the whole ol the rvork done is usedin producing heat or t'ice oersa. Here,W : JE rvhere J is theJoule'smechanical equivalent o[ heat. But in practice, rvfen a certainquantity of heat is supplied to a svstem the whole of ttr-e heat energymay not be converted into 'a'ork. Part of the heat may bc used indoing exrernal rvork and the rest ()[ the heat :nighr be used inincreasing the internal enerqv of the molecule(Qflet the quantityof heat supplied to a system be 8I/, the amorrntYF-external rvorkdonp be 87 and the increase in internaI energy of the molecules berlu)/Ine te rm U represenrs the internal energy"of a gas due to mole-cylar agitation as well as due to the forces of inter-molecular attrac-tion. lflathematically

\ w : aa 7m') ...(d)@

condition. Therefore for a cyclic process $aU : O

$ r': fa'

Thermodynarnics

and

22t

r 3'\ll'A:l 6lf:J l.{

For the path BrlB

;r-e -

.l re 8rr/ :

...(i)

. . .(i,)The values of II'e and Il'6 are not equal. Therefore work

cannct be expressed as a difference between the values of someproperty of the system in the two states. ThereJore, it is rtot cotectlo represettt

Ir, : []l:, BtF : ty, _ wlJW' ... i;ii)It may be pointed out rhat it is meaningless to say "rvork in a

system or rvork of a svstem". Work cann^t be interpreted similar totemperature or pressure of a system.

In terms of calculus 817 is an inexact differential. It meansthat F is not a property of the systen, una j 817 cannot be express.ed as the difference between two quantities that depend entirelyon the initial arrd the 6rral states.

_

Hence, heat and work are path functioas and they dependonly on the process They are not point functions such ai preisureor temperature. Work done in taking ttre slstem from state I tostate 2 will be different for differenr paths.6'7 Gomparisoo of Heat and l{ort

There are many similarities between heat and work. These.are :-

I. Heat and work are both transient phenomena. Systems donot possess heat or work.

2. When a system tindereoes a change, heat transfer or workdone may occur.

3. Heat and work are boundary phenomena. They are obser-ved at the boundary of the sysiem.

4. Heat and work represent the energy crossing the boundaryof the system.

5. Heat and work are path functions and hence they are inexactdifferentials. They are wriuin as 8,8 and 8I7.

6. (a) Eeut.is defined as the form of energy that is trarufrredacross a boundary by virtue of difference of iimpcrature or tem-perature gradient. -

(b) Work is said to be done by a system if the sole effect onthings external to the system could be the raising of a weight.

...(14lBc,th arc erpressed in l'reat urrits].'I-lris erluation represents Joule's law.For a svsrem carried through a cyclic process, its initial and

linal internal' energies are cqual.' From the first law of ti,.rmodv"a-mics, for a system undergoing any number of complete cvcles

ur-ut : sl" D,H : 16 rrrY:r

H : llt [Both are in heat units]6.9 First Law of Thermodynarnice for a Change ia State of

a Closed System,

Fo.r a clo.sed. system during a complete cycle, the first lau. ofthermodynamrcs rs lvrrttell as

" aa : rli st;,Yrwww.

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In practice,rathcr than a cYcle.

Heat and Thermodyrumiu

however, we are alco conccrned with a PJocclBLet thi system undergo a cycle, changing its

Tlxrmodgnamict

(8.e-8If) depends only on the initial and thc final states of thesvsrem and is independent of the path followed between the twostates,

Let d.E : (8H-Etr)From the above logic, it can be seen that

2

I an : constant and is indepcndent of the path.JI

This naturally suggests that E is a point function and dE rr an.cxact differential.

The point function .O is a iroperty of the system.Here dD is the derivative of E and. it is an eract differential.

8H-8W : dE ...(du)8H : d0qtW ...(r,)

Integtating equation (o), from the initial state I to the finalstate 2

tflt -

(E,_E)+LW,[Notc. 1EI1 carmot be writton ae (EI1-II1), bocsueo it dependa upouthe pathl.Similarly, ,}[. cannot be written as (W1-W1), because it also

dcpends upon the path.Here lEIs represents the heat transferred,

1fl1 represents the work done,E, represents the total encrgy of the system in

state 2,E, represints the total energy of the system in

state l.At this point, it is worthwhile discussing what this E c^n

possibly mean. With reference to the system, the^ energies_crossingi:he boundaries are all taken care of in the form of E and W. Foi.dimensional stability of Eq. (u), this f mustbe energy and this mustbelong to the system. Therefore,

E2 represents the energy of the system in state 2E, represents the energy of the system in state IThis energy E acquires a value at arry given equilibrium con-

dition by virtue of its thermodyna-mic state. The working substance,for example a gas, has molecules moving in all randbm fashion.The moleiules have energy associated by virtue of mutual attractionand this part is similar to the potential energy ofa body in macro.scopic terms.

_

They also have ve locities and hence kinetic energy.'fhis energy E therefore can be visualised as comprisine ctf moleculirpotential and kinetic energies in addition- to maLioscopic potentialind. kinetic energies. The first part, which owes its rxrstence ro the

223

H;-state from I to 2 along the path :{ and from ? to I aiong-the

-path B'This cyclic Process is iepresented in the P-Y diagcam (Fig' 6'2)'According to the first law of thermodyaamicl

ftr-{twFor the comPletc cYclic Process

2A r8 2A 18

I,rt'* J ta - {,t'* I,:'Now, consider the second cycle-in lvhich the-rystemchanges

from state'l to statc 2 along the paih /, and returns from state 2 tortate I along the Path C. For this cyclic process

2A. lo 2a lo

l,** [:': [,'** I;*Subtracting (id) from (;)

IB LO 18

I*'"- [:'- L''-l8 LO

or Ittr-tr):l(DE-Dr)l, zoIlere I and C represent arbitrary processe between the states

I and 2. ?herefore, it-can be concluded that thc quantity (88*8W)ig the same for all processes between the stdtes I and 2. The quantity

I

ro

[,,)ro

...(0

...(dr)

... (iii )

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224

lhermodynel,,tc r'l:l,rrre is often cailcd tl,c internal enercycomFletely deper,derrt on thc ihermodynamic stare. and'trvo depend o,t mecllarric;rl or phvsicaI surte oI thc system

E : U yKE + PE +Othcls whiclr depenrl uponnature etc.

For a closed s),stem inon-chemical) rhe clranges inexcept U are insignificant and

dE:dUFrom equation (r)

8H : dU +EWFlere all thq quantities are in consisrenr unirsEra:nple 6 l. ll,hen o .ey.\tem is taken, lrom the state A to tLe

stalc B, alonT the Vath ACB,80"ioules oJ heat f.ows ilto thr systrm,anil the system dois S0joules o/Lortc 1i,;g. A.S\.(a) y7w much heat fl.owa into the syslem along llrc ltullt ADB,iJ the work ilone is J0 jouies.-

(6) The -system ia returned, lrom the state B lo th,c stute A alonglhe curaeil poth. The work d,one-ott. lhe syslem is 20 joules. Docs the

systenx absorb or liberale heat anil how mich ? -(c) l! Ua

--C, UD: 40 joules, finit the ltcat absurbed in the?roeess AD anil DB.

Tkrmodytwrtiu

Erpr : AB-U^+WE

-

50*10 : 60 jouler(D) For the curvcd path frcm B to A,

W : -2}joules

: -50-20 : -I0.Jouler(-ve sign shows that heat is liberated by thc syrteur)(c) Dr : 0, Up

-

40 joulca0s-U1 : 56

Or : 50 jouler

For DB

6.10

a:fiY,r)Differentiating equation (r)

In the Drocess /?A, l0 joules of wort is done. WorL danefrom.r{, to D is J-10 joules";d};;; J';; is zero.For A-D,

.EIrp: (Up_Ag{W-

40+I0 -

50 joules

trIpa -

Ac-Ao*W: 50-40+0 : l0 joutee

II eal and lhei.modynamics

which isthe other

chemical

all othcrs

. ..("i)

Ot! ta&d

ti

P

Applications of Firat Law of Thermodynqnic! qSpeei6c lleat of a Gae (T aod, v Independent)

Thc internal energy of a system is a single valued function ofthe state variables or:2., pressure, volume.case of a gas, any two

*ifffr,,,*.,,, *,o,li'Si*:# i.h-*T#.1'JHl *;

oo: (#), rr* (#),* ...(,')ff anamountof heat gEI is supplied to. a thcrmodyaamicallyrtem' say an idcar gas and if the'iorume lncrer*e! by ily at a

ffXo", pressure P, theu according to the 6rst law;i-6il;;.-

...(d)

Fig. 0.3AJong the parh ACB,

Ilacs : Uy*U1!WIlere A:+80joules

P : $30 joules.s. +80 : as-Ua*3O

U s-t t: 80-30 : 50 jouler(a) Along the Path -r{DB,

W : *l0joules

Here8E: da+EW

8W : P.d,V8E : dU+P.i|Y

Substituting the value of dU from equation (d0,u

- (#),ur*{#\,rr*r*.. . (r:r'd)ww

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^ -l

()r

Dividing both sidcs bY dT

#: (3.r)..(#),#*#(# ) : G+)" "[ '*(sur),){,

iI the gas is heated at constant volume,, EH\[77 .)u : t'

dv :odT(# ),: (-37;":,.\\Ihen the gas is heated at constant Pressure,(#),:",

H cot ond Thennodgnumict

...idr)

..'("t)

...(utO

?hermodynaniu

Herc Cp, Cy and I are expressed in the same units.From cquation ldir;

'r: ( #)"rr*lr* (#) ,)uoFor a process at constant temPeraturc

ilT:g(a^a)r

-

P(irltr+ (#),ruo, ...(t)

...(d")

...(rid)

"..(ri;i)...qrio|

From Joule's experiment, for an ideal gas on opcning thc stop'*n.k, i-ro *o"tk u'as done and no heat transfer took placc'

So 8.8 : 0 : dU +0. Thercfore, dA : g. Even thougt thco,.'-rlrrml- cnu'gea while the tcmperature is coostant, there is nocirange in internal energ'Y'(ao_\:0

\ aF-1, - "lirora ttre ideal gas eguation

PY ':' R?

i:r P (*F;,:ocp-.c1 : r, l# ),*(p), (# ),

This equation rePresents the amount of heat energy -supplied

to a system ii an isothlrmal reversible process and is equal -

to thcs,'.m of the work done by the systeE aird the increase in its internalenergy.

For a reversible adiabatic process8E :9,

Therefore, from equation (fr),o: cv ur+lr. (#) ,T,

cydr :- [r* (#),y,Dividing throughout by dV,

,,(#) :-[r*(*#),]The isobaric volume coefficient of expansion

": +(#),",: (#),

ce-.cv -' (-# ),

Cp-Cv D-

-:t

aV

'* (*f ),: o * F--F(#), * ("#')-,

*(e,#'):-[on(iF;,1

...(u)

From equation (iu)',,: (#)"*[ r*(#),](#),c" : c"*f ,*(#),1(#),

c,-cv: [ ,*( #),](*f ),

/:lL \F'r,t i '^,, 1l(t T

L'P* {)!'. l,y *'- ,

=,.( ;i)u:o,*

"H .. . t.ilii iorww

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228

or

From equations (rid) and (rio)

"(#) :-(#)/ a" \ Cv-Qt\aTJ:-WThir expression holds good for an adiabatic reversible Proccss.

6'11 lrotLcrmal Procegeftf

" "rrt.- is pcrfectly conducting to the surroundings and

tl" t.g-p.rli"r. *-"i* cotirtarrt throughout the process, it is called

Heal ond ThcntoilY*amio

V--+frg. 0'a

anisotherma!process)_9'yidf 1,3i"i*,c--':b::1ll'"::";;fi;""*ir;-l.o4t"t. u"a nt"i"g a vol-ume represented

hrrroundings and itl temPeratureof the

substance, there is rise in temPerature because the extcrnal worltdone on the working substance increases its internal erre-rgy. \{}rr:nwork is done by thJworking substance, it is done at the cr:st of itsinternal energy. As the system is perfectly insulated frorn thesurroundings, there is fall in temPerature.

[Ttrrr, during an adiabatic process, the working substance r:rrerlettly insulated from the surroundings. AII along the process,ihere is change in temperatureJ A curve between pressure andvolume during the adiabatic pro"cess is called an adiabatic curve oran adiabatic.

Examples. l. The compression of the mi,xture of oil vapoureld air during compression stroke of ao internal combustioa is anadiabatic proCess and there is rige in temPeralure.

2. The expansion of the combustion product! during theworking stroLe of an engine is an adiabatic process and there is failin temperafitre.

3. The sudden bursting of a cycle tube is an adiabaticProcessr.

Apply the first law of thcrnodyuamics to an adiabatic Process,$.EI

-

0,

Tlwmodynomir"r 229

8.8: dU+Etr0 -du+8IF ...(d)

The procesd that takc place edlilenly or quickly are adiabatic

,..(rt)

certainby the

inint a (Fig. 6'a).Pressure is decreased and work is done_ by the w.orking sub'

.*r,".^"[f.-";;;i-i" i"t"t"tl tt'"tgy and *eie shor{d be farl instance at the cost oI- its rntrnal energy an

2E0 Eeol ond Thermodgnamice

.. . (i)

...(ii)

... (iii)

. ..(dd)

?hermodynarnic*"

C,.P.dV +C,.Y 'dPlCr.PdV -CoPdl: : o

Ce.P.dV+Co.V.dP -'0

Dividing by C,PY,c) ilv dPi'-r-+-P:u

But

8EI -

I xlrdz+P$But 8E

-

OydTP.dY : r'dTCaill

-

C..d7+ +o'-cn: +

Here C, and O, rePrescnt the specific heats for I gmm of a gasand r is the ordinarY gas constanL

If C, and Crtte the gram-molecular specifrc heatu of gas' theno,-c, : + ...(io)

Hffiis the universal gas corutant'-ttS/ Gas Equation Duriog rn Adlebedc Processv

Corrrid.r I gr-am cf the work-ing substance (ideal gas) perfectlyioruf"iJ-iro* tfrE r"r."""ai"gt. Lei the external work done by thegas be 87.

Applying the 6rst law of thermodynamics8.E[ : da +8w

But 8E: gand 8W : P'dYwhere P is the preslure of thc gas and dP is the change in volume'

o : da+!-*!

co : ,C,

dP dVJY --:: {)P II Y

Integrating, log P1Y log Y -

const.log PYt : coost.

PY't -

const. VT--'--"This is the equation ccnnecting pressure and volume tl 'rr:rr:

,W+\.fu)"*radiabatic process.

Taking PY:rTo-'?'-v

,"1 ";;#, d;;;i' i"u it temPeratdrc

bY??'dA

-

lx?,xilTc,dr+!+:9' r

Also

,t

Thu)during an adiabatic Process'$

,, PPI : const'

$rY TY-'."- const. andt .ft't

lit\/b: const.i,rzrrot" 6'2. A rtotor oar tYre

"niult tln-room ternperature of 27oC'

furd the reaulting tertPerolure.

/rT\(-r-,) ' i'Y: const'But r is const.

fTYr-t : Colutt.rYl-r : const.

r?.-_{: CCflSt.

: COIrSt,

: COISI.

...(4As the external work is done by the gas at thc cost of its inter'

Y

rT 1'r_lPJflr'fPFTPf .LT

haa o preslure of 2 ar,r;If the tgre audilenig bu'-| : c'-c,

For an idcal gas- Pv:q

^ "'(;;;1

Differcntiating,P.dYqv.dP : r'itT -'.1? -B#

Substituting thc valuc of dT it equation (di)'

a1%*ff=ocolP.&Y+Y;d4+r.'i' : o

But,

-a

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( f,)":0'4log (0.5) :

-0'1204 :l'4 log T, :

log ?. -

T,::

.,2-m:aple 6.3. A ou.antitg_o!.air at Z\"C anil atmoapheric pret-.,

.euilenly ccmpre.iaeil rc-naiy ;ta -or;a1i:not

ool.ume. Iind thc..i preasvre anil (ii) temperatire.td) Pr : I atarosphere ; ps : ?, 7 : 1,4Yt:Yi Yr- +Duriag sudden compression, the progess is adiabatic

P,Y,, : P,Y,TPr: Pr[ t)'

: I[2]1..: 2.636 atmoqpheres

iii)P1 -P; yr: +rr-

t:Trl7r;t-r

-

,r=

Eeal and ?he.rmodyaamict

2 atmospberes27? +27300 KI atmospheret1.4Prr -r-a-a-rt

( ?,\'\T" J/ 7', 1t'r\300/l'4 fiog

"1-log 300]I'4 los fi-3.40803.4680-0.12043.34763.3+76-lT-

2.39t I2.16.1 K-26.9.C

300 K i?t: I1.4Ir lYryt-t?r121t.t-t30q2y.t39s.9 Kl2i.9.C iJ

Tltcrrnodgnamiu giE

.

Erlmple 6 {. 4i, it. compreased adiabatically to ha$ itet'olume. Calculate the clmnge in ita lemperature. - @eth; lbAglLet the initial temperature be ?, K anC the 6nal tenperatureT,K,Initial volume : VtFinal volumc : V2

:vl2

During an adiabatic processTrYrt-r : Trl/rr-t

m ^f l', -17-rr2: r,Ly, JT, : Tr121t-t

_7 for air : l'40?, : Tr121r'ro-tT1 : Tr121o.to?t : l'319 ?t

Change in temperature* Tr-Tt: l.3lg rr_T,: 0.319 Tr K

n- Exampl"- 6.9. I g_ra1n molecule o! a monoatomic 1y : 5l3lperfect gas ot 2)"C ie adiabatically compieaaeil in a ,"r"rr;bi D"rx,costrom an initiol pressure o! 1 atmoaltierc to a final grcssr;c of d0otrnos7lherea,Calculatethe.resultingdifferenceintimp'eiotutc.

LDcthi (Eo*.11978)In a reversible adiabatic processPtz-r P't-r-Tr't : V;-or (+)'-':(+)'

Here, Pr: 50,&- l,Tt: 273*27

:300KTt:?Y- 5'3

(50;an_(#-)'"qIog (50) : -i- ft.S fl-log 3001

?r -

1,{3{ K: 11161r

Pr:,v_rl

-

DIt:qt_-t

-

,:Prr-,-TJ-

(+)'-'But

lltll,'

2T-ww

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234 Heat and ThermodYnamirsrXlanople 6'6. A quantity o! dry air at 27'C- is compresacd(il elowiuanld G;l auddenli to 113 r,7 its rolume. Finil the clnnge iniifrTature in each c,aae, a.*su.n1.ing 7 to be i'4 lor dry air.

-- .

-lAgra 1969 ; Delhi 71,7 i)(l) When the process is slovu'. the tenrperature of thejlslgm@ thCre is ng:1,"$-.:-.,temoerature.

L@;fifnermodynamia // z--V ufirlopee of Adiabatics and Isothermrls^ ,Y In an isothermal Process(v PV:const.

Differentiating, " vJt =-?A{PdV+VdP : O =)

or #: -+In an adiaba"" !17it .or,r,.

..(t(2) When the compressiou is sudden, tire process is aCiabatic..Here Y1 :V, nr:I

?1 :300K, ?r-?'t : l'4

^'Tr lrrlt'r

-

Trlrrlv-r12:,1;l;1,,?2 : 3oo I gll' '--- LV J

: 300 [3]t'r-t

^ =

i8i:i"tThe temperature of air increases by

192.5-27 : 165'5'C or f65'5 K

''Xi"*T|1)u, : o .=7v ? w =Px v{*t A{- dP ''1P

-dT : -,n-

T'herefore, the slope of an adiabatic is Yisothermal.

.. .(tr)times the slope of the

/Eranple 6'7. A aertain mass oJ gas cf NTP is -ea9tanded, lo

-threc times-ita t)olwme uniler adiabatio conditions. Calculote the"e;ffiiig femperoture onil preature. '( for the gas ia

t. 40.lDelhi (Eons.) i5!

Here, Y1o 7, -V, : 3l'Tr.: 273 K T2 * ?

?rrrt-r : IlYrt'r12: 11 [+]' 'rz: 27s[+]"-'Ts*176K:-97'C

Here, Y1 : Y, l't : 3VPr: latmosphere, Pr:'!

PrVrt -

P1V1l

P2: P1[+]',Pz: | (+I'P, : 0.21{8 atmosphere J

V*Fig. 6'6

Hence, tlre adiabatic curve is steeper- than the isothermal curve(Fig. 6$, utL poi"t where the two curves intersect each other' \ *.L7 Work Done During an Isotherrral P:eggss

When a gas is allowed to expand isothermally, work is donebv ir.k I t.o, the initial and final volumes be 71 and 73 respectively. Inri;.Y'6, ih. .t". of the shade{.s-1rip represents the work done for"'i"^fi'"to"g" in

volume d7. When the volume changes from 71to Yr,

Work dcne - [:t P . dY J arca aBba ...(r)^ )vrFis. 6'6 represents the indicator diagram' Considering one

gram m6lecule of the rii : A,oRTorv

(l)

(2)

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W-

V ----+Fig. 6'6

- RI log"

W = RTx 2.3026 log,,,

Also l)1v1 = Pllt'\'1 I'lor r\= h

.olume dV = P.dVby,

ptw = RTx2.3026* togro d \__--...(i\,) t

. Here, the change in the internal energy of the system is zero (becausethe temperature remains constant). So the heat transferred is equal to the work

...(id0. Here, heat transferred is zero because the systen 5 thernally

insulated fiom the surroundings. The decrease ir the internal*.rgy of the system (due to,fall in temperature) ir equal to the

> work done by thc system ^Dd

a$e !Qr8o.@ Irnrg. Relrtroa Bstween Adiabetlc end rrothcrrorl Etgtlctttcr

l. IeotherrnrlElesticitYDuring an irotherual Procers

PP : const

ThcdytonktDuring an adiabatic Proc:ts'

'o:: r': for ,__VT I

w : Kli:#-l-J. ! l- -l: r-716-F:rJ

Since / and B Iie on ttre same adiabaticP1Y1t: PrYra:K

w:1+l#^-#,=]I r PrVrY &717'1n : r=7-LVp=r--7rFi- J

: * [''n-'r']

Taking'f1 and ?ras the temPeratures u! q.-poincrepectively;nd considering one Sram molecule of the gas

P1Y1 : RI1and PlYs: BT7

Substituting these values in equation (di)w : #[nn-ar, ]

Diffeentiating,PitY+vitP

- o Q v ol'g = ? Avy.ilP D

-=aJF- : 'From the definition of elasticity of a gar?dp

Ei,- _ =trfV:#

Ecat ond llffiYrrr;mia

u\i"+

P

V.t

V,.. 1ii )

. (i,,)

...(d)

...(ii)AandB

lr2

T

done.nl) d.uwork Done During an Adiabatic Process

shown by the indicator diagram (Fig. 6.7) the work done for an increase tn

,--tFig. 6.7. c(

Work done when the gas expands fioln V1 to l/: rs uiven

l,a':tY=l PdV=AreaAlllta" I',

-..(i)

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238

From (d) and (di) lE6 n Pr/2. Adlebedc EleetlcttyDuring an adiabatic proces!

P77 : constDifferentiating, P'(Yt'rdV +Yt dP : O

YdP4V-1P

From the definition of elasticity of a gasEeitt: #_,:#

From (du) and (o),fr.4:7PComparing (iii) and (oi)

Er41 :7E1,sThus, the adiabatic elasticity of a gas is T times the

T oermodynamica 239rlre atmospheric pressure be Po. Tbe;ressure of air inside the verselis Pr.

The stopcock B is suddenly opened and closed just at themoment when the levels of the liquid on the two gid$ of the mano'tDeter are the sarne. Some quantiiy .of air escapes to the atm.osphere.The air insidc the vessel exfands adiabatically. The tenPerature ofair inside the vessel falls due to adiabatic expansion. The air inridethe vessel is alloli,ed to gain beat from the surroundingp and it finallyattains the temperaturelf the surr oundings. Let the pressure at thecnd be Pr and ihe diffe.ence in levels on the two sides of the mano'meter be [.

Theory. Consider a fixed mass of air left in the vessel in tlccnd. l'his mass of air has expanded from volume 71 (less than thevolume of the vessel) at preisure P, to volumeT3 at pressure P3.The process is adiabitic aishown by the curve /B (Fig. 6'9).

PrYrr -

PoYrt

. *i: (*)'

Finally thc poipt C is reached. The points A and C are at thcroom tempelaturi. Thcrefore AC can be 'considercd as an isother-rnal.

Eeol and Thcrmoilynamiae

...(di0

...(iu)

...(o)

...(ui)

isothermal---

...(0

...(r0elasticity. \.\_7-6'20 Clement aud Desorpes Method-Determinatioa of 1

, Clement and Desormes in 1Bl9 designed an experiment to find'f, the ratio between the two specifc heats of a gas.

P1Y1 : P2V2Y2 Plv;: -4

c(4,v2)

B (8,v2)

Fig.6.0

Substituting the value d + in equation (d),Y1

Taking Iogarithms,log P1-log .l3r : Y[ioq Pr.- log Pg]

.. iog /', *iot i-'o| * losT;1G"""

rig.0.8The vessel .d has a capacity of 20 to 30litres and is fitteC in

a iro:r containing cotton and wool. At the top end, three tubesare fitted as shown in Fig. 6'8. Through 8r, dry air is forced intothe vrssel d. Ttre stop cock B1 is closed when the pressure inside.4 i: ;li6htiy greater thau the atmospheric pressure. Let the,.{ifierence i* }evr;l arr the two sides of, the ruanometer be .& and

t:(*)'

'---:1 rj:--1 r-:--j.l E:-l L:

:::I r-----:I l:

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Ihcrmoilynamice 241E e,tt and ?hermodynont:t

But P, : Po1 H and Ps -

Po*lr. v_ _tggt&+_{l-_l9gj._-

' - Iog(Po1I/1-lcg (Po*t), /Po*E\

* -'tg \-7; /,_,o, (t#)

r.s (r. +)-;4;H-)

tApproximately, ,:&-#u

_P;

Hence y:=.8, ...(iu)- E-h

Similarly, 1 for any gas can be determined by this method.Ilrawbacks. When the stop-cock is opened, a series of oscilla-

tions are set up. This is shown by the up and down movement ofthe liquid in the manometer. Therefore, the exact momentwhen the stopcock should be closed is nor known. The pressure maynbt be equal to the atmospheric pressure when the stop-cock isclosed. It may be higher or iess than the atmospheric pressure.Thus the result obtained rvill not be accurate.6'2f Pertington'cMethod

Lummer, Pringsheim and Partington designed an apparatusto determine the value of 1. In this method, the pressure andtemperature are measured accurately beforc and after the adiabaticerParuron.

Fis.6.10Tne apparatus consisl of a vessel / hav{tg a capqcity between

130 and 150 litrcs. The valve .t can be opened and closed suCdenly.

It is controll..d. !y "

rpfuS.arrange'n-ent (Fig. 6.10). Dry air (or gas)at a pressurc higher than the arn cspheric plessure is loiced into'ihevessel ,{ and the srop-cock I is closed. - The oil manometer .ll{ isused to measure the pressure of air inside the vessel ;t. '.lhe bclo-T.Le_l .B (a platinum u,ire) and a sensitive galvanomerer are used inthe Wheatstone's bridgc arrangement

The vessel is surrounded by a constant temperature bath. Letthe -initial pressure -and temperature be P1 and'?1 (room tcmpera-ture). The bridge is kept slightly disturbed from the barancedposition.

,The valve .L is suddenly opened and closed. The wheat-stones bridge is at once adjusted for balanced position. Tire remrlera-ture of air inside ,{ has decreased due to adiibatic expansion oiair.Let the remperarure inside be ?o- and the atmospheric p.ess,.re Fo. rrthe ap-pararus is allowed to rcmain as such for some time. it will iai.nheat from the surroundings and the balance point gets aLtr.u"ll Inorder that the balance point

'emains undistuibed, iome piec.s oiiceare added into the watcr surrounding the vessel ,tl. whtn the icm-peraturc of water-bath is the same as that of air just

"rt.r rJi"b"ticexparuion, thc bridge will rcmain balanccd.. ^The-tempgTtu_re ?j of the bath represents the temperature ofair aftcr the adiabatic expansion

&, . Por-,T:-T;r1 p, \r-r_ ( T, \,\Tl : \7;/(r:lXlog P1-logPq) : y [og f1-log ?o]

v _ _ log P1-log Pot-.

As Pp P9, ",

:$- T, arc known, y can lre calculated. Thevalue of l for air at l7"C is found to Uc i.iO:+

4dqot$:s. - (l) Due to thc large volume of the vessel, theexpansion is adiabatic.-

r tl) .TA. ,,gTpT"turs are measured accurately just beforeano arter tne adlaDatrc expansion.Drewbechs. This method cannot be used to find the varue

of 1 at .$sh.. temperatur* because it is not possiblc t" a.tr-'-"the cooling correction accurately.

6'22 Ruclherdt's Experiodentr_ In 1929, Ruchhardt designgd an apparatus to find the value ofy. It is based on tbe principlJof mechan-ics. Air loiglj i, .""r"*ain a. big

-jar - (Fig. 6'i I ). h tube of uniform ".."

of ils. .""tio, isfitted and a ball of mass ry 6ts. in thS tube just like a pirt"n.-in tf,.equilibrium position, the baU h at the poin:t j. th.';;;;;.'p "fair inside the vessel, is given by

P : Poq !f-

AIR

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244 /q3/ krenersible Process

E eat tn iI Thernoilgn amict

The rhermodynamical state of a system can be defined rvirhthe help of the thermodynamical coordinates of rhe s)'srem. Theslate of a system can be changed by altering the thermodynamicalcoordinates. Changing from one state to the other by changing thethermodynamical coordinates is called a ptocess.

Consider two states of a system ie., state Aand state B.Change of state fiorn z{ to B or r:icc ocrsa is a process and the direc-tion of the process u'ill depend upon a new thermodynamical coor-dinatc called entropy. All processes arc not possible in the universe.

Consider the following processes :_(l) _Le1 two blocks .z{ and I at different tcmperatures ?, andTr(Tr;Tr\ be kept in contact but the system as a whole is insulat-

ed from the surroundings. Conduction of heat takes place berweenthe blocks, the temperaiure of I falls and rhe temperatur'e of Brises and thermodynamical equilibrium will be reached.

(2) Consider a flywheel rotati.ng with an angular velocitv -.Its initial kinetic energy is |1

246

must not dih-er appreciably from those of thestage of the_cycle of operation.

Jq)rAii p/ pro"or.s taking place in themust bi inEditely slor.'.

rLe-1c,11-\/ I o n- - z)(' (4 6 r4.,1-

l\'.-)rk done(.1\ namics

Heat atd Thermodunamica

But the second larv of thermo-v''hich heat can be converted

COI.IDUCTING;T-----'amAT Tt

ffiATT2

Fig. 8.12.A pcrfect non-conducting and frictionless piston is fitted into

the cylinder. The- working substance undergoes a complete cyclicoperation (Fig. 6'12).

A perfectly non.conducting stand is also provided so thatthe working substance can undergo adiabatic operation.

cf the working substance increases. lVork is done by -the working

substance. As the bottom is perfectly conducting to the source atternperature 7r, it absorbs heat. The process is completely isother-m;^1. The temperature remains constant. Let the amount of heat

76d the heat produced.ives the conditions undcr

Thetmoilynamics

absorbe.d by rhe rvorking substance be I/1 at theThe point J? is obtained.

Considcr one gram molecule of the workingWork done from A to B (isothermal process)

219

tcmperaturc ?,',.I

sub!tance.

having ""'

irrr,ur.a',o[lsubstance. The volume

/cgr-g!-B-er.er.ibl"&et..--.-' Heat cngines are used to convert heat into mechanical u'ork.Sadi Carnot (Frcnch) conceived a theoretical engine which is freefrom all the dclects of practical engines. Its e{Eciency is maximumand rt is an ideal heat engine.

For any e'rgine, there are three essential requisites :(l) Source. The source should bc at a fixed high temperature

?1 from u'Ii-rch tlie heat engine can draw heat. It has in6nite thermalcapacity and any amount of heat can be drawn from it at constanttemperature ?r.

(2) Sink. The sink should be at a fixed lower temperatureZ, to which any amount of heat can be rejected. It also has infinitetirermal capacity and its temPerature remains constant at fr.

(3) I{orking Substancc. A cylinder with non-conductingsides and conducting bottom contains the perfect gas as the wotkingsttbstanae.

CYLINDERWORKING

SUBSTANCE

I

I

P

(B , V3)

t,rEFG

.V.-..----.-.-}Fig.6.l3

increases. The process is completely adiabatic. Work is done bythe working substance at the cost of its ,:aternal energy. The tem-perat Te fa1ls. . Tle.working substance^undergoes- adiabatic changefrom I ro C. At C the temperature is

"1 (Fig. 6.13).

Work done from B to C (adiabatic process)

- \Yi': \i: P . dv 1 But PV't : constant :

'K

tvtd; i" Pzv'-Rra:ll

l.u_r_ Y' I Prv, -

RT,: 9"-'-RV't-r- 1

-- I Ptr"r:PzYzr-RE I+I{a|

-"t,RVr-Tr} _

RLTr-rzl: __T=r__ : *_ "/_l

ff1 : Area B0flG , ...(di)

aY,: \i',' dv : Rr, ',c,+: arca ABGD

(2) Place the engine on the standDecrease the pressure on the working

ffi C.a.leo.'T(rycle,Y\g-b tr{ Plaee the engine containing the workin} substance over

-2 D' fth" source at temperature Tr. .

The working sub.starice is also at atemperature, 1r. _]J. pressure is Pt and _volume is 71 as shown bythe point d in Fig. 6'13. Decrease the pressure. The volumeww

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250 Ecd and Thcrmodynamia(3) Place the engine on the sink at temperature ?;. Increare

the pressure. The s'ork is done on the working substance. Ar thebase is cgnducting to the sink, the proces is isothermal. A quantityof heat IIs is rejecled to the sink at temperature ?r. Finilly thtpoint D is reached.

Work dogc from C to D (isothermal process)\il: ff,' ,u,_ RTr,,s :+vt- -RTrbS+

Thermodynamia

The points I and C arc on the same adiabatic.

TrIrl-r -

rzyrY'rrr / 7, 1r-rT: \7;/

From (ui) and 1uid1(+ )": (#:)"Yr l',v;: Y"V, YsT:N;

nr,bslL-n?,log*l**I,,-,,Hr-H,Useful outDut W

, : ----------------',

: ---l;pui- T;

Heat is supplicd from the source from L to B only.E1: RTIdC +

W H,_8.rl E E;:-tr;-

RLr,-r,).-(+ ): "rlGG)-

,H212 : l-

-E;T"rl: !-T

261

...(uit)

r : area CEID(The

-ve sign indicates that work is done onsubstance.)

(4) Place the engine on the insulating stand. fncreace thepressure. The volume decreases. The process is completelyadiabatic. The temperature rises and finally the point d is reached.

Work done from D to u4 (adiabatic process).WI:

,f,, 'OU

._ _

R(Tt-T]l/'- Y-l n

.ufrr: ATIaDIEA ...(?).^ b:i"[]71and Waare equal and opposite and-cancel each othe..f 9HThe net work done by the working substance in one

"o*-pl.t.cycle ,: Area ABGEIAreaBCEO-A., O*!-OOrea

DIEA: Area ABCD

The net amount of heat absorbed by the working substance-

fl1-flgNet work : WL+Wy+W"*W,

- RrL br+*W-Rr,bs +,-^T={,,

w : nr, rcsft-ar,tor h ...(o)The points A atd D are on the same adiabatic

The Carnot's engine is perfectly reversiblc. It can be operatedin the revcrse dircction also. Then it works as a refrigcrator.The heat IIs is taken from the sink and external work is donJon theworking subitance and heat II1 is given to the sourcc at a highertemlrcrature.

The isothermal process will take place only when the pistonmcves very slowly to iive enough time for the heat transfer ti akcplacc. The aaiabatiC Prcccrs will take placc whcn the piston tnovet

...(rdd)

thc working

...(t i)

From equation (u)W:

Efficiency

w:W-

yrTl

-1, -

T u'-

: *-T-2 \-Z

tI

...(adrT)

TrYr:r'r : TrYr'r-r?, / 7t 1r'tE: \-%-/ww

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extremely fast to avoid heat transfer. Any practical engine cannotsatis[y t]rese conditions.

.

AII practical engines have an efficicncy less than the Carnot'sengrne.

6'27 Caroot's Engine and Refrigerator'Carnot's cycle is perfectly reversible, It can r,r'ork as a heat

engine and also as a refrigerator. When it works as a heat engine,it absorbs a quantit,v of heat .t/, from the source at a temperaturi 7r,does an amciunt of work lY and rejects an amount of heat .&, to thesink a_t temperature ?r. Wtren it rvorks as a refrigerator, it absorbsheat E2 from the sink at temperature Tr. W amount of work is doneon it by some external means and rejects'heat Hr to the source at atemperature 7r (FiS.6'14). Ir, the iecond case heat flows from abody at a lower temperature to a body at a higher temperature,with the help of external work done on the vvorking substance and itworks as a refrigerator. This will not be possible if the cycle is notcompletely reversible.

Coefficient of Perfotmance. The amount of heat absorbedat the lower temperature is 112. The amount of work done by theexternal process (input energy) : W and the amount of heat rejected: Hr Here f/, is the desired refrigerating effect.

o

26t Ecal and Thcrmoilgnanice {lrlrnodyramia 246

,-r{+: 1-# :0.25

0/6 efficiency :25%z' Exraple 6'11. A Carnot'a engine- !9 oyg'qted belween two

necntoirr oilemperattrea of 150 K anil J50 K. IJ the cngine reccilrtce1000 caloriee of heat troh the source in e-och cycle, c,alculate tha

omou,nt of fuat- rejecteil to the aink in eoch- cgcle. Calculate lheeficiency-ot the engine ond the work ilone bg thc engine in eoch cycle,(I calorie

-

4'2 joules).*450K; fr:350K: 1000 cal'i

-

E1 : 1Trll

,r* #,-tg# :272.77 Lrs.71'rr_ 3s0 100l-m-86

:0'22220/6 efrciency :22'22o/o

WorL done in each cYcle: Er-flt: t000-777.77-

222'23 cal:222'23x4'2 joules:9l?'rgjoutce

Errnplc 6.12. A Carnot'a enCill working aa- a refrigeratorbclwacn "ZN K and 800 K receioee ,500 calorics of luot lrotn theruscruoir dl thc bwer lemperatute. Calculate thc amount of heat rclicctcdAo tln rceatoir at the higher tempetalure, Oolculate aho thc amoualof worlc donc hr eaa,h cycle to operate thc rc7'rigeralq.

-- -

-lDelhi lflona.l 197aJE1: I E|: 5(X) calfr

-

300 E Ir: 260 KEr lrrT- n;;

&: Er. +z, 500x300ra,:

-

:576.92ca1' 260

lf : Er-Et: 76'92 cal76'92x4.2 joula

: 323.08jouluEra_mple-6.!3. A Carnol'a_retrigerator lahea heot lrom ualq al0'C anil iliacarda it to a room ot 22"C. 1 kg of woter at-A"C h to bc

clwngeil into ice ot.0'q. Eow mony caloriTe of heat ore diacorilcil tothc room I what is the uork-ilone by the retrigirator in thie proceaa IWhat is the coeficienl oJ perlormonie ol thi machine ? '

[Dclhi 19741Et:?IIr : l000x80 : 80,000 car?r:300K?t :273K

flr TLT:41E,

-

E#'Jl

80,000 x 300-

---2 -Er _ S7,9oo g$

Work done by the refrigerator:W:J1E\_ES

W : 4'2 (87,900-80,000)W

-

4.2x79A0F : 3.I8ilxl0.Joutc.

CoeGcient of perforu.ance,

90,000-- 97,900-80,000

80,000- 7900: 10.13

rrEr

E,E;Ez*

!:(l)

(2)

(3)Er: E;g-

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,tS ,/- Eeat onit ?hermoitynamica

'".)Unt"T_pIe 6I{. A carnot engine uhose lou temperalure rescr-coir ia ot 7"c t,,s on efi,ciency ,I 50%. It is desired ii ;iiir.rf, tn,:{-r!r:?v.to 7 0o/o; ns-how i""y.a"[]iu-_rmrW tii ir*i*i",i'

"tthe hqh temperatutc reaqroir be increased t (Oetht lgliyIn the first case

Thennodynamiu

Ii(Eciency of the engine d

-1:Er-8,--E;-

or

or

:? .- 50o/o : g'5, Tr : 273*2 : 2g0 K.It-?

"

: t-TJtl

0'5.: I-- 37r : 560 'K

In the se@nd case1'- 70!s: U7,Tt:280 K'rr':!,' : l-\ m,

0.? : r- '?g-. T,,'

T'' : 840 EIncreale in temperaturca Stl0-560

-

280 K

Fig.6.t6.

Efficicncy of the engine B

\ -

71': E"-E{ -\ flr,

Since I>rl';E{)ErAlso, 'W : Er-Ht: E{-E{

w:4

WEl

Consider two reversible engines d aud B, working between thetemperature limie ?, and ?r (Fig.6.15)- .d and A"are coupled-Suppose.r{-is more etcient than 8. The engine d workr ., . t..telglne 11rd^8 as a refrigerator. Thc engine? absorb,s an amouatoI hcat Irr ,rom the source at a teEperature [. It does exterua]work P and transfers it to 8. The heat r-jected to the sink is E. ata temperature &. The engine B absorbs Eeat Ea, from the sink' attemperature fi and. I[ amount of work is done on the workinf subs-tance. The heat given to the source at tempcraturc ?1 is .E1,."

Supposc tbe engine / is more efficient than B.

Thr.rs, for the two cngines A arrd lr vs tem, (4 +,i tr t L..q,.

"'ti t i;r r.i*fr#'i*t rfl },,i:oJ. :tcmperature ?2 and (Er,-Hrl i, tt. o"""i;

rytltl**i,"-.r,*.1$;ffi?i;[:":J".1_"#;i#:ffirll* lilthp', i::: i H:#,.x,# r*rH.Ifl{done on the svstem. This is "orrto.V'toihe secgnd law of thermo-dynamics. Thus, a

""r;"1 L;;;,'..*,iL n) The two ensines(reversibre) working between *,J*-.'t*o-tempe.ature timits iavethe same efficiencyf fl\{;r;;;;;;t" 1';.';"r;r h "; i, ; ;' i.,,'

"

d' r, 1", a ".

-,l - - i.i"ii;" ;""! i"I,tf TJ''*:lSi.;l'ff ETi: " efit? :$ ;:,* glr$ :,.e-.- i:# #:LT;the efficiency depends ."1'y ;p";ih; r-*I i._p"rarre ri,nits.

.

In a practical engine there is alwavstricticil;ft ,";ffi :i"La;",i"".r".:;;;.-*"':l.Jd:,."Hr_o"l;r;:I.wer thar. that of a Carnot's ."*,"r=f '";' 11G_) \

-n .*',u'r' Gt1''r\

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H eat aail Thermodynumicd

?h;rd Fourth

Thctmoilytlzmico

flt H,-T: Trl l)Er Tr:Er: r;

Er_E, Tr_T,-Z;: -T_

Here, flr -

L+ilL, E1 : L,T1 : T{iII, T1

-

lIEr-flr: LailL-L : ilLTr-?r-T+/IT_T:ilT.

dll itf_T-T

The area of thc figureABOD-fl1_82-ill

_ dP(Yr_Vrl.

dP (vt-vi drL:--T (VL-V)

2f3

For the same compression ratio, the efficiency of an Ottoengine_ is mor_e than a 'diesel engine. fn practice, th.

"ornpr.rrio.ratio for an otto engine is from i to g.rrh fo.. ai.r.r."sl"."it i.from l5 to 20. Due to-theJ-righer "o*p..*ror ratro, an actual dieseren,gine has higher efficiency lhan the' Otto (petrcrfl .nein.. -fhe

cyrrncter must bc strong enough to withstand very high p.6ss,r...6'37 Multicylin6.l Enginea -

With an engine,havin.g one cylinder, the engine works onlyd.uring the u'orking stroke. The piston movcs a,r.ing'ihe ,"rioi,t,.tlrree strokes due ro the momentum of the shaft. i""" -"iti""t-ina..engine (say 4-c_ylinder.engine) the four cyri*ders

".. "o"11.J1 --tir.

working of each cylinder is given below i-

-

Thc cycle AND reprcscnts a comptetc cycle and Carnot'stheorem can be

^ applied. Suppose the volume it the point z{ is p1

and temperarure h f 1d?. '[he pressure is just below iu raturationprcssure and- the liquid begins ro evaporate and at the point B thevolume_ is 72.

.The substance is in the uapour state. Suppose themass of the liquid at I is one gram. The amount of heat a-bsorbedis IIr, Here E1:L1d!, rvhere L+dL is the latent heat of thc liquidat temperaturc (T lit?1,

At the point B, the prcssure is decreased by dP. The vapourwill-expand and its tempcrature falls. The tcmperature at C it f .Ac this pressure and temperature f, the sas bceins to condense andis convcrted into the liquid stare. At thc-point D, the substance is inthe liquid srate. From-c to D, rhe amount of hlat reiected (civenout) is II1. Here II1 : .t where Z is the latcnt heat .i t".ociit,r."'/'. By in-crealtlS_t1r9 pressure a little, the original point z{. is iestored.The cycle ABCDA is co-mpletely reversible. -Applying thc principleof the Carnot's reversible -cycle

272

Eirst quartorSsooad quBrtor

Third quarterSourtb quartor

fdret

WorkilgErhausL

Charging :

Compreesion

8@nd

ErhaustCharging

Comprossion

YPorking

Yz

Compreasion

T9orking

ExheuaC

Charging

Charging

Coopreesioo

WorkingErheust

In this wa-y, -the poyqr of the engine increases and the shaftgets Eomentum during each qgarter cycl-e.

[,9#Clap_evroor,eteq-1H,9a!-F-.{gi-tio.ntT)-*""ider thei*ott..*"ii-iaen ., 'i.ilr.o,ure f+d? andauafl at temperature ?. }Ierc EA and.ED show the liquid state

P i,P@ ...(,

v'------;Fig.6.23

ol the substance- At I and D the substance is purely in the Iiqtridstate (Fig. 6'23). From I to I or:D to c trre subitance is in tra'nsi-IIT fr.T the liq.uid to-the gaseous state and o;cic ieiii.--ai'Cl"au the substance is purely in the gaseous state. From B to .F or Cto u the substance is in the gaseo,s state. Join I to D and B to cby dotted lines.

This is called thc Clapeyron's latent heat equation.. .

Applicetloar. $ eficct of clwnge o! gtreaeure on thc mcltingpoint.When a solid is convertcd into a liquid, therc is change in

volume.(i) If % is greatcr than 71dPfiV ls a positive quantity. It means that thc rate of change ofww

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274 E eal and Thermodynaniet

pressure with respect lo temperatrrre is positive." In such cases, rhemelting point of the substance rvill incr