Download PY2P10 Finn Problems Chap 4

226
Appendi.t D
Appantlix D
227
Figure D.2
Figure D.4
relatedas
tlt:t'ltltlz-4t42
4 ,9 A c a rn o t e n g i n e w orki ng on a satei l i tei n outer
spacehas to
deliver a fixed amount of power at the raLew. The
temperatureof
th e h e a t s o u rc ei s a l s o f i xed.at Tr. The l ow er temperature
reservoi r
at T, consistsof a largebody of area .4;its temperaiureis
maintained
at T2 becauseit radiates energy into spac. u, -u.h heat
as is
d e l i v e re dto i t b y th e engi ne.The rate of thi s radi ati on i s
oA rl
w h e reo i s a c o n s ta n t.The carnot engi nehas to be desi gnedso that,
fo r a g i v e n w a n d T r, A has a mi ni mum val ue.S how that ,4 has
a m i n i m u m v a l u e w h e n I, takes the val ue 3Tt
14.
4 .1 0 A h y p o th e ti c a l e ngi ne, w i th an rdeal gas as the w orki ng
s u b s ta n c eo, p e ra te srn t he cycl e show n i n Fi g. D .3. S how that the
of the engineis
cfficiency
4:l-
.
t(l_PrlP,\
" : I
|
V,lVt/
;'\l-
4ll A simplified representationof the diesel cycle, with just air
u the wor king subst ance.is as shown in Fig. D. 4. Show t hat t he
elliciencyof this engine is
:ll-ti
/1_1\
4:l-
r".: VtlVz, the expansionratio, and r,:
Ilt,.
q
t r n p r e s s l or na t i o .
:-T:1n.
v3f1,1, the
I f r " : 5 , r . : l 5 a n d ) , : 1 . 4 , e v a l u a t er l . N o t i c e
compressionraiio can be very much higher in a dieselengine
a petiolengine
because
diesels
do nor sufferfrompre-ignition,
lT.tl
I.ll"kilg, as ttrsfuel is sprayedin at rhe end of rhe compression
8te more effrcient
than petrol engin.s.
5 ENTROPY
Figure D.3
l .A bucket
cont aining 5kg of wat er at 2- . ic is put out sicr ea
t" t hat it coolstL o
v r
r rhe
rv
tr vem
r r r l / vper
r a t u l v at ur e
yvllij
of
L
rr t
L he
ll9
\our
J u t ) r uside
g
ar
d
t
J5 \c.
,.
what
lrtttc
Ll ^T
eD tt oPychangeof t he wat er ?( r '"f or
wat er : 4. 19 x 103J kg r . )
228
AppendixD
5.2 Calculatethe entropychangefor eachof the following:(a) l0g
of steamat 100"c and a pressureof one atmospherecondensing
into water at the sametemperatureand pressure.(The latent heat
of vaporizationof water is 2257Jg-t.) (b) 10g of water at 100"c
anda pressureof oneatmosphere
coolingto 0"C at the samepressure.
(The averagespecificheat of water between0'C and 100"C is
4.19J g- t.) (c) 10g of waterat 0'C and a pressure
of oneatmosphere
freezinginto ice at the samepressureand temperature.(The latent
heat of fusionof ice is 333J g- t.)
229
AppendixD
internalenergyof the sand?(c)What is the entropychangeassociated
with this LU at constant7? The sanddoesno externalwork as it denot its volume.)
formswhenit hits the pavement:only its shapechanges,
5.8 One mole of an ideal gas undergoesa free expansiontripling
its volume.What is theentropychangeof (a)thegas(b)theuniverse?
5.9 Two equal quantitiesof water, of massm and at temperatures
T, and Tr, areadiabaticallymixedtogether,the pressureremaining
constant.Show that the entropychangeof the universeis
53 The low temperaturemolar specifrcheat of diamond varies
with temperatureas:
c u :r . e 4 xr o 3 [ ; ] '
rmol-,K-1
where the Debye temperature0:1860K. What is the entropy
changeof I g of diamond whenit is heatedat constantvolumefrom
4K to 300K? (The atomicweightof carbonis 12.)
5.4 An electriccurrentof 10A flowsfor oneminutethrougha resistorof
20ohmswhichis kept at 10"c by beingimmersedin runningwater.
What is the entropy change of the resistor, the water and the
universe?
5.5 A thermally insulatedresistorof 20ohms has a current of 5 A
passedthrough it for ls. It is initially at 20"C.(a) What is rhe
temperaturerise?(b) what is the entropy changeof the resistorand
the universe?Mass of resistor is 5 g; cp for the resistor is
0.8 x 103Jkg-1K-1 (Hinfi In the actualprocess,
dissipative
work
is doneon the resistor.Imaginea reversibleprocesstaking it between
the sameequilibrium states.)
5.6 An ideal gas has a molar specificheat given by c,:A+BT
whereA and B areconstants.Show that the changein entropy per
mole in going from the state(V, Tr) to the state(V, Tr) is
As : Aln(TzlTr) + B( T, - Tr) + R ln(VrlVr)
5.7 A 50kg bag of sand at25"C falls 10m onto the pavementand
comesto an abrupt stop.What is the entropyincreaseof the sand?
Neglectany transferof heat betweenthe sandand the surroundings
and assumethat the thermal capacityof the sand is so large that
its temperatureis unchanged.(Hint: Ask yourself:(a) what is the
dissipativework done on the sand?(b) What is the changein the
Showthat
wherec" is the specificheatof waterat constantpressure.
AS > 0. (Hint: (a - b)t ) 0 for a and b real.)
5.10 Considertwo identicalbodiesof heat capacityC" and with
negligiblethermal expansioncoefficients.Show that when they are
placed in thermal contact in an adiabatic enclosuretheir hnal
temperature is ( Tt * Tr)12 where TL and T2 are their initial
temperatures.
Now consider these two bodies being brought to thermal
equilibriumby a Carnot engineoperatingbetweenthem.The size
of the bodiesdo not
of the cycleis small so that the temperatures
changeappreciablyduring one cycle; thus the bodies behaveas
reservoirsduring one cycle. Show that the final temperatureis
(7, Tr)tt'. (Hint: What is the entropychangeof the universefor this
secondprocess?)
membraneas one which allows
5.11 Let us definea semipermeable
the passage
of one type of molecule.At equilibriumthe gaspressures
exist.
on eithersideof sucha membraneareequal.Suchmembranes
Considera mixture of two ideal gasesA and B containedin the
left-handhalf of the box as shownin Fig. D.5(a).Thereis a vacuum
GasA
-€
(b)
Figure D.5
GasB
230
AppendixD
AppendixD
in the right-handhalf.The box is litted with a pair of coupledsliding
pistons;theleft-handoneis permeableto A only,whilethe right-hand
one is impermeableto both. The box is divided into two with a
partition permeableto B only. Now slidethe coupledpistonsslowly
to the right so that, eventually,the two gasesseparatereversibly.
They will finally eachoccupya volumeequalto the original volume
of the mixture.This is shownin Fig. D.5(c).Let this processoccur
isothermally.(a) By consideringthe pressuresdue to each gas on
eithersideof the membranes,
showthat the net forceon the coupled
pistonsis zero.(b)The heatflowinginto the systemin thisisothermal
reversibleprocessis Q : T(Sr- S,) where S, and S, are the initial
and final entropies.(c) By now applying the first law, show that
(b) Supposethat the gasesare identical.Clearly, on removing the
partition,therecan now be no entropychangeasthe physicalsystem
is unchanged,yet the result you have just proved in (a) gives
As_i*r,,g
# 0! This is known as the Gibbs paradox.Is the resultgiven
in (a) valid for identicalgasesand if not, why not? (Hint Consider
how Gibbs'theoremwasproved.)(c)Obtainthecorrectexpression
Si : Sr.
This result is known as Gibbs' theorem. It is saying that:
in a mixture of ideal gasesthe entropy is the sum of the entropies
that eachgas would haveif it alone occupiedthe whole volume, i.e.
Sn * " ( T , V ): S e(T, V ) + S s(T, V )
5.12 We have seen that for n moles of an ideal gas the entropy is
S : nCvln T* nR ln( Vln\+ So
velv"
v^+vB
naInt
TIT
GasA
[nn*nal
| - r ^ n h 4 - nnnAR
r^2
nB
for the entropy of mixing of identical gasesby applying equation
(5.11)to the threevolumesV^, Vsand Vo*Vr, all containingthe
samegas.(d) By using the fact that, for identicalgases,
v^+ vB: \ _ V B
ns* ns
lls
ns
show that the entropy of mixing given in (c) is indeedzero.(Gibbs'
paradox is discussednicely in the little book by Chambadal,see
Appendix F.)
(s.11)
nn moles of an ideal gas A of volume Vo and temperatureT are
separatedfrom n" molesof anotherideal gasB of volume v, at the
sametemperatureT (seeFig. D.6(a)).The partition is removedso
that the gasesmix isothermallyat the temperature?i the mixturethen
occupyingthe volume Vr* Vs (seeFig. D.6(b)).(a) Use Gibbs's
theorem,introducedin the previousquestion,to show that the
entropychangeoccurringin this mixing is
- ,^lvn:e}l
- "L"^'^'
As.i*ing:
^lrn n{v^:vtl -."t
t VA I- "'l Z, JJ
A s - i * i , :, (gn n * n " ) R h { y L l ! !
231
6 THE THERMODYNAMIC POTENTIALS AND
THE MAXWELL RELATIONS
6.1 It is a result of statisticalmechanicsthat the internal energyof
an ideal gasis
'''
u : u(s,v): q.Nk"r+)
",rr,rnr.,
\v/
where a is a constant and the other symbolshave their usual
meanings.Show that the equationof statePV : nRT follows from
this equation.
6.2 The Helmholtz function of one mole of a certaingasis:
f : Fln: - alu- RTln(u -b)+i(f')
T
MixtureA + B
Gasg
I
(a)
(b)
FigureD.6
wherea and b are constantsand 7 is a function of T only. Derive
an expression
for the pressureof the gas.
6.3 The table givesthe valuesof some thermodynamicproperties
of a substance
at two differentstates,both at the sametemperature.
232
Appendix D
AppendixD
Temp.usPv
"C
kJkg-lkJK-lkg I Nm-2
mtkg-t
Initial state 300
2727 6.364
4 x 106 0.0588
Final state 300
2816 8.538
0.05x 1065.29
^readenoreswork
t.rZ{
\\r
\
what is the maximum amount of work that can be extractedfrom
one kilogramof this substance
in taking it from the initial stateto
the final state?You will have to selectthe relevantdata from the
table.This substance,incidently,is superheatedsteam.
233
\To
V
6.4 The Gibbs function of one mole of a certaingasis given by
FigureD.7
O: RTINP * A + BP + CP2I2+ DP313
whereA, B, C and D are constants.Find the equationof stateof
the gas.
(Hints:(a)considertheargumentleadingto equation(6.28)(b)Hence
show W^^*< - AU + T.AS (c)Useequation(5.11).)
6.5 Derivethe followingequations'
6.8 one mole of an ideal gasexpandsat the constanttemperature
% of the surroundingsfrom a pressurep, to a pressurepr. The
atmosphericpressureis Po. (a) By consideringthe total work done
in a reversibleexpansionand subtractingoff the useless
work, show
that the maximum usefulwork done by the gasis
,,
(a)(I:F-r(+):-12(99
\0r/,
\ 0r t,
(b)cr:-r(#\
\ot-/v
(c)H:G-r(:;":- r,(m,
(d)cr - / :p _- r(
, \ o tS\
r,
l,
6.6 In the presenceof a catalyst,one mole of NO decomposes
into
nitrogen and oxygen.The initial and firnaltemperaturesare 25"C
and the processoccursat a pressureof one atmosphere.
The entropy
change is As : 76JK- 1mol- I and the enthalpy change is
L,H: -8.2 x lOsJmol-r. What is the changein the Gibbs free
energyand what is the heat producedin the decomposition?
6.7 A gascoolsfrom a temperatureT to the temperature% of the
surroundings.There is no change between the initial and hnal
volumes,A,V:0, but the volumemay uary duringthe processand
so the gas may perform work. This is indicatedin Fig. D.7. Show
that the maximumamount of work obtainablefrom the gasis
W^ ,: Cv(T - TJ * Cv Toln TolT
Rr, h(;)
",^.
utPz- ttP,)
(b) How is this work relatedto the changein the Gibbs function?
6.9 In a simple form of fuel cell, which is a devicefor producing
electricityfrom a chemicalreaction,hydrogengas is fed in at one
electrodeand oxygenat the other.water is producedaccordingto
the reaction
2H2+Oz-2H2O
The cell operatesat the pre"ssure
and temperature(298K) of the
atmosphere.
Assumingthat the cell operatesreversibly,
calculateits
EMF, giventhe followingmolar valuesfor S andH:
sh
(JK-rmol-1)
(Jmol-1)
02
H2
H, O
201
tzg
66.7
17.2x 103
8 . 1x 1 0 3
-269 x 103
(Hint: considerone mole of Hro beingproduced.The usefulwork
AppendixD
234
done by the cell is then 2FoE. For this isobaricisothermalprocess,
we may useequation(6.49).)
6.10 Showthat the equilibriumconditionslistedbelowfollow from
equation6.54for the availability:
System
Equilibriumcondition
Totally isolated
Thermally isolated,held at
constantP
Thermallyisolated,held at
constantZ
S a maximum
I/ a minimum
U a mimum
7 SOME GENERAL THERMODYNAMIC RELATIONS
7.1 Derivethe relation
(':t\ : -r(' t!\
\ a p) , -
\ar') ,
7.2 Showthat Cn for a vander Waalsgasis a functionof temperature
only.
Appendix D
235
7.6 Show that the differencebetween the isothermal and the
adiabaticcompressibilities
is
K T- * r : T V !
CP
7.7 For eachof the followingprocesses,
statewhetherthe process
is reversibleor irreversibleand statewhich of the quantitiesS, H,
U, F and G areunchanged:(a) An isothermalquasistaticexpansion
of an ideal gasin a cylinderfitted with a frictionlesspiston.(b) As
(a),but for a non-idealgas.(c) A quasistaticadiabaticexpansionof
a gasin a cylinderfitted with a frictionlesspiston.(d) An adiabatic
expansionof an ideal gas into a vacuum (a free expansion).(e) A
throttlingprocessof a gasthrougha porousplug (theJoule-Kelvin
effect).
7.8 we have seenthat a carnot cycletakesa particularlysimple
rectangular
form on an s-T plot (seeFig.5.11).An s-F/ plot is also
useful in engineering.Show that, for an ideal gas as the working
substance,
a Carnotcycleis againrectangularin this representation.
Hint: First derivethe generalthermodynamicrelations:
7.3 Derive the secondenergyequation
'9!\:-{
r(\ +P(Y\ }
\urt,
t \try, \oP/,)
7.4 Considern molesof a van der Waalsgas.Show that (0UIAV), n2afV2.Henceshow that the internalenergyis
u:
f
Lto
cvdT- an21v+
whereUe is a constant.
{Hint: ExpressU :U(T,V)}.
7.5 As in the previousquestion,considern molesof a van der Waals
gas.Showthat (a)
s- f zdr+nRln(v-nb)+so
JoT
whereSe is a constant.{Hint: Use dS : llT(dU + PdV\}. (b) The
equationfor a reversibleadiabaticprocessis
fV-nb\n1c":a constant
if Cv is assumedto be independentof T.
(*-),:"{T),
7.9 Derive the so-called TdS equations:
rds: cvdr+ r(9f),av
rds: cPdr- ,({)
,a,
rds:,,(Tr) * r,(T)
no,
,0,
7.10 A block of metal of volume Z is subjectedto an isothermal
reversibleincreasein pressurefrom P, to P, at the temperatureT.
(a)Showthat the heatgivenout by the metalis TVp(Pz- Pr).(b)
Show that the work done on the metal is V(P|- PI|2K.(c) By
usingthe first law, calculatethe changein U. (d) Obtain the same
236
AppendixD
resultas in (c) by writing U : U(T,P) so
du:rqq\ dr + (ry\ dP
\0r/,
7.ll A block of metal is subjectedto an adiabaticand reversible
increaseof pressurefrom Pr to Pr. Show that the initial and final
temperaturesT, and T2 are relatedas
l nT r l T ,: Y e r - P ,)
LP
You may assumethat the volumeof the block staysapproximately
constantduring the compression.
7.12 Assumingthat helium obeysthe van der Waals equationof
state,determinethe changein temperaturewhen one kilomole of
heliumgas,initiallyat 2}"Cand with a volumeof 0.12m3,undergoes
a freeexpansionto a final pressureof one atmosphere.You should
frrst show that
\*) ,:
a (nY
-e\r)
a : 3 . 4 4 x l 0 3 J m 3 k m o l - 2 ;b : 0 . 0 2 3 4 m 3 k m o l - r ; c v l R - 1 . 5 0 6 .
(Hint: You may approximate.First show that Pt >>P2.Then you
may take Vr>>Vr.)
is zerofor an idealgas.
7.13 Showthat theJoule-Kelvincoeflicient
7.14 One kilomole of an ideal gas undergoesa throttling process
of the gasis
from Pr:4atm to Pr: I atm.The initial temperature
(b)How muchwork would
change?
50'C.(a)Whatis the temperature
haveto be done on the gasto take it reversiblybetweenthe initial
and final states?(c) What is the entropychangeof the gas?(Hint:
You shouldcalculatethe entropy changein two ways:(i) Imagine
a reversibleprocessin which the gasis taken isothermallybetween
the initial and hnal states.(ii) Apply the generalapproachof writing
S : S(7, H) and then imagininga reversibleprocessin which the
pressureis changedat constantH. You will need to show that
(AVA\H: - VIT in general,from dI{: TdS + VdP, and this is
-nRlP for an idealgas.)
237
7.15 Show that the temperature and volume of the points (Tin, Vin)
on the inversion curve for a van der Waals gas undergoing a
Joule-Kelvin expansion are related as
\0P/,
You may assumethat p, K and V areapproximatelyconstantduring
the compression.
(aA
Appendix D
4n : 2a( vr n- nQ 2( nbPr ) - '
Assuming that, at the maximum inversion temperature, Vrn>>nb,
show that Ti|"' x2alRb.
7.16 Equation 7.42 gives the Joule-Kelvin coeffrcient
/ar\ ttr(u!\ _v1
\a,) ,: c"L \ar/" I
so that the coolingin a throttling processwith a pressurechange
from P, to P, is
lP, / Ar\
L,r: I l::lap
tp,\0P/,
In a similar way, show that the cooling in an adiabatic reversible
expansion from a pressureP, to P, is
Lr:l;'(#),'"
where
/ap\
r /aA
\-/,: ,,\n),
Hence show that, for a given pressurechange,the adiabatic
expansionproducesmore coolingthan a throttling process.
(Hint: considerthe difference
in the integrands(ATIAP)S-(ATIAP)H
and showthat this is positive.)
8 MAGNETIC SYSTEMS,RADIATION, RUBBER BANDS
AND ELECTROLYTIC CELLS
8.1 A paramagneticsalt is magnetizedisothermallyand reversibly
from zeroappliedinductionto a hnal inductionlield of Bo. It obeys
is
the Curie law X^:GlT. Showthat the heat of magnetization
83
Q: -GV
Tpo 2
where Iz is the volumeof the salt.Is this heatabsorbedor rejected?
(Hint: ExpressS : S(T,Bd.)
-)