Download PY2P10 Finn Problems Chap 1

2r9
Appendix D
APPENDIX
D
1.4 Differentthermometersdisagree,exceptat the fixed points by
defrnition.When usingthe moderndefinitionof temperatureusing
a singlefixed point at the triple point of water(273.16K)with
' r - ' r' T P X
'x-
Questions
I TEMPERATURE
1.1 The length of the mercury column in a mercury_in_glass
thermometeris 5 cm whenthe bulb is immersedin waterai its iripte
point. what is the temperatureon the mercury-in-glass
scalewhen
thelengthof thecolumnis 6.0cm?what will thelengthof thecolumn
be whenthe bulb is immersedin a liquid at 100degiees,
asmeasured
on the mercury-in-glass
scale,abovethe ice poini? If the length of
the column can be measuredto within onty O.Otcm, cai this
thermometerbe usedto distinguishbetweenthe ice point and the
triple point of water?You may take the temperatureoithe icepoint,
as measuredon the mercury-in-glass
scale,as 273J5degrees.
1.2 The resistanceof a wire is given by
R : Ro(l -f at + Btz)
where t is the temperaturein degreesCelsiusmeasuredon the
ideal gas scaleand so Ro is the resistanceat the ice point. The
constants
aandp are3.8x 10-3K- 1and - 3.0x 10-6K-2 respectively.calculatethe temperatureon the resistance
scaleat a temperature of 70'C on the ideal gasscale.
1.3 The table below lists the observedvaluesof the pressurep of
a gasin a constant-volume
gasthermometerat an unknowntemperatureand at thetriplepoint of waterasthemassof gasusedis reduced.
Pr"(torr) 100 200 300 400
P (torr)
t27.9 256.5 385.8 516
By considering
thelimit Prp-e @lprr) determineT to two decimal
places.What is this in "C? (l torr is a pressureof 1mm of Hg).
v
ATP
these dillerencescan be quite marked. To see how big these
can be,completethe tablebelowfor the temperaturesI
differences
of the boiling points of variousliquids (at an externalpressureof I
and the meltingpoint of one solid,usingthermometers
atmosphere)
with differentthermometricvariablesX. The valuesof X* for the
differentthermometersare given in the bottom row' Quote your
valuesfor the temperatureto the nearestdegreeonly.The temperature
givenfor Sn is for the meltingpoint.
Copper
nickel
thermocouple
TE
E (mV)
Platinum
resistance
thermometer
R (ohm) T"
Constant
volume H,
thermometer
P (a0
Tp
Constant
volume Ht
thermometer
Tp
P (at)
Liquidi
Solid
N2
02
HrO
Sn
- 0.10
o.0o
5.30
9'02
1.96 ?
2.50 ?
13.65
18.56 ;
1.82
2.13
9.30
12-70
0.29
0.33
t.37
1.85
At T.P.
2.98
?
?
?
,|
273 9.83
zt5
6.80
?
?
?
?
273 1.00
',!
?
?
?
273
Commenton the temperaturevaluesfor the two constantvolume
Which onegivesvaluescloserto theidealgasscale?
thermometers.
2 REVERSIBLE PROCESSESAND WORK
2.1 10 moles of an ideal gas are compressedisothermallyand
reversiblyfrom a pressureof 1 atmosphereto 10 atmospheresat
300K. How much work is done?
2.2 An ideal gas undergoesthe following reversiblecycle:(a) an
isobaricexpansionfrom the state(Pr, Vr)to the state(Pr,l/2) (b)
an isochoricreductionin pressureto the state(Pr, Vr) (c)an isobaric
reductionin volume to the state(Pt,Vt) (d) an isochoricincrease
220
Appendix D
in pressureback tothe original state(p1, Vr)
again.What work is
donein this cycle?If p, :_3 a1p, pz: ! itm, Vr]
t [tr. ina ir': Z
litres,how much work is done by t-hegasin traversing
the cycler00
times?
2.3 During a reversibleadiabatic expansion
of an ideal gas, the
pressureand volume at any moment
are related by pVr :i-rn1r"r"
c and y are constants.Show that the work
done by the gur'in
expandingfrom a state(pr, Vr) to a state(pr,
Vr) is
v'- Pz vz
14t:Pt
y-l
2.4 lce at 0"C and at a.pressure1atm, has
a densityof 916.23kg
m- 3,whilewaterundertheseconditions'has
a Oensityl99.A+tg;:..
How much work is done againstthe atmosphere
when iotg'.ii.,
melt into water?
2.5 A metal container,of volume v and
with diathermalwalls,
containsn molesof an idealgasat high pressure.
The gasrs alrowea
to leak,outslowryfrom the containeithrough
u ,rui uuru"io ii"
atmosphere
at a pressurepo. The processoccursisothermalry
at ihe
temperatureof the surroundings.Show that
the work d;;;by ;il"
gasagainstthe surroundingatmosphereis
W : Ps(nos- V)
where uo is the molar volume of the gas at
atmosphericpressure
and temperature.
2'6 show that the coerficientof rinear expansion
a is related to
the expansivityp as
f :3a
2'7 A hypotheticalsubstancehas an isothermal
compressibility
*:-a/l and an expansivityfl:2bTlu wherea
andb areconstants
and u is the molar volume.Show that the equation
of stateis
u-bT2 IaP=aconstant
2.8 A weldedrailway line, of length l5km, is
laid without expansionjoints in a desertwherethe nrght ana day
tempera,"*, &nl,
by 50K. The cross-sectional
area-of the rail is 3.6 x 10-3m2.
(a)what is the differencein the night and
day tensionin the rail if it is
kept at constantlength?(b) If theiail is freeio
expand,by how much
Appendix D
221
doesits lengthchangebetweennight and day?(a:8 x 10-6K-1'
Y:2x l011Nm-2.)
2.9 Find an expressionfor the work done when a wire of length
L is heatedreversiblyfrom a temperature Tt to a temperature T2
under conditions of constant tension9.
2.10 The equationof stateof a rubber band is
-F : "arl
- LLr o f\ !n)'l
r/ J
where ,L6 is the original length and a is a constant equal to
1.3x 10-2NK-r. How muchwork is performedwhenthe band is
stretchedisothermallyand reversiblyfrom its original length of 10cm
to 20cm, the temperaturebeing20"C?
is initially at
2.ll A block of metalat a pressureof one atmosphere
a temperatureof 20'C. It is heatedreversiblyto 32'C at constant
If the heatinghad beencarriedout
volume.Calculatethefinal pressure.
irreversibly, would this affect your answer?(The expansivity p:
5.0x 10-sK-r: the isothermalbulk modulus K: 1.5x 101r
Nm-'.)
3 THE FIRST LAW OF THERMODYNAMICS
3.1 Liquid is stirred at constant volume inside a containerwith
adiabaticwalls.The liquid and the containerare regardedas the
system.(a) Is heat beingtransferredto the system?(b) Is work being
(c)What is the signof theinternalenergychange
doneon thesystem?
of the system?
3.2 Water inside a rigid cylindrical insulated tank is set into
rotation and left to cometo rest under the action of viscousforces.
Regardthe tank and the water as the system.(a) Is any work done
by the systemas the water comesto rest?(b) Is thereany heatflow
to or from the system?(c) Is there any changein the internal energy
of the system?
3.3 A combustionexperimentis performedon a mixture of fuel
and oxygencontainedin a constant-volumecontainersurrounded
by a water bath. The temperatureof the water is observedto rise.
Regardthe matter insidethe containeras the system.(a) Has work
(b) Has heatbeentransferredbetweenthe
beendoneon the system?
222
AppendixD
systemand the surroundings?(c) what is the sign of the internal
energychangeof the system?
3.4 A gas is containedin a cylinder fitted with a frictionresspiston
and is taken from the statea to the stateb along the path acu
st o*n
in Fig. D.1. 80J of heatflow into the systemand the.yrt"r does
lol
of work. (a) How much heat flows into the systemalong the path
adb if the work done by the gassystemis r0Ji (b) when irr"
,vJtris returnedfrom b to a arongthe curvedpath, the work done
on the
systemis 20J. What is.the heat transfer?lc)If U : 0 and U
o
o: 40],
find the heat absorbedin the processesad and db.
3.5 An electrical resistancecoil, wired to the surroundings, is
placedinsidea cylinderfitted with a frictionlesspistonand
cont"]ning
an ideal gas'The walls of the cylinder and thepiston are adiabatic.
A current of 5 A is maintainedthrough the resiitance,acrosswhich
thereis a voltagedrop of 100v. The pistonis opposedby a constant
external force of 5000N. At what speedmust the piston
move
outwardsin order that there is no changein the tempeiature
of the
gas?Is theelectricalenergytransferredtothe gasasheator
work?
supposenow that the wansare diathermai and the resistancecoil
is wrappedround the outsideof the cylinder. Regardtt syrtea.
-coil.
the cylinder and the gas,excrudingthe heating
tr "ttt.
transfernow heat of work?
"n"rgy
3.6 Two moles of a monatomic ideal gas are at a temperature
of
309 The gas expands reversibly and isothermally to twice
its
5.
original volume. Calculatethe work done by the gas, the
heat
suppliedand the changein the internal enersv.
Appendix D
223
3.7 An ideal gas is containedin a cylinder fitted with a frictionless
piston at the pressureP and volume I/. It is heatedquasistatically
and at constantvolumeso that its temperatureis doubled,and then
it is cooled at constant pressureuntil it returns to its original
temperature.Show that the work done on the gasis PZ.
3.8 Find the change in the internal energy of one mole of a
monatomicidealgasin an isobaricexpansionat I atm from a volume
of 5m3 to a volumeof 10m3.y for a monatomicideal gasis 5/3.
3.9 The molar specificheat of many materialsat low temperatures
is found to obeythe Debyelaw c,: AlTl|f3 wherer4 is a constant
equalto 1.94x 103Jmol-rK-1 and with the Debyetemperature
0 taking dillerent valuesfor different materials.For diamond, it is
1860K. (a) Evaluatec, at 20K and 100K. (b) How much heat is
requiredto heat one mole of diamond between20K and 100K?
(c) What is the averagemolar specificheat in this range?
3.10 Show that the adiabaticcurve for an ideal gas is steeperby
a factor of y than the isothermthrough a point on the P-Iz indicator
diagram.
3.11 Showthat the following relationshold for a reversibleadiabatic
expansionof an ideal gas:
TVY-r: a constant
T
anotherconstant
pt - ur:
The fireball of a uranium fission bomb consistsof a sphereof gas
of radius 15m and temperature300000K shortly after detonation.
Assumingthat the expansionis adiabaticand that the fireballremains
spherical,estimatethe radius of the ball when the temperatureis
3000K.(Take7:1.4.)
3.12 An interstellarcloud, made up of an ideal gas,collapseswith
its radiusdecreasingas
^: ,0,.(#)"'*
FigureD.l
with r measuredin years.The time t is taken to be zero at zero
radiusso that t is alwaysnegative.The cloud collapsesisothermally
at 10K until its radius reaches1013m.It then becomesopaqueso
224
Appendix D
that, from then on, the collapsetakesplace adiabatically(y:
Sll)
and reversibly.How many yearsdoesit take for the t"-prrutur"'to
rise-by800K measuredfrom the time the cloud reachesa radius
of
10r3m?
3.13 A thick-walled insulating chamber contains n, moles
of
heliumgasat,a h-ighpressurep, and temperaturefr. ti is
allowed
to leak out slowly to the atmosphereat a pressurepo through
a
smallvalve.show that thefinal temperatureolthe n, -ote, ott
eiiuo,
left in the chamberis
- with nz:n,(fr)"'
rz:r,(fr):
(Hint: consider as your systemthe gas that is ultimatery
left in
the chamber.)
3.14 Calculate the work done by a van der Waals gas,
with
equation(3.14)as the equationof state,in expandingfrom-avolume
Vt to a volume Vr: (a) at constant pressurep (b) at constant
temperature T
3.15 A magneticsalt obeysthe Curie law
ItoM _€
BoT
where M is the magnetization,Bo is the applied induction in
the
abserceof the specimen,€is a constantund-pois the permeability
of free space. The salt is magnetized isothermaily from
;
magnetza
izationMrto ,,,2.
Mr.y Lww
ou !u<rJ
may .lsJurug
assumethat
tnat Ine
the magnenzatlon
magnetizati
is uniform over the volume of the salt. Show that thi work ol
of
magnetizationis
*:-;
Vu^T
( M ; _M i )
3.16 The infinitesimalwork done in charginga cell is dW: EdZ
so the rate of doing work is EdZldt: Ei where1 is the current
supplied.A batteryis chargedby applyinga currentof40A at l2y
for 30 minutes.During this chargingpro"i* the batteryloses200kJ
ofheat to the surroundings.By how much doesthe iniernalenergy
of the cell change,assumingthat there are no forms of work othir
than electricalwork?
Appendix D
225
3.17 A steamturbine takesin steamat the rate of 6000kg hour-l
and its power output is 800kW. Neglect any heat loss from the
turbine. Find the changein the specifrcenthalpy of the steamas it
passesthrough the turbine if (a) the entranceand exit are at the
and exit velocitiesare negligible
sameelevation and the entrance
-1
m
s
and the exit velocityis 200m s 1,
is
50
velocity
(b) theentrance
with the outlet pipe being2 m abovethe inlet.
4 THE SECOND LAW OF THERMODYNAMICS
4.1 Heat is suppliedto an engineat the rate of 106J minute-r
What is the effrciency
and theenginehasan output of 10horsepower.
per
minute?
heat
output
is
the
of the engineand what
4.2 A storage battery delivers a current into an external circuit
and performs electrical work. The battery remains at a constant
temperatureby absorbingheat from the surrounding atmosphere.
Heat then appearsto be completelyconvertedinto work. Is this a
violation of the secondlaw?
4.3 Show that two adiabaticscannot intersect.(Hint: Imagine that
they do; completea cycle with an isotherm and operate an engine
round this cycle.)
4.4 An inventor claims to have developedan engine which takes
in 11 x 10?Jat 400K, rejects5 x 107Jat 200K anddelivers16.67kW
hours of work. Would you adviseinvestingmoneyin this project?
4.5 Which gives the greater increasein the efliciencyof a Carnot
engine:increasingthe temperatureof the hot reservoiror lowering
the temperatureof the cold reservoirby the sameamount?
4.6 50kg of water at 0'C have to be frozeninto ice in a refrigerator.
The room temperatureis 20'C. What is the minimum work input
to the refrigerator to achieve this. (Latent heat of fusion of
t.)
water: 3.33x 10sJ kg4.7 It is proposedto heat a house using a heat pump operating
betweenthe houseand the outside.The houseis to be kept at 22"C,
the outsideis at - 10'C and the heat lossfrom the houseis 15kW.
What is the minimum power requiredto operatethe pump?
4.8 Show that the elliciencies of the three Carnot engines,
operatingbetweenthe three reservoirsas illustrated in Fig. D.2 are