01 Thermal Expansion Theory1

8/20/2019 01 Thermal Expansion Theory1

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56 Thermometry, Thermal Expansion andCaloriemetr

genius by Pradeep Kshetrapal

12.1 Heat.

The energy associated with confguration and random motion o the atoms and molecules

with in a body is called internal energy and the part o this internal energy which is transerred

rom one body to the other due to temperature dierence is called heat.

!" #s it is a type o energy, it is a scalar.

$" %imension & '( $$ −T ML .

)" *nits & Joule +.." and calorie Practical unit"

-ne calorie is defned as the amount o heat energy reuired to raise the temperature o one

gm o water through !/C more specifcally rom !0.1oC to !1.1/C".

0" #s heat is a orm o energy it can be transormed into others and vice-versa.

e.g. Thermocouple con2erts heat energy into electrical energy, resistor con2erts electrical

energy into heat energy. 3riction con2erts mechanical energy into heat energy. 4eat engine con2erts

heat energy into mechanical energy.4ere it is important that whole o mechanical energy i.e. wor5 can be con2erted into heat

but whole o heat can ne2er be con2erted into wor5.

1" 6hen mechanical energy wor5" is con2erted into heat, the ratio o wor5 done W " to

heat produced Q" always remains the same and constant, represented by J.

JQ

W = or W 7 JQ

J is called mechanical eui2alent o heat and has 2alue 0.$ J8cal. J is not a physical uantity

but a con2ersion actor which merely express the eui2alence between Joule and calories.

! calorie 7 0.!9: Joule ; 0.!$ Joule

:" 6or5 is the transer o mechanical energy irrespecti2e o temperature dierence,

whereas heat is the transer o thermal energy because o temperature dierence only.


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9" The heat lost or gained by a system depends not only on the initial and fnal states, but

also on the path ta5en up by the process i.e. heat is a path dependent and is ta5en to be positi2e

i the system absorbs it and negati2e i releases it.

12.2 Temperature.

Temperature is defned as the degree o hotness or coldness o a body. The natural >ow o

heat is rom higher temperature to lower temperature.

Two bodies are said to be in thermal euilibrium with each other, when no heat >ows rom

one body to the other. That is when both the bodies are at the same temperature.

!" Temperature is one o the se2en undamental uantities with dimension (θ '.

$" t is a scalar physical uantity with +.. unit 5el2in.

)" 6hen heat is gi2en to a body and its state does not change, the temperature o the body

rises and i heat is ta5en rom a body its temperature alls i.e. temperature can be regarded as

the eect o cause ?heat@.

0" #ccording to 5inetic theory o gases, temperature macroscopic physical uantity" is a

measure o a2erage translational 5inetic energy o a molecule microscopic physical uantity".

Temperature ∝ 5inetic energy

= RT E

$

) #s

1" #lthough the temperature o a body can to be raised without limit, it cannot be lowered

without limit and theoretically limiting low temperature is ta5en to be Aero o the 5el2in scale.

:" 4ighest possible temperature achie2ed in laboratory is about !B9K while lowest possible

temperature attained is !B9 K .


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Thermometry, Thermal Expansion and

Caloriemetr 58

genius

To construct a scale o temperature, two fxed points are ta5en. 3irst fxed point is the

reeAing point o water, it is called lower fxed point. The second fxed point is the boiling point o

water, it is called upper fxed point.

Name of the

scale

Symbol for

each degree

Loer !"ed

po#$t %L&'(

)pper !"ed

po#$t %)&'(

Number of d#*#s#o$s

o$ the scale

Celsius /C B/C !BB/C !BB

3ahrenheit /F )$/F $!$/F !9B

Geaumer /R B/R 9B/R 9B

Gan5ine /Ra 0:B Ra :


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5, Thermometry, Thermal Expansion andCaloriemetr


n modern thermometry instead o two fxed points only one reerence point is chosen triple

point o water $


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Caloriemetr 6

genius

d" These are more sensiti2e and accurate than liuid thermometers as expansion o gases is

more than that o liuids.

)" es#sta$ce thermometers Gesistance o metals 2aries with temperature according to

relation.

"!B cT RR α += where α is the temperature coeNcient o resistance.

*sually platinum is used in resistance thermometers due to high melting point and large

2alue o α .

i" 3ormula & centigraRR

RRT c °×−

−= !BB

B!BB

B or kelvi

R

RT

Tr

K !:.$


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b" +lope o line $% is 18

c" +lope o line $% is !8

d" +lope o line $% is )8

'olution & b" Gelation between Celsius and 3ahrenheit scale o temperature isE

)$

1

−= F C

Dy rearranging we get, C 7E

!:B

E

1−F

Dy euating abo2e euation with standard euation o line cmx += we getE

1=m and

E

!:B−=c

i.e. +lope o the line $% isE

1.

P roblem 2. The reeAing point on a thermometer is mar5ed as $B/ and the boiling point at as !1B/. #temperature o :B/C on this thermometer will be read as

a" 0B/ b" :1/ c" 9/ d" !!B/

'olution & c" Temperature on any scale can be con2erted into other scale byLFP(FP

LFP )

−−

7 Constant or

all scales

∴ o

C )

B!BB

B

$B!1B

$B

−°°−

=°−°

°− ⇒ ) 7 °+

°°×

$B!BB

!)BC 7 °=°+

°°×°

E9$B!BB

!)B:B

P roblem 3. # thermometer is graduated in mm. t registers )mm when the bulb o thermometer is inpure melting ice and $$mm when the thermometer is in steam at a pressure o one atm. Thetemperature in /C when the thermometer registers !)mm is

a" !BB$1

!)× b" !BB

$1

!:× c" !BB

$$

!)× d" !BB

$$

!:×

'olution & b" 3or a constant 2olume gas thermometer temperature in /centigra"e is gi2en as

CPP

PPT c °×−

−= !BB

B!BB

B ⇒ !BB

$1

!:!BB

")$$

")!)×=°×

−−−−

= CT c

P roblem +. The temperature coeNcient o resistance o a wire is B.BB!$1 *er +C. #t )BBK its resistanceis !Ω. The resistance o wire will be $Ω at

a" !!10K b" !!BBK c" !0BBK d" !!$


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Caloriemetr 62

genius

!" Thermal expansion is minimum in case o solids but maximum in case o gases becauseintermolecular orce is maximum in solids but minimum in gases.

$" +olids can expand in one dimension linear expansion", two dimension superfcialexpansion" and three dimension 2olume expansion" while liuids and gases usually suers

change in 2olume only.

)" The coeNcient o linear expansion o the material o a solid is defned as the increase inits length per unit length per unit rise in its temperature.

T L

L

∆×

∆=

!α

+imilarly the coeNcient o superfcial expansionT $

$

∆×

∆=

!β

and coeNcient o 2olume expansionT #

#

∆×

∆=

!γ

The 2alue o α , β and γ depends upon the nature o material. #ll ha2e dimension '( !−

θ andunit per /C.

0" #sT L

L

∆×

∆=

!α ,

T $

$

∆×

∆=

!β and

T #

#

∆×

∆=

!γ

∴ T LL ∆=∆ α , T $ $ ∆=∆ β and T # # ∆=∆ γ

3inal length "! T LLLL ∆+=∆+=′ α ..i"

3inal area "! T $ $ $ $ ∆+=∆+=′ β ..ii"

3inal 2olume "! T # # # # ∆+=∆+=′ γ ..iii"

1" L is the side o suare plate and it is heated by temperature ∆T , then its side becomesL.

The initial surace area $L $ = and fnal surace $L $ ′=′

∴ "$!"!"! $

$$

T T L

T L

L

L

$

$∆+=∆+=

∆+=

′=′

α α α

(*sing Dinomial

theorem'

or "$! T $ $ ∆+=′ α

Comparing with euation ii" we get β 7 $α

+imilarly or 2olumetric expansion ")!"!"! ))) T T

L

T L

L

L

#

# ∆+=∆+=

∆+=

′=′ α α α (*sing

Dinomial theorem'

or "! T # # ∆+=′ γ

Comparing with euation iii", we get α γ )=

+o )&$&!&& =γ β α

i" 4ence or the same rise in temperature

Percentage change in area 7 $ × percentage change in length.

Percentage change in 2olume 7 ) × percentage change in length.

ii" The three coeNcients o expansion are not constant or a gi2en solid. Their 2aluesdepends on the temperature range in which they are measured.


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iii" The 2alues o α , β , γ are independent o the units o length, area and 2olume respecti2ely.

i2" 3or anisotropic solids . x α α α γ ++= where α x , α , and α represent the mean coeNcients

o linear expansion along three mutually perpendicular directions.

ater#al K 1 or %C(19 K 1 or %C(19

+teel !.$ × !B1 ).: × !B1

Copper !.< × !B1 1.! × !B1

Drass $.B × !B1 :.B × !B1

#luminium $.0 × !B1


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Caloriemetr 6+

genius

Deore expansion & n triangle $0C

$!$

$$

$"

−= L

L0C .....i"

#ter expansion &

$

!

!$

$$

$

"!$"'!("

+−+= t

L

t L0C α α .....ii"

Euating i" and ii" we get

$

!!$

$$

$

!$$ "!

$"'!(

$

+−+=

− t L

t LL

L α α

⇒ t LL

t LLL

L ××−−××+=− !$!

$!

$$$

$$

$!$

$ $00

$0

α α (Feglecting higher terms'

⇒ "$"$0

$$$!

$! t Lt L

α α = ⇒ !

$

$

! $α

α =L

L

P roblem 7. # iron rod o length 1B cm is Ooined at an end to an aluminium rod o length !BB cm. #ll

measurements reer to $B/C. The coeNcients o linear expansion o iron and aluminium areC°× − 8!B!$ : and C°× − 8!B$0 : respecti2ely. The a2erage coeNcient o composite system

is

a" C°× − 8!B): : b" C°× − 8!B!$ : c" C°× − 8!B$B : d" C°× − 8!B09 :

'olution & c" nitially at $BoC" length o composite system L 7 1B J !BB 7 !1B cm

Hength o iron rod at !BB/C cmB09.1B"'$B!BB!B!$!(1B : =−××+= −

Hength o aluminum rod at !BB/C cm!E$.!BB"'$B!BB!B$0!(!BB : =−××+= −

3inally at !BBoC" length o composite system LQ 7 cm$0.!1B!E$.!BBB09.1B =+

Change in length o the composite system ∆L 7 LQ L 7 !1B.$0 !1B 7 B.$0 cm

∴ #2erage coeNcient o expansion at !BBoC T L

L∆×

∆=α 7"$B!BB!1B

$0.B−×

7

C°× − 8!B$B :

P roblem 8. # brass rod and lead rod each 9B cm long at B/C are clamped together at one end with theirree ends coinciding. The separation o ree ends o the rods i the system is placed in a

steam bath is "8!B$9and8!B!9 :: CC lea" !rass °×=°×= −− α α

a" B.$ mm b" B.9 mm c" !.0 mm d" !.: mm

'olution & b" The Drass rod and the lead rod will suer expansion when placed in steam bath.

∴ Hength o brass rod at !BB/C "!Q T LL !rass!rass!rass ∆+= α 7 '!BB!B!9!(9B: ××+ −

and the length o lead rod at !BB/C "!Q T LL lea" lea" lea" ∆+= α 7 '!BB!B$9!(9B : ××+ −

+eparation o ree ends o the rods ater heating 7 QQ !raslea" LL − 70!B'!9$9(9B −×−

mmcm 9.B!B9 $ =×= −

P roblem ,. The coeNcient o apparent expansion o a liuid in a copper 2essel is C and in a sil2er 2essel'. The coeNcient o 2olume expansion o copper is Cγ . 6hat is the coeNcient o linear

expansion o sil2er

a" )8" 'C C ++ γ b" )8" 'C C +− γ c" )8" 'C C −+ γ d" )8" 'C C −− γ

'olution & c" #pparent coeNcient o 2olume expansion or liuid sLa** γ γ γ −= ∴ sa**L γ γ γ +=

where γ s is coeNcient o 2olume expansion or solid2essel.

$ %

C

L$

L$

L!8$ L

!8$0


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6hen liuid is placed in copper 2essel then γ L 7 C J γ co**er ....i" (#s γ app. or liuid in copper

2essel 7 C'

6hen liuid is placed in sil2er 2essel then γ L 7 ' J γ silver ....ii" (#s γ app. or liuid in sil2er

2essel 7 ''

3rom euation i" and ii" we get C J γ co**er 7 ' J γ silver

∴ 'C co**er silver −+= γ γ

CoeNcient o 2olume expansion 7 ) R CoeNcient o linear expansion

⇒ ))

coppersil2ersil2er

'C −+== γ γ

α

P roblem 1. # uniorm solid brass sphere is rotating with angular speed Bω about a diameter. itstemperature is now increased by !BB/C. 6hat will be its new angular speed. =i2en

C% °×= − per!BB.$ 1α "

a" B!.! ω b" BB!.! ω c" BEE:.B ω d" B9$0.B ω

'olution & c" %ue to increase in temperature, radius o the sphere changes.

Het RB and R!BB are radius o sphere at BoC and !BBoC '!BB!(B!BB ×+= α RR

+uaring both the sides and neglecting higher terms '!BB$!($B$!BB ×+= α RR

Dy the law o conser2ation o angular momentum $$!! ω ω 11 =

⇒ $$!BB!

$B

1

$

1

$ω ω MRMR = ⇒ $

1$B!

$B '!BB!B$$!( ω ω ×××+=

−RR

⇒ BB

)

!

$ EE:.BBB0.!'!B0!( ω

ω ω

ω ==×+= −

12.8 4"pa$s#o$ of L#-u#d.

Hiuids also expand on heating Oust li5e solids. +ince liuids ha2e no shape o their own, they

suer only 2olume expansion. the liuid o 2olume # is heated and its temperature is raised by

∆θ then

"!Q θ γ ∆+= LL # # (γ L 7 coeNcient o real expansion or coeNcient o 2olumeexpansion o liuid'

#s liuid is always ta5en in a 2essel or heating so i a liuid is heated, the 2essel also gets heated

and it also expands."!Q θ γ ∆+= '' # # (γ ' 7 coeNcient o 2olume expansion or solid 2essel'

+o the change in 2olume o liuid relati2e to 2essel.

θ γ γ ∆−=− '(QQ 'L'L # # #

θ γ ∆=∆ a**a** # # ( =−= 'La** γ γ γ #pparent coeNcient o 2olume expansion or

liuid'

'L γ γ > B>a**γ =∆ a**#

positi2eHe2el o liuid in 2essel will rise on heating.

'L γ γ < B


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Caloriemetr 66

genius

'L γ γ = B=a**γ B=∆ a**# le2el o liuid in 2essel will remain same.

12., 4


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=i2en # 7 !BBB cc, α g 7 B.!R!B08/C ∴ CCgg °×=°××==

−− 8!B).B8!B!.B)) 00α γ

∴ ∆I 7 !BBB (!.9$ R !B0 B.) R !B0' R !BB 7 !1.$ cc

P roblem 12. Hiuid is flled in a >as5 up to a certain point. 6hen the >as5 is heated, the le2el o theliuid

a" mmediately starts increasing b" nitially alls and then rises

c" Gises abruptly d" 3alls abruptly

'olution & b" +ince both the liuid and the >as5 undergoes 2olume expansion and the >as5 expands frstthereore the le2el o the liuid initially alls and then rises.

P roblem 13. The absolute coeNcient o expansion o a liuid is < times that the 2olume coeNcient o expansion o the 2essel. Then the ratio o absolute and apparent expansion o the liuid is

a"<

!b"

:

<c"

<

:d" Fone o these

'olution & b" #pparent coeNcient o Iolume expansion γ a**. 7 γ L γ s 7 < γ s γ s 7 :γ s gi2en γ L 7 < γ s "

Gatio o absolute and apparent expansion o liuid:

<

:

<

.

==s

s

a**

L

γ

γ

γ

γ .

P roblem 1+. n cold countries, water pipes sometimes burst, because

a" Pipe contracts b" 6ater expands on reeAing

c" 6hen water reeAes, pressure increases d" 6hen water reeAes, it ta5es heat rompipes

'olution & b" n anomalous expansion, water contracts on heating and expands on cooling in the rangeB/C to 0/C. Thereore water pipes sometimes burst, in cold countries.

P roblem 15. # solid whose 2olume does not change with temperature >oats in a liuid. 3or twodierent temperatures !t and $t o the liuid, ractions !3 and $3 o the 2olume o thesolid remain submerged in the liuid. The coeNcient o 2olume expansion o the liuid iseual to

a"$!!$

$!

t 3 t 3

3 3

−−

b"$$!!

$!

t 3 t 3

3 3

−−

c"$!!$

$!

t 3 t 3

3 3

++

d"$$!!

$!

t 3 t 3

3 3

++

'olution & a" #s with the rise in temperature, the liuid undergoes 2olume expansion thereore the

raction o solid submerged in liuid increases.

3raction o solid submerged at !! 3 Ct =° 7 Iolume o displaced liuid "! !B t # γ += .....i"

and raction o solid submerged at $$ 3 Ct

=° 7 Iolume o displaced liuid"!

$B t # γ +=.....ii"

3rom i" and ii"$

!

$

!

!

!

t

t

3

3

γ

γ

++

= ⇒ $!!$

$!

t 3 t 3

3 3

−−

=γ

12.11 4"pa$s#o$ of /ases.

=ases ha2e no defnite shape, thereore gases ha2e only 2olume expansion. +ince theexpansion o container is negligible in comparison to the gases, thereore gases ha2e only realexpansion.

0oe?c#e$t of *olume e"pa$s#o$ #t constant pressure, the unit 2olume o a gi2en mass

o a gas, increases with !/C rise o temperature, is called coeNcient o 2olume expansion.

T #

#

∆×

∆=

!α ∴ 3inal 2olume "! T # # ∆+=′ α


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Caloriemetr 68

genius

0oe?c#e$t of pressure e"pa$s#o$ T P

P

∆×

∆=

!β 3inal pressure "! T PP ∆+=′ β

3or an ideal gas, coeNcient o 2olume expansion is eual to the coeNcient o pressureexpansion.

i.e.!

$


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2" +ince coeNcient o linear expansion α " is 2ery small or in2ar, hence pendulums are

made o in2ar to show the correct time in all seasons.

)" Thermal stress #$ a r#g#dly !"ed rod 6hen a rod whose ends are rigidly fxed such

as to pre2ent expansion or contraction, undergoes a change in temperature, due to thermal

expansion or contraction, a compressi2e or tensile stress is de2eloped in it. %ue to this thermal

stress the rod will exert a large orce on the supports. the change in temperature o a rod o

length L is ∆θ then

Thermal strain θ α ∆=∆

=L

L

∆×

∆=

θ α

! #s

L

L

+o Thermal stress θ α ∆= 4

=

strain

stress #s4

or 3orce on the supports θ α ∆= 4$F

0" 4rror #$ scale read#$g due to e"pa$s#o$ or co$tract#o$ a scale gi2es correct

reading at temperature θ , at temperature " θ θ >′ due to linear expansion o scale, the scale willexpand and scale reading will be lesser than

true 2alue so that,

True 2alue 7 +cale reading "'!( θ θ α −′+

i.e. '!( TI θ α ∆+= 'R with

" θ θ θ −′=∆

4owe2er, i θ θ


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Caloriemetr 7

genius

i" 6hen rails are laid down on the ground, space is let between the ends o two rails.

ii" The transmission cable are not tightly fxed to the poles.

iii" Pendulum o wall cloc5 and balance wheel o wrist watch are made o in2ar an alloy

which ha2e 2ery low 2alue o coeNcient o expansion".i2" Test tubes, bea5ers and crucibles are made o pyrexLglass or silica because they ha2e

2ery low 2alue o coeNcient o linear expansion.

2" The iron rim to be put on a cart wheel is always o slightly smaller diameter than that o

wheel.

2i" # glass stopper Oammed in the nec5 o a glass bottle can be ta5en out by warming the

nec5 o the bottle.

Sample problems based on Application of thermal expansion

P roblem 16. # bimetallic strip is ormed out o two identical strips, one o copper and other o brass.

The coeNcients o linear expansion o the two metals are Cα and %α . -n heating, thetemperature o the strip goes up by ∆T and the strip bends to orm an arc o radius o

cur2ature R. Then R is BBTAC44 %Scree$#$g( 1,,,9

a" Proportional to ∆T b" n2ersely proportional to ∆T

c" Proportional to C% α α − d" n2ersely proportional to C% α α −

'olution & b, d" -n heating, the strip undergoes linear expansion

+o ater expansion length o brass strip "!B T LL %% ∆+= α and length o copper strip

"!B T LL CC ∆+= α

3rom the fgure θ " " RL% += ......i"

and θ RLc = ......ii" (#s angle 7 #rc8Gadius'

%i2iding i" by ii"T

T

L

L

R

"R

C

%

C

%

∆+∆+

==+

α

α

!

!

⇒ !"!"!! −∆+∆+=+ T T R

" C% α α 7 "!"! T T C% ∆−∆+ α α 7 T C% ∆−+ "! α α

⇒ T

R

" C% ∆−= " α α or

T

" R

C% ∆−

=" α α

(*sing Dinomial theorem and neglecting

higher terms'

+o we can say R "

!

C% α α −∝ and R

T ∆∝

!

P roblem 17. Two metal strips that constitute a thermostat must necessarily dier in theirBBTAC44 1,,29

a" Mass b" Hength

c" Gesisti2ity d" CoeNcient o linear expansion

'olution & d" Thermostat is used in electric apparatus li5e rerigerator, ron etc or automatic cut o. Thereore or metallic strips to bend on heating their coeNcient o linear expansion should

be dierent.P roblem 18. # cylindrical metal rod o length BL is shaped into a ring with a small gap as shown. -n

heating the system

"

Rθ

)

r

"


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a" x decreases, r and " increase

b" x and r increase, " decreases

c" x , r and " all increase

d" %ata insuNcient to arri2e at a conclusion

'olution & c" -n heating the systemU x8 r8 " all increases, since the expansion o isotropic solids is similar

to true photographic enlargement

P roblem 1,. Two holes o uneual diameters !" and $" " $! " " > are cut in a metal sheet. thesheet is heated

a" Doth !" and $" will decrease

b" Doth !" and $" will increase

c" !" will increase, $" will decrease

d" !" will decrease, $" will increase

'olution & b" the sheet is heated then both "! and "$ will increase since the thermal expansion o isotropic solid is similar to true photographic enlargement.

P roblem 2. #n iron tyre is to be ftted onto a wooden wheel !.B m in diameter. The diameter o thetyre is : mm smaller than that o wheel. The tyre should be heated so that its temperatureincreases by a minimum o

CoeNcient o 2olume expansion o iron is C°× − 8!B:.) 1 "

a" !:

⇒ CT o1BB=∆

P roblem 21. # cloc5 with a metal pendulum beating seconds 5eeps correct time at B/C. it loses !$.1seconds a day at $1/C, the coeNcient o linear expansion o metal o pendulum is

a" C *er o9:0BB

!b" C *er °

0)$BB

!c" C *er °

!00BB

!d" C *er °

$99BB

!

'olution & a" Hoss o time due to heating a pendulum is gi2en as

∆ T 7 T θ α ∆$!

⇒ !$.1 7 9:0B"B$1$

!

×°−×× Cα ⇒ α C *er °= 9:0BB!

"!

"$


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Caloriemetr 72

genius

P roblem 22. # wire o length BL is supplied heat to raise its temperature by T . γ is the coeNcient o 2olume expansion o the wire and 4 is the Voungs modulus o the wire then the energydensity stored in the wire is

a" 4 T $$

$

!γ b"

)$$

)

!4 T γ c" 4

T $$

!9

! γ d" 4 T

$$

!9

!γ

'olution & d" %ue to heating the length o the wire increases. ∴ Hongitudinal strain is produced ⇒

T L

L∆×=

∆α

Elastic potential energy per unit 2olume E 7 +trai+tress$

!×× 7 $"+train

$

!×× 4

⇒ E 7 $$$

$

!

$

!T 4

L

L4 ∆×××=

∆×× α

or E 7$

$

)$

!T 4 ×

×× γ

7$$

!9

!4T γ (#s α γ )= and ∆T 7 T gi2en"'

P roblem 23. +pan o a bridge is $.0 km. #t )B/C a cable along the span sags by B.1 km. Ta5ing

C *er o:!B!$ −×=α , change in length o cable or a change in temperature rom !B/C to

0$/C is

a" . m

b" B.B m

c" B. m

d" B.0 km

'olution & c" +pan o bridge 7 $0BB m and Dridge sags by 1BB m at )B/ gi2en"

3rom the fgure LPRQ 7 m$:BB1BB!$BB$ $$ =+

Dut "!B t LL ∆+= α (%ue to linear expansion'

⇒ ")B!B!$!$:BB :B ××+= −L ∴ Hength o the cable mL $1EEB =

Fow change in length o cable due to change in temperature rom !BoC to 0$oC

"!B0$!B!$$1EE : −×××=∆ −L 7 B.m

12.13 Thermal 0apac#ty a$d ;ater 4-u#*ale$t.!" Thermal capac#ty t is defned as the amount o heat reuired to raise the

temperature o the whole body mass m" through B/C or !K .

Thermal capacityT

QCmc

∆=== µ

The 2alue o thermal capacity o a body depends upon the nature o the body and its mass.

%imension & '( !$$ −− θ T ML , *nit & cal8/C practical" Joule8k +.."

$" ;ater 4-u#*ale$t 6ater eui2alent o a body is defned as the mass o water which

would absorb or e2ol2e the same amount o heat as is done by the body in rising or alling

through the same range o temperature. t is represented by W .

m 7 Mass o the body, c 7 +pecifc heat o body, ∆T 7 Gise in temperature.

P / Q

1BB m

!$BB mP Q

/

R


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Then heat gi2en to body T mcQ ∆=∆ .. i"

same amount o heat is gi2en to W gm o water and its temperature also rises by ∆T

Then heat gi2en to water T W Q ∆××=∆ ! (#s !water =c ' ..ii"

3rom euation i" and ii" T W T mcQ ∆××=∆=∆ !

∴ 6ater eui2alent W " 7 mc gm

*nit & Kg +.." %imension & '( BBT ML

5ote & *nit o thermal capacity is J9kg while unit o water eui2alent is kg.

Thermal capacity o the body and its water eui2alent are numerically eual.

thermal capacity o a body is expressed in terms o mass o water it is called waterL

eui2alent o the body.

12.1+ Spec#!c Heat.

!" /ram spec#!c heat 6hen heat is gi2en to a body and its temperature increases, the

heat reuired to raise the temperature o unit mass o a body through !/C or K " is called specifc

heat o the material o the body.

Q heat changes the temperature o mass m by ∆T

+pecifc heatT m

Qc

∆= .

*nits & Calorie8gm × /C practical", J8kg × K +.." %imension & '( !$$ −− θ T L

$" olar spec#!c heat Molar specifc heat o a substance is defned as the amount o

heat reuired to raise the temperature o one gram mole o the substance through a unit degree

it is represented by capital" C.

Dy defnition, one mole o any substance is a uantity o the substance, whose mass M

grams is numerically eual to the molecular mass M.

∴ Molar specifc heat heaspecifc=ram×= M

or cMC =

T Q

T mQMC ∆=∆= µ ! =∆= M

mT m

Qc µ and#s

∴T

QC

∆= µ

*nits & calorie8mole × /C practical"U J8mole × kelvin +.." %imension & '( !!$$ −−− µ θ T ML

mportant points

!" +pecifc heat or hydrogen is maximum ( )Cgmcal o×81.) and or water, it isCgmcal °×8! .


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Caloriemetr 7+

genius

3or all other substances, the specifc heat is less than Cgmcal °×8! and it is minimum or

radon and actinium ( )Cgmcal °×− 8B$$.BW .$" +pecifc heat o a substance also depends on the state o the substance i.e. solid, liuid

or gas.

3or example, Cgmcalc °×= 81.Bice +olid", Cgmcalc °×= 8!water Hiuid" andCgmcalc °×= 80


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The 2ariation o specifc heat with temperature or water is

shown in the fgure. *sually this temperature dependence o

specifc heat is neglected.

3rom the graph &

Temperature %C( B !1 )1 1B !BB

Spec#!c heat %cal E gm

C(

!.BB9 !.BBB B.< B.9 !.BB:

#s specifc heat o water is 2ery largeU by absorbing or

releasing large amount o heat its temperature changes by small amount. This is why, it is used

in hot water bottles or as coolant in radiators.

5ote & 6hen specifc heats are measured, the 2alues obtained are also ound todepend on the conditions o the experiment. n general measurements made atconstant pressure are dierent rom those at constant 2olume. 3or solids and liuids

this dierence is 2ery small and usually neglected. The specifc heat o gases are

uite dierent under constant pressure condition cP" and constant 2olume c# ". n

the chapter ?Kinetic theory o gases@ we ha2e discussed this topic in detail.

Sample problems based on Specic heat! thermal capacity and "ater equi#alent

P roblem 2+. Two spheres made o same substance ha2e diameters in the ratio ! & $. Their thermalcapacities are in the ratio o

CB'4 1,,,9a" ! & $ b" ! & 9 c" ! & 0 d" $ & !

'olution & b" Thermal capacity 7 Mass R +pecifc heat

%ue to same material both spheres will ha2e same specifc heat

∴ Gatio o thermal capacity ρ

ρ

$

!

$

!

#

#

m

m== 9&!

$

!

)

0)

0))

$

!

)$

)!

=

=

==r

r

r

r

π

π

P roblem 25. 6hen )BB J o heat is added to $1 gm o sample o a material its temperature rises rom$1/C to 01/C. the thermal capacity o the sample and specifc heat o the material are

respecti2ely gi2en by

a" !1 J8/C, :BB J8kg /C b" :BB J8/C, !1 J/8kg oC c" !1B J8/C, :B J8kg /C d"

Fone o these

'olution & a" Thermal capacity 7 mc 7 C JT

Q°==

−=

∆8!1

$B

)BB

$101

)BB

+pecifc heat 7Mass

capacit Thermal7 Ckg J °=

× −8:BB

!B$1

!1)

P roblem 26. The specifc heat o a substance 2aries with temperature t /C" as

"8B$).B!0.B$B.B $ Cgmcalt t c °++=

The heat reuired to raise the temperature o $ gm o substance rom 1/C to !1/C will be

a" $0 calorie b" 1: calorie c" 9$ calorie d" !BB calorie

$B 0B :B 9B !BBB.

:

!.BBB

!.BB0

!.BB9

+ p . h

e a t c a l 9 g

C +

Temp. in+C


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Caloriemetr 76

genius

'olution & c" 4eat reuired to raise the temperature o m gm o substance by "T is gi2en as

"Q 7 mc "T ⇒ ∫ = "T mcQ

∴ To raise the temperature o $ gm o substance rom 1/C to !1/C is

∫ ++×=!1

1

$"B$).B!0.B$.B$ "T t t Q

!1

1

)$

)

B$).B

$

!0.B$.B$

++×=

t t t 7 9$ calorie

12.17 Late$t Heat.

!" 6hen a substance changes rom one state to another state say rom solid to liuid or

liuid to gas or rom liuid to solid or gas to liuid" then energy is either absorbed or liberated.

This heat energy is called latent heat.

$" Fo change in temperature is in2ol2ed when the substance changes its state. That is,

phase transormation is an isothermal change. ce at B/C melts into water at B/C. 6ater at !BB/C

boils to orm steam at !BB/C.

)" The amount o heat reuired to change the state o the mass m o the substance is

written as & ∆Q 7 mL, where L is the latent heat. Hatent heat is also called as 4eat o

Transormation.

0" *nit & cal9gm or J8kg and %imension & '( $$ −T L

1" #ny material has two types o latent heats

i" Hatent heat o usion & The latent heat o usion is the heat energy reuired to change ! kg

o the material in its solid state at its melting point to ! kg o the material in its liuid state. t is

also the amount o heat energy released when at melting point ! kg o liuid changes to ! kg o solid. 3or water at its normal reeAing temperature or melting point B/C", the latent heat o

usion or latent heat o ice" is

kg ;oulekilomolkJgcalLLF 8)):8:B89Bice ≈≈≈= .

ii" Hatent heat o 2aporisation & The latent heat o 2aporisation is the heat energy reuired

to change ! kg o the material in its liuid state at its boiling point to ! kg o the material in its

gaseous state. t is also the amount o heat energy released when ! kg o 2apour changes into

! kg o liuid. 3or water at its normal boiling point or condensation temperature !BB/C", the

latent heat o 2aporisation latent heat o steam" is

kg ;oulekilomolkJgcalLL# 8$$:B89.0B810Bsteam ≈≈≈=

:" n the process o melting or boiling, heat supplied is used to increase the internal

potential energy o the substance and also in doing wor5 against external pressure while internal

5inetic energy remains constant. This is the reason that internal energy o steam at !BB/C is

more than that o water at !BB/C.


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" Hatent heat o 2aporisation is more than the latent heat o usion. This is because when a

substance gets con2erted rom liuid to 2apour, there is a large increase in 2olume. 4ence more

amount o heat is reuired. Dut when a solid gets con2erted to a liuid, then the increase in

2olume is negligible. 4ence 2ery less amount o heat is reuired. +o, latent heat o 2aporisation

is more than the latent heat o usion.

!B" #ter snow alls, the temperature o the atmosphere becomes 2ery low. This is because

the snow absorbs the heat rom the atmosphere to melt down. +o, in the mountains, when snow

alls, one does not eel too cold, but when ice melts, he eels too cold.

!!" There is more shi2ering eect o iceLcream on teeth as compared to that o water

obtained rom ice". This is because, when iceLcream melts down, it absorbs large amount o

heat rom teeth.

!$" 3reeAing mixture & salt is added to ice, then the temperature o mixture drops down to

less than B/C. This is so because, some ice melts down to cool the salt to B/ C. #s a result, salt

gets dissol2ed in the water ormed and saturated solution o salt is obtainedU but the ice point

reeing point" o the solution ormed is always less than that o pure water. +o, ice cannot be in

the solid state with the salt solution at B/C. The ice which is in contact with the solution, starts

melting and it absorbs the reuired latent heat rom the mixture, so the temperature o mixture

alls down.

Sample problems based on $atent heat

P roblem 27. 6or5 done in con2erting one gram o ice at !B/C into steam at !BB/C is

' '4T E'T 1,88> 4=04T %ed.( 1,,5> ' 'T239

a" )B01 J b" :B1: J c"


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Caloriemetr 78

genius

⇒ Lmcm ×+×× Q"$B 7 '$B(


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where L is latent heat. 4eat is absorbed i solid con2erts into liuid at m.pt." or liuidcon2erts into 2apours at b.pt." and is released i liuid con2erts into solid or 2apours con2ertsinto liuid.

0" two bodies $ and % o masses !m and $m , at temperatures !T and $T " $! T T > andha2ing gram specifc heat !c and $c when they are placed in contact.

4eat lost by $ 7 4eat gained by %

or "" $$$!!! T T cmT T cm −=− (where T 7 Temperature o euilibrium'

∴ $$!!

$$$!!!

cmcm

T cmT cmT

++

=

i" bodies are o same material $! cc = then$!

$$!!

mm

T mT mT

++

=

ii" bodies are o same mass " $! mm = then$!

$$!!

cccT cT T

++=

iii" bodies are o same material and o eual masses ", $!$! ccmm == then$

$! T T T +

=

12.1, Heat#$g cur*e.

to a gi2en mass m" o a solid, heat is supplied at constant rate P and a graph is plotted

between temperature and time, the graph is as shown in

fgure and is called heating cur2e. 3rom this cur2e it is clear

that

!" n the region /$ temperature o solid is changing

with time so,

T mcQ '∆=

or T mct P '∆=∆ (as Q 7 P∆t '

Dut as ∆T 8∆t " is the slope o temperatureLtime cur2e

c' ∝ !8slope o line /$"

i.e. specifc heat or thermal capacity" is in2ersely proportional to the slope o temperatureL

time cur2e.$" n the region $% temperature is constant, so it represents change o state, i.e., melting o

solid with melting point T !. #t $ melting starts and at % all solid is con2erted into liuid. +o

between $ and % substance is partly solid and partly liuid. LF is the latent heat o usion.

F mLQ = orm

t t PLF

" !$ −= (as " !$ t t PQ −= '

or LF ∝ length o line $%

i.e. Hatent heat o usion is proportional to the length o line o Aero slope. (n this region specifc

heat ∝ ∞=Btan! '

T $

T !

/

$ Melting

Doiling

!. *t.

%

C0

t $

t !

t 0

t )

Time

T e m p .

m. *t.

E


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Caloriemetr 8

genius

)" n the region %C temperature o liuid increases so specifc heat or thermal capacity" o

liuid will be in2ersely proportional to the slope o line %C

i.e., cL ∝ !8slope o line %C"

0" n the region C0 temperature is constant, so it represents the change o state, i.e.,

boiling with boiling point T $. #t C all substance is in liuid state while at 0 in 2apour state and

between C and 0 partly liuid and partly gas. The length o line C0 is proportional to latent heat

o 2aporisation

i.e., L# ∝ Hength o line C0 (n this region specifc heat ∝ ∞=Btan

!'

1" The line 0E represents gaseous state o substance with its temperature increasing

linearly with time. The reciprocal o slope o line will be proportional to specifc heat or thermal

capacity o substance in 2apour state.

Sample problems based on Caloriemetry

P roblem 32. 1B g o copper is heated to increase its temperature by !B/C. the same uantity o heat isgi2en to !B g o water, the rise in its temperature is +pecifc heat o copper

!!L0$B −− °= Ckg Joule " 4=04T %ed.( 29

a" 1/C b" :/C c"


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P roblem 35. # bea5er contains $BB gm o water. The heat capacity o the bea5er is eual to that o $Bgm o water. The initial temperature o water in the bea5er is $B/C. 00B gm o hot water at

$/C is poured in it, the fnal temperature neglecting radiation loss" will be nearest to

NS4' 1,,+9

a" 19/C b" :9/C c"


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Caloriemetr 82

genius

n condensation i" +team release heat when it looses itQs temperature rom !1BoC to

!BBoC. '( T mcsteam∆

ii" #t !BBoC it con2erts into water and gi2es the latent heat. '(mL

iii" 6ater release heat when it looses itQs temperature rom !BBoC to BoC.

'( T ms

01 Thermal Expansion Theory1

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