Heat Transfer Calorimetry

cpp(Calorimetry &Thermal ElCpansion) 6th March It

Cpp

, Cal.II'imetrv,& Thermal EXpa'nSion'~

. .. 0;:/':". ';'7,,'1.

;:;:::'X;;--PART.

I : SUBJECTIVE. QUESTIONS

S,~jFJJ'J~~!

.-.sL I M I'T.E'D'

Section (A) ,: Calorimetry

A 1. ' ,In the following equation calculate the value of H.

1 kg steam at ZOO.C

=

H+1 kg water at 100.C

(5.-

= Constanl = 0.5 Callgm.C)

A 2.' From what heightsh~Uld a piece of ice-(O.C)full so that it melts complelely? Only one-quarter of the energy

produced is absorbed by the ice as heat. (Latent heal of ice = 3.4x l()5J kg-', g.= 10 m/s2)

A 3. A copper cube of mass 200g slides down on a rough inclined plane of inclination 37. at a constant speed.

Assume that any loss in mechanical energy goes into the copper block as thermal energy. Find the increase in the temperalur~ of the block as it slide~ down 60 em. Specific heat capacity of coP,:~:=\20J/kg.K

A 4. A paddle wheel is connected with a block of mass 10 kg as shown In figure. ~,

The wheel is completely immersed in liquid of heat capacity 4000 JIK. The (-

,if/\

container is adiabatic. For the time interval in which block ~oes down;~t m

,,"U/

sloWlycalculate

(',eo,'

\

,::..,.A

(a) Work done on the liquid '. - , ,.'

(b) Heat supplied to the liquid ' ~:,(\",

t

(c) Rise in the temperature of the liquid . ", " \

Neglect the heat capacity of the container and the paddle. (g =10 nillf) "

Seetron (B)

:

Tlierm&lExpa~SI6~

> \

j~~

.

.',.,.

.

B 1. :"he temp~rature of a m~tal ban!s ~~;Arrange the ~reenla9cechange in volume, surface area and radius

In ascendIng order. ~ '\

'¥" \ "\

,';p '_

B 2. A brass disc-fits in a'holein\a steel plate. Would youh'eal or cool the system 10 loosen the disc from

the hOle?A-i''nte lh~t\ci.

<0).

\', "

S

A ,,,- \ \\ ..

\r

B 3. [emperatu~ of plate is increaSlld-by Mlhen find new

G

~~~:~=~l~:\

V,;L)

'_ .

.

R :

,lcJ'i'\ lI}'ediffel-ence houler and inner radius and show ljItt

it

is positive

"{dr i:~=:uO~~I:e:~~~f expansion is0) " ". ,

•• W~ hC!.~e\hollow sphere and a solid sphere of equal radii and of the same material. They are heated to

, raiseiheir temperature by equal amounts. How winthe change in their volumes, due to volume expansions, be related? Consider two cases (I) hollow sphere is ,filledwith air, (ii) there is vacuum inside the hollow sphere.

Sect/on eC): Temperature

f' 1. The figure shows three temperatUre scales wilJ1the freezing and boiling points of water indicaled.

9O'Y Bolfing Point

O"Y Freezing Point'

,(a) Rank the size of'a degree on lhese seales, greatest first.

.(b) Rank the.followinglemperatures, highest first: 50.X, 5O"W and50"Y. ~,

C 2 . What is the temperature at which we get the same reading on both the centigrade and Fahrenheit scales?

PART. II : OBJECTIVE QUESTIONS'

• Marked Questions are, having more tllan one correct optlo'n.

SectIon (A): CalorImetry

A 1. Asr.lall quanlily, mass m, of water-at a temperaluree (inOC)is poured on loa'iarge mass M of ice which is

al its mel~ng point If c is the specific heat capacity of waler and lthe lalent heat of fusion of ice. then the

mass of ice melled is given by'; ,

(A) Ml

mc6 (8)mce

ML

(C) Mc6L (D) mce L

A2." When m gm of waler at 10'C is mixed with m gm of ice at O'C, which of the following slatements are'

false? ' ' ,

tit''''

(A) The temperature. of Ihe syslem will be given by the equation ' ..

',"'.;".",)l!>

m x 80 + m x 1 x (T - 0)

=

m x 1 x (10 - T)

".,J!lf

p

,

(B) Whole of ~cew~1Imelt and temperature w~1Ibe more than O'C b.utlesser tha,~;10'C

Jl

(C) Whole of Ice Will melt and temperature Will be O'C .

t; ',.,

(0) Whole of ice will not melt and temperature will be O'C . \

SectIon (8) : Thermal expansion

.'

'G,

,' ~~/~' ,

B 1. Two large holes are cut in a metal sheetlfthls'is heated, distancesABand BC, (as shown)

,

....

~

\~t:!\, ~

~B~)t...

ic~~~;;:r~,"\.~

.~~~~;::"~_

• "1# ~

Y

\

.J~

'

B 2. A-steel scale is to be prepared'such thaUhe millimeter intervals are to be accurate within 6 x 10" mm.

Th~'.maxi';;um temperatute variation from the temperature of calibration during the reading of the millimeter

arksJs (a~ 12 x

10-<

I*C)"Y

;.{(A)i~~'\

(B)

h'c

(C) 5.0'C ' (D) 5.5'C

Ex~Sion duringljeating -

I

. (Aj"occurs only in a solid , (B) increases the density of tile material

~ ..~ '_"0, '"

(C) decreases the density of the material (D) occurs at the same rate for all liquids lind solids.

\IV

"

,

B 4. If a bimetallic strip Is heated, itwill

(A) bend towards the metal with lower thennal expansion coefficient. , (B) bend towards the melal with higher thennal expansion coefficient.

(C) twisl itself into helix.

-(D) have no bending,

B 5*. Two identical beakers with negligible Ihennal expansion are fitted with water 10 the same level at 4'C.

If one says A is heated while the other says 8 is cooled. then:

(A) water level in A must rise (8) waler level in B must rise

(C) water level in A must fall (D) water level in 8 must fall

Section (C) : Temperature

C 1. A difference of temperature of25'C is equivalentto a difference of :

(A)45' F (8) 72. F (C)32°F (D) 25° F

..

----.---

..

2.

1.

3. ~'

..

EXERCISE#2

PART -

I :

SUBJECTIVE QUESTIONS

A thermally Isolated vessel contain~ 1009of water at OOC.When air above the water is pumped out, some

of the water freezes and some evaporates at OOCitself. Calculate the mass of the ice formed if nO,water is fell

in the vessel. latent heat of YaJlC!rizationof water at WC

=

2.10 x 10" Jlkg and latent heat of fusion of

ice

=

3.36 x 10" Jlkg.

A pitcher oontalns 20 kgof~ter. 0.5 gm of water comes out on the surface of the '0. , .',

pitcher every second through the pores and gets evaporated taking energy from the , remaining water. Calculate the approximate time in which temperature of the water decreases by 5"C :Neglect backward heat transfer from the atmosphere to the water.

Specific heat capacity of water

=

42ooJlKgoC. ,

latent heat of vaporization of water 2.27 : .10" JlKg ','

A~

A thermally insulated, closed copper vessel.conlains water at 15OC.When the vessel is shaken vigorously

for 15 minutes, the temperature rises to 17.C. The mass of the vessel is 1COgand that"Oftll8'Watefis 200g.

~. . ,.,:,." •.. , .•• .:.4f<: '....

The specific heat capacities of copper and water are 420 Jlkg-K and 4200 JlJ(g-K res~j( Neglect any

.k-.""., .•<",.~ ~ ..=e-, __._

."..--thermal expan~ion. (al How m,uch heat is transferred to the fiqUid-ve~s'!!l ~!em? (b) 'to:v ll1u~.!19rlchas

been.done on this system? (c) How much is the increa~n Intemalf~n~~the \yste~~i.:t1'. .' '),

The lime represented by the clock hands of a pendulum clock,depends on the number of OSCIllations

per-" ~"""," ..~..,.,, ~.>:<:~. ,'" .

formed by pendulum. Every time it reaches to its extreme position the seconi:l hand.of iIle clock advances by_{. """.~}

..' -,..;., ~-"..:..:..._..;.-... "',

one second that means second hand moves by two second when one Oscillation is completed.

o / 11.., • '-<0:, ,':,. '''1l,.,,' .. .,-- ,"

(a) How many number of OSCIllatIOnscompleted by pendulum of c1OCkjrl.15 minutes at calibrated

temperature 2O"C

:,~.!~.?"

\'" \

(bl How many number of oscillations are completed !1Y.a pendulum of clock in 15 minutes at

tempera-ture of 4O"C if"it".2'~ 103"'0' \'

\,,!, ..

"!/,

,

,

.

(el What time

"""~

s

represented by th~pendulumclock

,...'

l

ai 4O"C after 15 minutes If the initial time shown by

'l!tt. v , ,

the'CIock is 12'; 00 PITh? ' ", i .• '",~

,~ :~~~~~:~i!~~~~5, ~uteSthen.~d. (il Number of extra osdllations (if) New time

5. '

4'

gnsid~r,a cylindriCal containerof~s section area W, Hlngth 'h' having coefliclent of linear expansion

/Xc-l-,The con'tliner isfined

by

I~uid

&

volume expansion coefficient Tlup to height h" When temperature of the

."'C'... " ....••.• ,. ,

",system ~increased by.4llthen

~"

.~\-''.-

I

, l'f-i

.::: t

'1

.~~

..

(a) Find out new height area and volume of cyfindrical container and new vQlume of liquid.

(b) . Find the height of liquid level when expansion of container is neglected.

(el Find the relation between TLand0.forWhich volume of container above the liquid level.

(i)increases (ii) decreases (iii) remains constant. ,

(d) If T,> 3a. and h

=

h, then calculate, the volume of liquid overflow, ,

(e) If the surface of a cylindrical container is marked with numbers for the measurement of liquid level of

liquid filled inside it Assuming correct marking at inilia!' temperature if we increase the temperature

of the system by 46 then .

(i),' Find height of liquid level as shown by the scale on the vessel. Neglect expansion of liquid

(Ii) Find height of liquid level as shown by the scale on the vessel~'Neglect expansion of

container.

(iii) Find relation between TLand llcso that height of liquid level with respec-l to ground

(1) increases (2) decreases (3) remains constant.

,.

.

,

(,

6. A metal piece weighing 15g is heated to 100.C and then immersed in a mixture of ice and water at the

thermal equilibrium. The volume of the mixture is found to be reduced by 0.1S em" with the temPerature of mixture remaining constant. Find the specific heat of the metal. Given specific graVity of ice " 0.92, latent .heat of fusion of ice = 80 callgm.

'7- 20 gm ice at-10.C is mixed with m gm steam at 100 .C.Find the minimum valut' ofm so thalfinal1y all ice.

and steam converts into waler. (Use s•• =O.S callgm.C,s_=1 cal/gm.C.L (melting)=80 callgm and L

(vaporization)=540 callgm)

8. . A simple seconds pendulum is constructed out of a very thin string of thermal coefficient of linear expansion

a" 20 x1(t4/"C and a heavy particle attached to one end. The free end of the siring Is suspended from the

ceiling of an elevator at rest. The pendulum keeps correct time at DOC:When me temperature rises 10SOOC, the elevator operator of mass 60kg being a studenl of Physics accelerates the elevator vertically, to have the pendulum correct time. Find the apparent weighl of the operator When the pendulum keeps correct time at SO"C. (Take g = 10 m/s')

PART. II : OBJECTIVE QUI;STIONS

Singie choice type

....~

.,

.

. ~.

1. A melal ball of specific gravity 4.5 and specific heat 0.1 callgmp"C is placed on a larg~sl~2~.~tlCtl al

O.C. Half of the ball sinks in the ice. The inilial temperalure of !he ball i:~ "--

-41~'1rAlf),

(Latenl heal capacity of ice = 80 cal/g, specific ~.r;lIy,ifY\ofice "'.0,9 ./ ~

(A)100.C' . (B)90.C {~)80.C" {;(Q)JO'~G' . .

. ,~ . \ l' 0;... "; A .

2. . A sleel rod.25 cm long has a cross-sectiqnal area of 0,8 cl'l1" Force. requiredtostfetch this rod by.the

. 'same amounl as the expansion produCed-by heatingitthro'ligh 10.Cis:~ . . .

.-J'--'''2---=",'''' '~ \.. .•••.~., .' ~

(Coefficient of Iinearexpansicln ()f,steelis 10""C and Yourig'sinodulus of steel is 2 x 10'°N/rTi'.) .

(AI

160 N .

/~~)36~,..t\

(C)

\06)r'~'

(D)

260

N. .

. 3. If I is the moment of inertia of a solid.body having a -coefficient of linear explinslon then Ihe change in

_,-:

_~""'.~:''.

~:'~

~,-~,...-.

..

I corresponding to asniall change in temperature IITis

(~laIll~i.

\;ri~lllT' \!Acl2aIllT .

(Of3aUT

.

4.;*

AJoiq)idWith qoeffiCie!11of ~Ii~~ansion y is filled in a container of s.material having the coefficient of

." .linearexpansiona .lIthe IiquiCloverflows on

heating,then-,,~Affly

\,P(Blr <3a ., (C1..= 3a (D) none of these . .

5. "Two rods having lengthi,and i.,.made of materials with the linear coefficienl of expansion a, and

0,.

were

welde~,together. The equivalent coefficients of linear expansion for the ol>lained rod :~ .

~"W .

6. The volume thermal expansion coefficienl of an ideal gas at conslant pressure is

(A)T (6)1" 1

(D) T2 .

(Here T

=

absolute lemperalure of gas) ,

7. A metal ban immersed in water weighs w, aIS"C and

w.

at SO"C.The coefficiento~cubical expansion of metal

is less than that of water. Then

(AI w, >w, (B) w, <

w.

(Cl w,

=

w.

(D) dala is insufficient

..'

More th'an one choice type

8. When two nonreactive samples atdiffe~nttemperatures are mixed In an isolated'containerotnegligible

heat capacity the temperature of the mixture can be: .' . .

(A) lesser than lower or greater than higher temperature' (8) equal to fower or higher temperature

(Clgreater than lower but lesser than higher temperature

(D) average of lower and higher temperatures

9._ There is a rectangular metal plate in which two cavities in the shape of rectangle and circle are made,

as shown with dimensions. P and

a

are the centres of these cavities. On healing the plate, which of

the foliowing quantities increase?

(C) Height of cylinder outside the liquid remairlS constant

.

_.'

(A) volume of cylinder inside the liquid remains constant_'\:,.•-- .

(B) volume of cylinder outside the liquid remains constant .

(A) nr_{. ,}2 (8) ab

+0-f

p Column II (PlY=0 (q)y

=

2a d

(rly=

3a

p

(0) Height of cylinder irlSide the liquid remain constant _(s)y

= (

. _{2a+a - )}d

p

In the following question column - I represents some physical quantities & column-ll represents their units, match them

Column I Column II

(A) . Coefficient of linear expansion (p) CaIrC .,

(B) Water equivalent (q) gm

(C) heat capacity (r) (OCr'

(D) Specific heat (5) CaUgOC

.'

PART ~ \I : COMPREHENSION

Comprehension #1 "

A 0.60 kg sample of water and a sample of ice are placed in two compartments A and B that are , separated by a conducting wall, i,na thermally insulated container. The rate of heat transfer from the water to the ice though the conducting wall is constant p. until thermal equilibrium is reached. The temperature T of the liquid water and the ice are given in graph as functions of time l. Temperature of the compartments remain homogeneous during whole heat transfer process.

Given specific heat of ice

=

2100 jlkg-K Given specific heat of water =4200 J/kg-K Latent heat of fusion of fce

=

3.3 x 10s J/kg

(0)100.C 20 (C) 94.C

--

.

---~-

..

'

.

--- •••• B',

---

..

'

....

-.water--- • .Ice ••

--

_{---f'Z. •••.•••}

----...

'.

______ ~ .•.• ,,' ,t .• 3.

4.

6. .••.•.••Insulating conducting wali -20

Th'~I~~rn":4

,: ~

(A) 42.0 W -"'. (B).~6'~""fi \. (C)

~J~;~

Theinitial mnss oHhe ICe In.thecontalner ISequal to, '

(A) 0.36 ~ \ (~( 1.2 kg

'~~I!\

(Cl'2,:i<9, (D) 3,6 kg

5. The mass of the ice'formed'due to conversion from the water till thermal equilibrium is reached, is equal

Q;.:t \~ ..

V

(e)O.25"

(OlD.""

..

'"'',:.: In a CO'iltaine'r-ofnegligible heat ca'pacity, 200 gm illll at O.C and 100 gm steam at 1OO.Care.added to .~. 200 gm of waler.thilt has temperature 55.C. ASsume no heal is lost to the surroundings and the • preSsure in the container is constant.1.0 atm. (Latent'heat of fusion of ice=80 caVgm, Latent heat of

.

.

.

vaporization of water

=

540 caVgm, Specific heat capacity of ice

=

0.5 caVgm-K, Specific heat capacity of

--water =1 caVgm-K)

What is the final temperature of the system? (A) 48.C (B) 72.C

7. At the final temperature, mass of the total water present in the system, 'is

(A) 472.6 gm' (B) 483.3 gm (Cj 493.6 gm (0) 500 gm 8. Amount of the stel;)m left in the system, is equal to

(A) 16.7 gin (B) 12.0gm

(C) 8.4 gm (0),0 gm, as there Is no steam left.

PART -III:

ASSERTION IREASONING

9. STATEMENT-1 : Gas thermometers are more sensitive than fiquid thermometers. STATEMENT-2 : Coefficient of thermal expansion of gases is more lhan liquid.

(A) Stalemenl-1 Is True, Statement-2 is True; Statement-2 Is a correct explanation for Statement-1 (B) Statement-1 Is True, Slalement-2 is True;, Statement-2 Is NOT a correct explanation for Slaterilent-1 (C) Statement-1 is True, Stalement-2ls False, .

(0) Statement-1 is False, Slatement-2 is True~

,,;~',Il:"!t•.,t,/, <'o!.' . ,',

1-',,'\

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t. '.",

STATEMENT-1 : Water is considered unsuilable for use in thermometers.

. STATEMENT -2.: This is due to small specific heat and high thermal conductivitY.

(A) stalement-1ls True, statement-2ls True; Statemenl-2ls a correcl explanation for Stalemenl-1

(8) statemenl-1 is True, statemerit~2 is True; SU!tement.21S NOTa correct explanation for statement-1 (C) Statement-1 is True, Statement-2 is False

(D).Statement-1is False, stalemenl-2 is True ..

STATEMENT.1 : When water Is healed by a bumer in metallic container Its level firsl decreases then

increases. . . .

STATEMENT -2.: Thermal conductivitY of metalis very large compared to water. .

(A)stalemenl-1Is True Slatement.2 is True; stalement-2ls a correct explanation for Stalement-1

(B) Siatement-1 is True, Stalement.2 is True; Statement.21s NOTa correct explanation for St8tement-1

(C) statement.1 is True, Statement-2 is False

(0) Slatement-1 is False, Slatement-2 is True.

PART.

IV:

TRUE

I

FALSE

Slate the following statements are true or false:

(I) Latent heat of vaporization is greater then latent heat of fusion.

(II) The temperature of a body may not change••.when it Is sUP~ied

~~01..

~~\ . " 'llf.

r ~

.

. (U1). Normal temj)erature of human body is around 310 degree above thecabsOlute zero.

(Iv) Density of ice is lower than,~~water

\Z"'\

~1V

(r ~

'rD~.

., /P~RT;;:V:\:

~LL IN TFtfBt!ANKS

Fill In

thl!.~'tn!<,:\

I\A

\

\.)V'

(I) ~vegative tempe~ture is possible on \ .

,'lli. ~ fhJ~eig~~ef fiv:r~r o~~nzene ~~ in summer than in winter.

it

(1I~:;ne s~m of.le~gths «(~alumiflium and steel rod at O"C Is __ so that al all temperatures their

~J ./'~,

di~eren«8 irilength is a.25m. (Take coefficient of linear expansion lor aluminium and steel at O"Cas

'1l"tr22 x

W1.z9

an~ 11 x 10'" OCrespectively.) . . ...

,(Iv)~\A steellape is correctly calibrated at 20 OCand is uJ to measure the length of a lable at 30 'C. The

~per:enlage error in.the measurement of length is.__ [a_= 11 x 10"".C)

(v) The variation of temperature of material as heat is given 10 it at conslant rate Is shown in the

figure. The malerialls in solid stale at the point O. The stale of the material al the poinl P is

••.•• ,... (198&; 2M]

o.

13. 11. 12.

1

T . 0 " Nut ectcteer--+

(Vi) 300 9 of water at 2S.C is added to 100 g of ice at O.C. The finaltemperalure of the mixture

is ...• C. '.. (1989; 2M] .

(vii) A substanc~ of mass M kg requires a power input of P walts to remain in the molten slate at its

melting point. When the power source is tumed off, the sample completelY solidifies in time I

seconds. The latenl heat of fusion of the substance is (1983; 1M]

_.ExerCise#4_

PART. I : liT..lEE PROBLEMS (PREVIOUS YEARS)

[JEE.2003 (SCr.).3184,-1J

. !

_(A)

(B)

&

S

(C~ \

\.

(t..

~m_~

;7~,''\

,\~~'-l7~1UPPed

When a block of iron f1oatsJnmercury.afO.C a fraction kidf its volume is submerged, while at the

,_ .• "•. , .. , ;C=-, .-"\ \-, ,.:0 •

temperature 50.C, a fraelion'!. iSseeil lobe submerged. If lJJecoefficient of volume expanSlOnof Iron

~"\

'"",,/. ~ . k '>(~~;1

.

.

.

.

is TFei(lha~ of

m~~7}\

.••

'

then~~ ~li\ k; ca~

be

_expressed as:

.' .M~.?\\':1~F'

V

f+6OyF.' [JE~::::~scr.), 1/35,-1)

.. (e;~{OTH~,1ij~\ (Bh\t\lOiHv. (} 1-6OyHv (D) 1+6OyFe '. .

An lca'c\lbe of'mass 0,1 kg at O'C is placed In an isolated container which Is at 227'C. The specific

"j'lleal S of the con\alnlir varies with temperature T according to the empirical relation S

=

A + BT, where

"A" l'OO.cal/kg-Kand B = 2 x 1lt" callkg-K" . If the finalteinperature of the containerls 27.C, Determine

._""""'"

'the mas~ oflhe conlainer. (Lalent heat offusion of water=8 x 10' callkg, sp. heat of waterc 10' cal/kg-I<).

. ~;r' :.. . [JEE '2001 (Malns),5/100) Two rods, one of aluminium and the other made of steel, having initial length l,and l.are connected together

to form a single rod of length l,+

t,.

The coefficients of linear expansion for aluminium and steel are a. and.

a. respectively. If the length of each rod increases by the same amount when their temperature are raised by

II

re.

then find the f!llio (t1+l2) .

*

Marked Questions are having mo.re than one correct option.

1". A bimetallic strip is formed out of two identical strips, one of copper and the other of brass. The

coefficients of linear expansion of the two metals are ac and aB .On healing, the temperature of the strips goes uP by 4Tand the strip bends to form an arc o! radius of curvature R. Then R is:

[JEE 1999, 2/200)

(A) Proportionallo 4T (B)inversely proportionallo 4T

(C) proportional to laB - ac!. (D) inversely proportional to laB - acl

.

.

:.,A.

A block of Ice at-1 O.C Is slowly heated and converted to steam at 1000C. Which of the folloWing curves

. represents the phenomena qualitatiVely: [,JEE-200C( 1135)

r .

I

~

..

2.

5,

3.

a (A) -'. a. a. (B)- . as

.6. 2 kg Ice at- 20 "C Is mixed with 5 kg water at 20 .C . Then final a!Oount of water In the mixture will be :

[SpecifiC heat of ice

=

0.5 caVgm "C, Specific heat of. waler" 1 cal/gm "C, Latent heat of fusion'of Ice

" 80 cal/gm

I

[JEE.2003 (SCr.),3184,-1J

(A) 6 kg (B) 7 kg (e) 3.5 kg (0) 5 kg

7. . A Cube of coefficient of linear expansion a, is floating in a bath containing a1iquid of coefficient of volume

expansion TL• When the temperature Is raised by 4T, the depth upto which the cube is submerged in the

liquid (emains the same. Find the relation between(1,and TLshowing all the sleps. ,

. . [JEE-2004 (Mains), 2160)

'~

3.

4.

- .:::','• ':, ~-,'" ., Ii, :.:',1 .,.,'f

2 liters water at 27"C is heai~d by ~ 1 kIN heai~r

'in

an

open container. On an ••verage heat is lostlo

surroundings at the rate 160 J/s. The time required for the temperature to reach

n"C

is .,

. " {JEE-200S

(Scr.l,

3/t!4.

~1J

(A) 8 min 20 sec (8) 10 min (e) 7 min (0) 14 min

In an insulated vessel, 0.05 kg steam at 373 K ,and 0.45 kg of ice at 253 K are mixed. Find :lIle final

temperatureof the mixture (in Kelvin).. {JEE 2006, 61184.-1]

Given, L""", = 80 c<aVgm = 336 J/gm, L_ =540 caVgm = 2268 J/gm,

S•• = 2100 J/kg K = 0.5 caVgm K and S•••••.= 4200 Jlkg K = 1 caVgmK

A piece of ice (h~at capacity = 2100 J kg-I .C-I and latent heat =3.36 x 1()5J kg-I) of mass

m

grams is at

-5 .C at atmospheric pressure. II is given 420 J of heat so that lIle ice starts 'melting. Final/y'Whenthe ice.

water mixture is in equilibrium, it is found that

1

gm of ice has melted. Assuming there is no

other

heat

exchange in the process, the value of m is: ' {JEE 20:10,3/163)

'PA'RT -.":

AIEEE PROBLEMS (PREVIOUS YEA~~

Heat given t~ a body which raises its temperatureby 1"C is : .,~.•.x~.'. '[AI~~ ..E...200...2..•.1313.00]

(1)water eqUlvafent (2) thermalcapae~£::/ ~~, ""'.

(3) specific heal . . (4)temperature9,:adi~.~\',~;:/

,II mass - energy ~uiv3lence is tak~n into accoul)tr~n water i~C:ied

tqt6~

ice, the m~~~""ofwater

sh~uld:. " ,~~ ." .. ' [AiEEE 2002,3/300)

(1)Increase /' (2)~mainUnchange<ll;i ,

(3)decrease

A ....

~/

(4)fif's~;~~telllendecrease .' .!.

The lengthof a simple pendulumexecutingsimple harmonicmotion is increased by 21%.The percentage

increasein the time Ill;riOd'o.fthlpend~I~T'61 increasefrength is : (AtEEE2003,WO)

(1)11% \iW.,Y.'.(.2)21%

~i;H1

,\(3)42%J'" (4)10.5%

...•

v" ,\,....

-'.

Time taken~y,a 836 W'h~aler iq,heat one iiter of water from 100Cto 400Cis : {A/EEE2004,31300)

(')~'"..

(2\~i~t\

0

3)15OS (4) 200 s

''\

0/

Answers

PART.III

9. (A) 10. (C) 11. (A)

PART-IV

12- (I) True (Ii) True . (iii) True

(Iv) True

I

PART.V

13. <I) Kelvin scale. (Ii) Less

(iii) 0.75m (Iv) . .1.1x 10"

(v) Partly solid and partly liquid. P1

. (vi) OOC (vii)

M

(c) (i)3h a, >h, Y, (iij 3h a~ <'h, Y,

(iiij 3h a.

=

h, Y,

(d) 6.V=All (YL- 30.c)69

(e) (i) h,= h, (1-3 actoe).

(iij II, = h, (1 +YLtoe )

(iii)(1JrL> 20., (2JrL< 20.. (3JrL

=

20..' .85 6. 0.092 caVgm.C 7. 32 gm 8. 660N PART .11 1. (C) 2. (A)

3:/1.C)

4. (A) 5. (C) ~. ,; (C) 7. . (8) 8. (B)(C)(q) I 9. (A)(B)(C)(O)~

A4

~~~~.~I' (,~(A) - (p) ;(B)-(r) ;o(C) - (s) ; (0)- (q)

2~ ....

'~A:-{•.;'{~~T~"); (D)-{.) 3.J'(A) 4. . (C). 5. (B) 6. ,~(D) . 7. (B) 8. (A)

Exercise' # 1

PART-I Section (A): A 1. H=5510kcal. A2. 136km 3 A 3. 350

=

8.6K 10-'.C A 4, (a) 100J, (b) 0, (c) 1/40.C section (8) : 81. %R<%A<%V

, 8 2. We will cool the system.

83. (a) R; =R, (1+a.:\.9)

(b) R~

=

R, (1+a.M)

(c) R~ - R; = (1\ - R,)(1+ a.:\.9)

(d)A' =(ltR~ -ltR~ )(1+2aa9)=A(1 +2aa9)

8 4. (i) hollow sphere> solid sphere' (ii) hollow sphere =solid sRhere

Section

(e): ~ .~

C 1. (a) All tie (ti)50~X.50OY. 50.W;

\~,if

<' - &:4iJ. Section (8) : '" 81.\(Ai.;\.,'\82. (C) 83. (e) B4. \(A)',',.>B 5. (A)B) .si~"'*:-. Sectlon'(C) : C 1. (A) (A) (C) (A) (4) 3.

5.

8.

2.

Exercise

#

4

PART. I (A) (2) (3) (B)(D) 2. 107

2i6

kg;; 0.5 kg (A) 7.. YL= 2a•. 273 K 10. '8gm PART -II (1) . 3. 1. 4, 6. 9. 1. 4.

Exercise

#

2

PART -1 2100 =86.2 g . 21+3.36 6.16min (a) zero : (b) 1764J (c) 1764J (a)450 (b) 449 (e) 12:14:59 pm

("J

898 ("'J 10' .C . (d)(i) 1 II 450 S III 450 (a) h, ~ h { 1+ a,1>9) A, " A( 1+2a<",9) V,=Ah (1 + 30..6e)

volume of liquid Vw "Ah, ( 1+ YL6.e)

(b) h,=11,{ 1+YL6.e} 5. 1. 2. 3. 4.

..,"

"

MG8':

PART - I : OBJECTIVE QUESTIONS

'.

I'C

Sle~m at 100' C is 'passed'into 1.1 kg of water contained In a calorimeter of water equivalent O.02 kg at 15'C till the temperatUre of the calorimeter and its contents rises to SO'C . The mass of the steam

condensed in kilogram is : [JEE '8B, 2)

(A) 0,130 (8) 0.065 (C) 0.260 (0) 0.135

A piece of metal floats on mercury. The coefficient of volume expansion of the metal and mercury are

1, & 12 respectively . If the temperatilres of bOth mercury and the metal are increased by an amount ,

AT , the fraction of the volume of the metal submerged in mercury changes by the factor _' __

(Ratio of final fraction to the initial fraction) pEE '81,2)

Two vertical glass tubes filled with a liquid are connected a.ttheir lower /

r

'&.. ' _

ends by,a horizontal capillary tube. One tube Is surrounded by a bath • i .'

rnr :

C?ntaining }ce and water atO.~and ~e other by hot water at t.C. The I : ~ ~

dIfference In the height of the liquid In the two colum~ Is All, and the

',II :

height of the column at O.C is h". Coefficient of volume expansionof

'JttlD :

the liquid 1II l :

, AI.

A gas thermometer is used as a standard thermometer fOr measurJmentof temperature. When the

. ~ ~ .. , ',-". ~. It ).

gas container of.lhe thermometer Islmniersed in water at ils triple point 273.16 K; the pressure In the

A' ~ --- '~ ,~ ~-- ~

gas thermometer reads 3 x 1O'WiTt':WIleilthe gas container of the same thermometer Is Immersed in

"another system, the gas preuuTe7e2cls3.5 x 10' N/~emperature of this system is therefore. ,

" __ 'C.

A, ~

~,

[JEE ,87, 2J .

Earth receiv~.s 140Q}'-"/~of solar 'p,ower ..If all the so.lar energy faliing on a lens of area 0.2102is

focused on'to a block:pf ice of mass 280 g~niS, the time taken to melt the ice will be minutes.

(Latentlieat affusion ofice-:~.3 x 105''J~. [JEE '87, 2]

,. .PA.

SUBJE:CTlVE QUESTIONS

A 50~"tead 'b~liet. ;sp,eclfic.heat 0.02 callgm Is mllially qt'30' C . It is fired vertically uPr'3rds with a

~peed

.0J

840m/sec &'9n returning to the starting level strikes a cake of ice at 00C. How much ice is

melted. Assumi;'that all energy is spent in m~lting only.[Latent heat of ice

=

SOcallgm

J

'[REE~8,6.J

'The

temperature of 1OOgm of water is to be raised from 24. C to 90'C by adding steam to it~CalCulate

the mass of the steam required for this purpose.', [JEE '9B,21 '

An":Iectricai heating coil was placed In a calorimeter containing 360 gm of water at 100C'. The coil

consumes energy at the rate of 90 watt. The water equivalent of the calorimeter and the co" is 40 gm.

Calculate what will be the temperature of water after 10 minutes, [REE '85, 7].

J

=

4.2 Joules/cal. '

One gram of water (volume = 1 cm') becomes 1671 em' of steam when boiled at a pressure of one

atmosphere. Latent heat of vaporization afthis pressure Is 539 callgm . Compute the worK done.

'[ 1 atm = 1.013 x 10' Nrn-2j ,{REE'8B,3)

A sleal rod 25 em long has a cross-sectional area of O.Scm2• What force would be required to stretch

this rod by the same'amount as the expansion produced by heating it through 10.C? (Coefficient of

linear expansion of steel Is 10"/"C and Young's modulus of steel is - 2 x 10" N/m2.) [JEE '89, 3]

The hrass scale 0'a barometer gives correcl reading at O.C. Coefficient of thermal expansion of brass

Is O.lIOOO2JOC. The barometer reads

i5

ern at 27" C. What is the currecl atmospheric pressure at .

27.C~' '[ JEE 'B9. 21

3.

z.

I.

5.

2.

4.

1.

6, 3.

5.

4.

.'

7. A clock with an iroi, penilulum keeps co,rrect time at 20. C. How much will It lose or gain in a day if the

tem~:erature changes to 40' C? (Coefficient of cubicai expansion of iron

=

0.000036FC) IJEE '80, 3,1

8. Two r~~. of different.'TIelals h.avingsamearea ~f cross section A are placed end to end betwee~~ m?sslVe platforms, as shown In the figure.T~e.f,rstro~ ~asa length l" coefficient of.liriear ex ails' ,.'

. aland Young'lrmodUIUs V,. ~he cor~espondlO9qu;mlitl~s for the seco~d rod are L.,•.a,;.and~: :fD,n

temperallJre of both the rods.ls now Increased.byT"C. Find the force with which the rods act 0 ~,' ..,~e other ( at the higher temperature) Intermsof given quantities. Also find the lengths of the rOdsna~-:;;h higher temperature. Assume lhat thereis nochange In lhe cross sectional area of the rods and that th:

rods do nol bend. Thete is no deformationof lhe walls. (JEE 'so, s 1

9. 10. 11. 12. 14. 15.

A compos~e rod is made by joining a copperrod end to end with a second rod of different,material but of the same Ctoss section. At 25'C lhe compositerod is 1 m in length of which the length~~fthe COpper rod is 30cm.At 125'C the length of the composHerod Increases by 1.91 mm. When ••tI\ecoliiiiOsile rod' is not allowed to expand by holding Hbetween two rigid walls ilis found that the length~of.the two

const~uenls do not change with the rise of temperature: Find the ~~!ts mOdGIUS:~rl!ih'e'iinear

expan~ion of ~he second rod given thai Young's modulus of.for~, PPElr=, 1.3,x 1011,N,Im~ahd,the

coeffiCient ofllnear expansion of copper=1.7 x 10"rC. ~ ..., . (JEE'SO, 4l

A piece of metal weighs 46 9 In air. When ~ Is immersed In a IiqJjdo!.sp.e~gravity1.~4 at 27!C'it

weighs 30 g. When the temperature of liquidis rai!l~~2'C the metal J:!i~cew~ghS 30.5g:Specific

gravity 01 Iiqllid at42'C is 1,20. Calculatethe coeff~e=~inear\e~a,::s:r~TOflhlj metal.[JEE 'Sl,3]

A one !Her flask contains some mercurY.':l!is found that at differenl'tEl.mperallJ~Els.the volume of air .

inside the flask remains. the sam--e;Wtiat!sthe VOlume-€~cury in tlie'f1as,k? Coefficient of linear:

expansion of glass

=

9 x 1O"~~C!Coefficient ofvolume;:expansion of mercury = 1.8x 10" ,. C.

Two Aluminium rod~tee3s.s-secti.~ equal length .'.( JEE 'S1, 3]

t.

are joined'rigidly sidebYsid~as shown'!"figure:.lnltiallythe rods are at

we.

Find . .Aluminium

the tenQlh

af

!hI!.rod al'ttieteltlperature OWyom;9:smodulus of elas~cily of the Steel

aliln:tinium anc stee! are Yrailc y'lrespectivlily'iJid coefficient 01linear expansion Aluminium

• of aluminiutn 'ana,steel ar: ••'tand ~ respectively.

-

~".

~

....,

The attemate diSCSof iron andiitrb'tn. havingsamearea of cross-5ection , are cemented together to make

a.;..cyfn~~r ~n~t:1~ose iempe~ure coe~cient of.res!s~vity is zero. If th~ change in temperature in

~~male dISCS~\!1esame, determinetheratio~elr thickness and the ratiOof heat.produced in them.

crheresistivity of iron and carbon at20"C are 1 x 10 and 3 • 10-5Q-m and their temperature coefficient of

'~siSti'nc;e are 5x 10--'and _ 7.5

;'10.'

per'C, respectively.Neglect thermal expansion. tREE 1988)

4kfir a melal scale of length 30 em erd an object The scale is cafibrated for temp 20'C:

(a) What Is the aetuallength of division which is shown as 1 cm by scale at40OC.

Given a. = 2.10-'

re.

(b) What will be the reading of scale at 40'C if the actual length of object is 10 cm.

(c) What will be the actual length of object at;40 'C if its measured length is 10 em.

(d) What is%error In measurement for part (b) and (c) .

. (e) If the linear expansion coeffiCientof object is a. =4 • 10-' and neglecting the expansion Ofscale

then an~rs of (b) and (c) parts. •.

(~ If a.

=

4x 10-' and a.

=

2 • 10-' then find answers.of (b) and (c) part.

The apparatus shown in the figure consists of lour glass columns 'connected byhorizontal sections

. The height of two central columns B&

c:

are 49 cm each. The two outer columnS A&D are open to th~

atmosphere. A&Care mainlained at a temperature of 95'C While the columns B&D are maintained

at 5'C ..The height of the liquid in A& D mea~ured fromth~ base line are 52.8 em & 51 em respectively.

Determme the coefficient of thermal expanSion olthe liquId. (JEE '97.~!

A OS'

c 0

95- 5'

'-.1

_'----An-s-w-er-s-_

~.

(A)

2.

3.

y=-

Ah

hot

5.

5.5 min

=

330 sec .. : PART -II

.,

42.14"C 75.0405 em

3.

6.

/

129m 1.6x1Q2N

. 2.

5.

52.875gm. 169.171 J 10.3685

AT{Lt«, +l2a2)Y'Y2 .' l + L1l2T{Y1a,

pYia2)

F

=

l Y +l v •length. of the first rod

=,

l v +J,,,v ,

, 2

2',

,'2

•. "

t\= 10 { 1

.,.

+20 "4" 10 -5}

t;

= 10 { 1- 20" 4 " 10-5} (~ .I,= 10{ 1 +40" 10.5}

.t.

= 10 {1-40" .10.'} . y=2"10-4'"C.'

8.

14.

't

4.

7. 11.

9.

15•

MOBSolution

PART .1

\

o'c

I

O'~c

gy !liven by the sun= 14C'0• 0.2 • t

Emr!Jy~rr;qu!rr;d 1,3rneilihe ice" 260 • 10'" • 3.3. 1Q5

\ ~... .." 1~.00'x0.2 • t" 280 " 10~'" 3.3 • 10' . t = 330 s~r.. f{ 1+Y2aT

..

~ =

1+V1AT In equilibrium, •.

.Pressure atA =Pressure :al B

h-t-pg .h) Pog

o 0 . 0 1+yl

3.

5.

1. Heal released by sleam = heal gained by waler and calorimelry

ml+m"S (100-80)=(1.1+ 0.02)"S.(80-15)

m.540+20m=1.12 "65

m = 0.130 kg.

2. For floating condition

mg = B

Vdog.=v••d,g'

where do= density of object d, = densily of liquid

fraction of volume submerged in liquid (f,) = v; do

f,=

d;:-after increasing lemp, by aT

P~RT.1l

1. Heal released by bulleI = heal gained by ice

1 . .~ -mv'+ m SAG= ml 2 I

.!..

(50" 10-') • (840)' "

-.:!..-

+ 50 • 0.02 • (30 - 0)

=

m • 80 2 . U . rn=52.875 gm

2. Heat releasedbysteam = Heal absorbe~ by water

rn,L+ 111,"S (100-90) =m, "S(90-24)

540

rn,

+10

m,

=

66

m,

'66"100

~ ;.1,=' 550- "12r;:m

-'

I, . ,,'.

3.

Energy supplied by coil

=

heat gain by water and calorimetry

=> Pt

=

(m,

+ w) " 8 A9 => 90 x (10 " 60) = (360+40) x 4.2 x(Cl - 10) 9=42.14.C .

I

. (1) Process is isobaric

w=

P(V -V)

=

1.013" 10' x (1671-1) x 10'" . 2 1 . W = 169.171 J F/A . V=-Mil. A[ T=aAO F=VaAA9 :, 2 x 10'. x'(0.8 x 10-')" 10-' x 10 F= 160 N . FL,

e

=

L,a, T - x

= vi;.

6: h

=

h. (1 +aAO) h

=

75(1+0.00002 " 27} h = 75.0405 cm

5.

7.

4.

Gain or loss in time due to thermal expansion

1 . . (. 0.000036 /)~ .1I='2.xA9xt"a (1- 3 I"~ ~ ~~~~~t~~n ~i:G(j sec~' .\ 1 0.000036 \ .~ .JiB';'\ :. At = -2 x '\' i2..3~.. x'20 "~4...".3600 ... illF.'\ \ .•

iJI'\

At=c_4&:!";'"

1

0..3681~~"\_{',....;.,:'. . <tt#._}Yffff •_{.- '.-.}. _• .

Note: If\we increase temperature then.time period increases and watch becomes slow.

8.(l~\+)\:';' \ \. ~

V'

.. l =l"-x0,... 2 ~ \;' ~_{_ F".i~ .} . V - ••

",e

.

\

\

Fd

e~VA'- =extension or compression due to force

.

.

FL2

e

=

L,a, T + x

=

Y A : (2)

• 2

By adding eqmition (1) and equation (2)

We get

Ell' oividif1(!{l) ?nd (2)

. We get

.'

8o,

9.

At = 1.91.mm =At, + At.

=:> 0.191 em

=

t,a,

Ae+

t,a,

Ae.

=:) 0.191 =(30 ~ 17 ~ 10'"+70a,) ~100.

a.=2 ~ 10..0

F

Y

=

-;:;;xe

F,

=

F.

Y,Aa,Ae;= Y.Aa. Ae.

Y,a, 1.3x10" x1.7~10-s Y ---

=

1.105 x 10"N/m' :.• = ~

=

2x10-5 "';

I

I _a'

•

\

i/JI

c:

•

---t,---1--+

!Illffili

B, = (46 - 30)gm =:> S,

=

15.5 gm wt

=

V.P,9 15.5 . - 12

16

=

(1 +3a, ~ 15) ~ 1.24 if a,>~,

.

,

\.

aUI .•C.

-If

rOcls

are free to expand.

t,

=

to(1+a,O) t. =to (1+a,O) F/A y= xlt

Fxi

x=-. AY _ fl(15.5 ~ 1.24)_1lx~ a, - ~ 16 1.2

J

45 Wo

=

mg

=

46 9 wI alO

,

=

27. C W,

=

30 gm

wt

=

Wo- B, alO

=

42.'C

•

W.

=

30.5 9

wt =

Wo- B. B. V,P2

..

~

=

V1P1 ~

12.

11. 10. 2Fx to AY . =io(l +a,AO)-i

•

... (i) Fxto AY

=

i-io (1 + a,AO)

•

bysolving (i)and (ii)

wege!

....:(ii)

3. both discs are in series combination R=R,+R, _PF_t_F+ Pc Ie =~p~Fl_1_+_U~F_T]t~F+ ,-P",e,-11_+_u~c~T)~t~e A A A' 'f"""" A. '" Peue

-=---

=45 te . PFuF . HF PFIF 3 He

=

Pele = 20 114. (a) (b)

(c)

(el) (e) .. i

=

1 { 1+2 x 10-5x20} l:010{1-4x 10-4) t= 10{ 1 +4x10-4} %it

= -

4 X 10-2% . -4x10-2 %l2= 1+4x10-4 %;: -4x 10-2% In the figure-h,

=

52.8 em, .h2

=

51 em and h = 49 em

Now pressure at B = pressure at

c.

Therefore .• p.+ h, Pw g • h PS' g

=

Po+ h2 PS'9 -.h PRS'g

==-

.

PRS' (h, + h )= PS' (h2+ h) PO'" (1+.95r) h2+h'

~

='

h,+h (1+5'() . 1+5y _ 51+49 .100

==-

1+95'1 - 52.8+49 = 101.8

Solving Ihis equation, we gel . \

r= 2 x 10'" per "C.

FliT-JEE Ltd., 48, GlIrlll.ripa Complex, M.P. NagaI; Zone-II, Ilhopal (M.!'.), Ph.: 4253355, 4253455 17

-.-CPP (Heat Transfer) 6th March 11

, HealTranS'er.

Exercise:

.

.

.

#

1

..

Section (A) : Thermal conduction In linear conductors at steady state

A 1. A uniform slab ofdimension 1Oemx10emx1em is kept be.tweentwo Iie<ltreservoirs'aUemPEiiatures 10.C

and 90"C. The larger surface areas touch tI1ereservoirs. Thethermal condllCtivity olthe malerial is 0.80 WI m-.C. Find the amount of heat flowing through the slab per second.

A 2. One end of a steel rod(K=42 J/m-s-.C) of lengUi'1.0 m is kept in ice ~t O.C and the othelJtnd is kept in

boningwater at 1DO.C.The area of cross-section of the rod is O.04cm'. Assuming no heafloss. to the

.' . ,F; ".;, .'.

atmosphere. find the mass of the ice melting per second. Lalent heat of fusi.on of icef3.~;,1 0' J/kg.

. . A I ' / .•..••8

~ 100'C

c.•~

A 3.' ~ A rod CO of thermal resistance 5.0 KJW is joined at the middle of an ). \.:;: •

identical rodAB as shown in figure. The endsA. B andO are maintained atl ~ \, _.._..':.

; 100'C, O.C and 2S.C respectively. Find the heatc~o.

t.,

.'.~e2~C'

~i.~r'

A 4.. A semicircular rod is joined at its ends to a.straight rod;ofthe same material and sameposs.sectional area.

The straight rod forms a diameter,oflhe other rod. The junCti~ns are m'aintalmld at'different temperatures. Find the ratio of the heatlransfelted througfi a cross.setuori of tt,e semicircular rod to the heat bansferred

through a cross-section of th~Straight main a'givan time1' , .' . . .

A 5. Three slabs of Sa~~rla'q ar':'bJt~l~~tcondut~ti~~ktl\" II, and different

tJjjK,

K,

thickne;l,./t.(,.•..t,. t, are;Pla.ce.d,•..,.in••.•'.close CCin!S.,'ct...'\fter s~eadystate this combination t,'1, I,

behaves as a single slab. Find.jts effective thermal conductivity. .

.

.' \j" \\.'~'\

\,T

'.

.

.

Se~B) : Thermal conductlon In nonlinear conductors at steady state

I

:*' ,,/.

"

\

\ '.~

..

.

.

13

1.,;'J,

A hollOwmetallic sp~ere of radius 20 cm surrounds a concentric metallic sphere of radius 5 em. The space

~;f~;;

..

b-e.twee..r'n.the~ sP!l:[llS i.llfilled with a nonmetallic mate)ial. The inner and outer spheres are maintained at

.'~50'C and 10'C respectively and it is found that 16011Joule of heat passes from the Inner sphere to the outer

<""O~;., ..'1, .•••'

'~sphe~e per second. Find the thermal conductivity of the material between the spheres.

B 2. .

A.

hO~ tube has Cllength I, inner radius R, and outer radius R,. The material has thermal conductivity K.

F:indthe heat flowing through the walls of the lube per secorid if the inside of the tube is maintained at

. , te.mperalure T, and theoulside is maintained atT, {assume T

.

2>.T,1

. .' .

Section (C) : Thermal conductIon througll conductors which have not echleved steady, steta

C 1. A metal rod of cross-sectional area 1.0 em2is being healed at one end. At one lime, the temperature gradient

is 5.0'Clem afcross-sec,lionAand is 2.6 .C/em alcross-section B: Calcullite the rate atwllich thetemperature

is increasing in the part~B 01the rod. The heal capacity 01the partAB

=

0.40

Jrc.

thermal conductivity of

the material 01the rod=200.W/m-.C. Neglect any loss of tieat to the atmosph~e. .

SectIon (0) : Radiation, stefen's law and weln's law.

D 1. When q, joules of radiation is incident on a body it reflects and transmIts total of q, joules. Find the

emissivity olthe body. "

D 2. Ablackbody of surfaCe area 1 em' is placed inside an enclosure. The enclosure has a constant temperature

27"C and the blackbody is maintained at 327.C by heating it electrically. What electric pOweris needed to.

maintain the temperature? "

=

6.0 x 10" W/rn" .:.j('. . .

D 3. Estimate the temperature at which a body may appear blue or red. The values of 1._ for these are 5000 and

7500 A respectively. { Given Wein's constant b=0.3 em K I

cpp~

[).4 The tempera\)Jre of.!! hot liquid i~a container o{ negligible heat capacity falls at the rate of.3 Klm!n due:tq heat emissiOn to the surroundings, Just before it begijis to solidifY.l:he temperature.then remains constant for 30 min, by which lime the liquid h'asall slliidified. Find the ratio of specific heat. capacity .of liquid to.

specific latent heat of fusion. .' . .. . ..

Sectlon(E) : Newton's Law.of cooling

E 1.. A liquid cools from ;cioC to S0"6i'k5 minutes. Find tJie lime in which it will further cool down to 50~C, if its

.surrounding is held.at a constant temperature of 30.C. .

,

d

(D) R +. R2

, 2.

.(B) bad absorber is good reflector

(D)bad emitter is good absorber .

(b)

*

Marked Questions are having more than one corrett option.

Section'

(A)::

Thermal conduction In linear conductors "at steady state

A 1. A wall has two layers A and B, each made of different material. Both the layers have the same thickness.

The thermal conductivitY for Ais twiCe that of B. Under steadystate, the temperatureili~ere~e across

~~ ;~~Ie wall is 3S.C. ~~~~;~~ temperature di~;)~~~ across the lay~~ ~~.C

F't~~

. A 2., Two metal cubes with 3 cm-edges of copperandaluminium are arra4~~' . \\~~} ~()...;

as shown in figure '(Ke" =385 Wlm-K, KAt=' 209 W/m~K) ','.' •. ,..' :. ''''''. '

(a) The total thermal C\lrrent from one reservoir-t61he other is :. 100"C

,;;:M;a 2O:C

.

(A) 1.421<jO'W '.' . (B)2.53'{1O'W

\if~t

.

(C) 1.53. ~()4W.4t~)2.53.\~~W~' '-...'

~J7'

. .

The ratio of the t~mal curreptcarried{by th8\. copper cube'lo that carried by the

aluminium cube is :.-. '

-r

'\it-~

(A) 1.79.

A

~(B)~1.69'.. ",r""(C)1.54 . (I?) 1.84

.

~, \

X,

.\ -,?,

• .

.

A 3. A Viall consists of aRernaling blocks, with length 'd' and coeffICient of thermal

,ccinduqlivitYR, and ~'rheCross secli'bmi!ar~ of the blocks are the same. The

eq,uivaient coefflcilinl

Of

thenl\al conductivityoflhe wall between left and right is

.~.~~i\.

~~\~~.,)

.

(e) :~~ .. (D)~K;~, ...

\_]L}:/l~

"> '\ "\\. ...., 2 . . .1 2 ' ..

A 4, -I,~_~~iler il; made~f} copper plate 2.4 mm Ihick with ao (nside coating of a 0,2 mm thick layer oftin: The

\¥=.u.

r.fv.a••.ce.'area exposed IQgases at 700.C is 400C"l'. The amount of sleam that could be generaled per ~ hOIJLat atmospheric pressure Is (K", = 0.9 and K"" = 0.15 .cal/cm/srC and Lite.,. = 540 cal/g)

(A)SOOOKg_,....J . _.(B) 1000 kg . (C)4000 kg (D) 200 kg .

A 5. A la~e surface is exposed to an atmospher:e where the temperature is < O.C. if the thickness of the ice

layer formed on the surface grows from 2 cm.to 4 em in 1 hour, The almospherlc t~mperature, T. will

be-(Thermal conductivity of ice K = 4 x 10" cal/cm/arC; density of ice = 0.9 gmlcc. Latent heal of fusion

of ice = 80 cal/gm. Neglect the change of density during the siale change. Assume Ihat the water below the ice has O. temperature every where)

. (A) - 20 .C (B) 0 .C (C)..: 30 .C (D) -15 .C

Section (8) : Thermal conduction In nonlinear conductors at st,ady. state

B 1. Heat flows radially outward through a spherical shell of outside rSdlusR2 andinnllr radius R,. The

temperature of inner su'rface of shell is 9, and that of outer is 9" The radial distance from cenlre of shell where the temperature is just half way between' 9, and 92 is:

'(A) R,+R, (B) R,R. (C) 2R,R.

. 2 . ~+~~,+~

SectIon. (C) : Racilatlon, stefen's law and weln's law

C 1*. Assume transmllivity t-+ 0 for all the cases:

.(A) bad absorber is bad emitter . (C) bad reflector is good emitter

(0) 0.03 .C/sec (C) .044 'C/sec

C 2*, A hollow and' a solid sphere 01 same material and having identical outer slirface are heated under

. identical conditiori .lotlie same temperature at the same time (toth h~ve same e, iI) : . .'

(A) in the beginning both will emit e~ual amount 01 radiation per unit time . .

. (B) in the beginninll both will absorb equal amount of radiation per unit time

(C) both spheres will have same rate offall oftemperature (dT/dt)

(0) both spheres will have equal temperatures at any moment

C 3. A metallic sphere having radius 0.08 m and mass iii=10kg is'heated to a temperature

of

227'C and

suspended Inside a box whose walls are at iI temperature of 27*C. The maximum rate at which its temperature will falUs

:-(Take

e"

1, Stefan's'constant (J= 5.8 x 10"Wm" K" and specific heat oithe metal s =90callkg/deg

J

=

4.2 Joules/Calorie)

(A) .055 'C/sec (B) .066 'C/sec

Section (0) : Newton's L!1w of cooling

0-1._ Which of the law can be understood in terms of Stefan;s'law'

-ij

(A) Wien's displacement law (B) Kirchoffs law

A

z'

(C) Newton's law of cooling . (0) Planck's law

I'.~~?;~.

0-2,_ A hot liquid is kepI in a big room. Rate 01cooling of liquid (represented as y) is plotted against its temperature

iilI/~' .••.•..':illi;

'.,d'

T.WhichOfthefOIlOwingCurvesmayrepresentthep(~ ~

Jf'

~

y~

Yi

At:; .

T~';~ '

~i

_? (A)

Ll- .(~)

W'

(C)

M

(D)

'L

+T . T~! ~. \~

~.YV

T

\;.~i~

\~(,;"

V

bercis9#2

.' ".. .,~... .

1, '"Figure shows a'sleel rdd joined to a brass rod. Each of the rods has length of 31 cm and area of

""cross-(ection 0.20'cm" ~The junction is maintained at a constant temperature 50~C and the two ends are

maintained at 100'C. Calculate the amount of heat taken out from the cold junction in 10 minutes after the

.~ea.dy stille is reached. The thermal conductivities are

K_

= 46 W/m-'C and K",." =109 W/n .•..••C.

~ . 50'C

.100'C

I

.steel 1--B-ra-SS--\100'C

2, Consider the situation shown in figure. The frame is made of the same malerial and has a uniform

.cross-sectional area everywhere. II amount of heat flowing per second througti a cross~sectionolthe benl <'part is 60 J. Calculate the amount of total heat taken out per second from the end at50"C.

50em

O'CI 5cml

10cm

I

10cm

150'C

3.

Seven rods A. B, C, 0, E,F and G are joined as shown in figure. All the

rods have equal cross-sectional area A and length I. The thermal

qanductivilies of the rods are K. = 2K,;

=

31<"= 61<0

=

K" . The rod E is

kept at a constant temperature 01 and the rod G is kept at a constant temperature 0,(0, > 0,). (a) Show that the rod F has a uniform temperature 0='(30, + 0,)14. (b) Find the rate of he~t flow from the source which

maintains the temperature 0,; . .

9,

E

A

B

0,

F

G

C 0

4. Four lhin identical rods

AB.

AC, J3D and EF made of the, same malerial are joined as

shown. The !ree-end$ C. 0 anili' !Ire l]1i1inl?ined at temperatures

t,;

T

2 and

T~

respectively. Assiim,ing that there is no loss of heat to the surroundings, find the'

temperature at joint E when the steady state is attained.

_S-J17.C

I

O.C 27.C 5. 6.

7.

8.

9. 10.. 11.

12.

One end of copper rod,of uniform cross-section and of length 1:45 m is in contact with ice at O.C and ' the other end with water at 100'C, Find the position of point along its length where a temperature of 200.C should !Je maintained so that in steady state tlie mass of ice melling is equal to that of steam

produced in The same inlerval oftime [Assume thatttie wholesystem'is l,nsulated from surroundings).

(take

l;

= 540 cal/gL,

'=

80 cal/g) " '

Find the rate of heat flow through a cross~sec~on <;Ifthe rod shown in figure (TH> T c), Thermal conductivity of

the material of the rod is K

R~a_)__ L__

ffi:~

" T">Te H ~

A hollow spherical 'conducting shell of inner radius R, = 0.25 m an~ 'oH!e~~ir;,

radius R, = 0.50 m is placed inside a heat reservoir oft~perature T.~ 1090

"C. The shell is initially filledwiih water at O.C. The thermal conductivitY of

the

material is k = ~: W/m-K and itshe~~iadty is nJig;~I~ind th~ine"-.. '.~

, . A,,/'

\,' .\

'

K

'.-required to raise the temperature of water ,to-(1OO.C.Take sRBcific heat of ,".

'.-,

"','~

V'.-;;

okc\

'A,

--'Ii; f R,

4. '-

'"

'\'

,

,,:)j,~22,

water s = 4.2 kJ/kg .•

Gi,

de~sity of water ~;=1000kg/m~, ..Jt=

"'7

A\,

,;~S \~\

" ' '

A cylindrical roo of lengthJ' m is.filted beme,en a large ice chamber at O.C and an evacuated chamber

,!ffaii\~ined ~m;c~sshciv;n(~l'~u,re: Only

~mall

portio~s of ~e ~ are inside llie chambers arid the rest

, '" ,1,5thermally IOsulated from the surrounding. The cross-,seCtiongOing IOtothe evacuated chamberis blackened

'f_4'~ . 'l'-'-"-"i"-.'. " "'" -~_"" :.'0"',- • •

',50that itcompletely'alisorDs any radiation falling on it. The temperature of the blackened end is 17.C when

-'i"'; ': '\. 'l- _,: .,,-'t., '".. . •

sle?dy state is reacl1ed!Stefan constanla = 6 x 1O"W/m'-l(4. Find the thermal conductivity of the material

> --:"'10<.:-- -oW'" " -~"--' 'W-'-._ •

oflhe,rod. (29' =707c281) ,

"-';~~~

~

V

1 ;.

A spherical tungsten piece of radius 1.0 em is suspended in an evacu'!tedchamber maintained at 300 K. The

'piece is maintained at1 000 K by heating it electrically. Find the rate at which the electrical energy must be

supplied: The emissivity of tungsten is 0.30 and the slefan constant ais 6.0 x 10-" W/m2-K4.

A solid aluminium sphere and a soHdcopper sphere of twice the radius of aluminium are heated to the same temperature and are allowed to cool under identical surrounding temperalOres. Assume that the emissivity of both the sphere is the same. Find the initial ratio of (a) the rate of heat loss from the aluminium sphere to the , rate of heat loss from the cOpper sphere and (b) the rate of fall of temperature of the aluminium sphere to the rate of fall of temperature of the copper sphere. The specific heat capacity of aluminium = 900 J/kg- OCand

that of copper

'=

390J/kg-.C. The density of copper';' 3.4 times the density of aluminium. ' '

A hot body placed in a surrounding oftemperalure T •. lts temperature all =0 is T,. The'spedfiel1eatcapa~

olthe body is s and Its mass is m. Assuming Newton's law of cooling 10be valid, find (a) ~ maximum heat

,thalthe body can lose and (b) the time starting from t = 0 in,which it will lose 50% of this maximum heal

Find the iotai tilhe elapsed for a hollow copper sphere of ;nner radi~s 3 em outer iadiiis

6

em; density p = 9

x 10' kg/m', specific heals

=

4 x 10' J/kgK and emissivitye = 0.4 to cool from 727"C to227'C when the

surrounding temperature is 0 K(for inner suiface e

=

1 Stefan's constant a

=

5:6 x 1D" W/m2 K4)

~t.,_!:I.~;;".'~'~.-.," ., ••.. t, ; B 4"C Insulating \Nalls Source . of Energy --l t-8cm iO.C (D) 31/656 (D) 86

'c

and 57

'c

(A) 0.75 min

4.

.

..,. 2.

13._ A metal ball of mass 2 k9.Is'healed by means of a 40

W

heater.in.a room at 25'C. The temperature.of the ball

becomeS steady at60'C: (a) Find the rate of loss of heat to the surrounding When the' ball is at 60'.C. (b)

Assuming Newton's law.of cooling, calculate the rate of loss Qf heat to the surrounding ~en the ball is at

39'C. (c) Assume that the temperature of the ball rises uniformly from 25'C 1039'C in~ minutes. Find the totallo~s of heat to thE!surrounding during this period. (d) Calculate the specific heatc;apacitY of the metal.

14•...

> ..

Airietalblock of heat capacity 90 JiC placed in a rOOl!)at 25'C Is heated eleCtrically. The heater Is switched

", .','offwhenlhe temperature reaches 35~G.The temperature of the block rises at the rate of2'C/s'justaflerthe

" .•heater-is'switched on and falls althe rate of 0.2 'Cis just after the healeris switched off. Assume Newton's .. law.of cooling ~ hold. (a) Find the power of the heater. (b) Find the power radiated by the block just after the heater Is switched off. (c) Find the power radiated by the block when the temperatUre of the block is 30.C. (d) Assuming that the power radiated at 30'C respresents the average value in the heating process, find the time

for which the heater Was kflpton. .'

~Wi1t.il»lt!gmmiQDm_1II

Single

choice

type

."

At. ..)

1. Two identical square rods of metal are welded end to end as shown in figure (a), Assume that 10 cal of

heat flows through the rods in 2 min. Now the rods are welded as shown,in'figure, (b).The'ti~il~ould

take for 10 cal to flow through the rods now, is . .

6sf'

"i..'"

"

. O'C

, (b) .

. (8) 0~5 mi; ~ (C) 1.5 min? ) (0) 1 min

~ """'" F, ./

,v~

Three metal rods made of. copper, aluminium and brass, each 20 cm long and 4 em in diameter, are

"\ 'l\.. 't'- ~ ~ or-.

placed end to end with aluminium between the other, two. Tne free ends of copper and brass are

maintaine<fat'100 and O'C respectively; Assume that the thermal conductivity of copper is twice that of

. aJuminiiYm and four tirrles tt1~t'ot brass.'rheJipproximately equilibrium temperatures of the copper.

...•. . \_.;~... "*i: -i\, -wr- .

. :, alum.inium,and ailJmlnium.brassjunctions are respectively .

.1~,:tP,',,',.", '\~(\,','

'"

~J:'

T, . Ai. T, brass

./< ~ ,\OOOCI • A

•• ~{ .~)' ~A ';'

t

(A.L68.C and 75 'C (B) 75 'C and 68

'c

(C) 57"C and 86

'c

A closed cubical box is made of a perfectly insulating material walls of thiCkness 8 cm and the only way for heat to enter or leave the box is

through two solid metallic cylindrical plugs, each of cross-sectional

area 12 em' and length 8 em, fixed in the opposite walls of the box.

".,.,> outer surfaceAon one plug is maintained at100'C while the outer A

~urface'8 of the other plug is maintained at 4'C. The thermal 100'C

conductivity of the material of each plug is 0.5 calrC/cm. A source of

energy generating 36 calfs is enclosed inside the box. Assuming the temperature to be the same at ali points on the inner surface, the equilibrium temperature of the Inner surface of the box is

W~~

~~~

~m~

~~~

Two models of a windowpane ani made. In one model, two identical glass panes of thickneSs. 3 mm are se'parated'iVith an air gap of 3 niin. This composite system Is fixed in the window of a room. The other

n" .,/. }'~dn'sists of a single glass pane of thickness 6 mm, the temperature difference being the same

t.:';'\';'!le first model. The ratio of the heat flow for-the double pane to that for the single pa'ne is

(K•••••.=2.5 x1~ caVs,m .."C andK ••.=6,2 x 1~ caVs.m.'C)

(A) 1/20 '. . (8) 1nO . (C)31/1312

O. , tceWater Chamber ,C , 'A 1,00,. Steam Chamber

keq1pe,r rod\l~iti~J1}\at iOJlntemperature 20.C}of non.uniform ~~oss section is placed between a

team chambeh~j)o:c and ice-water chamber at D.C. ,', ,

, 'Conduclingrod

5. A spherical solid black body of radius 'r' tl1d.iates:po~er 'H' a,nd its rate of coC1lingis 'C'. If,density Is,

constant then whlCh,of thefoll~wlngiStari!'iNe: .' " . '", " ,

"

,"

'1'

"

1

(A) H a::rand i: a::r2 (B) H a::r' and c ",' -' (C}H'

ci:

rand ca:: 2' (D) Ha::,r' and c a::r'

,r

r

More than one choice type

6. T~ bOdies Aand S'have thernlal emissMlles of0.01 and O~81respectiVely. Thesuitai:e areas of the

two bodies ar~ the same. The two b~ies emit total nidi~nt power at the same rate. The Wavelength i.e corresponding to maximum spectral radiancY in the radiation from'S is shifted from the wavelength

, corresponding to maximum spectral radiancy in the radiation frolil,Aby 1.00 11M. If the temperature of

A 11\5802 K," , , ' [JEE't4~21

(A) the temp~rature of B is 1934 K " (8) A.

=

,1.5II~ '

(C) the temperature of 81s11604 K: (O) the temperature ,of Sis 2901 K

4

'

7. The solar constant is the amount of heat energy received per second pet unitarea of a perfectly blacll

• - ~ ..2IfllIt"

surface placed at a mean distance of the Earth from the Sun, in the absence of Earth's,atmosph!lre,

the'surface bei,ng held perpendicular to !he direction of sun's rays. I~v~ue is 13~ri1~ '

If the solar constantfor the earth is's'. The surface temperature of the siln,is TK. The sun subtends II'

small angle 'e' at the earth. Then correct options is/are. :-:- '~." ...'

(A) s a::T'(B) s a::T' (C) s ,a::e' (D)"s a::e ~

8, A heated body emits radiation whjc~ has maximlliTi'ftnslly at frllql,lenc"m' IUhe temperatur~ of thi!

body is doubled: " ,~,' ~ ' ' , ,

(A) the maximum intensity radiationviillbe at frequency 2 v,i, ,

(8) the maximum intensityradiatlE! 'wmbeat frequelil:y'"'';,

(C) the total emitted power tiirincrease'by_{• . AlA ,<-. ---...} IIfactor 16_.

(0) the total emitted'power Willlricf!lase, bya factor'2.

, _{. .} ' _. '<r(p7O :'_{~- .} •_liBIU"...:.. ,

t . _ : . _

_

:.b.lse#3_

. ~ 'r,«~"''''A. . 'r'-~~."- . ,. "'

, ", (A) Initially rate o~ heat flow (d~) will be ,

(B)AtS~eadY state rate ~f heat flow (~)~lIbe','

,(~i

At steady stat~ lemperature gradient

\(:)1

wiUbe '

. . . .' .

, ' (0) At steady state, rale of change'of

-'.

".

(p) maximum at sectioriA

'(q) maximum at section B

(r) minimum at sectl~n A (s)' minimum at section B

" 'temperature (:) ala certain point will be (t) same ror all section '

".,':I.,?,,('.",'.I~"':W:.',.'CT.: t,--,

-~,:;c.:;;.7i-,.Ti""';T-~:~'~---,<,'.";

"1.< ..•.,'~""..' -.,'jl~'~-.~

(s) Transmission(refraction) of radiation will always 'Occur

!o .: Matchthe statementsin~olumn-l with the statementsin column-n.

Coillmn-l Column-U'

(A) For a perfect black body (p) Absorption of radiationoccurs

(6) For a perfectly polishedwhite body (q) Emission of radiation occurs

(C) When radiation frem air is incident (r) Reflection of radiation will always occur

on a perfectlytransparent medium .

:of greaterrefraclive Index'

,.{O) When radiationmoves from a transparent mediumof greater refractive index to air

(All conditions are for temperatureT>0 K)

,.

_i'ltiiilMPJlijBti-

.

Cempreh!!RSion # l' .'

4:.

Figure showsin'cross section a wall consistingof four layerswith ttiermalconductivitle$ Kt;=Q.06WI

mK;K;= 0.04 W/r'nKand K4= 0.10 W/mK. The layer thicknesses areII = 1.5 cm.;'L':'=,2.8.cm and

L4= 3'.S'cm. The temperature of interfaces is as shown in figure. Energy'transfer tlirougnih~ w~lI is

steady. ~

, layer 1 , layer 2, Layer"3. layer '4 , .~

:

:

:

.

,:

'

i

K,\ K,' K,\'K.itJ

30.~

iil ;:

pO.C

.;,

..,

~,..

*t:;--+:~+-Ll~J!.-3~,f;i.'{,q'M temperature-ofthe mterface belween layers3 and 4IS: .

~~ifj~1~~a

..

'

".-,(C)2.C

..

(O)O.C

4., "he temperature 6f the interface betweenlayers2 and 3 is: .

'(A)1;1.C.(6)~

(C)7.2.C . (0)5.4.C .

5. 'If layerthlckness 1.!2 is:'1.4 cm, then its thermal conductivity~ will have value (In W/mK) :

(A)f;<1o-2 ,..,. (6) 2.104 (C) 4x 10-2 (0) 4x 10"" \

Comprehension II 2

~'~y coolsin a surroundingof constanttemperature30-C.Its heat capacityis 2JI"C.lnttialtemperature

of lhe body is 40'C . Assume Newton's law of cooling is vaiid. The body cools to 38'C in 10 minutes.

6. In further 10 minutes it will cool from 38'C to:

(A) 36'C (B)36.4'C (C) 37'C (0) 37.5'C

7. - The temperature of the body in 'C denoled by 0 lhe variation of 0 versus time t is best denoted as.

40'C _t4cfC~ 40'C 4r!c t

t

t

•

~30'C ...•....••...••• 9 30'C ...•....•.•. 3cfc ...•.•...•.. (A)

(6)

(C)

(0) \- \- I....• (0,0

I-8. When the body temperature has reached 38 .C, it is heated again so that it reaches to 40'C in 10

minutes .Thetotal heat required from a heater by the body is:

(A)3.6J (B50.364J (C) 8 J (0) 4 J