Page - 1

QUEST TUTORIALS

Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439

HEAT & THERMODYNAMICS

PROBLEMS ASKED IN JEE & REE

Q 1. State whether True of False : (with reason) [ JEE '87, 2 + 2 ]

(a) At a given temperature, the specific heat of a gas at constant pressure is always

greater than its specific heat at constant volumes .

(b) The root mean square (rms) speed of oxygen molecules (O2) at a certain temperature

T (degree absolute) is V . If temperature is doubled and oxygen gas dissociates

into oxygen atoms, the rms speed is unchanged .

Q 2. An ideal gas has a specific heat at constant pressure Cp =

5

2

R . The gas is kept in a closed

vessel of volume 0.0083 m3 at a temperature of 300 ºK & a pressure of 1.6 x 106 N/m2 .

An amount of 2.49 x 104 Joules of heat energy is supplied to the gas . Calculate the final

temperature & pressure of the gas . [ JEE '87, 7 ]

Q 3. A container is divided into two equal portions . One portion contains an ideal gas at

pressure P and temperature T, while the other portion has a perfect vaccum . If a hole is

opened between the two portions, find the change in energy and change in temperature

of the gas . [ REE '87, 3 ]

Q 4. 2 M3 volume of gas at a pressure of 4 x 105 Nm−2 is compressed adiabatically so that the

volume becomes 0.5 M3 . Find the new pressure that would result, if the compression

was isothermal . Calculate work done in each process . [ γ = 1.4 ] [ REE '87, 7 ]

Q 5. Two moles of helium gas γ =

5

3 are initially at temperature 27º C and occupy a volume

of 20 litres . The gas is first expanded at constant pressure unitl the volume is doubled .

Then it undergoes an adiabtic change until the temperature returns to its initial value .

(i) sketch the process on a p-V diagram .

(ii) what are final volume and pressure of the gas ?

(iii) what is the work done by the gas ? [ JEE '88, 6 ]

Q 6. State whether True or False : (give reasons) [ JEE '88, 2 ]

2 spheres of the same material have radii 1 m & 4 m and temperature 4000º K & 2000º K

respectively . The energy radiated per second by the first sphere is greater than that by

the second .

Q 7. In a certain region of space there are only 5 molecules per cm3 on an average . The

temperature there is 3º K . What is the average pressure of this very dilute gas ?

[ REE '88, 5 ]

Q 8. A 50 gm lead bullet, specific heat 0.02 is initially at 30º C . It is fired vertically upwards

with a speed of 840 m/sec & on returning to the starting level strikes a cake of ice at 0º C.

How much ice is melted . Assume that all energy is spent in melting only .

[ latent heat of ice = 80 cal/gm ] [ REE '88, 5 ]

Q 9. hydrogen atom in its ground state is excited by means of a monochromatic radiation of

wavelength 970.6 Aº . How many different wavelengths are possible in the resulting

emission spectrum ? Find the longest wavelength amongst these .

[ REE '89, 7 ]

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QUEST TUTORIALS

Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439

Q 10. An ideal monoatomic gas is confined in a

cylinder by a spring loaded piston of cross-section

8 x 10−3 m2 . Initially the gas is at 300 K and occupies

a volume of 2.4 x 10−3 m3 and the spring is in its

releaxed (unstretched, uncompressed) state as in

figure . The gas is heated by a small electric heater

until the piston moves out slowly by 0.1 cm .

Calculte the final temperature of the gas & the heat

supplied (in joules) by the heater . The force constant of spring is 8000 N/m, atmospheric

pressure is 1.0 x 105 N/m2 . The cylinder & the piston are thermally insulated . The

pistion is massless & there is no friction between the piston & the cylinder . Neglect heat

loss through the lead wires of the heater. The heat capacity of the heater coil is negligible

. Assume spring to be massless.

[ JEE '89, 8 ]

Q 11. Two perfect gases at absolute temperatures T1 & T

2 are mixed . There is no loss of energy.

Find the temperature of the mixture if the masses of the molecules are m1 & m

2 and the

number of molecules in the gases are n1 & n

2 respectively . [ REE '89, 5 ]

Q 12. Hydrogen atom in its ground state is excited by means of a monochromatic radiation of

wavelength 970.6 Aº . How many different wavelengths are possible in the resulting

emission spectrum ? Find the longest wavelength amongst these . [ REE '89, 3 ]

Q 13. An ideal gas having initial pressure P, volume V and temperature T is allowed to expand

adiabatically until its volume becomes 5.66 V while its temperature falls to T

2 .

(i) How many degrees of freedom do the gas molecules have ?

(ii) Obtain the work done by the gas during the expansion as a function of the initial

pressure p and volume V . [ JEE '90, 7 ]

Q 14. 3 moles of an ideal gas at 300º K are isothermally expanded to five times its volume and

heated at this constant volume so that the pressure is raised to its initial value before

expansion . In the whole process 83.14 kJ heat is required . Calculate the ratio C

C

p

v

of the

gas . [ ln 5 = 1.61 ] [ REE '90, 5 ]

Q 15. A piston divides a closed gas cylinder into two parts . Initially the piston is kept pressed

such that one part has a pressure P & volume 5 V and the other part has pressure 8 P &

volume V . The piston is now left free . Find the new pressures & volumes for the adiabtic

and isothermal processes . [ for this gas γ = 1.5 ] [ REE '90, 7 ]

Q 16. 3 moles of an ideal gas CR

p =

7

2 at pressure P

A & temperature T

A is isothermally expanded

to twice its initial volume . It is then compressed at constant pressure to its original volume.

Finally the gas is compressed at constant volume to its original pressure PA .

(a) Sketch P−V & P−T diagrams for complete process .

(b) Calculate the net work done by the gas and net heat supplied to the gas during the

complete process . [ JEE '91, 8 ]

Q 17. 0.014 kg of nitrogen is enclosed in a vessel at a temperature of 27º C . How much heat

has to be transfered to the gas to double the r.m.s. velocity of its molecules .

[ REE '91, 7 ]

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QUEST TUTORIALS

Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439

Q 18. 10−4 kg oxygen is contained in a non-conducting cylinderical vessel of cross sectional

area 29.15 x 10−5 m2 & volume 10−3 m3 . It is closed by a frictionless non-conducting

pistion which is free to move in a vertical direction . If the piston is slightly depressed

and then released, show that the piston will execute simple harmonic motion . Find its

frequency . Ignore atmospheric pressure & take C

C

p

v

= 1.4 for oxygen .

[ REE '91, 7 ]

Q 19. A solid copper sphere (density d & specific heat C) of radius r at an initial temperature

200 K is suspended inside a chamber whose walls are at almost 0 K . The time required

for the t emperature of the sphere to drop to 100 K is :______ . [ JEE '91, 2 ]

Q 20. A piece of metal floats on mercury . The co-efficients of volume expansion of the metal

and mercury are Y1 & Y

2 respectively . If the temperatures of both mercury and the metal

are increased by an amount T, the fraction of the volume of the metal submerged in

mercury changes by the factor ______ . [ JEE '91, 2 ]

Q 21. A point source of heat of power P is placed at the centre of a spherical shell of mean

radius R . The material of the shell has thermal conductivity K . If the temperature difference

between the outer & inner surface of the shell is not to exceed T, the thickness of the shell

should not be less then ______ . [ JEE '91, 2 ]

Q 22. Two moles of helium gas undergo a cyclic process as shown

in figure . Assuming the gas to be ideal , calculate :

(a) the net change in the heat energy .

(b) the net work done .

(c) the net change in internal energy .

[ JEE '92, 8 ]

Q 23. How much energy is absorbed by 10 kg mole of an ideal gas if it expands from an initial

pressure of 8 atmosphere to 4 atmosphere at a constant temperature of 27º C ?

[ REE '92, 5 ]

Q 24. A cylindrical block of length 0.4 m and area of cross-section 0.04 m2 is placed co-axially

on a thin metal disc of mass 0.4 kg & of the same cross-section . The upper face of the

cylinder is mainitained at a constant temperature of 400 K & the initial temperature of the

disc is 300 K . If the thermal conductivity of the material of cylinder is 10 Watt/m−K &

the specific heat of the material of the disc is 600 J/kg−K , how long will it take for the

temperature of the disc to increase to 350 K ? Assume , for purposes of calculation , the

thermal conductivity of the disc to be very high and the system to be thermally insulated

except for the upper face of the cylinder . [ JEE '92, 6 ]

Q 25. A container of volume 1 m3 is divided into equal parts by a partition . One part has an

ideal gas at 300 K and the other part is vacuum . The whole system is thermally isolated

from the surroundings . When the partition is removed the gas expands to occupy the

whole volume . Its temperature will now be ______ . [ JEE '93, 1 ]

Q 26. One mole of mono-atomic ideal gas is taken through the cycle

shown in the figure .

A → B Adiabatic expansion

B → C Cooling at constant volume

C → D Adiabatic compression

D → A Heating at constant volume .

×

Ú

B

C

2 a

tm

Ù

Ø- - - - - -A

D- - - - - -

- - -

- -

- -

- -

- - -

- -

- -

- -

→ P

1 a

tmT300 K 400 K

→ P

→ V

ÙÚ

×

ØA

D

B

C

Page - 4

QUEST TUTORIALS

Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439

The pressure & temperature at A, B, etc are denoted by PA, T

A, P

B, T

B etc. respectively .

Given that TA = 1000 K , P

B =

2

3 P

A & P

C =

1

3 P

A

, calculate the following quantities :

(i) The work done by the gas in the process A → B

(ii) The heat lost by the gas in the process B → C

(iii) The temperature TD . [ JEE '93, 4 + 4 + 2 ]

Q 27. State whether True or False : (give reason) [ REE '93, 2 ]

If an ideal gas expands adiabatically, it does positive work and its internal energy

decreases .

Q 28. In an isobaric process , the change in internal energy is dU = n Cp dT .

Q 29. One mole of a mono-atomic gas is mixed with 3 moles of a diatomic gas . The molar

specific heat of the mixture at constant volume is ______ . [ REE '93, 1 ]

Q 30. In the figure are shown three isothermals at temperatures

T1 = 4000º K , T

2 = 2000º K . When 1 gm mole of an ideal

monoatomic gas is taken through the paths A → B ,

B → C , C → D & D → A , find the change in

internal energy U , the work done by the gas W & the heat

absorbed by the gas Q in each path . Also find these

quantities for the complete cycle ABCDA. Given VA = 1 m3

and VB = 2 m3 .

[ REE '93, 6 ]

Q 31. The kinetic energy, due to translational motion, of most of the molecules of an ideal gas

at absolute temperature T is ______ . [ REE '94, 1 ]

Q 32. A vessel of volume 2 x 10−2 m3 contains a mixture of hydrogen and helium at 47º C

temperature and 4.15 x 105 N/m2 pressure . The mass of the mixture is 10−2 kg . Calculate

the masses of hydrogen and helium in the given mixture . [ REE '94, 4 ]

Q 33. There are two vessels . Each of them contains one mole of a mono-atomic ideal gas .

Initial volume of the gas in each vessel is 8.3 x 10−3 m3 at 27º C . Equal amount of heat is

supplied to each vessel . In one of the vessels, the volume of the gas is doubled without

change in its internal energy, whereas the volume of the gas is held constant in the second

vessel . The vessels are now connected to allow free mixing of the gas . Find the final

temperature and pressure of the combined gas system . [ REE '94, 6 ]

Q 34. An ideal gas with pressure P, volume V & temperature T is expanded isothermally to a

volume 2 V and a final pressure Pi . If the same gas is expanded adiabatically to a volume

2 V, the final pressure is Pa . The ratio of the specific heats for the gas is 1.67 . The ratio

P

P

a

i

is ______ . [ JEE '94, 2 ]

Q 35. An ideal gas is taken through a cyclic thermodynamic process through four steps . The

amounts of heat involved in these steps are Q1 = 5960 J, Q

2 = − 5585 J, Q

3 = − 2980 J

and Q4 = 3645 J respectively . The corresponding works involved are W

1 = 2200 J ,

W2 = − 825 J , W

3 = − 1100 J and W

4 respectively .

(i) Find the value of W4 . (ii) What is the efficiency of the cycle ?

[ JEE '94, 6 ]

Q 36. A closed container of volume 0.02 m3 contains a mixture of neon and argon gases, at a

temperature of 27º C & pressure of 1 x 105 Nm−2 . The total mass of the mixture is 28 gm.

B

→ P

→ V

→

←

↑ ↓

A

D C

T3

T2

T1

VA V

B

Page - 5

QUEST TUTORIALS

Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439

If the gram molecular weights of neon and argon are 20 & 40 respectively , find the

masses of the individual gases in the container, assuming them to be ideal .

[ Universal gas constant R = 8.314 J/mol K ] [ JEE '94, 6 ]

Q 37. A gaseous mixture enclosed in a vessel of volume V consists of one gram mole of a gas

A with τ = =C

C

p

v

5

3 & another gas B with τ =

7

5 at a certain temperature T . The gram

molecular weights of the gases A & B are 4 & 32 respectively . The gases A & B do not

react with each other and are assumed to be ideal . The gaseous mixture follows the

equation ; PV19/13 = cont. in adiabatic processes .

(A) Find the number of gram moles of the gas B in the gaseous mixture .

(B) Compute the speed of sound in the gaseous mixture at T = 300 K .

(C) If T is raised by 1 K from 300 K, find the percentage change in the speed of

sound in the gaseous mixture .

(D) The mixture is compressed adiabatically to 1

5 its initial volume V . Find

the change in its adiabatic compressibility in terms of the given quantities.

[ JEE '95, 1 + 1 + 1 + 10 ]

Q 38. At 27º C two moles of an ideal monoatomic gas occupy a volume V . The gas expands

adiabatically to a volume 2V . Calculate :

(i) the final temperature of the gas , (ii) change in its internal energy &

(iii) the work done by the gas during the process . [ JEE '96, 5 ]

Q 39. The temperature of 100 gm of water is to be raised from 24º C to 90º C by adding steam to

it . Calculate the mass of the steam required for this purpose . [ JEE '96, 2 ]

Q 40. Two metal cubes A & B of same size are arranged as shown in

figure . The extreme ends of the combination are maintained at

the indicated temperatures . The arrangement is thermally

insulated . The co-efficients of thermal conductivity of A & B

are 300 W/mº C and 200 W/mº C respectively . After steady state

is reached the temperature T of the interface will be ______ .

[ JEE '96, 2 ]

Q 41. There is a soap bubble of radius 2.4 x 10−4 m in air cylinder which is originally at the

pressure of 105 Nm−2 . The air in the cylinder is now compressed isothermally until the

radius of the bubble is halved . Calculate now the pressure of air in the cylinder . The

surface tension of the soap film is 0.08 Nm−1 . [ REE '96, 5 ]

Q 42. A vertical hollow cylinder contains an ideal gas . The gas is enclosed by a 5kg movable

piston with an area of cross-section 5 x 10−3 m2 . Now, the gas is heated from 300 K to

350 K and the piston rises by 0.1 m . The piston is now clamped at this position and the

gas is cooled back to 300 K . Find the difference between the heat energy added during

heating process & energy lost during the cooling process . [1 atm pressure = 105 N m−2]

[ REE '96, 5 ]

Q 43. A double-pane window used for insulating a room thermally from outside consists of two

glass sheets each of area 1 m2 & thickness 0.01 m separated by a 0.05 m thick stagnant air

space . In the steady state, the room-glass interface and the glass-outdoor interface are at

constant temperatures of 27º C & 0º C respectively . Calculate the rate of heat flow through

10

0º C

0º C

T

A B

Page - 6

QUEST TUTORIALS

Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439

the window pane . Also find the temperatures of other interfaces . Given, thermal

conductivities of glass and air as 0.8 & 0.08 Wm−1 K−1 respectively . [ JEE '97, 5 ]

Q 44. A sample of 2 kg of monoatomic Helium (assumed ideal) is taken

through the process ABC and another sample of 2 kg of the

same gas is taken through the process ADC as in figure . Given,

molecular mass of Helium = 4 .

(i) what is the temperature of Helium in each of the states

A, B, C & D ?

(ii) Is there any way of telling afterwards which sample of

Helium went through the process ABC and which went

through the process ADC ? Write Yes or No .

(iii) How much is the heat involved in each of the processes ABC & ADC .

[ JEE '97, 5 ]

Q 45. A gas thermometer is used as a standard thermometer for measurement of temperature .

When the gas container of the thermometer is immersed in water at its triple point

273.16 K, the pressure in the gas thermometer reads 3 x 104 N/m2 . When the gas

container of the same thermometer is immersed in another system, the gas pressure reads

3.5 x 104 N/m2 . The temperature of this system is therefore ______ ºC . [ JEE ,97, 2 ]

Q 46. The average translational kinetic energy of a molecule in a gas becomes equal to 1 eV at

a temperature ______ . [ REE '97, 1 ]

Q 47. Two moles of an ideal monoatomic gas are confined within a cylinder by a massless &

frictionless spring loaded piston of cross-sectional area 4 x 10−3 m2 . The spring is,

initially in its relaxed state . Now the gas is heated by an electric heated, placed inside

the cylinder, for some time . During this time, the gas expands and does 50 J of work in

moving the piston through a distance 0.10 m . The temperature of the gas increases by

50 K . Calculate the spring constant & the heat supplied by the heater . [ REE '97, 5 ]

Q 48. Two vessels A & B, thermally insulated, contain an ideal monoatomic gas . A small

tube fitted with a valve connects these vessels . Initially the vessel A has 2 litres of

gas at 300 K and 2 x 105 N m−2 pressure while vessel B has 4 litres of gas at 350 K &

4 x 105 N m−2 pressure . The valve is now opened and the system reaches equilibrium in

pressure & temperature . Calculate the new pressure & temperature . [ REE '97, 5 ]

Q 49. One mole of a diatomic ideal gas (γ º = 1.4) is taken through a cyclic process starting from

point A . The process A → B is an adiabatic compression . B → C is isobaric expansion.

C → D an adiabatic expansion & D → A is isochoric . The volume ratios are V

V

A

B

= 16

and V

V

C

B

= 2 & the temperature at A is TA = 300º K . Calculate the temperature of the gas

at the points B & D and find the efficiency of the cycle . [ JEE '97, 5 ]

Q 50. The apparatus shown in the figure consists of four glass

columns connected by horizontal sections . The height of

two central columns B & C are 49 cm each . The two

outer columns A & D are open to the atmosphere . A & C

are maintained at a temperature of 95º C while the columns

B & D are maintained at 5º C . The height of the liquid in

A & D measured from the base line are 52.8 cm & 51 cm respectively . Determine the co-

efficient of thermal expansion of the liquid . [ JEE '97, 5 ]

A B C D95º 5º 95º 5º

C

D

- - - - - - -

- - -

- -

- -

- -

10 20

- - -

- -

- -

- -

- - - - - - -

B

A5

10

P (104 N/m2)

V (m3)

Page - 7

QUEST TUTORIALS

Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439

Q 51. Earth receives 1400 W/m2 of solar power . If all the solar energy falling on a lens of area

0.2 m2 is 25 X focussed on to a block of ice of mass 280 grams, the time taken to melt the

ice will be ______ minutes . (latent heat of fusion of ice = 3.3 x 105 J/kg)

[ JEE '97, 2 ]

ANSWER SHEET

Q 1. (i) T (ii) F Q 2. T = 675 K , P = 3.6 x 106 Q 4. WiSo

= −1.109 x 106 J ;

WAd

= −14.8 x 105 J Q 5. (ii) V = 113.14 litres , P = 0.440 atm (iii) 124450 J

Q 6. False Q 8. 52.875 gm Q 10. (i) 800 K (ii) 720 J Q 13. (i) 5 (ii) 5

4 PV

Q 14. V = 1.42 Q 15. For adiabatic , P = 1.84 P, V1 =

10

3 V ; V

2 =

8

3 V &

For isothermal , P = 13

6 P , V

1 =

30

13 V ; V

2 =

48

13 V Q 16. (B) , W = 0.58 RT

A ,

Q = 0.58 RTA

Q 18. 0.318 per sec Q 19. t = 7

72

1

106

Crd

σ

where d is density

Q 21. 4 πR2 KT/P Q 22 (a) 1153 J (b) 1153 J (iii) 0 Q 23. Q = 1.728 x 107 J

Q 24 t = 2 min. 46.3 sec. Q 25. 300 K Q 26. (i) 1869.75 J (ii) 5297.6 J (iii) 500 K

Q 31. 3

2 KT Q 33. 96.3º C , 2.462 x 106 atm Q 35. (i) 765 J (ii) 10.83 %

Q 36. 23.928 g ; 4.072 g Q 37. (d) nB = 2 ; 401 ms−1 ;

1

6 ; − 0.0248

V

T

Q 38. (i) 189 K (ii) − 2767.23 J (iii) 2767.23 J Q 39. 12 gm Q 40. 60º C

Q 41. 8.08 x 105 Pa

Q 42. 55 J Q 43. 41.5 W , θ1 = 26.48º C , θ

2 = 0.52º C

Q 44. (i) TA = 120.33 K , T

B = 240.66 K , T

C = 481.32 K , T

D = 240.66 K (ii) No

(iii) ∆ QABC

= 3.25 x 106 J ; ∆ QADC

= 2.75 x 106 J Q 45. 45.536º C Q 46. 7730 K

Q 47. K = 2000 N/m , dQ = 1295 J Q 48. P = 3.3 x 105 N/m2 , T = 338.71 K

Q 49. TB = 909.3 K , T

D = 791.09 K , η = 61.4 % Q 50. 5.46 x 10−5 /º C

Q 51. 5.5 min