P0IITU10 - Heat & Thermodynamics Qns Asked in IIT

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 =

5R . The gas is kept in a closed 2

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 =  are initially at temperature 27º C and occupy a volume

 

5 3

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

heater

rigid support

Q 10. An ideal monoatomic gas is confined in a 123456789012345678901234567 123456789012345678901234567 cylinder by a spring loaded piston of cross-section 123456789012345678901234567 123 123 123456789012345678901234567 123 −3 2 123456789012345678901234567 123open atmosphere 8 x 10 m . Initially the gas is at 300 K and occupies 123 12 123456789012345678901234567 123 12 123456789012345678901234567 123 12 −3 3 123 12 123456789012345678901234567 a volume of 2.4 x 10 m and the spring is in its 123 12 123456789012345678901234567 123 12 ← 123 12 123456789012345678901234567 123 12 releaxed (unstretched, uncompressed) state as in 123456789012345678901234567 123 12 123 123456789012345678901234567 123 123 123456789012345678901234567 123 figure . The gas is heated by a small electric heater 123456789012345678901234567 123 123456789012345678901234567 123456789012345678901234567 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 & T2 are mixed . There is no loss of energy. Find the temperature of the mixture if the masses of the molecules are m1 & m2 and the number of molecules in the gases are n1 & n2 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 (i) (ii)

T . 2

How many degrees of freedom do the gas molecules have ? 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 gas .

[ ln 5 = 1.61 ]

Cp Cv

of the

[ 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  C p =

7R  at pressure PA & temperature TA is isothermally expanded 2 

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

Cp Cv

= 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 & Y2 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 ]

-----D

B

Ø

Ù

Ú

---------

×

300 K

---------

2 atm

- - - -A --

1 atm

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 .

→ P

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 ]

C

400 K

T

[ 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 ]

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A → P

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 .

Ù D

Ø ×

→ V

B

Ú C

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, TA, PB, TB etc. respectively . Given that TA = 1000 K , PB = (i) (ii) (iii)

2 1 PA & PC = PA , calculate the following quantities : 3 3

The work done by the gas in the process A → B The heat lost by the gas in the process B → C 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 ] A → → P

Q 30. In the figure are shown three isothermals at temperatures T1 = 4000º K , T2 = 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 .

B

↓

↑ D VA

←

C

T1 T2

VB T 3

→ V

[ 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 Pa is ______ . Pi

[ 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, Q2 = − 5585 J, Q3 = − 2980 J and Q4 = 3645 J respectively . The corresponding works involved are W1 = 2200 J , W2 = − 825 J , W3 = − 1100 J and W4 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. Page - 4

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 =

Cp Cv

=

7 5 & another gas B with τ = at a certain temperature T . The gram 5 3

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 its initial volume V . Find 5

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 ]

A

B 0º C

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 ______ .

100º C

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 ]

T

[ 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

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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 P (10 N/m ) through the process ABC and another sample of 2 kg of the B C same gas is taken through the process ADC as in figure . Given, 10 - - - - - - molecular mass of Helium = 4 . 5 ------(i) what is the temperature of Helium in each of the states D A A, B, C & D ? V (m ) (ii) Is there any way of telling afterwards which sample of 10 20 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 ] ---------

2

---------

4

3

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 and

VA = 16 VB

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

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 A B C D 95º 5º 95º 5º 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 coefficient of thermal expansion of the liquid . [ JEE '97, 5 ] Page - 6

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 WAd = −14.8 x 105 J Q 6. False

Q 4. WiSo = −1.109 x 106 J ;

Q 2. T = 675 K , P = 3.6 x 106

Q 5. (ii) V = 113.14 litres , P = 0.440 atm

Q 8. 52.875 gm

Q 10. (i) 800 K

(ii) 720 J

Q 14. V = 1.42

Q 15. For adiabatic , P = 1.84 P, V1 =

For isothermal , P =

13 30 48 P , V1 = V ; V2 = V 6 13 13

Q = 0.58 RTA

Q 18. 0.318 per sec

Q 21. 4 πR2 KT/P

Q 22 (a) 1153 J

Q 24 t = 2 min. 46.3 sec. Q 31.

3 KT 2

Q 16. (B) , W = 0.58 RTA ,

(iii) 0

Q 23. Q = 1.728 x 107 J

Q 35. (i) 765 J

Q 37. (d) nB = 2 ; 401 ms−1 ; Q 42. 55 J

10 8 V ; V2 = V & 3 3

Q 26. (i) 1869.75 J (ii) 5297.6 J (iii) 500 K

Q 38. (i) 189 K (ii) − 2767.23 J (iii) 2767.23 J Q 41. 8.08 x 105 Pa

(ii) 10.83 %

1 V ; − 0.0248 6 T

Q 39. 12 gm

Q 40. 60º C

Q 43. 41.5 W , θ1 = 26.48º C , θ2 = 0.52º C

Q 44. (i) TA = 120.33 K , TB = 240.66 K , TC = 481.32 K , TD = 240.66 K (iii) ∆ QABC = 3.25 x 10 J ; ∆ QADC = 2.75 x 10 J 6

Q 47. K = 2000 N/m , dQ = 1295 J

5 PV 4

 7 Crd   1    where d is density  72 σ   10 6 

Q 33. 96.3º C , 2.462 x 106 atm

Q 36. 23.928 g ; 4.072 g

Q 13. (i) 5 (ii)

Q 19. t = 

(b) 1153 J

Q 25. 300 K

(iii) 124450 J

6

Q 45. 45.536º C

(ii) No

Q 46. 7730 K

Q 48. P = 3.3 x 105 N/m2 , T = 338.71 K

Q 49. TB = 909.3 K , TD = 791.09 K , η = 61.4 % Q 51. 5.5 min

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Q 50. 5.46 x 10−5 /º C