Heat and Thermodynamics

HEAT AND THERMODYNAMICS HEAT AND THERMODYNAMICS Thermal expansion of solids, liquids and gases and their specific heats, Relationship between Cp and Cv for gases, first law of thermodynamics, thermodynamic processes. Second law of thermodynamics, Carnot cycle, efficiency of heat engines. TRANSFERENCE OF HEAT Modes of transference of heat. Thermal conductivity. Black body radiations, Kirchoff’s Law, Wien’s law, Stefan’s law of radiation and Newton’s law of cooling. INITIAL STEP EXERCISE 1. For an ideal gas : (a) The change in internal energy in a constant pressure process from temperature T1 to T2 = nCV(T2 – T1) where CV is the molar specific heat at constant volume and n the number of moles of the gas. The change in internal energy of the gas and the work done by the gas are equal in magnitude in an adiabatic process. The internal energy does not change in isothermal process. all are correct (b) (c) (d) 2. A monatomic gas (γ = 5/3) is suddenly compressed to (1/8) of its initial volume adiabatically, then the pressure of the gas will change to : (a) (c) 24/5 40/3 (b) (d) 8 32 3. In an adiabatic change, the pressure P and temperature T of a diatomic gas are related by the relation P ∝ TC where c equals (a) (c) 5/3 3/5 (b) (d) 2/5 7/2 4. A Carnot engine working between 300 K and 600 K has a work output of 800 J per cycle. The amount of heat energy supplied to the engine from the source in each cycle is (a) (c) 800 J 3500 J (b) (d) 1600 J 6400 J 5. One mole of an ideal gas requires 207 J heat to raise the temperature by 10 K when heated at constant pressure. If the same gas is heated at constant volume to raise the temperature by the same 10 K, the heat required is : [R = 8.3 J/mol K] (a) (c) 198.7 J 215.3 J (b) (d) 29 J 124 J 6. A vessel contains a mixture of one mole of oxygen and two moles of nitrogen at 300 K. The ratio of the average rotational kinetic energy per O2 molecules to that per N2 molecule is (a) (b) (c) (d) 1:1 1:2 2:1 depends on the moment of inertia of the two molecules 7. The average translational kinetic energy of O2 (molar mass 32) molecules at a particular temperature is 0.048 eV. The translational kinetic energy of N2 (molar mass 28) molecules in eV at the same temperature is (a) (c) 0.0015 0.048 (b) (d) 0.003 0.768 8. The temperature of an ideal gas is increased from 120 K to 480 K. If at 120 K the root mean square velocity of the gas molecules is v, at 480 K it becomes (a) (c) 4v v/2 (b) (d) 2v v/4 9. The average translational energy and the rms speed of molecules in a sample of oxygen gas at 300 K are 6.21 × 10–21 J and 484 m/s respectively. The corresponding values at 600 K are nearly (assuming ideal gas behaviour) (a) (b) (c) (d) 12.42 × 10–21 J, 968 m/s 8.78 × 10–21 J, 684 m/s 6.21 × 10–21 J, 968 m/s 12.42 × 10–21 J, 684 m/s 10. At room temperature the rms speed of the molecules of a certain diatomic gas is found to be 1930 m/s. The gas is (a) (c) H2 O2 (b) (d) F2 Cl2 11. If the rms velocity of oxygen molecule at certain temperature is 0.5 km/s, the rms velocity for hydrogen molecule at the same temperature will be (a) (c) 2 km/s 9 km/s (b) (d) 4 km/s 16 km/s 12. A vessel contains 1 mole of O2 gas (molar mass 32) at a temperature T. The pressure of the gas is P. An identical vessel containing one mole of He gas (molar mass 4) at a temperature 2T has a pressure of (a) (c) P/8 2P (b) (d) P 8P 13. Two gases A and B having the same temperature T, same pressure P and same volume V are mixed. If the mixture is at the same temperature T and occupies a volume V, the pressure of the mixture is (a) (c) 2P P/2 P1 > P2 P1 < P2 P1 = P2 P1 ≥ P2 (b) (d) P 4P 14. For V versus T curves at constant pressure P1 and P2 for an ideal gas are shown in fig. (a) (b) (c) (d) 15. Let , vrms and vP respectively denote the mean speed, root mean square speed, and most probable speed of the molecules in an ideal monatomic gas at absolute temperature T. The mass of a molecule is m. Then : (a) (b) (c) (d) no molecule can have a speed greater than 2v rms . the average kinetic energy of a molecule is ¾ mvp2 v P < v < v rms both (b) and (c) are correct 16. A spherical black body with a radius of 12 cm radiates 450 W power at 500 K. If the radius were halved and the temperature doubled, the power radiates in watt would be (a) (c) 225 900 (b) (d) 450 1800 17. The intensity of radiation emitted by the Sun has its maximum value at a wavelength of 510 nm and that emitted by the North Star has the maximum value at 350 nm. If these stars behave like black bodies, then the ratio of the surface temperature of the Sun and the North Star is (a) (c) 1.46 1.21 (b) (d) 0.69 0.83 18. A sphere, a cube and a thin circular plate all made of the same mass and finish are heated to a temperature of 2000C; which of these objects will cool slowest when left in air at room temperature (a) (b) (c) (d) The sphere The cube The circular plate All will cool at the same rate 19. A ball A has twice the diameter as another ball B of the same material and with same surface finish. A and B are both heated to the same temperature and allowed to cool radiatively; then (a) (b) (c) (d) Rate of cooling of A is same as that of B Rate of cooling of A is twice that of B Rate of cooling of A is half that of B Rate of cooling of A is four times that of B 20. The temperature of a body is increased from 270C to 1270C. The radiation emitted by it increases by a factor of (a) (c) (256/81) (4/3) (b) (d) (15/9) (12/27) 21. A black body emits (a) (b) Radiation of all wavelengths No radiations (c) (d) 22. Radiations of only one wavelength Radiations of selected wavelength Ice starts freezing in a lake with water at 00C when the atmospheric temperature is –100C. If the time taken for 1 cm of ice to be formed is 12 minutes the time taken for the thickness of the ice to change from 1 cm to 2 cm will be (a) (b) (c) (d) 12 minutes Less than 12 minutes More than 12 minutes but less than 24 minutes More than 24 minutes 23. A wall has two layers A and B, each made of a different material. Both the layers have the same thickness. The thermal conductivity of the material of A is twice that of B; if under thermal equilibrium the temperature difference across the wall is 360C, the temperature difference across the layer A is (a) (c) 6 0C 18 C 0 (b) (d) 120C 240C 24. Two ends of rods of length L and radius r of the same material are kept at the same temperature. Which of the following rods conducts most heat? (a) (b) (c) (d) L = 50 cm, r = 1 cm L = 100 cm, r = 2 cm L = 25 cm, r = 0.5 cm L = 75 cm, r = 1.5 cm 25. Heat is flowing through two cylindrical rods of the same material. The diameter of the rods are in the ratio 1 : 2 and their lengths are in the ratio 2 : 1. If the temperature difference between their ends is the same, then the ratio of the amounts of heat conducted through them per unit time will be (a) (c) 1:1 1:4 (b) (d) 2:1 1:8 26. Two identical objects A and B are at temperature TA and TB respectively. Both objects are placed in a room with perfectly absorbing walls maintained at a temperature T(TA > T > TB). The objects A and B attain the temperature T eventually. Select the correct statements from the following (a) (b) (c) (d) Each object continues to emit and absorb radiation even after attaining the temperature T A loses more heat by radiation than it absorbs, while B absorbs more radiation than it emits, until they attain the temperature T Both A and B only absorb radiation, but do not emit it, until they attain the temperature T. both (a) and (b) are correct 27. When m gm of water at 100C is mixed with m gm of ice at 00C, which of the following statements are false (a) (b) (c) (d) The temperature of the system will be given by the equation m × 80 + m × 1 × (T – 0) = m × 1 × (10 – T) Whole of ice will melt and temperature will be more than 00C but lesser than 100C Whole of ice will melt and temperature will be 00C all are correct 28. A 2 gm bullet moving with a velocity of 200 m/sec is brought to be sudden stoppage by an obstacle. The total heat produced goes to the bullet. If the specific heat of the bullet is 0.03 cal/gm C0 the rise in its temperature will be (a) (c) 158.0 C0 1.58 C 0 (b) (d) 15.80 C0 0.1580 C0 29. A gas is contained in a metallic cylinder fitted with a piston. The piston is suddenly moved in to compress the gas and is maintained at this position. As time passes the pressure of the gas in the cylinder (a) (b) (c) (d) increases decreases remains constant increases or decreases depending on the nature of the gas 30. The molar heat capacity of oxygen gas at STP is nearly 2.5 R. As the temperature is increased, it gradually increases and approaches 3.5 R. The most appropriate reason for this behaviour is that at high temperatures (a) (b) (c) (d) (a) oxygen does not behave as an ideal gas oxygen molecules dissociates in atoms the molecules collide more frequently molecular vibrations gradually become effective. Each object continues to emit and absorb radiation even after attaining the temperature T 31. One gm of ice at 00C is added to 5 gm of water at 100C. If the latent heat is 80 cal/gm, the final temperature of the mixture is (a) (c) 5 0C –50C (b) (d) 0 00C None of the above 32. 200 gm of a solid ball at 20 C is dropped in an equal amount of water at 800C. The resulting temperature is 600C. This means that specific heat of solid is (a) (b) (c) (d) One fourth of water One half of water Twice of water Four times of water 33. The work performed for the process shown in figure is (a) (b) (c) (d) 1 (P1 + P2 )(V2 − V1 ) 2 1 (P1 − P2 )(V2 − V1 ) 2 1 (P1 − P2 )(V2 + V1 ) 2 1 (P1 + P2 )(V2 + V1 ) 2 34. A liquid cools in 6 minutes from 800C to 600C. Take the temperature of surrounding to be 300C and assume that Newton’s law of cooling is applicable throughout the process. Its temperature after 10 minutes is (a) (c) 35. 48.20C 37.50C (b) (d) 42.80C 32.50C An air bubble of volume 1.0 cm3 rises from the bottom of a lake 40 m deep at a temperature of 120C. To what volume does it grow when it reaches the surface, which is at a temperature of 350C ? (a) (c) 1.0 cm3 1.75 cm3 (b) (d) 2.5 cm3 none of these 36. If the degree of freedom for the gas is f then choose the correct statement (a) (b) ⎛ 2⎞ ⎜1 + ⎟ the adiabatic exponent of the gas is ⎝ f ⎠ f R the molar heat capacity at constant volume is 2 ⎛f ⎞ ⎜ R +R⎟ ⎠ the molar heat capacity at constant pressure is ⎝ 2 all the above C* and C* P V (c) (d) 37. Let the specific heat capacity of a gas at constant volume and at constant pressure is C* − C* P V equals to (a) (c) R MR (b) (d) R/M then MR Here M is the molecular weight of the gas. 38. Let the adiabatic exponent of a gas is γ then the molar heat capacity at constant pressure of the gas is (a) (c) 39. (a) (b) (c) (d) 40. (a) (b) (c) (d) 41. R γ −1 (b) (d) Rγ γ −1 R γ +1 Rγ γ +1 For which process the molar heat capacity is zero? Isochoric process Adiabatic process Isothermal process Isobaric process internal energy work done amount of heat molar heat capacity of gases Which of the following quantity is path independent? The plots of intensity versus wavelength for three black bodies at temperatures T1, T2 and T3 respectively are as shown in figure. Their temperature are such that (a) (b) (c) (d) T1 > T2 > T3 T1 > T3 > T2 T2 > T3 > T1 T3 > T2 > T1 Solutions: 1. 7. 13. 19. 25. 31. 37. d c a c d b b 2. 8. 14. 20. 26. 32. 38. d b a a d b b 3. 9. 15. 21. 27. 33. 39. d d d a d a b 4. 10. 16. 22. 28. 34. 40. b a d d a b a 5. 11. 17. 23. 29. 35. 41. d a b b b d b 6. 12. 18. 24. 30. 36. a c a b d a FINAL STEP EXERCISE 1. Steam at 1000C passed into 1.1 kg of water contained in a calorimeter of water equivalent 0.02 kg at 150C till the temperature of the calorimeter and its content rises to 800C. If the enthalpy of vaporization of water at 1000C is 2.26 kJ/g, the mass of steam condensed is (a) (c) 2. 0.130 kg 0.260 kg (b) (d) 0.065 kg 0.33 kg A thermodynamical process is shown in fig. with PA = 3 × 104 Pa; VA = 2 × 10–3 m3; PB = 8 × 104 N/m2; VD = 5 × 10–3 m3. In the processes AB and BC, 600 J and 200 J of heat is added to the system respectively. The change in internal energy of the system in process AC would be (a) (c) 560 J 600 J (b) (d) 800 J 640 3. Two samples of air A and B having same composition and initially at the same temperature and pressure are compressed from a volume V to V/2, the sample A isothermally and the sample B adiabatically. The final pressure of (a) (b) (c) (d) A is greater than B A is lesser than B A is equal to that of B A is twice that of B 4. Two cylinders A and B fitted with pistons contain equal amounts of an ideal diatomic gas at 300 K. The piston of A is free to move, while that of B is held fixed. The same amount of heat is given to the gas in each cylinder. If the rise in temperature of the gas in A is 30 K, then the rise in temperature of the gas in B is: (a) (c) 30 K 50 K (b) (d) 18 K 42 K 5. If one mole of a monatomic gas (γ = 5/3) is mixed with one mole of diatomic gas (γ = 7/5), the value of γ for the mixture is : (a) (c) 1.4 1.53 (b) (d) 1.5 1.67 6. When an ideal diatomic gas is heated at constant pressure, the fraction of the heat energy supplied which increases the internal energy of the gas is (a) (c) (2/5) (3/7) (b) (d) (3/5) (5/7) 7. Three closed vessels A, B and C are at the same temperature T and contain gases which obey Maxwellian distribution of velocities. Vessel A contains O2, B only N2 and C mixture of equal quantities of O2 and N2. If the average speed of the O2 molecules in the vessel A is v1, that of N2 molecules in the vessel B is v2, the average speed of the O2 molecules in vessel C is (a) (c) (v1 + v2)/2 (b) (d) v1 4v2 3kT / m 8. Two identical containers A and B with frictionless pistons contain the same ideal gas at the same temperature and the same volume V. The mass of the gas in A is mA, and that in B is mB. The gas in each cylinder is now allowed to expand isothermally to the same final volume 2V. The changes in the pressure in A and B are found to be ΔP and 1.5 ΔP respectively. Then (a) (c) 4mA = 9mB 3mA = 2mB must be heated must be cooled must be first cooled and then heated must be first heated and then cooled (b) (d) 2mA = 3mB 9mA = 4mB 9. If the indicator diagram for expansion of a gas is as shown in figure, the gas (a) (b) (c) (d) 10. A vessel containing one gm-mol oxygen is enclosed in a thermally insulated vessel. The vessel is moved with a constant speed v0 and then suddenly stopped. The process results in a rise in the temperature of the gas 10C. What is the speed v0 (a) (c) 15 m/s 25 m/s (b) (d) 36 m/s 60 m/s 11. A blackbody is at a temperature of 2800 K. The energy of radiation emitted by this object with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien constant b = 2.88 × 106 nm K. Then (a) (c) U1 = 0 U1 > U2 (b) (d) U3 = 0 U2 > U1 12. A thermally insulated vessel contains 100 g of water at 00C. When air above the water is pumped out, some of the water freezes and some evaporates at 00C itself. Calculate the mass of the ice formed if no water is left in the vessel. Latent heat of vaporization of water at 00C is 2.1 × 106 J/kg and latent heat of fusion of ice is 3.36 × 105 J/kg (a) (c) 62 g 54 g (b) (d) 86 g 78 g 13. The initial pressure and volume of a gas are Pi and Vi. The gas after expansion attains final volume Vf. Let W1, W2 and W3 are the corresponding work done under isothermal, adiabatic and isobaric pressure. Then (a) (c) W1 = W2 = W3 W1 > W2 > W3 (b) (d) W2 > W1 > W3 W3 > W1 > W2 14. A cylinder of radius R made of a material of thermal conductivity K1 is surrounded by a cylindrical shell of inner radius R and outer radius 2R made of a material of thermal conductivity K2. The two ends of the combined system are maintained at two different temperatures. There is no loss of heat across the cylindrical surface and the system is in steady state. The effective thermal conductivity of the system is (a) (c) K1 + K2 (b) (K1 + 3K2)/4 (3K1 + K2)/4 K1K2/(K1 + K2) (d) 15. One end of a copper rod of length 1.0 m and area of cross-section 10–3m2 is immersed in boiling water and the other end in ice. If the coefficient of thermal conductivity of copper is 92 cal/m s C0 and the latent heat of ice is 8 × 104 cal/kg, then the amount of ice which will melt in one minute is (a) (c) 9.2 × 10–3 kg 6.9 × 10 kg –3 (b) (d) 8 × 103 kg 5.4 × 10–3 kg 16. Two bodies A and B have thermal emissivities of 0.01 and 0.81 respectively. The outer surface areas of the two bodies are equal. The two bodies emit total radiant power at the same rate. The wavelength λB corresponding to maximum spectral radiancy in the radiation from B is shifted from the wavelength corresponding to maximum spectral radiancy in the radiation from A, by 1.00 µm. If the temperature of A is 5802 K (a) (b) (c) (d) The temp. of B is 1934 K λB = 2.5 µm The temp. of B is 1160 K The temp. of B is 2901 K 17. The amount of heat involved for the cyclic process shown in figure is (a) π (P2 − P1 )(V2 − V1 ) 4 (b) (c) (d) π − (P2 − P1 )(V2 − V1 ) 4 π − (P2 + P1 )(V2 − V1 ) 4 π − (P2 + P1 )(V2 + V1 ) 4 18. For the thermodynamic processes shown in figure, 440 J of work are performed by the system along the diagonal path AC and 320 J of work are done by the system along the path ADC. The work does the system do along the path ABC is (a) (c) 320 J 560 J (b) (d) 440 J 120 J 19. (a) (c) 20. A gas mixture consists of 2 moles of oxygen and 4 moles of argon at temperature T. Neglecting all vibrational modes, the internal energy of the system is 5RT 13RT (b) (d) 11RT 17RT The molar heat capacity of a thermodynamic process, performed on a system conaining one mole of monoatomic gas, is 2R. The equation of the process on P – V diagram is (a) (c) P=α P = α√V (b) (d) P = αV2 P = αV 21. An ideal gas is taken through a cyclic thermodynamic process through four steps. The amount of heat involves in these steps are Q1 = 5960J, Q2 = –5585J, Q3 = –2980J and Q4 = 3645J respectively. The corresponding quantities of work involved are W1 = 2200 J, W2 = –825J, W3 = –1100 J, and W4 respectively. The value of W4 is (a) (c) 265 J 765 J 10.82 % 8.55 % (b) (d) (b) (d) 575 J 975 J 7.65 % 9.75 % 22. In the above problem, the efficiency of the cycle is (a) (c) 23. A cylindrical pipe consists of a material of thermal conductivity k having length L, and the inner and outer radii are R1 and R2, respectively. The pipe conducts heat radially outward at a constant rate dQ/dt. The temperature difference between the inner and outer radii is (a) 1 dQ R 1 ln (2πl )K dt R 2 1 dQ R 1 ln (3πl )K dt R 2 1 dQ R 2 ln (2πl )K dt R1 1 dQ R 2 ln (4πl )K dt R1 (b) (c) (d) 24. A source of power P is placed at the centre of a spherical shell of coefficient of thermal inner conductivity k with inner radius r1 and outer radius r2. The temperature difference between surface and outer surface is (a) P(r1 + r2 ) 4πkr1r2 2 2 (b) P( r2 − r1 ) 4πkr1r2 P( r1 + r2 ) 4πkr1r2 (c) 25. P( r2 + r1 ) 6πkr1r2 (d) A parallel beam of nitrogen molecules moving with velocity v impinges on a wall at an angle θ to its normal. The concentration of molecules in the beam n. The pressure exerted by the beam on the wall assuming the molecules to scatter in accordance with the perfectly elastic collision law (a) (c) 2nmv2cosθ 2nmv2sinθ (b) (d) 2nmv2cos2θ 2nmv2sin2θ 26. On a cold water winter day, the atmospheric temperature is – θ (on Celsius scale) which is below 00C. A cylindrical drum of height h made of a bad conductor is completely filled with water at 00C and is kept outside without any lid. Thermal conductivity of ice is K and its latent heat of fusion is L. Neglect expansion of water on freezing. The time taken for the whole mass of water to freeze is (a) (c) ρLh 2 2Kθ ρLh 4Kθ 2 (b) (d) ρLh 2 3Kθ ρLh 2 5Kθ 27. Consider a polytropic process PVα = constant is perform on an ideal gas (adiabatic exponent γ). In this process the change in temperature is ΔT. Let the work done, change in internal energy, amount of heat and molar heat capacity are given by W, ΔU, Q and CM respectively. Choose the incorrect option W= (a) (b) nR ΔT α −1 nR ΔT γ −1 ΔU = (c) ⎛ 1 1 ⎞ Q = nRΔT⎜ ⎜ 1− α + γ −1 ⎟ ⎟ ⎝ ⎠ ⎛ 1 1 ⎞ CM = R ⎜ ⎜ 1 − α + γ −1 ⎟ ⎟ ⎝ ⎠ (d) 28. In the P-V diagram shown in figure, the process I is performed on one mole of monoatomic gas, process II is performed on two moles of diatomic gas and process III is performed on three moles of monoatomic gas. Let the change in internal energy, work done and amount of heat in the I, II and III. Processes are ΔU1, W1 and Q1, ΔU2, W2 and Q2 and ΔU3, W3 and Q3 respectively. Which of the following must be incorrect? (a) (b) (c) (d) W1 > W2 > W3 Q1 > Q2 > Q3 ΔU1 = ΔU2 = ΔU3 ΔU1 = ΔU3 < ΔU2 29. Three rods made of the same material and having the same cross-section have been joined as shown in the figure. Each rod is of the same length. The left and right ends are kept 00C and 900C respectively. The temperature of the junction of the three rods will be (a) (c) 450C 300C (b) (d) 600C 200C Solutions:  1. 7. 13. 19. 25. a b d b b 2. 8. 14. 20. 26. a c b d a 3. 9. 15. 21. 27. b d c c a 4. 10. 16. 22. 28. d b a a c 5. 11. 17. 23. 29. b d b c b 6. 12. 18. 24. d b c b ANALYSIS 1. “Heat cannot be itself flow from a body at lower temperature of a body at higher temperature” is a statement or consequence of (a) (b) (c) conservation of mass first law of thermodynamics second law of thermodynamics (d) [Ans. : c] 2. conservation of momentum The earth radiates in the infra-red region of the spectrum. The spectrum is correctly given by (a) (b) (c) (d) [Ans. : b] Stefan’s law of radiation Wien’s law Rayleighjeans law Planck’s law of radiation 3. n According to Newton’s law of cooling, the rate of cooling of body is proportional to ( Δθ) , where Δθ is the difference of the temperature of the body and the surroundings, and n is equal to (a) (c) [Ans. : b] 4. four two (b) (d) one three During an adiabatic process, the pressure of a gas is found to be proportional to the cube of its absolute, temperature. The ratio Cp/Cv for the gas is (a) (c) [Ans. : b] 5/3 4/3 (b) (d) 3/2 2 5. Which of the following parameters does not characterize the thermodynamic state of matter ? (a) (c) [Ans. : a] Work Temperature (b) (d) Volume Pressure 6. A Carnot engine takes 3 × 106 cal. of heat from a reserviour at 6270C, and gives it to a sink at 270C. The work done by the engine is (a) (c) 16.8 × 106 J 4.2 × 106 J (b) (d) zero 8.4 × 106 J [Ans. : d] 1. One mole of ideal monoatomic gas (γ = 5/3) is mixed with one mole of diatomic gas (γ = 7/5). What is γ for the mixture? γ denotes the ratio of specific heat at constant pressure, to that at constant volume. (a) (c) 3/2 35/23 (b) (d) 23/15 4/3 [Ans. : a] 2. If the temperature of the sum were to increase from T to 2T and its radius from R to 2R, then the ratio of the radiant energy received on earth to what it was previously will be (a) (c) [Ans. : d] 3. Which of the following statements is correct for any thermodynamic system? (a) (b) The internal energy changes in all purposes Internal energy and entropy are state functions 4 32 (b) (d) 16 64 (c) (d) [Ans. : b] 4. The change in entropy can never be zero The work done in an adiabatic process is always zero Two thermally insulated vessels 1 and 2 are filled with air at temperatures (T1, T2), volume (V1, V2) and pressure (P1, P2) respectively. If the valve joining the two vessels is opened, the temperature inside the vessel at equilibrium will be (a) (b) T1 + T2 (T1 + T2)/2 (c) T1 + T2 ( P1V1 + P2 V2 ) P1V1T2 + P2 V2 T1 T1 + T2 ( P1V1 + P2 V2 ) P1V1T1 + P2 V2 T2 (d) [Ans. : c] 5. The temperature of the two outer surfaces of a composite slab, consisting of two materials having coefficients of thermal conductivity K and 2K and thickness x and 4x, respectively are T2 and T1 ⎛ A (T2 − T1 ) K ⎞ ⎜ ⎟f x ⎠ , with f equal to (T > T ). The rate of heat transfer through the slab, in a steady state is ⎝ 2 1 (a) (c) [Ans. : d] 6. 1 2/3 (b) (d) 1/2 1/3 Which of the following is incorrect regarding the first law of thermodynamics? (a) (b) (c) (d) It introduces the concept of the internal energy It introduces the concept of the entropy It is not applicable to any cyclic process It is restatement of the principle of conservation of energy [Ans. : d] 7. The temperature-entropy diagram of a reversible engine cycle is given in the figure. Its efficiency is (a) (c) [Ans. : a] 1/3 1/2 (b) (d) 2/3 1/4 8. The figure shows a system of two concentric spheres of radii r1 and r2 and kept at temperature T1 and T2, respectively. The radial rate of flow of heat in the substance between the two concentric spheres is proportional to (a) r1r2 (r2 − r1 ) (b) (r2 – r1) (c) [Ans. : a] 9. (r2 – r1)/(r1 r2) (d) ⎛r ⎞ Ln ⎜ 2 ⎟ ⎜r ⎟ ⎝ 1⎠ A system goes from A to B via two processes I and II as shown in figure. If ΔU1 and ΔU2 are the changes in internal energies in the processes I and II respectively, then (a) (b) (c) (d) [Ans. : c] ΔU2 > ΔU1 ΔU2 < ΔU1 ΔU1 = ΔU2 Relation between ΔU1 and ΔU2 cannot be determined Cp 10. A gaseous mixture consists of 16 g of helium and 16 g oxygen. The ratio C v of the mixture is (a) 1.4 (b) 1.54 (c) 1.59 (d) 1.62 [Ans. : d]