Solved Examples
EXAMPLE 26.1
A gas is contained in a vessel fitted with a movable piston. The container is placed on a hot stove.A total of 100 cal of heat is given to the gas and the gas does 40 J of work in the expansion resulting from heating.Calculate the increase in internal energy in the process.
EXAMPLE 26.2
Calculate the work done by a gas as it is taken from thestate a to n, b to c and c to a as shown in figure.
Sol. The work done by the gas in the process a to b is the area of adobe. This is
Wab = (120 kPa) (250 cc)
= 120 × 103 × 250 × 10–6 J = 30 J
In the process b to c the volume remains constant and the work done is zero.
In the process c to a the gas is compressed. The volume is decreased and the work done by the gas is negative.
The magnitude is equalt to the area of caed. This area is cab + baed.
= (80 kPa) (250 cc) + 30 J
= 10 J + 30 J = 40 J.
Thus, the work done in the process c to a is – 40 J.
QUESTIONS FOR SHORT ANSWER
1. Should the internal energy of a system necessarily increase if heat is added to it?
2. Should the internal energy of a system necessarily increase if its temperature is increased?
3. A cylinder containing a gas is lifted from the first floor to the second floor. What is the amount of work done on the gas?
What is the amount of work done by the gas? Is the internal energy of the gas increased? Is the temperature of the gas increased?
4. A force F is applied on a block of mass M. The block is displaced through a distance d in the direction of the force. What is the work done by the force on the block? Does the internal energy change because of this work?
5. The outer surface of a cylinder containing a gas is rubbed vigorously by a polishing machine. The cylinder and its gas become warm. Is the energy transferred to the gas heat or work?
6. When we rub our hands they become warm. Have we supplied heat to the hands?
7. A closed bottle contains some liquid. The bottle is shaken vigorously for 5 minutes. It is found that the temperature of the liquid is increased. Is heat transferred to the liquid? Is work done on the liquid? Neglect expension on heating.
8. The final volume of a system is equal to the initial volume in a certain process. Is the work done by the system necessarily zero? Is it necessarily nonzero?
9. Can work be done by a system without changing its volume?
10. An ideal gas is pumped into a rigid container having diathermic walls so that the temperature remains constant. In a certain time interval, the pressure in the container is doubled. Is the internal energy of the contents of the container also doubled in the interval?
11. When a tyre bursts, the air coming out is cooler than the surrounding air. Explain.
12. When we heat an object, it expands. Is work done by the object in this process? Is heat given to the object equal to the increase in its internal energy?
13. When we stir a liquid vigorously, it becomes warm. Is it a reversible process?
14. What should be the condition for the effciency of a carnot engine to be equal to 1?
15. When an object cools down, heat is with drawn from it. Does the entropy of the object decrease in this process? If yes, is it a violation of the second law of thermodynamics stated in terms of increase in entropy?
Objective - I
1. The first law of thermodynamic is a statement of
(A) conservation of heat (B) conservation of work
(C) conservation of momentum (D) conservation of energy
2. If heat is supplied to an ideal gas in an isothermal process,
(A) the internal energy of the gas will increase
(B) the gas will do positive work
(C) the gas will do negative work
(D) the said process is not possible
5. Consider the process on a system shown in fig. During the process, the work done by the system
(A) continuously increases (B) continuously decreases
(C) first increases then decreases (D) first decreases then increases
6. Consider the following two statements.
(a) If heat is added to a system, its temperature must incresase.
(b) If positive work is done by a system in a thermodynamic process, its volume must increase.
(A) Both a and b are correct (B) a is correct but b is wrong
(C) b is correct but a is wrong (D) Both a and b are wrong
7. An ideal gas goes from the state i to the state f as shown in fig. The work done by the gas during the process -
(A) is positive (B) is negative (C) is zero
(D) cannot be obtained from this information
9. 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) increases (B) decreases (C) remains constant
(D) increases or decreases depending on the nature of the gas.
Objective - II
1. The pressure p and volume V of an ideal gas both increase in a process.
(A) Such a process is not possible
(B) The work done by the system is positive
(C) The temperature of the system must increase
(D) heat supplied to the gas is equal to the change in internal energy.
2. In a process on a system, the initial pressure and volume are equal to the final pressure and volume.
(A) The initial temperature must be equal to the final temperature
(B) The initial internal energy must be equal to the final internal energy.
(C) The net heat given to the system in the process must be zero
(D) The net work done by the system in the process must be zero
5. The internal energy of an ideal gas decreases by the same amount as the work done by the system
(A) The process must be adiabatic (B) The process must be isothermal
(C) The process must be isobaric (D) The temperatuer must decrease
Worked Out Examples
Q.1 A sample of an ideal gas is taken through the cyclic process abca (figure. It absorbs 50 J of heat during the part ab, no heat during bc
and rejects 70 J of heat during ca. 40 J of work is done on the gas during the part bc.
(a) Find the internal energy of the gas at b and c if it is 1500 J at a.
(b) Calculate the work done by the gas during the part ca.
Sol. (a) In the part ab the volume remains constant. Thus, the work done by the gas is zero. The heat absorbed by the gas is 50 J.
The increase in internal energy from a to b is
Q.2 A thermodynamic system is taken through the cycle abcda (figure)
(a) Calculate the work done by the gas during the parts ab, bc, cd and da
. (b) Find the total heat rejected by the gas during the process.
Sol. (a) The work done during the part ab
Q.3 Calculate the increase in internal energy of 1kg of water at 100ºC when it is converted into steam at the same temperature and at 1 atm (100 kPa).
The density of water and steam are 1000 kg/m3 and 0.6 kg/m3 respectively. The latent heat of vaporization of water = 2.25 × 106 J/kg.
Sol. The volume of 1kg of water
= 1/1000 m3 and of 1 kg of steam = 1/0.6 m3.
The increase in volume
= 1/0.6 m3 – 1/1000 m3
= (1.7 – 0.001) m3 = 1.7 m3.
4. The internal energy of a monatomic ideal gas is 1.5 nRT. One mole of helium is kept in a cylinder of cross-section 8.5 cm2.
The cylinder is closed by a light fractionless piston. The gas is heated slowly in a process during which a total of 42 J heat is given to the gas.
If the temperature rises through 2ºC, find the distance moved by the piston. Atmospheric pressure = 100 kPa.
5. A steam engine intakes 100g of steam at 100ºC per minute and cools it down to 20ºC. Calculate the heat rejected by the steam engine per minute.
Latent heat of vaporization of steam = 540 cal/g.
Sol. Heat rejected during the condensation of steam in one minute
= (100 g) × (540 cal/g) = 5.4 × 104 cal.
Heat rejected during the cooling of water
= (100 g) × (1 cal/g-ºC) (100ºC - 20ºC)
= 8000 cal.
Thus, heat rejected by the engine per minute
= 5.4 × 104 cal + 0.8 × 104 cal
= 6.2 × 104 cal.
6. Figure shows a process ABCA performed on an ideal gas. Find the net heat given to the system during the process.
Sol. As the process is cyclic, the change in internal energy is zero. The heat given to the system is then equal to the work done by it.
The work done in part AB is W1 = 0 as the volume remains constant. The part BC represents an isothermal process so that the
work done by the gas during the part is
W2 = nRT2 In (V2/V1)
During the part CA,
So, V/T is constant and hence,
p = nRT/V is constant
The work done by the gas during the part CA is
W3 = p (V1 – V2)
= nRT1 – nRT2
= –nR (T2 – T1)
The net work done by the gas in the process ABCA is
W = W1 + W2 + W3 = nR[T2In V2/V1–(T2 – T1)]
The same amount of heat is given to the gas.
7. Consider the cyclic process ABCA on a sample of 2.0 mol of an ideal gas as shown in figure. The temperatures of the gas at A and B
are 300 K and 500 K respectively. A total of 1200 J heat is withdrawn from the sample in the process. Find the work done by the gas in part BC. Take R = 8.3 J/mol–K
Sol. The change in internal energy during the cyclic process is zero. Hence, the heat supplied to the gas is equal to the work done by it. Hence,
WAB + WBC + WCA = –1200 J. .......(i)
The work done during the process AB is
WAB = PA (VB – VA)
= nR(TB – TA)
= (2.0 mol) (8.3 J/mol–K) (200 K)
= 3320 J
The work done by the gas during the process CA is zero as the volume remains constant. From (i),
3320 J + WBC = –1200 J
or WBC = –4520 J.
= –4520 J.
8. 2.00 moles of a monatomic ideal gas (U = 1.5 nRT) is enclosed in an adiabatic, vertical cylinder fitted with a smooth, light adiabatic piston.
The piston is connected to a vertical spring of spring constant 200 N/m as shown in figure. The area of cross-section of the cylinder is 20.0cm2.
Initially, the spring is at its natural length and the temperature at the gas is 300K. The atmospheric pressure is 100 kPa. The gas is heated slowly
for some time by means of an electric heater so as to move the piston up through 10cm. Find
(a) the work done by the gas
(b) the final temperature of the gas and
(c) heat supplied by the heater.
Sol. (a) The force by the gas on the piston is
F = p0A + kx
where p0 = 100 kPa is the atmospheric pressure, A = 20cm2 is the area of cross-section, k=200 N/m is the spring constant and x is the
compression of the spring. The work done by the gas as the piston moves through l = 10cm is
= (100 × 103 Pa) × (20 × 10–4 m2) × (10 × 10–2 m)
+ (200 N/m) × (100 × 10–4 m2)
= 20 J + 1 J = 21 J.
(b) The initial temperature is T1 = 300 K. Let the final temperature be T2. We have
9. A sample of an ideal gas has pressure p0, volume V0 and temperature T0. It is isothermally expanded to twice its original volume.
It is then compressed at constant pressure to have the original volume V0. Finally, the gas is heated at constant volume to get the original temperature.
(a) Show the process in a V-T diagram
(b) Calculate the heat absorbed in the process.
Sol. 
(a) The V–T diagram for the process is shown in figure. The initial state is represented by the point a. In the first step, it is isothermally
expanded to a volume 2V0. This shown by ab. Then the pressure is kept constant and the gas is compressed to the volume V0. From the ideal gas equation,
V/T is constant at constant pressure. Hence, the process is shown by a line bc which passes through the origin. At point c, the volume is V0. In the final step, the gas is heated at constant volume to a temperature T0. This is shown by ca. The final state is the same as the inital state.
(b) The process is cyclic so that the change in internal energy is zero. The heat supplied is, therefore, equal to the work done by the gas. The work done during ab is
W1 = nRT0 In = nRT0 In 2 = p0 V0 In 2.
Also from the ideal gas equation,
pa Va = pb Vb
or, pb = paVa/Vb.= p0Va/2V0=p0/2
In the step bc, the pressure remains constant. Hence the work done is,
W2 = p0/2 (V0 – 2V0) = – p0V0.
In the step ca, the volume remains constant and so the work done is zero. The net work done by the gas in the cyclic process is
W = W1 + W2
= p0 V0 [In 2 – 0.5]
= 0.193 p0V0.
Hence, the heat supplied to the gas is 0.193 p0V0.
10. A sample of 100 g water is slowly heated from 27ºC to 87ºC. Calculate the change in the entropy of the water. Specific heat capacity of water = 4200 J/kg-K.
Putting T1 = 27ºC = 300 K and T2 = 87ºC = 360 K,
S2 – S1 = (0.1 kg) (4200 J/kg - K) In 360/300
= 76.6 J/K.
11. A heat engine operates between a cold reservoir at temperature T2 = 300 K and a hot reservoir at temperature T1. If takes 200 J of heat from the hot reservoir and delivers 120J of heat to the cold reservoir in a cycle. What could be the minimum temperature of the hot reservoir?
Sol. The work done by the engine in a cycle is
W = 200 J – 120 J = 80 J.
The efficiency of the engine is
.
The minimum temperature of the hot reservoir may be 500 K.
Exercise
1. A thermally insulated, closed copper vessel contains water at 15ºC. When the vessel is shaken vigorously for 15 minutes, the temperature rises to 17ºC. The mass of the vessel is 100g and that of the water is 200g. The specific heat capacities of copper and water are 420 J/kg-K and 4200 J/kg-K respectively. Neglect any thermal expansion. (a) How much heat is transferred to the liquid-vessel system? (b) How much work has been done on this system? (c) How much is the increase in internal energy of the system?
Ans. (a) zero (b) 1764 J (c) 1764 J
2. Figure shows a paddle wheel coupled to a mass of 12 kg through fixed frictionless pulleys. The paddle is immersed in a liquid of heat capacity 4200 J/K kept in an adiabatic container. Consider a time interval in which the 12 kg block falls slowly through 70cm. (a) How much heat is given to the liquid? (b) How much work is done on the liquid?. Calculate the rise in the temperature of the liquid neglecting the heat capacity of the container and the paddle.
Ans. (a) zero (b) 84 J (c) 0.02ºC
3. A 100kg block is started with a speed of 2.0m/s on a long, rough belt kept fixed in a horizontal position. The coefficient of kinetic friction between the block and the belt is 0.20. (a) Calculate the change in the internal energy of the block-belt system as the block comes to a stop on the belt. (b) Consider the situation from a frame of reference moving at 2.0 m/s along the initial velocity of the block. As seen from this frame, the block is gently put on a moving belt and in due time the block starts moving with the belt at 2.0 m/s. Calculate the incrase in the kinetic energy of the block as it stops slipping past the belt. (c) Find the work done in this frame by the external force holding the belt.
Ans. (a) 200 J (b) 200 J (c) 400 J
4. Calculate the change in internal energy of a gas kept in a rigid container when 100J of heat is supplied to it.
Ans. 100 J
5. The pressure of a gas changes linearly with volume from 10 kPa, 200cc to 50 kPa, 55cc. (a) Calculate the work done by the gas. (b) If no heat is supplied or extracted from the gas, what is the change in the internal energy of the gas? Ans. (a) –4.5 J (b) 4.5 J
6. An ideal gas is taken from an initial state i to a final state f in such a way that the ratio of the pressure to the absolute temperature remains constant. What will be the work done by the gas?
Ans. zero
7. Figure shows three paths through which a gas can be taken from the state A to the state B. Calculate the work done by the gas in each of the three paths
Ans. 0.30 J in AB, 0.450 J in ACB and 0.150 J in ADB
8. When a system is taken through the process abc shown in figure, 80 J of heat is absorbed by the system and 30 J of work is done by it. If the system does 10 J of work during the process adc, how much heat flows into it during the process?
Ans. 60 J
9. 50 cal of heat should be supplied to take a system from the state A to the state B through the path ACB as shown in figure. Find the quantity of heat to be supplied to take it from A to B via ADB
.
Ans. 55 cal
10. Calculate the heat absorbed by a system in going through the cyclic process shown in figure.
Ans. 31.4 J
11. A gas is taken through a cyclic process ABCA as shown in figure. If 2.4 cal of heat is given in the process, what is the value of J?
Ans. 4.17 J/cal
12. A substance is taken through the process abc as shown in figure. If the internal energy of the substance increases by 5000 J and a heat of 2625 cal is given to the system, calculate the value of J.
Ans. 4.19 J/cal
13. A gas is taken along the path AB as shown in figure. If 70 cal of heat is extracted from the gas in the process, calculate the change in the internal energy of the system.
Ans. -241 J
14. The internal energy of a gas is given by U = 1.5 pV. It expands from 100 cm3 to 200 cm3 against a constant pressure of 1.0 × 105 Pa. Calculate the heat absorbed by the gas in the process.
Ans. 25 J
15. A gas is enclosed in a cylindrical vessel fitted with a frictionless piston. The gas is slowly heated for some time. During the process, 10 J of heat is supplied and the piston is found to move out 10 cm. Find the increase in the internal energy of the gas. The area of cross-section of the cylinder = 4cm2 and the stmospheric pressure = 100 kPa.
Ans. 6 J
16. A gas is initially at a pressure of 100 kPa and its volume is 2.0m3. Its pressure is kept constant and the volume is changed from 2.0m3 to 2.5m3. Its volume is now kept constant and the pressure is increased from 100 kPa to 200 kPa. The gas is brought back to its initial state, the pressure varying linearly with its volume. (a) Whether the heat is supplied to or extracted from the gas in the complete cycle? (b) How mych heat was suplied or extracted?
Ans. (a) extracted (b) 25000 J
17. Consider the cyclic process ABCA, shown in figure, performed on a sample of 2.0 mole of an ideal gas. A total of 1200 J of heat is withdrawn from the sample in the process. Find the work done by the gas during the part BC. 
Ans. –4520 J
18. Figure shows the variation in the internal energy U with the volume V of 2.0 mole of an ideal gas in a cyclic process abcda. The temperature of the gas at b and c are 500 K and 300 K respectively. Calculate the heat absorbed by the gas during the process.
Sol. The total heat absorbed by the gas = total work done by the gas
= Wab + Wcd work done by gas in bc & da = 0
= nRTb ln 2v0/v0 + nRTc ln v0/2v0 = nRTa ln 2 – nRTc ln 2
= nR ln 2 (Tb – Tc) = 400 R ln 2
= 2300 J
[Ans. 400 R ln 2, Joules = 2300 J]
19. Find the change in the internal energy of 2kg of water as it is heated from 0ºC to 4ºC. The specific heat capacity of water is 4200 J/kg-K and its densities at 0ºC and 4ºC are 999.9kg/m3 and 1000 kg/m3 respectively. Atmospheric pressure = 105 Pa.
Ans. (33600 – 0.02) J
20. Calculate the increase in the internal energy of 10g of water when it is heated from 0ºC to 100ºC and converted into steam at 100 kPa. The density of steam = 0.6 kg/m3. Specific heat capacity of water = 4200 J/kg–ºC and the latent heat of vaporization of water = 2.5 × 106 J/kg.
Ans. 2.5 × 104 J
21. Figure shows a cylindrical tube of volume V with adiabatic walls containing an ideal gas. The internal energy of this ideal gas is given by 1.5 nRT. The tube is divided into two equal parts by a fixed diathermic wall. Initially, the pressure and the temperature are p1, T1 on the left and p2, T2 on the rigth. The system is left for sufficient time so that the temperature becomes equal on the two sides (a) How much work has been done by the gas on the left part? (b) Find the final pressures on the two sides. (c) Find the final equilibrium temperature. (d) How much heat has flown from the gas on the right to the gas on the left?
Ans. (a) zero
22. An adiabatic vessel of total volume V is divided into two equal parts by a conducting separator. The separator is fixed in this position. The part on the left contains one mole of an ideal gas (U = 1.5 nRT) and the part on the right contains two moles of the same gas. Initially, the pressure on each side is p. The system is left for sufficient time so that a steady state is reached. Find (a) the work done by the gas in the left part during the process, (b) the temperature in the two sides in the beginning, (c) the final common temperature reached by the gases, (d) the heat given to the gas in the right part and (e) the increase in the internal energy of the gas in the left part.
