10.11. Heat, Q, required to raise the temperature of a substance by 1 K is called the heat

capacity, C, of the substance. The relationship can be expressed as

2. The unit for C is J K-1.3. Heat, Q, required to raise the temperature of one kilogram of a substance by 1 K

is called the specific heat capacity, c, of the substance. The relationship can be expressed as

4. The unit for c is J kg-1 K-1.5. The relationship between C and c is C = me .6. Specific heat capacities of some common substances are shown in the table

below.

EXAMPLE300 g of water at a temperature of 40 °C is mixed with 900 g of water at a temperature of 80 °C. If there is no heat loss to the surroundings, what is the final temperature when thermal equilibrium is achieved by the mixture of water?

Molar Heat Capacity of GAS .1. Heat, Q, required to raise the temperature of one mole of gas by 1 K at constant

volume is called the molar heat capacity at constant volume, C.2. The heat required to raise the temperature of n moles of gas at constant volume

through temperature AT is given by

3. Heat, Q, required to raise the temperature of one mole of gas by 1 K at constant pressure is called the molar heat capacity at constant pressure, cm.

4. The heat required to raise the temperature of n moles of gas at constant pressure through temperature AT is given by

Exercise 10.11. When an electric heater is supplied with an electric power of 2.0 kW to heat 4.0

kg of water for 1 minute, calculate the increase in temperature of the water. Assume that the specific heat capacity of water is 4 200 J kg-' °C-1 and there is no heat loss to the surroundings.

2. When a metal cube of sides 2.0 cm is immersed into a perfectly insulated container filled with 1.0 kg of water at 5 °C, the temperature of water rises to 7 °C. Assuming there is no heat loss to the surroundings, calculate the original temperature of the metal cube. [Density of the metal cube = 8 900 kg m-3, specific heat capacity of water = 4 180 J kg' K-1 and specific heat capacity of the metal cube = 385 J kg-1 K-1]

10.2 WORK DONE BY GAS

1. Figure 10.1 shows work is done when a gas in a cylinder of cross-sectional area, A, expands when it pushes the piston through a small distance, Ax.

2. Work done, Hence, ∆W = p∆V, where ∆V = A∆x is the change in the volume of the gas.

3. When a gas expands from a volume of V, to a volume of V2 at constant temperature, the total work done is

4. When a gas expands, V2 > V1, work is done by the gas, W is positive.

When a gas is compressed, V2 < V1, work is done on the gas, W is negative.

When V2 = V1, In = 0. Hence, work done, W, is zero.5. When the gas expands under constant pressure, work done

The work done is represented by the area under the p—V graph as shown by the shaded area in Figure 10.3.

6. The graph in Figure 10.4 shows a gas expanding from A to B, compressed at

constant pressure from B to C, and then the pressure of the gas increased at constant volume from C to A. The net work done is represented by the shaded area in the p—V graph.

Example 10.2 A sample of gas at an initial volume of 250 cm3 expands at constant pressure of 1.50 x 105 Pa until it reaches a final volume of 320 cm3. Find the work done by the gas.

Exercise 10.21. A helium gas of mass 2.50 g is heated and expands at constant pressure so that its

temperature increases from the ice point to the steam point. Assuming that helium gas behaves like an ideal gas, what is the total work done by the gas?[R = 8.31 J mot-1 K-1, molar mass of helium = 4.00 g mol-']

2. When 2.60 mol of an ideal gas is heated, its temperature increases from 25.0 °Cto 30.0 °C as it expands at constant pressure. Find the work done by the gas.

10.3 First Law of Thermadynamics1. The first law of thermodynamics states that the heat energy supplied to a gas is

equal to the increase in the internal energy and the work done by the gas. This is equivalent to the total energy in a closed system being constant. Mathematically, it can be expressed as

2. Consider a gas that is compressed without heat being supplied to it. From the first

law of thermodynamics, the internal energy, ∆U = Q — ∆W = —∆W > 0. This is because AW is negative when work is done on the gas. Hence, this results in an increase in the internal energy of the gas.

3. The sign convention for the quantities used in the first law of thermodynamics are as shown in the table below.

EXAMPLEA cylinder of gas with a volume of 200 cm3 is supplied with heat energy of 45.00 J. The gas expands at a constant pressure of 1.05 X 105 Pa until a final volume of 250 cm3. Calculate(a) the work done by the gas,(b) the increase in internal energy of the gas.

Relationship Between CP and CV

1. Consider Figure 10.6 (a) whereby 1 mole of gas in a cylinder is heated at constant volume (isochoric process) so that its temperature is increased by 1 K.

2. The heat required, Q = nC„mAT, where Cvm is the molar heat capacity at constant volume.

3. Since n =1 and AT = 1, Q = Cvm.4. By the first law of thermodynamics, Q = AU + pAV. At constant volume, AV =

0, hence, the heat supplied, Q, is solely used to increase the internal energy, U, of the gas, Q = AU. Therefore, AU = Cvm.

5. Consider Figure 10.6 (b), whereby 1 mole of gas in a cylinder is heated at constant pressure (isobaric process) so that its temperature is increased by 1 K and the gas volume is increased from V, to V2.

6. Since the temperature is increased by 1 K in both cases, the increase in internal energy in both cases, AU = C,,m. Similarly, the total energy supplied,

7. Using the first law of thermodynamics, we have

8. For 1 mole of gas,

9. If M is the molar mass of the gas, then

1. When one mole of ideal gas is heated at constant volume so that its temperature increases by AT, then by the first law of thermodynamics, Q = AU + pAV

2. The internal energy for 1 mole of ideal gas is given by U = NA(kT) where NA

is the Avogadro's number, f is the number of degrees of freedom and k is the Boltzmann's constant. Hence,

3.

4. The table below shows the various values of y for different types of atom.

Exercise 10.31. An insulated cylinder contains an ideal gas. The initial volume, pressure and

temperature of the gas are 3.20 X 10' m3, 1.01 X 105 Pa and 320 K respectively. Assume that there is no heat transferred through the piston.(a) Calculate the number of moles of the gas.(b) The gas is compressed until its volume becomes 4.50 X 10-5 m3 and the temperature becomes 750 K. Calculate the final pressure of the gas.(c) The work done on the gas is 120 J. Calculate the increase in the internal energy of the gas.

2. A gas is compressed at constant pressure of 340 kPa as its volume decreases from 2.50 m3 to 1.20 m3 while 250 Id of heat is released from the gas. Calculate the change in internal energy of the gas.

3. Work of 230 kJ is done on a thermodynamic system while 650 kJ of heat is supplied to it. Determine the change in internal energy of the system.

4. The density of an ideal gas is 1.775 kg m-3 at temperature 27 °C and pressure 1.0 X 105 Pa. The specific heat capacity of the gas at constant pressure is 846 J kg-1 K-'. Find the ratio of its specific heat capacity at constant pressure to that at constant volume.

5. When 12.0 g of helium gas is heated, its temperature increases from 27.0 °C to 57.0 °C. Determine the heat absorbed and the work done by the gas when it expands(a) at constant volume,(b) at constant pressure.[For helium gas, c, = 3.14 kJ kg-1 K-'; cp = 5.22 kJ kg-1 K-']

6. Determine the value of Cvm and Cn,m for a gas which has y = 1.29. [R = 8.31 J mol-' K-']

7. The specific heat capacity at constant pressure of a gas is 1.05 kJ kg-1K-1. Given that the molar mass of the gas is 28.0 g mol-' and molar gas constant, R, is 8.31 J mol-' K-'. Calculate y of the gas.

10.4 Isothermal and Adiabatic Changes1. A gas that undergoes changes at constant temperature is known as isothermal

change.2. Since at constant temperature, there is no change in internal energy; hence, the

pressure and volume of the gas have to vary according to Boyle's law, i.e. pV = constant.

3. Figure 10.7 shows the p—V graphs for Boyle's law at different temperatures

whereby each curve is known as an isotherm.4. When a gas is compressed, work is done on the gas and the temperature and

internal energy of the gas increase. Hence, in order to maintain no change in temperature or internal energy in isothermal compression, the extra internal energy acquired during compression has to be dissipated as heat to the surroundings.

5. Conversely, when a gas expands, work is done by the gas and the temperature and internal energy of the gas decrease. Hence, in order to maintain no change in temperature or internal energy in isothermal expansion, the loss in internal energy during expansion has to be compensated by receiving heat from the surroundings.

Work Done During Isothermal Change1. When an ideal gas expands from volume V, to V2 at constant temperature, T, the

total work done is

2. Since p,V, = p2V2 = nRT, we have

3. For isothermal expansion, the change in internal energy, AU = 0, AW > 0 since

V2 > V. From the first law of thermodynamics, Q= AU + AW, Q> 0 which means that the work done by the gas is equal to the heat, Q, supplied to the gas.

4. Conversely, for isothermal compression, the change in internal energy, ∆U = 0, AW < 0 since V2 < V,. From the first law of thermodynamics, Q = ∆U + ∆W, Q < 0 which means that the work done on the gas is dissipated as heat to the surroundings.

Adiabatic Change1. Expansion or compression of an ideal gas without heat entering or leaving the gas

system is called adiabatic change.2. From the first law of thermodynamics, Q = AU + AW, Q = 0; hence, the work

done, 4W, by the gas is at the expense in the decrease in internal energy, AU, of the gas. That is, AW = —AU. This means that there is a decrease in temperature for adiabatic expansion.

3. Conversely, for adiabatic compression, work is done on the gas and AW< 0. Hence, —AW = AU. This means that there is an increase in the temperature of the gas.

Equation for Adiabatic Change1. Assume 1 mole of an ideal gas is compressed adiabatically; so there is work done

on the gas and it causes an increase in internal energy of the gas.2. From the first law of thermodynamics, Q = AU + AW = 0 and

3. From the equation of state, pV = RT

Since pV = RT and ∆p∆V can be ignored as it is very small, we have

Substitute (i) into (ii), we have

4. In the limit when Ap —' 0 and AV — 0, we have

5.

6.

Example 10.4 A mole of hydrogen gas with a volume of 0.080 m3 is compressed adiabatically to half of its original volume. If the initial average kinetic energy per hydrogen molecule is 1.38 X 10-2' J. what are the initial and the final pressure of the hydrogen gas? Assume that hydrogen is a diatomic gas.

Work Done During Adiabatic Change

1. Consider a gas that expands adiabatically from a volume of V1 at ,a temperature

of T2 to a volume of V2 at a temperature of T1 as shown in Figure 10.8.2. The work done by the gas in expanding is given by

3. Since pi V,' = p2V2Y = k, the equation can be rewritten as

4. Furthermore,

5. Since the change in temperature, AT = T2 — T1, we have

6. The positive work done by the gas in an adiabatic expansion has caused a

reduction in its internal energy as shown by the negative sign of AU.7. Conversely, in an adiabatic compression, the negative work done on the gas will

cause an increase in its internal energy.

Reversible Process1. A reversible process is a process which can be made to retrace its path from one

equilibrium point to another through small changes at every step.2. In a reversible process, the gas is always in equilibrium state and obeys the ideal

gas equation of state T = constant at every instant throughout the changes.3. The process is irreversible if the gas does not obey the equation of state

throughout the changes.4. For a reversible isothermal compression or expansion, the walls of the heat

conducting container containing the gas must be very thin to enable dissipation of heat to the surroundings or absorption of heat from the surroundings quickly. Furthermore, the isothermal change must be performed as slow as possible so that there is ample time for the transfer of heat.

5. For a reversible adiabatic compression or expansion, the walls of the heat insulating container containing the gas must be very thick to prevent the transfer of heat. Furthermore, the adiabatic change must be performed as quickly as possible so that there is very little time for the transfer of heat.

6. However, perfect reversible changes do not occur. This is due to the fact that there is no perfect conductor and perfect insulator.

Isothermal and Adiabatic Changes on p-V graph

1. The p—V graph in Figure 10.9 shows a gas expanding isothermally from A to B, compressed adiabatically from B to C and then compressed further at constant pressure from C to A.

2. For the isothermal expansion from A to B, there is no change in internal energy of the gas. From the first law of thermodynamics, Q = ∆U + ∆W, Q = AW since ∆U = 0. The positive work done, AW, in expansion is equal to the heat supplied, Q, to the gas.

3. The adiabatic compression from B to C results in an increase in temperature or an increase in internal energy, as the adiabatic curve moves from a lower isotherm to a higher isotherm. Since no heat (Q = 0) enters or leaves the gas, the work done on the gas (∆W < 0) results in an increase in internal energy (∆U > 0) of the gas. Furthermore, the gradient at any point on the adiabatic curve is always higher than that on the corresponding isotherm.

4. The isobaric (constant pressure) compression from C to A results in a decrease in temperature or a decrease in internal energy (∆U < 0), as the isobaric curve moves from a higher isotherm to a lower isotherm. The work done on the gas, ∆W < 0. From the first law of thermodynamics, Q = ∆U + ∆W, results in Q < 0. This means that heat is dissipated to the surroundings.

5. The net work done is represented by the area enclosed by the curve ABCA.

Exercise 1 0.41. An ideal gas of 0.50 mol expands at a constant temperature of 30 °C from 50 cm3

to 65 cm3. Find the work done by the gas.2. 1.50 mol of an ideal gas has a pressure of 60.0 kPa at 27 °C. The gas is

compressed isothermally until its pressure reaches 200 kPa. Calculate the quantity of heat transferred in the gas.

3. An oxygen gas of mass 40.0 g is compressed adiabatically at s.t.p. to a pressure of 450 kPa. Assuming that oxygen gas behaves as an ideal gas, determine(a) the initial volume,(b) the new volume, of the gas.[R = 8.31 J mol-'K-1; molar mass of oxygen = 32 g mol-1, y = 1.40; standard pressure = 101 kPa]

4. If 5.00 X 10-3 m3 of an ideal gas at a pressure of 125 kPa and a temperature of 300 K is compressed until its pressure is doubled and y = 1.40, determine(a) the new volume,(b) the new temperature.

5. An ideal gas expands adiabatically from a pressure of 76.0 cm Hg and a volume of 500 cm3 to a pressure of 40.0 cm Hg. Calculate(a) the new volume,(b) the work done.[y = 1.40, density of Hg = 13.6 X 103 kg m3]

6.

Figure 10.10 shows a p—V graph of an ideal gas with a temperature of 27 °C at point D.(a) Deteimine the temperature of the gas at point

(i) A,(ii) B.

(b) the net work done by the gas.

7. A fixed mass of gas undergoes changes in pressure and volume as shown in Figure 10.11.

When the gas changes from state P to state R through the stages PQ and QR, 8 J of heat is absorbed and 3 J of work is done. If the same resultant change is achieved through stages PS and SR, calculate the amount of heat absorbed if 2 J of work is done by the gas.

Summary: Chapter 101 The heat required to raise the temperature of a substance by 1 K is called the heat capacity of the substance.2 The heat required to raise the temperature of one kilogram of a substance by 1 K is called the specific heat capacity of the substance.

3 When a gas expands under constant pressure, work done,4 The first law of thermodynamics states that the heat energy supplied to a gas is equal to the increase in internal energy and the work done by the gas.

5 Since the kinetic energy of a gas molecule depends on the temperature of the gas, hence the internal energ` of an ideal gas depends on the temperature of the gas.6 Heat, Q, required to raise the temperature of one mole of gas by 1 K at constant volume is called the molar heat capacity at constant volume, Cv,m.7 Heat, Q, required to raise the temperature of one mole of gas by 1 K at constant pressure is called the molar heat capacity at constant pressure, C.

9 The internal energy for 1 mole of ideal gas,8 Relationship:

10 The ratio of the molar heat capacity at constant pressure to the molar heat capacity at constant volume is

11 The changes a gas undergoes at constant temperature is known as isothermal change.12 The changes a gas undergoes at constant pressure is known as isobaric change.13 The changes a gas undergoes at constant volume is known as isochoric change.14 When an ideal gas expands from volume V1 to V2 at constant temperature, T, the total work done,

15 Expansion or compression of an ideal gas that involves no heat entering or leaving the gas system is called an adiabatic change.

16 Adiabatic change equations :

17 Work done during an adiabatic expansion, W = —AU. The positive work done by the gas in an adiabatic expansion has caused a reduction in its internal energy as shown by the negative sign of AU.18 The positive work done by the gas in an adiabatic expansion has caused a reduction in its internal energy.

19 Work done during an adiabatic expansion,20 A reversible process is a process that can be made to retrace its path from one equilibrium point to another equilibrium point through small changes at every step.

Revision ExerciseObjective

1. A system consists of two bulbs, G and H, which are maintained at constant temperature in a water bath. G and H are filled with ideal gases at a pressure of 200 kPa and 600 kPa respectively and the volume of G is two times that of H.When the valve is opened and the system is allowed to achieve equilibrium, what is the final pressure of the gas system?

2. Argon and neon are two monatomic gases which have relative atomic mass of 40

and 20 respectively. number of atoms in one mole of argon gas The ratio number of atoms in one mole of neon gas

3. If there exists 2.70 X 1019 atoms in 1 cm3 of an ideal gas at 273 K and 105 Pa,

what is the number of atoms in 1 cm3 of the gas at 273 K and 10-s Pa?

4. If the molar heat capacity at constant volume for a gas is kR, where k is a constant, what is the ratio of the molar heat capacity at constant pressure to the molar heat capacity at constant volume?

5. Heat energy of 70 J is supplied to an ideal gas of volume 1.5 X 10-3 m3 and

pressure 1.0 X 105 Pa so that the volume is increased to 1.7 X 10-3 m3 but the pressure remains unchanged. The internal energy of the gas isA raised by 50 JB raised by 60 J C raised by 80 JD lowered by 30 J

6. The equation, pV = nRT, is applicable for a real gas only ifA the density is smallB the temperature is very low C the pressure is very lowD the volume is large and the pressure is high

7. The first law of thermodynamics can be written as

where Q is the heat energy supplied to a system of gas, AU is the increase in internal energy and AW is the external work done by the gas system. Which of the following is correct when an adiabatic expansion occurs?

8. Gas is pumped into a balloon and this causes its diameter to increase to 3 times its

original diameter. What is the work done to overcome the atmospheric pressure, p?

9. What is meant by the internal energy of an ideal gas?

10. When heat is supplied to an ideal gas in a cylinder, the gas expands at constant

pressure, p, until its volume becomes 3 times the original value, V. The work done by the gas is

11. The equation pV„, = RT refers to A one kilogram of an ideal gasB one mole of an ideal gas C one kilogram of a diatomic gasD one mole of a real gas

12. The pressure, p, of an ideal gas can be expressed as p = akT, with k as Boltzmann's constant and T as the thermodynamic temperature of the gas. Symbol a representsA the number of moleculesB the number of molecules per mol C the number of molecules per unit massD the number of molecules per unit volume

13. One mole of an ideal gas undergoes an adiabatic change which causes its temperature to increase by AT. Which of the following statements is true about the gas?A The volume of the gas increases.B The pressure of the gas decreases. C Heat is dissipated from the gas.D The change in internal energy of the gas is CAT.

14. Figure 10.14 shows a thermodynamic cycle for an ideal system.

Heat enters the gas system at stagesA PQ and SP C QR and SPB PQ and RS D QR and RS

15. The first law of thermodynamics explains the conservation of energy of a system. Based on this law, which of the following statements is correct? A The decrease in internal energy is the same as the work done by the

system when the system expands adiabatically. B The increase in internal energy is the same as the heat absorbed by the

system when the system expands isothermally.C The decrease in internal energy is the same as the heat removed from the

system when the system expands at constant pressure.D The increase in internal energy is the same as the work done by the system

when the system expands at constant pressure.

16. The graphs below show the isothermal processes of an ideal gas at absolute temperature, T1 and T2 respectively.

What is the value of T2 in terms of T1?A T1 C 3.0 T1

B 1.5T1 D 6.0T1

17. Which of the following equations cannot be used in an adiabatic process of an ideal gas?A dQ = 0 C pV = constantB pV = nRT D = constant

18. Which of the following is true about an adiabatic expansion of an ideal gas?A The temperature of the gas does not change.B The internal energy of the gas increases. C The r.m.s speed of the gas decreases.D The external work done is more than the change in internal energy.

19. When N moles of an ideal gas is heated at a constant pressure, p, its volume increases from V, to V2 and its temperature increases by T K. If the internal energy of the gas increases from U1 to U2, the principal molar heat capacity of the gas at constant pressure, Cp,„, can be expressed as

20. An ideal gas expanded at constant pressure, cooled at constant volume and finally

compressed adiabatically until it returned to its original state. Which graph shows correctly the changes that occur to the gas?

21. The ratio of the molar heat capacity of an ideal gas is a. What is the number of degrees of freedom of the gas?

22. The graph below shows the process undergone by an ideal gas. The gas which

initially had an internal energy, U1, expanded isothermally and the work done is W. The gas is then compressed at constant pressure until its internal energy became U2.

The change in internal energy of the gas isA W C U2— U1

B U1 — W D U1 — W — U2

23. The temperature of two moles of an ideal diatomic gas is raised by 8 °C from room temperature. The increase in internal energy of the gas is

24. The diagram below shows the change in pressure and volume of an ideal gas

through two paths, PQR and PSR. For the change through the path PQR, the heat absorbed by the gas is 32 J.

What is the heat absorbed by the gas for the change through the path PSR?A 10J C 17 JB 15J D 47J

25. The work done to compress adiabatically one mole of an ideal diatomic gas is 100 J. What is the rise in temperature of the gas?A 2.00K C 4.81KB 3.54K D 5.20 K

26. Which statement is true of an adiabatic expansion? A Charles's law is obeyed.B The temperature is always constant.C No heat enters or leaves the system.D The internal energy increases.

27. The graph below shows the variation of pressure, p, with volume, V, for a mass of gas.

What are the changes to the heat transfer, internal energy and work done when the gas changes from R to S?

28. When a gas undergoes an isothermal expansion, 40 J of heat is supplied to it.

What is the change in the internal energy, AU and the work done, W, by the gas in this process?

29. A well insulated container fitted with a piston contains a diatomic gas. Which of the following describes the r.m.s. speed and the degrees of freedom of the gas molecules if the gas expands and pushes back the piston?

Structure Question1. 2.90 X 10-4 m3 of an ideal gas at a pressure of 1.04 X 105 Pa and a temperature

of 314 K is kept inside an insulated cylinder to prevent heat loss. The gas undergoes adiabatic compression until its volume and temperature become 2.90 X 10-5 m3 and 790 K respectively.(a) Calculate the value of y and the ratio of Cp' Cvni(b) Calculate the pressure of the gas after the compression.(c) Calculate the work done to compress the gas.

2. The gas in a heat pump can be considered as undergoing a cycle of changes of pressure, volume and temperature. One such cycle, for an ideal gas, is shown in the graph below.

The table below shows the changes in the internal energy along the path A to B, B to C and C to D. It also shows that along the path A to B and C to D, no heat is supplied to the gas.

Using the first law of thermodynamics and the data from the graph, determine(a) the work done by the gas from A to B,(b) the work done by the gas from C to D(c) the work done by the gas from B to C,(d) the heat supplied to the gas from B to C,(e) the work done by the gas from D to A,(f) the increase in the internal energy from D to A,(g) the heat supplied to the gas from D to A.

3. The molar heat capacity at constant volume of diatomic hydrogen gas and oxygen gas are each of value 2 R.(a) If two moles of hydrogen is mixed with one mole of oxygen gas,

deteunine the molar heat capacity at constant volume of 0.018 kg of the mixture.

(b) If an explosion occurs, determine the heat capacity at constant volume of the new gas. (Molar masses: hydrogen = 2 g mo1-1; oxygen = 32g mo1-1)

4. One mole of a monatomic ideal gas is at a temperature of 0 °C and a pressure of 1.01 X 105 Pa. How much work is needed to compress the gas to 10.0 litres if the process is perfoinied(a) isothermally,(b) adiabatically?

5. A cylinder, fitted with a frictionless piston, contains an ideal gas with initial volume of 5.0 X 10-i m3, pressure of 1.0 X 105 Pa and temperature of 300 K. The gas is(i) heated at constant pressure to 450 K, and then(ii) cooled at constant volume to the original temperature of 300 K. The heat

dissipated from the gas during stage (ii) is 63 J.(a) Illustrate these changes on a p—V graph labelled with the appropriate

values of pressure and volume.(b) How much work is done by the gas in pushing back the piston in stage (i)?(c) What is the total heat absorbed in stage (i)?(d) Find the ratio of the principal specific heat capacities, y, of the gas.

6. A quantity of 0.200 mol of air enters an engine at a pressure of 1.04 X l05 Pa and at a temperature of 297 K. Assume that air behaves like an ideal gas.(a) Find the volume of this quantity of air.(b) The air is then compressed to one twentieth of its volume and the pressure

is raised to 6.89 X 106 Pa. Find the new temperature.(c) Heating of the air then takes place at constant pressure of 6.89 X 106 Pa

by burning a small quantity of fuel in it to supply heat energy of 6 150 J. As the volume of the air increases and the temperature rises to 2 040 K, find

(i) the molar heat capacity of air at constant pressure,(ii) the volume of the air after burning the fuel,(iii) the work done by the air during this expansion,(iv) the change in the internal energy of the air during this expansion.

7. An ideal gas of volume V and at a pressure of p undergoes the cycle of changes as shown in the graph below.

Determine the coolest point and the hottest point.

8. A cylinder fitted with a smooth piston contains 0.050 mol of a monatomic ideal gas at a temperature of 27 °C and a pressure of 1.00 X 105 Pa. Calculate

(a) the volume of the gas,(b) the internal energy of the gas.

The temperature of the gas is then raised to 77 °C and the pressure remains constant. Calculate

(c) the change in the internal energy,(d) the external work done,(e) the total heat energy supplied.

9. (a) What is meant by an adiabatic change for an ideal gas?(b) A diesel engine does not need a spark plug for the combustion of a mixture of diesel and air in a cylinder. Spontaneous combustion occurs when a high temperature is achieved through the compression of air. If air with an initial temperature of 27 °C is compressed adiabatically to a final temperature of 681°C, what is the ratio of the initial volume to the final volume of the air? [Assume that air is a diatomic ideal gas]

10. The variation of internal energy, U, of one mole of an ideal gas with temperature, T, is shown in Figure 10.19.

(a) Calculate the molar heat capacity at constant volume.(b) Calculate the molar heat capacity at constant pressure.(c) Calculate the ratio of the molar heat capacity at constant pressure to the molar heat capacity at constant volume.(d) What can you conclude from your answer in (c)?

11. (a) The specific heat capacity at constant volume of an ideal gas is 2.4X102Jkg-1 K-1. Calculate the change in the internal energy of 5.0 X 10-3 kg of the gas when the temperature of the gas is increased from 27 °C to 337 °C.(b) The graph in Figure 10.20 shows the variation of pressure, p, with volume, V, for 0.04 mol of a diatomic ideal gas.

The gas expands adiabatically from state X to state Y. Determine the change in the internal energy during the process XY.

12. (a) Explain what is meant by a reversible isothermal change.(b) Discuss if the compression and rarefaction of air along the direction of propagation of a sound wave of frequency of 400 Hz and wavelength of 0.80 m can be considered as an adiabatic change.(c) (i) Explain why the temperature of a gas decreases during an adiabatic

change. (ii) Discuss if the decrease in temperature is the same for an ideal gas and a real gas. With the aid of labelled diagrams, describe briefly an experiment to verify your answer.

(d) A fixed mass of a monatomic ideal gas has a volume of 1.70 X 10-4 m3 and a pressure of 250 kPa.

(i) Calculate the internal energy of the gas.(ii) The gas expands at constant pressure to a volume of 4.50 X 10-4 m3. What is the increase in the internal energy of the gas and the work done by the gas?

13. (a) (i) Write an equation to represent the first law of thermodynamics. State the symbols used in your equation.

(ii) Use the equation above to explain the energy used by an ideal gas when it expands isothermally and adiabatically.

(b) A mole of argon gas with an initial volume of Vo expands isothermally at temperature To until its volume is doubled. Derive an expression for the work done by the gas if the gas obeys the equation of state(p + a v2)(V — b) = RT where a and b are constants and R is the molar gas constant.(c) The value of Rm. for oxygen gas is 2.48, where R is the gas constant and Cym is the molar heat capacity at constant volume. Deduce the value of c,,, the specific heat capacity of oxygen gas at constant volume.

14. (a) One mole of a diatomic ideal gas is heated at a constant pressure of 1.01X 105 N m-2 from 25.0 °C to 45.0 °C. Calculate

(i) the heat absorbed by the gas,(ii) the change in the volume of the gas,(iii) the work done by the gas,(iv) the change in the internal energy of the gas.

(b) 2 moles of a diatomic ideal gas in a container has a pressure of 1.01 X 105 N In-2 at 27.0 °C. The gas undergoes an adiabatic compression until its volume becomes a quarter of its original volume.

(i) What is meant by an adiabatic compression?(ii) Determine the final absolute temperature of the gas.(iii) Calculate the work done on the gas.

15. (a) (i) What is meant by work done in an isolated gas system?(ii) Differentiate between the internal energy and the thermal energy of a gas system.(iii) State the first law of thermodynamics and the meaning of each symbol used.

(b) An isolated system of 5.0 moles of an ideal gas is initially at a pressure of p, and a volume of V,. It is then allowed to expand at constant temperature, T = 400 K to a new pressure of p2 and a new volume of V2 which is twice the initial volume of V,.

(i) Sketch the p—V diagram to show the expansion process and shade the region representing the work done during the process.(ii) Calculate the work done during the process.