422
10 Heat and Matter
(iii) What is the resultant change in the internal energy over the whole
three-stage cycle?
(At STP, temperature = 0 ◦ C = 273.15 K and pressure = 1 atm =
1.01 × 10 5 Pa, R = 8.31/J/K/mol.)
10.45 The initial values for the volume and pressure of a certain amount of nitrogen
gas are V 1 = 0.06 m 3 and p 1 = 10 5 N/m 2 , respectively.
First, the gas undergoes an isochoric process (process 1–2), which triples
the pressure; then it is followed by an isobaric process (process 2–3), which
reduces the volume by a factor of three; finally, the volume of the gas is
tripled by an isothermal process (process 3–4).
(a) Give the initial and final temperatures, T 1 and T 4 , of the nitrogen gas if
the temperature after the first (isochoric) process is T 2 = 1083 K.
(b) Find the volume, V 4 , and pressure, p 4 , at the final state of the gas, then
sketch the three processes in a p–V diagram.
(c) How much heat is gained by the nitrogen gas during the first (isochoric)
process and how much heat is given away by the nitrogen gas during
the second (isobaric process)? The amount of heat required to raise the
temperature of 1 mol of nitrogen by 1 K while the gas pressure is kept
constant is c p = 29.12 J/(mol K).
(d) Find the change in the internal energy of the nitrogen gas by the end of
the final process compared to the initial value.
[University of Aberystwyth, Wales]
10.46 When a gas expands adiabatically, its volume is doubled while its absolute
temperature is decreased by a factor 1.32. Compute the number of degrees
of freedom for the gas molecules.
10.47 A heat engine absorbs heat of 10 5 k cal from a source, which is at 127 ◦ C and
rejects a part of heat to sink at 27 ◦ C. Calculate the efficiency of the engine
and the work done by it.
[Osmania University 2004]
10.48 A reversible engine has an efficiency of 1/6. When the temperature of the
sink is reduced by 62 ◦ C its efficiency gets doubled. Find the temperatures of
the source and the sink.
10.49 Assuming that air temperature remains constant at all altitudes and that the
variation of g with altitude is negligible
(a) show that the pressure P at an altitude h above sea level is given by
p = p 0 exp(−Mgh/RT ), where M is the molecular weight of the gas.
(b) show that n = n 0 exp(−Mgh/RT ) where n is the number of molecules
per unit volume.
(c) taking the average molecular weight of air to be 29 g, calculate the height
at which the air pressure would be half the value at sea level.
10 Heat and Matter
(iii) What is the resultant change in the internal energy over the whole
three-stage cycle?
(At STP, temperature = 0 ◦ C = 273.15 K and pressure = 1 atm =
1.01 × 10 5 Pa, R = 8.31/J/K/mol.)
10.45 The initial values for the volume and pressure of a certain amount of nitrogen
gas are V 1 = 0.06 m 3 and p 1 = 10 5 N/m 2 , respectively.
First, the gas undergoes an isochoric process (process 1–2), which triples
the pressure; then it is followed by an isobaric process (process 2–3), which
reduces the volume by a factor of three; finally, the volume of the gas is
tripled by an isothermal process (process 3–4).
(a) Give the initial and final temperatures, T 1 and T 4 , of the nitrogen gas if
the temperature after the first (isochoric) process is T 2 = 1083 K.
(b) Find the volume, V 4 , and pressure, p 4 , at the final state of the gas, then
sketch the three processes in a p–V diagram.
(c) How much heat is gained by the nitrogen gas during the first (isochoric)
process and how much heat is given away by the nitrogen gas during
the second (isobaric process)? The amount of heat required to raise the
temperature of 1 mol of nitrogen by 1 K while the gas pressure is kept
constant is c p = 29.12 J/(mol K).
(d) Find the change in the internal energy of the nitrogen gas by the end of
the final process compared to the initial value.
[University of Aberystwyth, Wales]
10.46 When a gas expands adiabatically, its volume is doubled while its absolute
temperature is decreased by a factor 1.32. Compute the number of degrees
of freedom for the gas molecules.
10.47 A heat engine absorbs heat of 10 5 k cal from a source, which is at 127 ◦ C and
rejects a part of heat to sink at 27 ◦ C. Calculate the efficiency of the engine
and the work done by it.
[Osmania University 2004]
10.48 A reversible engine has an efficiency of 1/6. When the temperature of the
sink is reduced by 62 ◦ C its efficiency gets doubled. Find the temperatures of
the source and the sink.
10.49 Assuming that air temperature remains constant at all altitudes and that the
variation of g with altitude is negligible
(a) show that the pressure P at an altitude h above sea level is given by
p = p 0 exp(−Mgh/RT ), where M is the molecular weight of the gas.
(b) show that n = n 0 exp(−Mgh/RT ) where n is the number of molecules
per unit volume.
(c) taking the average molecular weight of air to be 29 g, calculate the height
at which the air pressure would be half the value at sea level.