[SOLVED] Phys 362 statistical and thermal physics: homework 10

1 Gould and Tobochnik, Statistical and Thermal Physics, 2.63, page 106.Consider a heat engine that absorbs heat Qh from a reservoir that is initially at temperature
Thi and ejects heat Qc to a reservoir that is initially at temperature Tci. In class we analyzed
the efficiency of such an engine with reservoirs which remained at a constant temperature.Now suppose that the reservoirs each have a finite but constant heat capacity and that these
are the same. Thus the temperatures of the reservoirs will change. In one cycle the high
temperature reservoir will drop to temperature Thf and the low temperature reservoir will
rise to Tcf . If c is the heat capacity of the higher temperature reservoir, then
Qh = c(Thf Thi)
and similarly, if the low temperature reservoir has the same heat capacity,
Qc = c(Tcf Tci).a) Starting with
dSh = c dT
T
determine an expression for the change in entropy for the hot reservoir in terms of
Thi, Thf and c. Rewrite Thf in terms of Qh and c to obtain expressions for the change
in entropy in terms of Thi, c and Qh. Repeat this for the cold reservoir.b) Use the second law to show that this implies that
Qc
Qh
!
Tci
Thi
+ Qc
cThi
= Tcf
Thi
and use this to determine a bound on the efficiency of the heat engine.c) Check that when the heat capacity becomes infinite, that one obtains the original efficiency that we obtained in class. How does this compare to the efficiency when the heat
capacity is finite?d) As time passes and the engine runs through many cycles, what will happen to its efficiency?Consider a Carnot engine using an ideal gas and operating between heat baths at temperatures Thigh and Tlow.
a) Assume that the temperature of the hotter reservoir is fixed but the temperature of the
cooler reservoir can vary. Which of the following is true?i) The efficiency of the engine increases as the temperature difference between the
baths increases.
ii) The efficiency of the engine decreases as the temperature difference between the
baths increases.iii) The efficiency of the engine stays the same as the temperature difference between
the baths increases.
Explain your answer.b) Consider two Carnot engines, denoted A and B. The temperatures of the heat baths
which they access are different but they are such that for each heat engine the temperature difference for the baths is the same. Which of the following is true?i) The efficiency of the engine A will be the same as that of engine B regardless of
any other facts.
ii) The efficiency of the engine A might be larger than that of engine B depending
on the circumstances.
Explain your answer.Consider two Carnot engines which operate
between the same two isothermals. The cycles are as illustrated. How do the efficiencies
of the two engines compare (i.e. is one larger,
smaller, etc. )? Explain your answer.
P
V
P
VDuring the Carnot cycle, the engine extracts heat from a high temperature bath and loses
heat to a low temperature bath. How does the entropy change of the high temperature bath
compare to that of the low temperature bath?One can imagine using the work output of one heat engine as the input to another heat
engine. By connecting abstract engines together in this way, you can arrive at conclusions
about thermodynamic processes.Suppose that an engine exists which takes heat from a high
temperature reservoir and converts all of this into work with no waste heat. Show that by
connecting this to a refrigerator, one could produce a device whose sole effect is to transfer
heat from a low temperature bath to a high temperature bath.What does this say about
the possible existence of an engine exists which takes heat from a high temperature reservoir
and converts all of this into work with no waste heat?