114 3 The Second and Third Laws of Thermodynamics: Entropy
3.7 a.Assume that natural gas is pure methane (it is actually
90–95% methane). Find the amount of heat put into a
house by the combustion of 100 cubic feet of natural
gas at 20.0◦C and 1.00 atm if 20.0% of the heat is
wasted (hot gases go up the flue).
b.Find the amount of heat put into a house if 100.0 cubic
feet of natural gas at 20.0◦C and 1.00 atm is burned in
an electric generating plant that is 80.0% as efficient as
a Carnot engine operating between 2000◦C and 800◦C
and if 90% of this energy is delivered to a heat pump
operating between 20.0◦C and 0.00◦C. Assume that this
heat pump has a coefficient of performance that is
80.0% of that of a Carnot heat pump.
c.Find the amount of heat put into a house if 100.0 cubic
feet of natural gas is burned in an electric generating
plant that is 80.0% as efficient as a Carnot engine
operating between 2000◦C and 800◦C and if 90% of
this energy is delivered to a resistance heater that has
100% efficiency.
3.8 a.Calculate the coefficient of performance of a Carnot
food freezer with an interior temperature of− 18 ◦C and
an exterior temperature of 25◦C. 1 watt1Js−^1.
b.Calculate the amount of electrical energy in
kilowatt-hours necessary to freeze 1.000 kg of water in
the Carnot freezer of part a.
3.9A reversible heat engine accepts heat from a hot reservoir
at temperatureT 1 but exhausts part of the heat (−q 2 )attemperatureT 2 and part (−q 3 ) of it at temperatureT 3 ,
whereT 2 >T 3. Find the efficiency ifq 3
T 2
T 3q 2.3.10 a.Assume that the human body has the same efficiency in
doing mechanical work as a Carnot engine with an
upper temperature equal to human body temperature,
37 ◦C, and a lower temperature equal to 25◦C. Find the
efficiency.
b.The actual efficiency of the human body is
approximately equal to 20%. How can you explain this,
since no heat engine can be more efficient than a Carnot
engine?
3.11 It has been proposed that a heat engine might economically
operate using the temperature difference between sea water
near the surface and at a depth of several hundred feet.
a.Assume that such a heat engine has 50% of the
efficiency of a Carnot engine and operates between
30 ◦C and 20◦C. Find the efficiency of the
engine.
b.Assume that the heat engine drives an electric generator
that produces 100 Mwatt (100 megawatts). Find the
volume of sea water that must pass through the
high-temperature heat exchanger per second if the
heat exchanger lowers the temperature of the sea
water from 30◦Cto20◦C. Assume that the sea
water has the same heat capacity as pure water at
298.15 K, 75.351 J K−^1 mol−^1 , and density equal to
1. 00 × 103 kg m−^3.3.2 The Mathematical Statement of the Second
Law: Entropy
The second law of thermodynamics can be stated mathematically in a way that defines
a new state function:If the differentialdSis defined bydSdqrev
T(definition of the entropyS) (3.2-1)thendSis an exact differential andSis a state function called the entropy.We now
show thatdqrev/Tis an exact differential. The integral of an exact differential is path-
independent, so that its integral around a closed path (a path that starts and ends at
the same point) vanishes. The converse is also true: If the integral around an arbitrary
closed path vanishes, the differential is exact. We need to show that
∮
dqrev
Tsurr