(a) Show that under conditions of standard pressure and temperature,
the volume of a sample of an ideal gas depends only on
the number of molecules in it.
(b) One mole is defined as 6.0×1023 atoms. Find the volume of one
mole of an ideal gas, in units of liters, at standard temperature and
pressure (0 °C and 101 kPa).
A gas in a cylinder expands its volume by an amount ΔV ,
pushing out a piston. Show that the work done by the gas on the
piston is given by ΔW = PΔV .
(a) A helium atom contains 2 protons, 2 electrons, and 2 neutrons.
Find the mass of a helium atom.
(b) Find the number of atoms in 1 kg of helium.
(c) Helium gas is monoatomic. Find the amount of heat needed
to raise the temperature of 1 kg of helium by 1 degree C. (This is
known as helium's heat capacity at constant volume.)
Refrigerators, air conditioners, and heat pumps are heat engines
that work in reverse. You put in mechanical work, and it the effect
is to take heat out of a cooler reservoir and deposit heat in a warmer
one: QL +W = QH. As with the heat engines discussed previously,
the efficiency is defined as the energy transfer you want (QL for a
refrigerator or air conditioner, QH for a heat pump) divided by the
energy transfer you pay for (W).
Efficiencies are supposed to be unitless, but the efficiency of an air
conditioner is normally given in terms of an EER rating (or a more
complex version called an SEER). The EER is defined as QL/W, but
expressed in the barbaric units of of Btu/watt-hour. A typical EER
rating for a residential air conditioner is about 10 Btu/watt-hour,
corresponding to an efficiency of about 3. The standard temperatures
used for testing an air conditioner's efficiency are 80 °F (27 °C)
inside and 95 °F (35 °C) outside.
(a) What would be the EER rating of a reversed Carnot engine used
as an air conditioner?
(b) If you ran a 3-kW residential air conditioner, with an efficiency
of 3, for one hour, what would be the effect on the total entropy
of the universe? Is your answer consistent with the second law of
(a) Estimate the pressure at the center of the Earth, assuming
it is of constant density throughout. Use the technique of example
5 on page 140. Note that g is not constant with respect to depth
- it equals Gmr/b3 for r, the distance from the center, less than b,
the earth's radius 1). State your result in terms of G, m, and b.
(b) Show that your answer from part a has the right units for pressure.
(c) Evaluate the result numerically.
(d) Given that the earth's atmosphere is on the order of one thousandth
the thickness of the earth's radius, and that the density of
the earth is several thousand times greater than the density of the
lower atmosphere, check that your result is of a reasonable order of
1)Derivation: The shell theorem tells us that the gravitational field at r is the same as if all the mass existing at greater depths was concentrated at the earth's center. Since volume scales like the third power of distance, this constitutes a fraction (r/b)3 of the earth's mass, so the field is (Gm/r2)(r/b)3 = Gmr/b3.
(a) Determine the ratio between the escape velocities from the
surfaces of the earth and the moon.
(b) The temperature during the lunar daytime gets up to about
130 °C. In the extremely thin (almost nonexistent) lunar atmosphere,
estimate how the typical velocity of a molecule would compare with
that of the same type of molecule in the earth's atmosphere. Assume
that the earth's atmosphere has a temperature of 0 °C.
(c) Suppose you were to go to the moon and release some fluorocarbon
gas, with molecular formula CnF2n+2. Estimate what is
the smallest fluorocarbon molecule (lowest n) whose typical velocity
would be lower than that of an N2 molecule on earth in proportion
to the moon's lower escape velocity. The moon would be able to
retain an atmosphere made of these molecules.
Most of the atoms in the universe are in the form of gas that
is not part of any star or galaxy: the intergalactic medium (IGM).
The IGM consists of about 10-5 atoms per cubic centimeter, with
a typical temperature of about 103 K. These are, in some sense, the
density and temperature of the universe (not counting light, or the
exotic particles known as dark matter). Calculate the pressure of
the universe (or, speaking more carefully, the typical pressure due
to the IGM).
A sample of gas is enclosed in a sealed chamber. The gas
consists of molecules, which are then split in half through some
process such as exposure to ultraviolet light, or passing an electric
spark through the gas. The gas returns to thermal equilibrium with
the surrounding room. How does its pressure now compare with its
pressure before the molecules were split?
The figure shows a demonstration performed by Otto von Guericke
for Emperor Ferdinand III, in which two teams of horses failed
to pull apart a pair of hemispheres from which the air had been
(a) What object makes the force that holds the hemispheres
(b) The hemispheres are in a museum in Berlin,
and have a diameter of 65 cm. What is the amount of force holding
them together? (Hint: The answer would be the same if they were
cylinders or pie plates rather then hemispheres.)