Problems
The symbols√
, , etc. are explained on page 351.
1 (a) Show that under conditions of standard pressure and tem-
perature, 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).√2 A gas in a cylinder expands its volume by an amount dV,
pushing out a piston. Show that the work done by the gas on the
piston is given by dW=PdV.
3 (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 1.0 kg of helium.√(c) Helium gas is monoatomic. Find the amount of heat needed to
raise the temperature of 1.0 kg of helium by 1.0 degree C. (This is
known as helium’s heat capacity at constant volume.)√4 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?
5 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 10^3 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).
√6 Estimate the pressure at the center of the Earth, assuming it
is of constant density throughout. Note thatgis not constant with
respect to depth — as shown in example 19 on page 105,gequals
Gmr/b^3 forr, the distance from the center, less thanb, the earth’s
radius.
(a) State your result in terms ofG,m, andb.√(b) Show that your answer from part a has the right units for pres-
sure.
(c) Evaluate the result numerically.√(d) Given that the earth’s atmosphere is on the order of one thou-
sandth 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 magnitude.Problems 347