[SOLVED] Phys 362 Statistical and Thermal Physics: Homework 2

The density of and pressure due to Earths atmosphere vary with distance above sea level.
The purpose of this exercise is to determine precisely how these depend on height above sea
level.Throughout this exercise, you can assume that the air is an ideal gas and that there
are no currents in the air. This is a classic problem and is a starting point for understanding
anything to do with a planetary atmosphere.a) Denote the height above sea level by z. Consider a thin rectangular slab of air whose
top and bottom are parallel to Earths surface. Denote the vertical thickness of the slab
by dz. Use the fact that the slab is at rest to determine an expression for the pressure
difference dP between the top and bottom of the slab. Use this to show that
dP
dz = g
where is the density of the air in the slab.b) Assume that the air is an ideal gas, consisting of molecules of mass m. Show that
dP
dz = gm
kT P
and solve this to get P as a function of z. Use the result to determine an expression for
as a function of z.c) Air consists of a mixture of N2 (78% by volume), O2 (21% by volume) and argon (1%
by volume). Determine the average mass of one molecule of air. This will be the value
of m.d) Determine the air pressure in Grand Junction and on top of the Grand Mesa (about
11000 ft above sea level), assuming that the air pressure at sea level is 1.01 105 Pa and
that the temperature at both locations is 293 K (this is not exactly true and a more
sophisticated model is needed to describe temperature variations).e) Determine the altitude at which the density of the atmosphere would be 0.10 of what it
is at sea level.A van der Waals gas with a fixed number of molecules is held at constant volume. The
pressure of the gas is doubled. Which of the following is true, assuming that a > 0?
i) The temperature increases by exactly a factor of two.
ii) The temperature increases by more than a factor of two.
iii) The temperature increases by less than a factor of two.
iv) The temperature decreases.
Explain your answer.In a real gas, molecules can exert attractive forces on each other as they approach each other.
Consider two such gases, A and B. The number of molecules in A is the same as in B, the
temperature of A is the same as that of B and the volume of A is the same as that of B.Suppose that the molecules of gas A exert larger attractive forces on each other than the
molecules of B.a) Would you expect the pressure of A to be the same as that of B? If not how would it
differ? Explain your answers.b) What does the ideal gas law predict regarding the pressure of these gases? Could it be
correct?c) Could the van der Waals equation of state predict a difference in pressure? To simplify
your answer suppose that the constant b is the same for both gases and focus on the
constant a. Would a be the same for the gases? Explain your answers.