Physics 3760 Spring 2002
Final Exam –
Due Wed, May 8, before 5pm.
P. Sokolsky
1. Consider the following “Sadly Carnot” cycle. Path 1 lies along a straight line in
the P-V plane, having an equation P = aV+b. The end points of the straight line
are: P0 =32 Pa, V0=8 m3; P1 = 1 Pa, V1 =64 m3. Path 2 is adiabatic, connecting P0,
V0 to P1, V1.
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
What are the values of a and b?
Find T as a function of V along path 1
What is the maximum value of T?
Find the values of T0 and T1.
What is the total heat flow along path 1?
What is the total work along the adiabat ( path 2 )?
What is the net work done in the cycle?
What is the net heat transferred to the gas during the cycle?
What is the efficiency of the cycle?
What is the change of the enthropy of the gas along path 1
What is the change of enthropy of the gas along path 2?
What is the change of enthropy of the heat reservoirs over the cycle?
2. Prove that two isentropic curves do not intersect for systems where P and V are
the independent variables.
3. A gas obeys the equation of state: Pv= RT + B(T)P
Where B(T) is a function of T only. Calculate Cp and Cv as a function of B(T), P, T
and Cp(P=0) ( the heat capacity at constant pressure evaluated at very small
pressures).
(Hint: show that ∂Cp/∂P)T = - T(∂2V/∂T2)P)
4. Mercury is poured into a U- tube open at both ends. The total length of the
mercury is h.
a. Suppose the level of Mercury is pushed down on one side. Calculate the period of
oscillation of the resulting motion.
b. Suppose one end of the U-tube is closed so that the air inside is trapped. If the
length of the trapped air is L and we assume that the air behaves like an ideal gas,
calculate the period of oscillation of the Mercury, assuming the compression of
the air is adiabatic.
c. How would you use this set-up to measure the ratio of specific heat at constant
volume to specific heat at constant pressure?
5. Estimate the probability that a postage stamp ( mass = 0.1 gm) resting on a desk
top at room temperature ( 300 K) will spontaneously fly up to a height of 10-8 cm
above the desk top. Hint: Think of an infinite number of noninteractng stamps
placed side-by-side. Formulate an argument showing that these stamps obey the
Maxwell-Boltzmann distribution..
6. A gas consists of N point particles in an infinite space. Each particle is attracted to
the origin by a force proportional to the distance from the origin, F = - kr, so in
addition to its translational kinetic energy, each particle has a potential energy U =
.5k(x2+ y2 + z2)
a. What is the Maxwell-Boltzman distribution function for this gas?
b. Show that the probability of finding a particle in dr at distance r is given by:
Ar2e-cr*r/2kT dr, and evaluate the constant A.
c. Find the average values of x, x2, r and r2.
d. What is the molar heat capacity at constant volume of this gas?
e. At what radius would a spherical shell contain 90% of the gas molecules?