Heat
Physics 306
Homework Assignment 8
(Statistical)
1) Classical and Statistical Thermodynamics, Carter Chapter 11
11.1 The distribution of particle speeds of a certain hypothetical gas is given by
N(v) dv = Ave-v/v0 dv,
where A and v0 are constants.
a) Determine A so that f(v) = N(v)/N is a true probability density function;
i.e. R 8
0
f(v) dv = 1. Sketch F(v) versus v.
b) Find ¯v and vrms in terms of N0
c) Differentiate f(v) with respect to v and set teh result equal to zero to find
the most probable speed vm
d) The standard deviation of the speeds from the mean is defined as
s =
h
(v - v¯)
2
i1/2
where the bar denotes the mean value. Show that
s =
¯v
2 - (¯v)
2
1/2
in general. What is s for this problem?
2) Classical and Statistical Thermodynamics, Carter Chapter 12
12.5) For N distinguishable coins the thermodynamic probability is
? =
N!
N1!(N - N1)!,
where N1 is the number of heads and N - N1 the number of tails.
a) Assume that N is large enough that Stirling’s approximation is valid. Show
that ln ? is a maximum for N1 = N/2.
b) Show that ?max ˜ e
N ln 2
.
12.8 Two distinguishable particles are to be distributed amoung nondegenerate energy
levels 0, and2 such that teh total energy U = 2.
a) What is the entropy of the system?
b) If a distinguishable particle with zero energy is added to teh system, show
that the entropy of the assembly is increased by a factor of 1.63.
3) Classical and Statistical Thermodynamics, Carter Chapter 13
13.4 Show that for a system of N particles obeying Maxwell-Boltzmann statistics, the
occupation number for the j
th energy level is given by
Nj = -N kT
? lnZ
?j
T
.
4) Assume that thermal neutrons emerging from a nuclear reactor have an energy distribution corresponding to a classical ideal gas at a temperature of 300? K. Calculate the
density of neutrons in a beam of flux 1013/m2
· sec.
Figure 1: In the July 14, 1995 issue of Science magazine, researchers from JILA reported
achieving a temperature far lower than had ever been produced before and creating an
entirely new state of matter predicted decades ago by Albert Einstein and Indian physicist
Satyendra Nath Bose. Cooling rubidium atoms to less than 170 billionths of a degree above
absolute zero caused the individual atoms to condense into a ”superatom” behaving as a
single entity. The graphic shows three-dimensional successive snap shots in time in which
the atoms condensed from less dense red, yellow and green areas into very dense blue to
white areas. JILA is jointly operated by NIST and the University of Colorado at Boulder.;
file from wikimedia commons