17.2 Statistics
17.1.How many ways are there of putting three identical
balls in four separate boxes, with one ball in each box? Does
the number of possibilities agree with equation 17.1? How
many different ways are there if there are no restrictions on
the number of balls in each box?
17.2.How many different ways are there of putting a red, a
blue, and a green ball in four separate boxes? Compare your
answer with exercise 17.1.
17.3.Estimate the value of 1,000,000! (that is, one million
factorial). Express your answer in terms of a power of 10 as
well as a power of e.
17.4.One form of Stirling’s approximation is N!
(2)1/2NN1/2eN. Show that equation 17.2 can be obtained
from this.
17.5.An even more exact form of Stirling’s approximation is
(N1)!
eNNN1/2(2)1/2 1 121 N 2881 N 2
where higher-order terms inside the parentheses are omitted.
Take the natural logarithm of this expression and evaluate
ln (5000!). Compare your answer with the value given in the
table below equation 17.2. How close are the different values?
17.6.Determine the average score on an exam two different
ways and show that the same average score is obtained. The
scores are 78, 44, 74, 92, 85, 50, 74, 80, 80, and 90.
17.7.For values of some observable that can be represented
by a function, the average value of that observable is the area
under that function (that is, the integral) divided by the inter-
val. Population densities of insects can often be expressed as
a function. Assume that in the first month of a calendar year
interval, two insects are released into a controlled system. As
the year progresses, they have some offspring, which in turn
have some offspring, until by the middle of the year there are
38 insects in the system. Then as the year progresses, insects
die off and at the beginning of the next calendar year there
are only two left. Plotting the number of insects versus the
month, it is found that the population follows the quadratic
equation
No. of insects (7 x)^2 38where xis the number of the month in the year (starting with
1). Determine the average number of insects per month in the
system.
17.8.If the nivalues are all the same, a shorthand way of
indicating a combination is C(x, y), which is read, “How
many combinations are there of xdistinguishable objects sep-
arated into systems, each of which have ythings?” Evaluate
(a)C(10, 2) (b)C(3, 1) (c)C(6, 3) (d)C(6, 2). (Hint:you
should determine the number of systems you need for each
case first.)
17.3 & 17.4 Ensembles; Most
Probable Distribution
17.9.A grand canonical ensemble is defined as an ensemble
whose microsystems all have the same volume, temperature,
and chemical potential. Rewrite equations 17.4–17.6 to relate
the states of the microsystems with the state of the overall
system.
17.10.Redo exercise 17.9, except for a microcanonical en-
semble. The definition of the microcanonical ensemble is in
the text.
17.11.What is the most probable distribution of a three-
particle system having four possible energy levels, as shown in
Figure 17.5, where the system has a total of 5 energy units?
Are the thermodynamic properties of such a system deter-
mined solely by considering that most probable distribution?
Why or why not?
17.12.A common thought experiment is to suppose that all
of the gas molecules in a room may instantaneously cluster in
one corner of the room, killing everyone in the room. Explain
in statistical thermodynamic terms why we don’t need to
worry about this ever happening.
17.13.The derivation of equation 17.15, in which derivatives
are applied to a summation and only a single term remains as
a more simplified expression, is best illustrated by example. A
function can be expressed in terms of three variables, 1 , 2 ,
and 3 as3
i 1Ciiwhere the Civalues are the coefficients multiplying each vari-
able .
(a)Write the expression for explicitly (that is, without the
summation sign) and verify that / 1 C 1 , / 2 C 2 ,
and / 3 C 3.
(b)Write a general expression for the derivative of with re-
spect to i, in which ican be either 1, 2, or 3. Compare this
general expression with equation 17.15 and explain how equa-
tion 17.15 was derived from the expression immediately pre-
ceding it.
17.14.Explain why qis a constant for a given system at a
specified temperature.
17.15.What is the ratio of ground-state nickel atoms (in
which Eis defined as zero) to nickel atoms that are in the first
excited state of 200 cm^1 at 298 K? (The spectrum of Ni
atoms is complicated by the existence of such an excited state.
Does your answer explain why?) Assume the two states have
the same degeneracies.
17.16.Using the fact that 1/kT, show that equations
17.29 and 17.30 are equivalent.614 Exercises for Chapter 17
EXERCISES FOR CHAPTER 17