Physics 264L: Assignment 6
Made available: Wednesday, September 30, 2015
Due:
Monday, October 5, 2015, by 7 pm
Problems
1. An optional review problem on complex numbers: as we start our discussion of quantum mechanics
next week (Monday, October 5), it will be important that all members of the class have a working
knowledge of how to work with complex numbers, especially the idea of the complex conjugate z ∗ of
a complex number z, the rules for working with complex conjugation (e.g., that (z1 + z2 )∗ = z1∗ + z2∗
and (z1 z2 )∗ = z1∗ z2∗ ) and Euler’s identity eix = cos(x) + i sin(x). If you are not familiar with complex
number or want some review, please work through my “Bootcamp on Complex Numbers”, which is
Problem 2 from this Physics 162 homework assignment from last semester:
www.phy.duke.edu/~hsg/162/files/homeworks/assignment-10-mar-29-2015.pdf
I will review some of these ideas in class but you will learn the ideas best by working through this
tutorial problem at your own pace. I will be glad to provide you with answers to this tutorial problem
if that would be useful.
2. Problem 1-1 on page 45 of French and Taylor. This is one of my favorite scientific insights, that by
measuring the area spanned by a tablespoon of oil on a lake, one can estimate the size of a molecule
to within a factor of ten.
3. Problem 2-1 on page 95 of French and Taylor.
4. Problem 2-3 on page 96 of French and Taylor
5. Problem 2-6 on page 97 of French and Taylor.
6. Problem 2-15 on page 101 of French and Taylor. For this problem,you may need to review some
material about the Bohr atom from Chapter 1 of French and Taylor or from your intro physics book
or from an intro chemistry book.
7. In this problem, you will use your knowledge of an ideal gas, of the de Broglie wavelength of a molecule,
and of the equipartition theorem to deduce at what temperature T something new and quantum
mechanical (and so presumably weird) should occur as one steadily reduces the temperature of a gas
of helium atoms. It is surprising and pleasing that one can make such an interesting prediction with
so little effort.
As you perhaps knew before taking Physics 264L and now know from your reading of Chapter 2 in
French and Taylor, atoms and molecules are not crisp geometric objects with a precise size but fuzzy
quantum mechanical things, with electron clouds that extend arbitrarily far from the nucleus. One
way to associate a length with a fuzzy molecule is by its de Broglie (pronounced approximately as
“de-broy”, not as “de-brog-lee) wavelength
λ=
1
h
h
=
,
p
mv
(1)
where p = mv is the momentum of the molecule and where h is Planck’s constant. (For most gases
on Earth, molecules move with speeds of order the speed of sound, which is so much smaller than the
speed of light that we can use our non-relativistic Newtonian formulas here, so m is the rest mass of
the molecule.)
Note that the wavelength Eq. (1) is usually not the same size as the so-called covalent radius, which
is defined to be one half of the distance between two nearest neighbor molecules in a crystalline solid
as determined by X-ray crystallography (with the data extrapolated to absolute zero to eliminate the
effects of thermal expansion).
Now consider an ideal gas, which empirically is a volume V containing N identical molecules with
pressure P and with absolute temperature T (in kelvin temperature units) such that the following
ideal gas equation is accurately satisfied:
P V = N kT,
(2)
where
m2 · kg
J
≈ 1.4 × 10−23 2
,
(3)
K
s ·K
is the so-called Boltzmann constant which is fundamental to all phenomena related to thermodynamics
and statistical physics. (Chemists often write the ideal gas in terms of moles, P V = nRT , where n is the
number of moles of the gas in the volume V , and where R is the so-called gas constant.) Physically,
a gas is ideal when the molecules in the gas do not affect each other, mainly because they are so
far apart compared to their size that they spend most of the time moving freely, without collisions.
Experimentally, all gases are found to be ideal in the limit of sufficiently low pressure or low gas density.
k ≈ 1.4 × 10−23
For any volume V containing N molecules (so for liquids and solids also), one can define an average
distance d between the molecules by observing that, if there are N molecules in a volume V and the
molecules are approximately uniformly distributed throughout the volume, there is an average volume
of V /N per molecule. If we think of this average volume as a little cube of size d, then V /N = d3 or
d=
µ
V
N
¶1/3
,
(4)
which indeed has physical units of length. Combining this with the ideal gas law Eq. (2), you should
be able to deduce that1
d ∝ T 1/3 ,
(5)
which implies that, as the absolute temperature T decreases for fixed pressure P , the mean distance d
between molecules decreases (the gas becomes denser).
On the other hand, the molecule’s de Broglie wavelength Eq. (1) increases with decreasing temperature
as
λ ∝ T −1/2 .
(6)
This is intuitively reasonable, at lower temperatures, molecules move more slowly so their typical
speed v decreases, causing λ to increase by Eq. (1). To see why the particular scaling relation Eq. (6)
holds, note that Eq. (1) implies λ ∝ p−1 ∝ v −1 and the equipartition theorem for molecules in a gas
says that the average kinetic energy is given by
r
¿
À
p
3kT
1
3
mv 2 = kT ⇒ vrms = hv 2 i =
∝ T 1/2 .
(7)
2
2
m
Then λ ∝ v −1 ∝ (T 1/2 )−1 ∝ T −1/2 which is Eq. (6).
1 Make sure that you know how to use the proportionality symbol ∝ for making scientific deductions: we say that some
quantity a is proportional to some quantity b, a ∝ b, if there is a positive constant c such that a = cb. The notation is valuable
when one wants to emphasize or deduce a functional relation involving power laws without worrying about the proportionality
constants. You should be able to verify that “being proportional” is commutative (a ∝ b implies b ∝ a), and associative (a ∝ b
and b ∝ c implies a ∝ c), also that a ∝ b implies aβ ∝ bβ for any non-zero power β. With these insights, the ideal gas law
P V = N kT implies V ∝ T for fixed N and fixed P , while V = N d3 implies d3 ∝ V ∝ T which implies d ∝ T 1/3 as claimed.
2
So as the temperature T of a gas decreases, is it then possible for the decreasing average distance d
between molecules in a gas to become about the same size as the increasing de Broglie wavelength? If
so, does anything interesting happen physically?
Answer this question for a gas of He4 atoms in a vessel of constant pressure P = 1.0 atm ≈ 105 Pa
whose volume can be varied by an experimentalist (say by moving a piston in a cylinder). For this
problem, assume that the He gas is ideal.
(a) At what temperature T in kelvin do you predict that the de Broglie wavelength becomes equal to
the average distance between molecules?
(b) What is the common value of these two lengths when they become equal? For comparison, the
covalent radius of a He atom is about 3×10−11 m ≈ 30 pm where 1 pm = 10−12 m is one picometer.
(c) Discuss whether your estimate for T corresponds to any interesting features in the phase diagram
for He4
http://ltl.tkk.fi/research/theory/helium.html
8. To the nearest integer, please give the time in hours that it took you to complete this assignment,
including reading in the text.
Also, if you got help with this assignment, please give the names of the people from whom you got
help (classmates, the TA, myself, etc).
3