PHY251 Homework Set 4
Reading:
Chapter 5
Homework:
Chapter 5, Questions 1,3,4 Problems 6,8,9,16,18
Hints and Solutions
Question V.1
In studying the Bohr model in this chapter, we ignored special relativity. Why, then, does the
speed of light appear in our expression for the Bohr radius, Eq. (5-9)? (Hint: If the speed of
light were 10 times what it is, by how much would the numerical value of the Bohr radius
change?)
Hints:
No hints
Solution:
The Bohr radius does
not
depend on the speed of light
c
: the appearance of
c
in the
formula for the Bohr radius is purely due to the presence of the dimensionless constant
α
,
which itself introduces
c
in its definition
=
e
²/(4
πε
0
) /
hc
; i.e. the product
c
appearing
in the Bohr radius, the orbital speed, the energy, and so on, do
not
depend on
c
.
Question V.3
Given that the atom's mass is roughly the mass of the nucleus, that the atom's size - about the
Bohr radius - is determined by the electronic structure of the atom, and that the density of
matter is in the range 10
3
kg/m
3
to 10
4
kg/m
3
, what would you estimate the nuclear mass to be
compared with the electron mass? Outline your assumptions and approach before you actually
make any calculations.
Hints:
No hints
Solution:
We are asked to estimate the average mass of an atomic nucleus in matter, using
the value of the atom's size as determined by the Bohr radius
a
0
, and the density of
ordinary matter 10
3
kg/m
3
. First, we will estimate the typical volume of an atom,
and then determine how many atoms would fit in an m
3
. Using the typical mass of
a m
3
of matter, we can then estimate the average mass of an individual atom.
Because the atomic mass is largely determined by its nucleus, we thus find the
typical nuclear mass.
The typical atomic volume is ~(2
a
0
)
3
= (0.1 nm)
3
= 10
-30
m
3
phy251_hw04
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