1. science
2. mathematics
3. geometry

HOMEWORK EXERCISES
The forces of covalent bonds usually act over what range of interatomic
4.1 How is “material” different from “matter?”
separations?
4.2 True
or false?
solids.
True or false? A single water molecule
is formed
by All
vansubstances
der Waalsform
bonding,
4.3
Interatomic
forces
are
usually
which
type of force?
while covalent bonding holds numerous water molecules together.
4.4
True
or
false?
On
a
plot
of
the
potential
True or false? The dispersion force can cause a nonpolar molecule to behave energy of two atoms versus their separation distance, the lowest energy point on the curve is the separation distance
like a dipolar molecule.
attractive
andpoints.
repulsive forces TTY
are equal./ Syksy 2014
FYS-1350
Nanofysiikka
Explain why smaller molecules
tend towhere
have
lower
boiling
4.5
True
or
false?
The
radius
of
an
ion
is
roughly
equivalent to that of the uncharged
Rank the following bonds from strongest to weakest and provide the bond
atom.
Explain
your molecule;
answer. the bond
energy: the bond between
hydrogen and
oxygen
in a water
Laskuharjoitus
4
4.6
The
forces
of
covalent
bonds usually
between sodium and chloride in the NaCl molecule; the bond between
atoms act over what range of interatomic
separations?
in a metal; the van derTehtävä
Waals bond
between adjacent
hydrogen atoms.
1 (Exercise
4.11).A
4.7
True or
molecule
In Back-of-the-Envelope 4.2, we determined
thefalse?
van der single
Waals water
attractive
force is formed by van der Waals bonding,
In Back-of-the-Envelope
4.2,
we determined
the van der Waals attractive force between a pair
while
covalent
bonding
holds
numerous
between a pair of hydrogen atoms. Of course, in practice we more often deal water molecules together.
of
hydrogen
atoms.
Of
course,
in
practice
we
dealmolecule
with devices
and objects, and
True or
false?
Thelead
dispersion
can more
cause aoften
nonpolar
to behave
with devices and objects, and van4.8
der Waals
forces
often
to partsforce
sticking
van
der between
Waalslike
often
lead
to parts
together. The attractive energy between a
a dipolar
molecule.
together. The attractive
energy
aforces
spherical
body
of radius
R and asticking
flat
4.9
Explain
why
smaller
molecules
tend
to
have
lower
boiling
spherical
body
of
radius
R
and
a
flat
surface,
separated
by a points.
distance x (see Figure 4.24) is
surface, separated by a distance x (see Figure 4.24) is given by
given by 4.10 Rank the following bonds from strongest to weakest and provide the bond
energy: the bond between hydrogen and oxygen in a water molecule; the bond
HR
sodium and chloride in the
NaCl molecule; the bond between atoms
E( x )sphere-surface =between
−
(4.7)
6x
in a metal; the van der Waals bond between adjacent hydrogen atoms.
4.11 In Back-of-the-Envelope
we determined
the van
der Waals
attractive
In this equation,
H is the so-called 4.2,
Hamaker
constant.
In the
case of
solidsforce
separated by air, H
between a pair of hydrogen atoms. Of course, in practice we more often deal
≈ 10−19 J.
with devices
and objects,
and attractive
van der Waals
forces
often
lead toto
parts
a) Determine
the equation
for the
force
with
respect
x. sticking
together.
The
attractive
energy
between
a
spherical
body
of
radius
R
and afrom
flat the surface, as
b) Now determine the force of gravity pulling the sphere down, away
surface,
by a distance
(see Figureis4.24)
is given
by
a function
of separated
R. Assume
that thex particle
made
of silicon,
with a density of 2330
kg/m3.
HR pulls it away from the surface if they are
c) How large can the sphere
before gravity
x Ebe
( x )sphere-surface
=−
(4.7)
6x
separated by 10 nm of air? By 1 nm?
R
A spherical body of radius, R, separated from a surface by a distance, x. (See
xercise 4.11.)
x
R
A spherical body of radius, R, separated from a surface by a distance, x. (See
Homework Exercise 4.11.)
FIGURE 4.24
Tehtävä 2 (Exercise 4.13).
A given crystal structure (such as NaCl) can be represented as consisting of planes of atoms,
as shown in Figure 4.25. A beam of x-rays can be reflected off the crystal, where some of the
beam penetrates through the atoms of the upper layer and strikes the atoms in the lower plane.
A pair of incident x-rays from an x-ray source is reflected from the crystal as shown and into
an x-ray detector.
a) How much farther does the beam reflected from the lower plane travel from the source
to the detector than the one reflected from the upper plane?
b) If the beams have wavelength, λ, under what conditions will the two reflected beams
constructively interfere with each other (i.e., have maxima at the same points)?
c) The relationship you derived in part (b) was first derived by W.L. Bragg (1890–1971).
If we can experimentally measure the diffraction angle and we know the wavelength
used, what feature of the crystal can we use this relationship to determine?
d) X-rays with a wavelength of 140 pm are reflected from an NaCl crystal and are found
Nanomaterials
◾ 119
to constructively interfere at an angle of incidence of 14.4°. Calculate
d.
D
t
ec
et
Xra
ys
ou
rc
e
or
q
d
Upper plane
q
Lower plane
X-rays reflected by the top two planes of a crystal. The gray dots represent atoms in
the crystal lattice. The planes are parallel. (See Homework Exercise 4.13.)
FIGURE 4.25
Tehtävä 3.
Palataan tehtäväänIn1.thisMuutetaan
nyt Hamaker
siten, että
pallonInsijaan
equation, Htilannetta
is the so-called
constant.
the casepinnassa
of solids on kiinni
−19
kuutiomainen kappale,
jonka
sivun
on L. Tämä kuutio (tilavuus L3) lepää lähellä
separated
by air,
H ≈ 10pituus
J.
a. Determine
thepintaa
equationsiten,
for theettä
attractive
forcejawith
respect(pintaa
to x.
äärettömän laajuista
tasomaista
pinnan
kuution
lähinnä olevan
b.
Now
determine
the
force
of
gravity
pulling
the
sphere
down,
away
from
the joka on
tahkon) välinen etäisyys toisistaan on x. Tässä tilanteessa van der Waals -voima,
3
surface,on
as aFfunction
of /R.(6
Assume
the particle
is made
with per pintakuution ja pinnan välillä,
π x ).that
Huomaa,
että
tämäof silicon,
on voima
vdw = H
3.
a
density
of
2330
kg/m
alayksikkö. H on jälleen Hamaker-vakio.
c. How large
thekuvattu
sphere bekuutio
before gravity
pulls it away from
if kattoon.
Voit nyt kuvitella,
ettäcan
yllä
on hyönteinen,
joka the
onsurface
tarttunut
they are separated by 10 nm of air? By 1 nm?
Kun hyönteinen on pieni, se pysyy katossa kiinni van der Waals –voiman ansiosta. Osoita
4.12 The spacing between atoms in a crystal is about 100 pm. What forms of elecskaalausrelaatioilla, että kun hyönteisen koko kasvaa, niin jossain vaiheessa väistämättä
tromagnetic radiation have wavelengths short enough to fit between the atoms
hyönteistä alaspäininvetävä
gravitaatiovoima voittaa hyönteistä katossa pitävän van der Waals
the crystal?
–voiman, ja hyönteinen
alas
lattialle.
skaalausrelaatioiden
4.13 A given humpsahtaa
crystal structure
(such
as NaCl)(Muista
can be represented
as consisting ofidea: tässä
tapauksessa kuinkaplanes
voima
riippuu
kappaleen
koosta.)
of atoms, as shown in Figure 4.25. A beam of x-rays can be reflected
off the crystal, where some of the beam penetrates through the atoms of
Tehtävä 4.
the upper layer and strikes the atoms in the lower plane. A pair of incident
x-ray source
is reflected from
the crystal
shown
and into an että niitä
Tarkastellaan kahtax-rays from
tyhjiössäanolevaa
vesimolekyyliä
(katso
kuvatasalla).
Kuvitellaan,
detector.
voidaan kuvata x-ray
yksinkertaistetusti
siten, että niitä kumpaakin kuvaa vain niiden
a. How
much farther
does the beam
reflected from the lower
plane travel from kohdalla.
dipolimomentti, jonka
keskipiste
sijaitsee
ko. vesimolekyylin
massakeskipisteen
the
source
to
the
detector
than
the
one
reflected
from
the
upper plane?
Tällöin näiden vesimolekyylien välinen potentiaalienergia on
b. If the beams have wavelength, λ, under what conditions will the two reflected
E = p1 p2 K / (4 π ε0 r3beams
), constructively interfere with each other (i.e., have maxima at the same
points)?
c. The relationship you derived in part (b) was first derived by W.L. Bragg
jossa K = sin θ1 sin θ(1890–1971).
– 2 experimentally
cos θ1 cos θ2measure
. Kulmat
esitetty angle
myösand
alla olevassa
2 cos (φ1 –Ifφwe
2) can
theon
diffraction
kuvassa kahden vesimolekyylin
dipoleille
p
ja
p
.
1
2
we know the wavelength used,
what
feature of the crystal can we use this
a) Missä asennossa
toisiinsa
nähden nämä kaksi vesimolekyyliä haluaisivat olla, jotta E
relationship
to determine?
minimoituu?
b) Jos nyt kyseessä olisikin 3 vesimolekyyliä (”1”, ”2”, ja ”3”), ja laskisit koko systeemin
yhteisen potentiaalienergian (vesimolekyylien ”1” ja ”2” välisen potentiaalienergian,
molekyylien ”2” ja ”3” välisen energian, ja vesien ”1” ja ”3” välisen
potentiaalienergian, ja kuvaten näiden energioita yhdessä), niin mikä olisi noiden
kolmen molekyylin keskinäinen orientaatio, jossa kokonaispotentiaalienergia
minimoituu?