MCEN 2024, Spring 2008
The week of Jan 28
HW 3–Final
A Quiz related to HW2 will be posted on CULearn on Friday, Feb 1st\. You must complete the quiz
before 2PM on Monday the 4th. You get only one shot to complete the quiz, that is, once you open it
you must complete it. You cannot open it a second time.
References to A&J:
Chapter 3: Definitions of stress and strain, and the elastic constants.
Chapter 4: Bonding between atoms.
Chapter 5; Packing of atoms in solids.
Chapter 6.1–6.3: Models for elastic moduli.
Overview:
Over the last two weeks we have studied how to bridge the length scales for designing fiber
reinforced polymer composites for the diving board application. We have learnt about how the
geometry, or the microstructure of the composite is related to its modulus, and then how the
modulus determines the performance of the diving board. The discussion has focused on developing
models that give the upper and the lower bounds for the modulus of the composite. We have found
that these bounds can be expressed in terms of the volume fraction of the fiber, and the elastic
modulus of the matrix material and the fiber material.
However, often the materials are selected in their
simple form. For example, the skyview at the
Grand Canyon has to be made from transparent
glass. Glasses can have many different
compositions. How do we understand, or predict
how the modulus of the glass can change with its
composition. Here we consider a binary
composition, made by mixing pure silica with
soda lime (sodium oxide). We shall use this
example to understand not only the relationship
between modulus and composition of glass but
also the chemical bonding between atoms of
different kinds, in this instance silicon, oxygen,
and sodium.
A diving board can also be made from aluminum
or from steel. These two metals have very
different moduli (aluminum a modulus of ~70
GPa, and steel a modulus of ~200 GPa). We will
understand why the moduli of metals can differ
from one metal to another.
Do keep in mind that this time we are learning to bridge the length scale from the atomic level (the
nanoscale) to the physical (i.e. the diving board). This problem entails describing the modulus in
terms of size of the atoms and the bonding among atoms.
Problem(s) related to the calculating the size of atoms and molecules in solids.
1. Calculate the volume occupied by a aluminum atoms in crystalline aluminum. Then
calculate the average distance between the aluminum atoms.
2. Calculate the volume occupied by SiO2 molecules in silica glass (assume a density of 2.2 g
cm-3), and then the average spacing between the molecules.
3. Calculate the volume occupied by one carbon atom in diamond, and then the average
spacing between the carbon atoms.
4. Calculate the volume occupied by one long–chain molecule in polyethylene. Assume that
the molecule has the following (approximate) composition: CH 3 − [ CH 2 ]10,000 − CH 3 . Why,
in this case, we cannot calculate the average spacing between the molecules?
The density, ρ , the molecular (or the atomic) weight, M w and the Avogadro’s Number,, N A
are related to the volume per molecule (or atom), Ω , by the following equation (be careful
with the units!):
M
Ω= W
ρN A
The average spacing between the atoms, ro , is equal to the cube root of Ω .
An excel file was used to calculate the answers, which are copied below:
At or Mol Avogad.
weight
Density
Number
g/mol
per mole
g/cm^3
Volume
Volume
Interatom
Interatom
Which
per atom
per atom
Molecular
Molecular
atom
spacing
spacing
molecule
nm
pm
cm^3
nm^3
27
6.02E+23
2.7
1.66E-23
1.66E-02
0.255153
255.153 Al
60
6.02E+23
2.2
4.53E-23
4.53E-02
0.356487
356.487 SiO2 (glass)
12
6.02E+23
3.5
5.70E-24
5.70E-03
0.178583
178.583 C (diamond)
140030
6.02E+23
1
2.33E-19
2.33E+02n/a
63.5
6.02E+23
8.96
1.18E-23
1.18E-02
0.227487
91
6.02E+23
2.5
6.04651E-23
0.060465
0.392496
n/a
PE
227.487 Copper
392.495762SiO_2.Na_2 O
Problem(s) related to the Periodic Table
5. Explain why the polyethylene molecule has the composition given in #4 above.
From the periodic table we note that H atoms have one electron (they would like to have 2 in
order to fill the shell), while C atoms have four electrons in the outermost shell. Thus carbon
wishes to share four electrons with other atoms into order to complete a shell of eight. The
groups, CH3– achieves this by sharing one electron each with three hydrogen atoms and one
with the neighboring carbon atom in the chain. The carbon atoms bonded as –CH2– share bonds
with two carbon atoms and two hydrogen atoms.
6. Explain why soda lime glass has the composition given by SiO2 .xNa2O .
The compositions involving more than two atoms must be stoichiometric, that is they must
satisfy the valences of the atoms according to the periodic table. Thus SiO2 satisfies the valency
requirement because the valency of four for silicon is met by two atoms of oxygen each having
a valency of two. Each bond between Si and O atoms leads to sharing of one electron, one from
silicon and the other from O. Similarly Na2O satisfies the valency since sodium has a valency of
one and oxygen a valency of 2. Note that one cannot simply add free sodium atoms to silica
glass.
7. What is meant by metallic, ionic and covalent bonding?
In metallic bonding, for example in aluminum, the atoms release the electrons in their outermost
shell to create “free” electrons that can be shared among all aluminum atoms. These so called
free electrons can conduct electricity.
In ionic bonding, as in one atom entirely gives up one electron to the other atom. For example
Na atoms which have one excess electron donate it to Cl which needs one electron to complete
the shell. This electron transfer leads to the sodium atom becoming positively charged and the
chlorine atoms becoming negatively charged. The Na+ and Cl– atoms are called ions, hence the
name ionic bonding. The electrostatic force between these charged atoms creates the bond.
In covalent bonding electrons are shared, more or less equally between two atoms. The bonding
is localized by this sharing of electrons; it is also directional since the electron orbits are usually
constrained along specific directions. SiO2 is an example of covalent bonding since electrons are
shared between the silicon and the oxygen atoms. Diamond made from carbon is also covalently
bonded, since each carbon atoms shares electrons with its four nearest neighbors.
8. Why do aluminum atoms in a crystal have 12 nearest neighbors?
The bonding among aluminum atoms is metallic, which is non–directional. The aluminum
atoms, therefore, like to surround themselves with as many neighbors as possible. We shall
see next week the highest number of nearest neighbors that is possible in the packing of
spheres is 12. That is why aluminum atoms have 12 nearest neighbors.
9. Why do carbon atoms in diamond have four nearest neighbors?
Carbon can form only four covalent bonds with its neighbors. This condition is satisfied by
having four nearest neighbors such that one electron is shared with each of the neighbors.
10. Why are silicon atoms in silica glass surrounded by four oxygen atoms, and why are
the oxygen atoms linked to two silicon atoms?
Because silicon wishes to share one electron each with four nearest neighbors (the oxygen
atoms), while oxygen wishes to share an electron with two nearest neighbors.
11. What is meant by van der Waal’s bonding?
In van der Waal’s bonding electrons remain within the atoms; they are not shared in any way.
Instead the proton and electron nature of the atoms, or a molecule, is such that it becomes like a
dipole with a positive and a negative charge (the magnitude of these charges is of course much
less than the charge on one electron). The attraction between these dipoles is the source of van
der Waal’s bonding. Examples are He, and also the bonding between H2O molecules. In the
H2O molecule the O-H pair forms a weak dipole.
12. Why is the bonding between H2O molecules in ice so weak (it breaks easily!)?
Please see the explanation given just above in the context of van der Waal’s bonding.
13. Construct a hypothetical (the simplest possible) structure for NaCl. Rock salt is
ionically bonded.
The simplest structure one can make (this is not the actual structure of NaCl) is shown below.
Note that each Na+ has one Cl– on one side and another on the other side. Each of these bonds is
equal to one half bond, therefore the net bonding is one whole bond between the sodium and the
chlorine ions. There are many different ways of constructing the structure, but the structure can
be valid only if , the aggregate, there is one sodium atom for every one chlorine atom in the
crystal.
19. A particularly simple soda–lime glass has a composition 2SiO2.Na2O. Show that for this
composition only three of the four silicon–tetrahedra are bonded to other
silicon–tetrahedra. The fourth corner to bonded to NaO0.5. Draw a schematic for the
structure. If the elastic modulus of pure silica is 94 GPa, what would you expect would
be the elastic modulus of this soda lime glass. Can you draw a curve from these two
data points showing how the modulus of the glass would change depending on the
weight fraction of Na2O that is added to it.