Materials 101 Problem Set #1
Answers
1.) The bonds between pairs of atoms in the solid can be
a.) ionic bonds, b.) covalent bonds, c.) metallic bonds, d.) van der Waals (induced dipole-induced
dipole) and e.) hydrogen bonds.
Which type(s) of bond can not occur in pure solid elements? ionic bonds and hydrogen bonds
since they occur between two different types of atoms.
Which type(s) of bond has, at long distances, an attractive force that varies as 1/r2 between pairs?
an ionic bond
Which type(s) of bond is highly directional? covalent bonds and hydrogen bonds
Which type of bonding occurs between Xe atoms in solid xenon? van der Waals bonding
Which type(s) of bond is found in solid Si? covalent bonds
Which type of bond is the weakest, i.e has the lowest spring constant? van der Waals bonds
Which type(s) of bonds are important in ice crystals? (Skiers take note!) hydrogen bonds
In solids with which type of bonding are valence electrons free to move quickly over large
distances? metallic bonding
2.) The bonds between two atoms in a solid have a potential energy En versus distance r curve
given by the equation:
where C = 3.2E-20 J/bond and a=0.38 nm.
a.) Plot the attractive and repulsive terms of this potential energy versus distance between the
nuclei of the
atoms.
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b.) Which of the terms in En (r) is dominant at large distances? Which of the terms is dominant at
short distances?
c.) Find the equilibrium distance ro between the atoms if no force is applied to the
bond.
d.) Find the energy that must be supplied to pull the two atoms from ro to a very large distance
apart. This energy is called the binding energy
Eb.
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e.) Compare the binding energy with thermal energy kBT at room temperature, where kB is
Boltzmann's constant. If kBT >> Eb the material will not be a solid or even a liquid at room
temperature (thermal agitation will break the bond). At what temperature is will thermal energy
be equal to Eb?
f.) Determine the force f versus distance curve for the bond and plot it. What is the "spring
constant" of the bond?
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g.) If no thermal energy is available (T = 0 K), what is the maximum force the bond can support?
h.) The strain in simple tension is usually defined as (L-Lo)/ Lo where Lo is the initial length and
L is the final length along the axis of tension. Apply this definition to the bond under tension and
determine the tensile strain to break the bond.
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3.) Aluminum metal has a face centered cubic crystal structure and an atomic radius of 0.143 nm.
Compute the volume of its fcc unit cell.
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4.) The unit cell of an elemental solid has the following structure (all angles of the corners are
90°):
a.) To what crystal system (Bravais lattice) does a crystal with this unit cell belong? body
centered tetragonal
b.) If the atomic weight of the material is 114 g/mol, compute the density of the solid with this
unit cell.
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5.) Atomic radii for elements can be determined by assuming that the atoms touch along the
closest packed directions in the unit cell of their crystal structures, e.g. along the face diagonal in
the face centered cubic (fcc) unit cell and along the body (cube) diagonal in the body centered
(bcc) unit cell. Rhodium has an atomic radius of 0.1345 nm and a density of 12.41 g/cm3.
Determine whether rhodium has a fcc or bcc crystal structure.
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6.) Draw the unit cell of the fcc structure. How many close packed planes with different
orientations are there in the unit cell? (Don't count planes that are parallel to one another.) Sketch
them on the unit cell surface.
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