Select and answer 5 of the following 9 questions. All questions carry 20 points. If you answer more than five questions, your score will be obtained by summing the best five. The exam is closed-book. Useful formulae and material properties are provided on the attached sheet.

  1. Dilithium is an element with an atomic mass of 300. Dilithium forms crystals which have a bcc structure. If the effective atomic radius of dilithium is 2nm, what is the density of a dilithium crystal (in kg/m3)? At 900 K, dilithium undergoes a phase change to an fcc structure. What is the percentage change in volume associated with this transformation?

    Model Answer to Question 1

    We begin by sketching the BCC Structure. It is straightforward to calculate the relationship of the lattice parameter a to the atomic radius R by considering the length of the major diagonal. From here, we go on to calculate the volume of a unit cell, recalling that the density of a unit cell will be the same as the density of the bulk material. Lastly, we calculate the change in density resulting from a change to FCC structure.

  2. A positively charged sodium ion is brought from infinity to a distance x from a negatively charged chlorine ion. Sketch two graphs, one of force vs x, showing the force required to bring the ion from infinity to a distance x, the second of energy vs x, showing the potential energy that the sodium ion possesses while at the distance x Explain the shapes of the two graphs. You need not give exact values on the two axes, but give an order-of-magnitude figure for the scales used -- e.g., is the x-axis labelled in metres, millimetres or microns? How would the graphs differ for

    1. two copper atoms
    2. two helium atoms?

    Consider a cubic salt crystal, 1 cm on a side. What compressive force would be required to change the length of one side by a micron? (Density of salt = 2000 kg/m3; sodium and chlorine are both monovalent.)

    Model Answer to Question 2

    The two graphs required are Callister Figure 2.8; it's important to get the distance corresponding to the equilibrium bond length and the energy corresponding to the bond energy clearly marked. The diagram for two copper atoms would look much like that for the sodium and chlorine ions, but the diagram for the two helium atoms would have a very small attractive force, so the resultant force curve would always be below the x-axis and there would be no equilibrium bond length.

    The calculation of the force needed to compress the salt crystal is given in Squashing Salt. and continued in Squashing More Salt. It is possible to complete the analysis by taking a simplified unit cell for salt (this was done in class), but a more rigorous approach is to use the full unit cell, as in Squashing Real Salt When we get to adding up the layers in the model, which we do in Squashing Real Salt (2) and Squashing Real Salt (3), we have to remember that the unit cell is two layers thick -- that is, there are two Na - Cl bonds in series between opposite faces of the cell.

    1. What is the linear density in the [100] direction for a bcc crystal?
    2. What is the planar density of the (110) plane for an fcc crystal?
    3. In what line do the (110) and (101) planes intersect?
    4. Suppose an fcc crystal of copper has 1 impurity atom [of tin] for every 10,000 atoms of copper. How many tin atoms will there be in a square metre of the (110) plane of the crystal? (Density of copper = 8600 kg/m3.)

    Model Answer to Question 3

    The first three parts of this question are pure geometry: we find the linear density of the [100] direction in linear density and the planar density of the (110) plane in planar density. Lastly, we find the intersection of the two planes in intersecting planes

    To find the number of tin atoms in the (110) plane, we follow counting tin atoms and counting more tin atoms

  4. Distinguish between a Schottky and a Frenkel defect.

    The number of vacancies in a material is given by the equation

    Nv= N exp(-Qv/kT)

    where N is the total number of lattice sites, Qv is the activation energy, and k is Boltzmann's constant. Use this formula to work out the number of vacancies in a cubic metre of iron at 900 K, given that there is one vacancy per 1,000,000 atoms at 600 K. (Atomic weight of iron = 56 amu, density of iron = 7800 kg/m3.)

    Model Answer to Question 4

    The first part is straight from the textbook: both types of defect are found in ceramics, which are made up of ions, and their structure results from the requirement that any defect must maintain overall charge neutrality.

    The solution to the second part of the question is given in Vacancies in iron and More Vacancies in iron

  5. Distinguish carefully between the mass-average and the number-average molecular weights of a polymer. Which of these two numbers will be greater?

    A sample of 10 gms of polyethylene (C2H4)n is analysed, and found to contain 1 gm of molecules having lengths 50-150, 2 gms of molecules having lengths 150-250, 3 gms of molecules with lengths 250-350, and four gms with lengths in the range 350-450. Calculate the mass-average and number-average molecular weights, the average degree of polymerisation, and the polydispersity index.

    Model Answer to Question 5

    See polymer molecular weights, more polymer molecular weights, and polymer molecular weights (concluded).

  6. Copy Figure 1, and mark on it

    1. the solidus line
    2. the liquidus line
    3. the solvus line
    4. a eutectic point

    Suppose a sample having the composition A is cooled from 1000 K to room temperature. Sketch a graph of the variation in the sample's temperature with time, explaining any interesting features. Do the same for a sample of composition B.

    Referring to Figure 1, consider the cooling of a sample having the composition C. Explain the phenomenon of segregation, supporting your explanation with quantitative figures for the composition of the first and last solids to precipitate out. What consequences does segregation have for the bulk properties of the material? How could it be prevented? Are there ever conditions under which you would want segregation to occur?

    Still referring to the sample of composition C, what fraction of the final solid is made up of the alpha phase?

    Model Answer to Question 6

    Figure 1a shows the solidus, liquidus and solvus lines, also the eutectic point.

    Figure 1b shows the cooling curve for an alloy of composition A, Figure 1c shows the cooling curve for an alloy of composition B.

    We note that the cooling rate slows when a solid phase is precipitated from a liquid phase, as the latent heat of fusion is released. The cooling rate also slows as one solid phase forms within another. For an alloy of the eutectic composition, the temperature remains at the eutectic temperature from the beginning of solidification to the end.

    Segregation occurs when one phase is precipitated from a liquid or from a second solid phase, if the composition of the precipitate changes as the temperature of the system falls. For the system shown in Figure 1, for example, the composition of the first alpha-phase deposited from an alloy of composition C is rich in the metal at the left end of the x-axis, while the last alpha-phase to be deposited has a higher fraction of the metal from the right-hand end.

    In general segregation is bad -- an alloy with segregation may partially liquefy at a lower temperature than its overall composition would lead us to expect. However, it may be used to separate one metal from an alloy of the two, via the process of zone-melting, if the two metals have a suitable phase diagram.

    Using the lever rule, about 75% of the final solid of composition C is made up of the alpha phase.

  7. Figure 2 is an enlargement of a photograph of 10 mm2 of the etched surface of a metal. Calculate the ASTM grain-size number for this metal. Calculate also the mean chord length, pL.

    Alloys can be prepared either by precipitation or eutectoid decomposition . What differences are there between these two processes? The rate at which each process takes place depends on temperature. Sketch a graph showing the dependance of precipitation or decomposition rate on temperature and explain its shape.

    Model Answer to Question 7

    The ASTM grain size, n, is obtained by counting the grains, N, in one square inch of a 100-fold magnified photo of the grain structure, then applying Equation 4.16:

    N = 2n-1

    For the photograph shown, N < 1, so n=1.

    The mean chord length (which Callister calls the average grain diameter) is obtained by drawing a line of standard length on the photo, then dividing the length of the line by the number of grain boundaries it crosses.

    Figure 3 gives the main differences between precipitation and eutectoid decomposition.

    Figure 4 shows the dependance of decomposition rate on temperature. We see the conflicting effects of two processes: nucleation of the new phase proceeds more rapidly as the temperature falls, but transport of atoms to the newly formed nuclei proceeds more slowly as the temperature falls. The interaction of these two effects gives the knee-shaped curve shown in the figure.

  8. Sketch the stress-strain curve for a ductile metal. Mark on the graph, or show how you would use the graph to calculate:
    1. the ultimate tensile strength
    2. the yield strength
    3. Young's modulus of elasticity
    4. the toughness of the material
    5. the resilience of the material

    Describe the differences, at microscopic and macroscopic levels, between

    1. ductile fracture
    2. brittle fracture
    3. fracture resulting from fatigue.

    Model Answer to Question 8

    Stress-Strain curve

    Ultimate Tensile Strength

    Yield Strength

    Young's modulus

    Resilience

    Toughness

    Ductile, Brittle and Fatigue Failure

    1. Describe, giving full electrochemical equations, the process of galvanic corrosion.
    2. Describe, giving full electrochemical equations, the process of crevice corrosion, which occurs when different regions of the surface of a material are exposed to different concentrations of exposed oxygen.

    Model Answer to Question 9

    Galvanic Corrosion

    Crevice Corrosion

Useful Formulae

Avogadro's number: 6 * 1023 atoms/gram-mole

Boltzmann's constant: 8.12 * 10-5 eV/atom.K

Planck's constant: 6.63 * 10-34J-s

1 eV = 1.602 * 10-19 J

Some bond energies: C--C: 368 kJ/gram-mole; C=C: 719 kJ/gram-mole

Electrostatic attraction:

F = q1q2 /(4 pi epsilon0 x2)

where

1 / (4 pi epsilon0) = 9 * 109 farads/metre

TSR = sigmaf k/ (E alphal)