Why is undercooling required for solidification? Derive an equation showing the total free energy change as a function of undercooling when the nucleating solid has the critical nucleus radius $r^*$.

Calculate the number of vacancies per cubic meter at $1000^{\circ} \mathrm{C}$ for a metal that has an energy for vacancy formation of $1.22 \mathrm{~eV/atom}$, a density of $6.25 \mathrm{~g} / \mathrm{cm}^{3}$, and an atomic weight of $37.4 \mathrm{~g/mol}$.

(A) $1.49 \times 10^{18} \mathrm{~m}^{-3}$

(B) $7.18 \times 10^{22} \mathrm{~m}^{-3}$

(C) $1.49 \times 10^{24} \mathrm{~m}^{-3}$

(D) $2.57 \times 10^{24} \mathrm{~m}^{-3}$

What is a heat-affected zone?

(a) For the solidification of nickel, calculate the critical radius $r^*$ and the activation free energy $\Delta G^{*}$ if nucleation is homogeneous. Values for the latent heat of fusion and surface free energy are $-2.53 \times 10^{9} \mathrm{J} / \mathrm{m}^{3}$ and $0.255 \mathrm{J} / \mathrm{m}^{2}$, respectively. Use the supercooling value. (b) Now, calculate the number of atoms found in a nucleus of critical size. Assume a lattice parameter of $0.360 \mathrm{~nm}$ for solid nickel at its melting temperature.

Verify that distance defined by (1-20) with a11 = 4, a22 = 1, and a12 = −1 satisfies the first three conditions in (1-25). (The triangle inequality is more difficult to verify.)

Are the following distance functions valid for distance from the origin? Explain. (a) $x_{1}^{2}+4 x_{2}^{2}+x_{1} x_{2}=(\text { distance })^{2}$ (b) $x_{1}^{2}-2 x_{2}^{2}=(\text { distance })^{2}$

The following are five measurements on the variables $x_{1}$, $x_{2}$, and $x_{3}$: Find the arrays $\overline{\mathbf{x}}$, Sn, and R.

$$\begin{matrix} \text{}{x_{1}} & \text{9} & \text{2} & \text{6} & \text{5} & \text{8}\\ \text{}{x_{2}} & \text{12} & \text{8} & \text{6} & \text{4} & \text{10}\\ \text{}{x_{3}} & \text{3} & \text{4} & \text{0} & \text{2} & \text{1}\\ \end{matrix}$$

The world’s 10 largest companies yield the following data:

$$\begin{matrix} \text{} & \text{}{x_{1}= sales} & \text{}{x_{2}= profits} & \text{}{ x_{3}= assets}\\ \text{Company} & \text{(billions)} & \text{(billions)} & \text{(billions)}\\ \text{Citigroup} & \text{108.28} & \text{17.05} & \text{1,484.10}\\ \text{General Electric} & \text{152.36} & \text{16.59} & \text{750.33}\\ \text{American Intl Group} & \text{95.04} & \text{10.91} & \text{766.42}\\ \text{Bank of America} & \text{65.45} & \text{14.14} & \text{1,110.46}\\ \text{HSBC Group} & \text{62.97} & \text{9.52} & \text{1,031.29}\\ \text{Exxon Mobil} & \text{263.99} & \text{25.33} & \text{195.26}\\ \text{Royal Dutch/Shell} & \text{265.19} & \text{18.54} & \text{193.83}\\ \text{BP} & \text{285.06} & \text{15.73} & \text{191.11}\\ \text{ING Group} & \text{92.01} & \text{8.10} & \text{1,175.16}\\ \text{Toyota Motor} & \text{165.68} & \text{11.13} & \text{211.15}\\ \end{matrix}$$

(a) Plot the scatter diagram and marginal dot diagrams for variables $x_{1}$ and $x_{2}$. Comment on the appearance of the diagrams. (b) Compute , $s_{11}$, $s_{22}$, $s_{12}$, and $r_{12}$. Interpret $r_{12}$. $\overline{x}_{1}, \overline{x}_{2}$

What is the difference between fusion welding and brazing and soldering?

Describe the microstructure present in a 10110 steel after each step in the following heat treatments: (a) heat to $900^{\circ} \mathrm{C}$, quench to $400^{\circ} \mathrm{C}$ and hold for $10^3 \mathrm{~s}$, and quench to $25^{\circ} \mathrm{C}$; (b) heat to $900^{\circ} \mathrm{C}$, quench to $600^{\circ} \mathrm{C}$ and hold for $50 \mathrm{~s}$, and quench to $25^{\circ} \mathrm{C}$; (c) heat to $900^{\circ} \mathrm{C}$ and quench to $25^{\circ} \mathrm{C}$; (d) heat to $900^{\circ} \mathrm{C}$, quench to $300^{\circ} \mathrm{C}$ and hold for $200 \mathrm{~s}$, and quench to $25^{\circ} \mathrm{C}$; (e) heat to $900^{\circ} \mathrm{C}$, quench to $675^{\circ} \mathrm{C}$ and hold for 1 s, and quench to $25^{\circ} \mathrm{C}$; (f) heat to $900^{\circ} \mathrm{C}$, quench to $675^{\circ} \mathrm{C}$ and hold for 1 s, quench to $400^{\circ} \mathrm{C}$ and hold for $900 \mathrm{~s}$, and slowly cool to $25^{\circ} \mathrm{C}$; (g) heat to $900^{\circ} \mathrm{C}$, quench to $675^{\circ} \mathrm{C}$ and hold for $1 \mathrm{~s}$, quench to $300^{\circ} \mathrm{C}$ and hold for $10^3 \mathrm{~s}$, and air cool to $25^{\circ} \mathrm{C}$; and (h) heat to $900^{\circ} \mathrm{C}$, quench to $300^{\circ} \mathrm{C}$ and hold for $100 \mathrm{~s}$, quench to $25^{\circ} \mathrm{C}$, heat to $450^{\circ} \mathrm{C}$ for $3600 \mathrm{~s}$, and cool to $25^{\circ} \mathrm{C}$.

Estimate the AISI-SAE number for steels having the following microstructures: (a) $38 \%$ pearlite $-62 \%$ primary ferrite; (b) $93 \%$ pearlite-7% primary cementite; (c) $97 \%$ ferrite-3% cementite; and (d) $86 \%$ ferrite- $14 \%$ cementite.

Determine the ASTM grain size number if 25 grains per square inch are measured at a magnification of 600*.

Using the densities in Appendix A, convert the heats of fusion in Table 9-1 from units of $\mathrm{J} / \mathrm{cm}^3$ to $\mathrm{kJ} / \mathrm{kg}$.

An aluminum alloy plate with dimensions $20 \mathrm{~cm} \times 10 \mathrm{~cm} \times 2 \mathrm{~cm}$ needs to be cast with a secondary dendrite arm spacing of $10^{-2} \mathrm{~cm}$ (refer to Figure 9-6). What mold constant B is required (assume $n=2$ )?