Find the constants $\mathrm{B}$ and $n$ in Chvorinov's rule by plotting the following data on a $\log \log$ plot:

Define the following terms: nucleation, embryo, heterogeneous nucleation, and homogeneous nucleation.

Give examples of materials based on inorganic glasses that are made by solidification.

How many grams of nickel must be added to $500\ \text{grams}$ of copper to produce an alloy that has a liquidus temperature of $1350^{\circ} \mathrm{C}$ ? What is the ratio of the number of nickel atoms to copper atoms in this alloy?

Locate the following points in the Bi-Sb phase diagram and indicate the phases present and their relative amounts: $(a)$ $60$ at $\%$ Bi at $250^{\circ} \mathrm{C}$; $(b)$ $30$ at $\% \mathrm{Bi}$ at $500^{\circ} \mathrm{C}$; and $(c)$ $50$ at $\%$ Bi at $600^{\circ} \mathrm{C}$.

The high-strength steel in Figure $7-20$, which has a critical fracture toughness of $80 \mathrm{MPa} \sqrt{\mathrm{m}}$, is subjected to an alternating stress varying from $-900 \mathrm{MPa}$Figure 7-20 Crack growth rate versus stress intensity factor range for a high-strength steel. For this steel, $C=1.62 \times 10^{-12}$ and $n=3.2$ for the units shown. (Repeated for Problems 7-35 and 7-36) (compression) to $+900 \mathrm{MPa}$ (tension). It is to survive for $10^5$ cycles before failure occurs. Assume that $f=1$. Calculate (a) the size of a surface crack required for failure to occur; and (b) the largest initial surface crack size that will permit this to happen.

Determine the degrees of freedom under the following conditions: $(a)$ $\text{T1}-20$ wt $\%$ Pb at $325^{\circ} \mathrm{C}$ and $400^{\circ} \mathrm{C}$; $(b)$ $\text{T}1-40$ wt $\%$ Pb at $325^{\circ} \mathrm{C}$ and $400^{\circ} \mathrm{C}$; $(c)$ $71-90 \mathrm{wt} \% \mathrm{~Pb}$ at $325^{\circ} \mathrm{C}$ and $400^{\circ} \mathrm{C}$.

A ceramic part for a jet engine has a yield strength of $75,000 \mathrm{psi}$ and a plane strain fracture toughness of $5,000 \mathrm{psi} \sqrt{\mathrm{in}}$. To be sure that the part does not fail, we plan to ensure that the maximum applied stress is only one-third of the yield strength. We use a nondestructive test that will detect any internal flaws greater than $0.05$ in. long. Assuming that $f=1.4$, does our nondestructive test have the required sensitivity? Explain.

For the nonequilibrium conditions shown for the $\mathrm{Nb}-80 \mathrm{wt} \%$ W alloy, determine (a) the liquidus temperature; (b) the nonequilibrium solidus temperature; (c) the freezing range; (d) the composition of the first solid to form during solidification; (e) the composition of the last liquid to solidify; (f) the phase(s) present, the composition of the phase(s), and the amount of the phase $(\mathrm{s})$ at $3000^{\circ} \mathrm{C}$; and $(\mathrm{g})$ the phase(s) present, the composition of the phase(s), and the amount of the phase(s) at $2800^{\circ} \mathrm{C}$. (See Figure $10-23$.)

Suppose that liquid nickel is undercooled until homogeneous nucleation occurs. Calculate (a) the critical radius of the nucleus required and (b) the number of nickel atoms in the nucleus. Assume that the lattice parameter of the solid $\mathrm{FCC}$ nickel is $0.356 \mathrm{~nm}$.

Calculate the fraction of solidification that occurs dendritically when iron nucleates (a) at $10^{\circ} \mathrm{C}$ undercooling; (b) at $100^{\circ} \mathrm{C}$ undercooling; and (c) homogeneously. The specific heat of iron is $5.78 \mathrm{~J} /\left(\mathrm{cm}^3+{ }^{\circ} \mathrm{C}\right)$.

In solids, the process of diffusion of atoms and ions takes time. Explain how this is used to our advantage while forming metallic glasses.

A carburizing process is carried out on a $0.10 \% \mathrm{C}$ steel by introducing $1.0 \% \mathrm{C}$ at the surface at $980^{\circ} \mathrm{C}$, where the iron is $\mathrm{FCC}$. Estimate the carbon content at $0.01 \mathrm{~cm}$, $0.05 \mathrm{~cm}$, and $0.10 \mathrm{~cm}$ beneath the surface after $1 \mathrm{~h}$.

Determine the electrical conductivity of silicon when $0.0001$ at $\%$ antimony is added as a dopant and compare it to the electrical conductivity when $0.0001$ at $\%$ indium is added.