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Materials Science and Engineering. An Introduction

William D. Callister

Chapter 5

Diffusion - all with Video Answers

Educators

Chapter Questions

01:32

Problem 1

Briefly explain the difference between selfdiffusion and interdiffusion.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:21

Problem 2

Self-diffusion involves the motion of atoms that are all of the same type; therefore it is not subject to observation by compositional changes, as with interdiffusion. Suggest one way in which self-diffusion may be monitored.

Manik Pulyani
Manik Pulyani
Numerade Educator
05:14

Problem 3

(a) Compare interstitial and vacancy atomic mechanisms for diffusion.
(b) Cite two reasons why interstitial diffusion is normally more rapid than vacancy diffusion.

Ameer Said
Ameer Said
Numerade Educator
02:35

Problem 4

Briefly explain the concept of steady state as it applies to diffusion.

Ameer Said
Ameer Said
Numerade Educator
01:28

Problem 5

(a) Briefly explain the concept of a driving force.
(b) What is the driving force for steady-state diffusion?

Ameer Said
Ameer Said
Numerade Educator
01:13

Problem 6

The purification of hydrogen gas by diffusion through a palladium sheet was discussed in Section 5.3. Compute the number of kilograms of hydrogen that pass per hour through a 6 -mm-thick sheet of palladium having an area of $0.25 \mathrm{~m}^2$ at $600^{\circ} \mathrm{C}$. Assume a diffusion coefficient of $1.7 \times 10^{-8} \mathrm{~m}^2 / \mathrm{s}$, that the concentrations at the high- and low-pressure sides of the plate are 2.0 and 0.4 kg of hydrogen per cubic meter of palladium, and that steady-state conditions have been attained.

Narayan Hari
Narayan Hari
Numerade Educator
01:38

Problem 7

A sheet of steel 2.5 mm thick has nitrogen atmospheres on both sides at $900^{\circ} \mathrm{C}$ and is permitted to achieve a steady-state diffusion condition. The diffusion coefficient for nitrogen in steel at this temperature is $1.2 \times 10^{-10} \mathrm{~m}^2 / \mathrm{s}$, and the diffusion flux is found to be $1.0 \times 10^{-7} \mathrm{~kg} / \mathrm{m}^2-\mathrm{s}$. Also, it is known that the concentration of nitrogen in the steel at the high-pressure surface is $2 \mathrm{~kg} / \mathrm{m}^3$. How far into the sheet from this highpressure side will the concentration be 0.5 $\mathrm{kg} / \mathrm{m}^3$ ? Assume a linear concentration profile.

Narayan Hari
Narayan Hari
Numerade Educator
02:30

Problem 8

A sheet of BCC iron 2 mm thick was exposed to a carburizing gas atmosphere on one side and a decarburizing atmosphere on the other side at $675^{\circ} \mathrm{C}$. After having reached steady state, the iron was quickly cooled to room temperature. The carbon concentrations at the two surfaces of the sheet were determined to be 0.015 and $0.0068 \mathrm{wt} \%$. Compute the diffusion coefficient if the diffusion flux is $7.36 \times 10^{-9}$ $\mathrm{kg} / \mathrm{m}^2$-s. Hint: Use Equation 4.9 to convert the concentrations from weight percent to kilograms of carbon per cubic meter of iron.

Narayan Hari
Narayan Hari
Numerade Educator
09:28

Problem 9

When $\alpha$-iron is subjected to an atmosphere of nitrogen gas, the concentration of nitrogen in the iron, $C_{\mathrm{N}}$ (in weight percent), is a function of hydrogen pressure, $p_{\mathrm{N}_2}(\mathrm{in} \mathrm{MPa})$, and absolute temperature ( $T$ ) according to

$$
C_{\mathrm{N}}=4.90 \times 10^{-3} \sqrt{p_{\mathrm{N}_2}} \exp \left(-\frac{37.6 \mathrm{~kJ} / \mathrm{mol}}{R T}\right)
$$

Furthermore, the values of $D_0$ and $Q_d$ for this diffusion system are $3.0 \times 10^{-7} \mathrm{~m}^2 / \mathrm{s}$ and $76,150 \mathrm{~J} / \mathrm{mol}$, respectively. Consider a thin iron membrane 1.5 mm thick that is at $300^{\circ} \mathrm{C}$. Compute the diffusion flux through this membrane if the nitrogen pressure on one side of the membrane is $0.10 \mathrm{MPa}(0.99 \mathrm{~atm})$, and on the other side 5.0 MPa (49.3 atm).

Ameer Said
Ameer Said
Numerade Educator
01:35

Problem 10

Show that

$$
C_x=\frac{B}{\sqrt{D t}} \exp \left(-\frac{x^2}{4 D t}\right)
$$

is also a solution to Equation 5.4b. The parameter $B$ is a constant, being independent of both $x$ and $t$.

Ameer Said
Ameer Said
Numerade Educator
01:18

Problem 11

Determine the carburizing time necessary to achieve a carbon concentration of $0.30 \mathrm{wt} \%$ at a position 4 mm into an iron-carbon alloy that initially contains $0.10 \mathrm{wt} \% \mathrm{C}$. The surface concentration is to be maintained at $0.90 \mathrm{wt} \%$ C , and the treatment is to be conducted at $1100^{\circ} \mathrm{C}$. Use the diffusion data for $\gamma-\mathrm{Fe}$ in Table 5.2.

Manik Pulyani
Manik Pulyani
Numerade Educator
03:38

Problem 12

An FCC iron-carbon alloy initially containing $0.55 \mathrm{wt} \% \mathrm{C}$ is exposed to an oxygen-rich and virtually carbon-free atmosphere at 1325 K $\left(1052^{\circ} \mathrm{C}\right)$. Under these circumstances the carbon diffuses from the alloy and reacts at the surface with the oxygen in the atmosphere; that is, the carbon concentration at the surface position is maintained essentially at $0 \mathrm{wt} \%$ C. (This process of carbon depletion is termed decarburization.) At what position will the carbon concentration be $0.25 \mathrm{wt} \%$ after a 10-h treatment? The value of $D$ at 1325 K is $4.3 \times 10^{-11} \mathrm{~m}^2 / \mathrm{s}$.

Ameer Said
Ameer Said
Numerade Educator
03:29

Problem 13

Nitrogen from a gaseous phase is to be diffused into pure iron at $675^{\circ} \mathrm{C}$. If the surface concentration is maintained at $0.2 \mathrm{wt} \% \mathrm{~N}$, what will be the concentration 2 mm from the surface after 25 h ? The diffusion coefficient for nitrogen in iron at $675^{\circ} \mathrm{C}$ is $1.9 \times 10^{-11} \mathrm{~m}^2 / \mathrm{s}$.

Ameer Said
Ameer Said
Numerade Educator
05:43

Problem 14

Consider a diffusion couple composed of two semi-infinite solids of the same metal, and that each side of the diffusion couple has a different concentration of the same elemental impurity; furthermore, assume each impurity level is constant throughout its side of the diffusion couple. For this situation, the solution to Fick's second law (assuming that the diffusion coefficient for the impurity is independent of concentration), is as follows:

$$
C_x=\left(\frac{C_1+C_2}{2}\right)-\left(\frac{C_1-C_2}{2}\right) \operatorname{erf}\left(\frac{x}{2 \sqrt{D t}}\right)
$$

In this expression, when the $x=0$ position is taken as the initial diffusion couple interface, then $C_1$ is the impurity concentration for $x<0$; likewise, $C_2$ is the impurity content for $x>0$.

A diffusion couple composed of two platinum-gold alloys is formed; these alloys have compositions of $99.0 \mathrm{wt} \% \mathrm{Pt}-1.0 \mathrm{wt} \%$ Au and $96.0 \mathrm{wt} \% \mathrm{Pt}-4.0 \mathrm{wt} \% \mathrm{Au}$. Determine the time this diffusion couple must be heated at $1000^{\circ} \mathrm{C}(1273 \mathrm{~K})$ in order for the composition to be $2.8 \mathrm{wt} \% \mathrm{Au}$ at the $10 \mu \mathrm{m}$ position into the $4.0 \mathrm{wt} \%$ Au side of the diffusion couple. Preexponential and activation energy values for Au diffusion in Pt are $1.3 \times 10^{-5}$ $\mathrm{m}^2 / \mathrm{s}$ and $252,000 \mathrm{~J} / \mathrm{mol}$, respectively.

Ameer Said
Ameer Said
Numerade Educator
01:02

Problem 15

For a steel alloy it has been determined that a carburizing heat treatment of 15 h duration will raise the carbon concentration to $0.35 \mathrm{wt} \%$ at a point 2.0 mm from the surface. Estimate the time necessary to achieve the same concentration at a $6.0-\mathrm{mm}$ position for an identical steel and at the same carburizing temperature.

Narayan Hari
Narayan Hari
Numerade Educator
02:50

Problem 16

Cite the values of the diffusion coefficients for the interdiffusion of carbon in both $\alpha$-iron (BCC) and $\gamma$-iron (FCC) at $900^{\circ} \mathrm{C}$. Which is larger? Explain why this is the case.

Ameer Said
Ameer Said
Numerade Educator
01:01

Problem 17

Using the data in Table 5.2, compute the value of $D$ for the diffusion of magnesium in aluminum at $400^{\circ} \mathrm{C}$.

Narayan Hari
Narayan Hari
Numerade Educator
01:01

Problem 18

At what temperature will the diffusion coefficient for the diffusion of zinc in copper have a value of $2.6 \times 10^{-16} \mathrm{~m}^2 / \mathrm{s}$ ? Use the diffusion data in Table 5.2.

Narayan Hari
Narayan Hari
Numerade Educator
01:30

Problem 19

The preexponential and activation energy for the diffusion of chromium in nickel are $1.1 \times 10^{-4} \mathrm{~m}^2 / \mathrm{s}$ and $272,000 \mathrm{~J} / \mathrm{mol}$, respectively. At what temperature will the diffusion coefficient have a value of $1.2 \times 10^{-14} \mathrm{~m}^2 / \mathrm{s}$ ?

Narayan Hari
Narayan Hari
Numerade Educator
01:20

Problem 20

The activation energy for the diffusion of copper in silver is $193,000 \mathrm{~J} / \mathrm{mol}$. Calculate the diffusion coefficient at $1200 \mathrm{~K}\left(927^{\circ} \mathrm{C}\right)$, given that $D$ at $1000 \mathrm{~K}\left(727^{\circ} \mathrm{C}\right)$ is $1.0 \times 10^{-14} \mathrm{~m}^2 / \mathrm{s}$.

Narayan Hari
Narayan Hari
Numerade Educator
02:02

Problem 21

The diffusion coefficients for nickel in iron are given at two temperatures:
$$
\begin{array}{lc}
\hline \boldsymbol{T}(\boldsymbol{K}) & \boldsymbol{D ( m ^ { 2 } / \mathrm { s } )} \\
\hline 1473 & 2.2 \times 10^{-15} \\
1673 & 4.8 \times 10^{-14} \\
\hline
\end{array}
$$

Narayan Hari
Narayan Hari
Numerade Educator
02:11

Problem 22

The diffusion coefficients for carbon in nickel are given at two temperatures:
$$
\begin{array}{cc}
\hline \boldsymbol{T}\left({ }^{\circ} \boldsymbol{C}\right) & \boldsymbol{D}\left(\mathrm{m}^2 / \mathrm{s}\right) \\
\hline 600 & 5.5 \times 10^{-14} \\
700 & 3.9 \times 10^{-13} \\
\hline
\end{array}
$$
(a) Determine the values of $D_0$ and $Q_d$.
(b) What is the magnitude of $D$ at $850^{\circ} \mathrm{C}$ ?

Narayan Hari
Narayan Hari
Numerade Educator
02:04

Problem 23

Below is shown a plot of the logarithm (to the base 10) of the diffusion coefficient versus reciprocal of the absolute temperature, for the diffusion of gold in silver. Determine values for the activation energy and preexponential.

Narayan Hari
Narayan Hari
Numerade Educator
01:32

Problem 24

Carbon is allowed to diffuse through a steel plate 10 mm thick. The concentrations of carbon at the two faces are 0.85 and 0.40 kg $\mathrm{C} / \mathrm{cm}^3 \mathrm{Fe}$, which are maintained constant. If the preexponential and activation energy are $6.2 \times 10^{-7} \mathrm{~m}^2 / \mathrm{s}$ and $80,000 \mathrm{~J} / \mathrm{mol}$, respectively, compute the temperature at which the diffusion flux is $6.3 \times 10^{-10} \mathrm{~kg} / \mathrm{m}^2-\mathrm{s}$.

Narayan Hari
Narayan Hari
Numerade Educator
01:17

Problem 25

The steady-state diffusion flux through a metal plate is $7.8 \times 10^{-8} \mathrm{~kg} / \mathrm{m}^2-\mathrm{s}$ at a temperature of $1200^{\circ} \mathrm{C}(1473 \mathrm{~K})$ and when the concentration gradient is $-500 \mathrm{~kg} / \mathrm{m}^4$. Calculate the diffusion flux at $1000^{\circ} \mathrm{C}(1273 \mathrm{~K})$ for the same concentration gradient and assuming an activation energy for diffusion of $145,000 \mathrm{~J} / \mathrm{mol}$

Manik Pulyani
Manik Pulyani
Numerade Educator
01:53

Problem 26

At approximately what temperature would a specimen of $\gamma$-iron have to be carburized for 4 h to produce the same diffusion result as at $1000^{\circ} \mathrm{C}$ for 12 h ?

Narayan Hari
Narayan Hari
Numerade Educator
01:57

Problem 27

(a) Calculate the diffusion coefficient for magnesium in aluminum at $450^{\circ} \mathrm{C}$.
(b) What time will be required at $550^{\circ} \mathrm{C}$ to produce the same diffusion result (in terms of concentration at a specific point) as for 15 h at $450^{\circ} \mathrm{C}$ ?

Anand Jangid
Anand Jangid
Numerade Educator
03:50

Problem 28

A copper-nickel diffusion couple similar to that shown in Figure $5.1 a$ is fashioned. After a $500-\mathrm{h}$ heat treatment at $1000^{\circ} \mathrm{C}(1273 \mathrm{~K})$ the concentration of Ni is $3.0 \mathrm{wt} \%$ at the $1.0-\mathrm{mm}$ position within the copper. At what temperature should the diffusion couple be heated to produce this same concentration (i.e., $3.0 \mathrm{wt} \%$ Ni ) at a $2.0-\mathrm{mm}$ position after 500 h ? The preexponential and activation energy for the diffusion of Ni in Cu are $2.7 \times 10^{-4} \mathrm{~m}^2 / \mathrm{s}$ and $236,000 \mathrm{~J} / \mathrm{mol}$, respectively.

Ameer Said
Ameer Said
Numerade Educator
01:42

Problem 29

A diffusion couple similar to that shown in Figure $5.1 a$ is prepared using two hypothetical metals A and B. After a 20 -h heat treatment at $800^{\circ} \mathrm{C}$ (and subsequently cooling to room temperature) the concentration of B in A is $2.5 \mathrm{wt} \%$ at the $5.0-\mathrm{mm}$ position within metal A. If another heat treatment is conducted on an identical diffusion couple, only at $1000^{\circ} \mathrm{C}$ for 20 h , at what position will the composition be $2.5 \mathrm{wt} \% \mathrm{~B}$ ? Assume that the preexponential and activation energy for the diffusion coefficient are $1.5 \times 10^{-4} \mathrm{~m}^2 / \mathrm{s}$ and $125,000 \mathrm{~J} / \mathrm{mol}$, respectively.

Narayan Hari
Narayan Hari
Numerade Educator
01:49

Problem 30

The outer surface of a steel gear is to be hardened by increasing its carbon content; the carbon is to be supplied from an external carbon-rich atmosphere that is maintained at an elevated temperature. A diffusion heat treatment at $600^{\circ} \mathrm{C}(873 \mathrm{~K})$ for 100 min increases the carbon concentration to $0.75 \mathrm{wt} \%$ at a position 0.5 mm below the surface. Estimate the diffusion time required at $900^{\circ} \mathrm{C}(1173 \mathrm{~K})$ to achieve this same concentration also at a $0.5-\mathrm{mm}$ position. Assume that the surface carbon content is the same for both heat treatments, which is maintained constant. Use the diffusion data in Table 5.2 for C diffusion in $\alpha$-Fe.

Narayan Hari
Narayan Hari
Numerade Educator
04:57

Problem 31

An FCC iron-carbon alloy initially containing $0.10 \mathrm{wt} \% \mathrm{C}$ is carburized at an elevated temperature and in an atmosphere wherein the surface carbon concentration is maintained at $1.10 \mathrm{wt} \%$. If after 48 h the concentration of carbon is $0.30 \mathrm{wt} \%$ at a position 3.5 mm below the surface, determine the temperature at which the treatment was carried out.

Ameer Said
Ameer Said
Numerade Educator