Foundations of Materials Science and Engineering
Foundations of Materials Science and Engineering
6th Edition
ISBN: 9781259696558
Author: SMITH
Publisher: MCG
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Question
Chapter 5.7, Problem 33SEP
To determine

The flux of Mn atoms between the surface and plane 2mm deep.

Expert Solution & Answer
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Answer to Problem 33SEP

The flux of Mn atoms between the surface and plane 2mm deep is 6.8625×105atoms/m2s.

Explanation of Solution

Write the formula for lattice constant (a) of unit of FCC iron.

  a=4R2        (I)

Here, the radius of the iron atom is R.

Write the formula for volume (V) of the unit cell.

  V=a3        (II)

Write the formula for Fick’s law of diffusion.

  J=DdCdx        (III)

Here, flux or flow of atoms is J in atomsm2s, constant of diffusivity is D in m2s , and concentration gradient is dCdx in atomsm4.

Conclusion:

Refer Table 3.2, “Selected metals that have the BCC crystal structure at room temperature (20°C) and their lattice constants and atomic radii”.

The atomic radius of iron is 0.124nm.

Substitute 0.124nm for R in Equation (I).

  a=4(0.124nm)2=0.351nm

Substitute 0.351nm for a in Equation (II).

  V=(0.351nm)3=(0.351nm×109m1nm)3=4.32×1024m3

The number of atoms per unit volume is expressed as follows.

  atoms per unit volume=no. of atoms in FCC crystalvolumeof unit cell(V)=4atoms4.32×1024m3=9.25×1028atoms/m3

The concentration of atoms at the surface Cs is 0.6a%.

  Cs=0.6a%=0.6100(9.25×1028atoms/m3)=5.55×1026atoms/m3

The concentration of atom at the distance of 2mm below the surface C2mm is 0.1a%.

  C2mm=0.1a%=0.1100(9.25×1028atoms/m3)=9.25×1025atoms/m3

Refer Table 5.2, “Diffusivities at 500°C and 1000°C for selected solute-solvent diffusion systems”.

The diffusivity of manganese in FCC iron at 500°C is 3×1024m2/s.

Here,

  dC=C2mmCs=(9.25×1025atoms/m3)(5.55×1026atoms/m3)=4.575×1026atoms/m3

Substitute 3×1024m2/s for D, 4.575×1026atoms/m3 for dC, and 2mm for dx in Equation (III).

  J=(3×1024m2/s)4.575×1026atoms/m32mm=(3×1024m2/s)4.575×1026atoms/m32mm×1m1000m=6.8625×105atoms/m2s

Thus, the flux of Mn atoms between the surface and plane 2mm deep is 6.8625×105atoms/m2s.

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  • Rank the magnitudes of the diffusion coefficients from greatest to least for the following systems:   N in Fe at 700°C Cr in Fe at 700°C N in Fe at 900°C Cr in Fe at 900°C   Now justify this ranking. (Note: Both Fe and Cr have the BCC crystal structure, and the atomic radii for Fe, Cr, and N are 0.124, 0.125, and 0.065 nm, respectively. You may also want to refer to Section 5.4.)
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