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Solutions to Homework #5
Problem 1
In the temperature range – 70 to 400
o
C the diffusion coefficient for C in aFe is D =
0.02exp(10120/T) cm
2
/s.
If the average vibration frequency of the carbon atoms in the
lattice is 5.0x10
13
/s, calculate the entropy of motion, S
m
, for carbon diffusion (Take the
lattice parameter to be 2.48x10
8
cm.)
Solution:
Since
α
Fe is b.c.c. structure, the diffusion coefficient for C in
α
Fe will be:
D
a
a
C
Fe
ss
==
×
×
×=
1
6
1
6
4
2
1
6
22
ΓΓ
λ
()
2
Γ
where a is the lattice parameter for
α
Fe.
Therefore,
)
/
(
10
88
.
1
)
10
48
.
2
(
10
5
02
.
0
6
ln
10
38
.
1
6
ln
)
exp(
6
1
)
exp(
)
exp(
)
exp(
6
1
23
2
8
13
23
2
0
2
0
0
2
K
J
va
D
k
S
k
S
va
D
kT
H
D
kT
H
k
S
a
v
D
m
m
m
m
m
Fe
C
−
−
−
×
=
×
×
×
×
×
×
=
=
∆
⇒
∆
=
⇒
∆
−
=
∆
−
∆
×
×
×
=
Problem 2
.
Discuss the influence of zinc chloride addition to the diffusivity of all ions (Zn, Na,
and Cl) in a singlecrystal NaCl from room temperature to the melting temperature.
Solution:
We assume diffusion of all the ions, Zn, Na and Cl, by the vacancy mechanism.
The diffusion
coefficients of each ion is given by
D
i
=
γ
a
o
2
w
i
X
i
where
γ
is the geometrical factor, a
o
is the lattice parameter of NaCl crystal structure,
w
i
is the
jump frequency of a given ion to a given vacancy,
X
i
is the vacancy concentration.
Suppose the
addition of zinc chloride does not change w
i
and one can neglect the defect association between
the Zn ion and Na vacancy (in general, defect association has to be considered because Zn ion
and Na vacancy are oppositely charged) then the only influence of zinc chloride addition on the
diffusivity of all ions in NaCl is through the change in the vacancy concentrations.
At very high temperatures (close to T
m
of NaCl), we assume that the defect concentration in the
crystal is controlled by the defect reaction,
1
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↔
V
Na
'
+
V
Cl
•
,
hence,
[]
∆
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 Spring '02
 CHEN

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