Solution to Homework #7
Problem 1.
a.
The effective charge Z* for carbon in
α
Fe is 12.
An electric field of
0.01V/mm is maintained in a 1mm long sample of an FeC alloy at a
temperature of 700
o
C.
What will be the concentration ratio between the
two ends at the steady state?
b.
If the initial atom fraction of carbon in the alloy was 0.0005, what fraction
of the length of the sample is at a concentration lower than 0.0005 at
steady state, if no carbide forms?
Solution
a.
The mass flux in a electrical field is given by
−
−
=
E
T
k
eZ
C
dx
dC
D
J
B
1
1
1
1
1
where C
1
is the carbon concentration.
At steady state,
0
1
=
J
, hence,
E
T
k
eZ
C
dx
dC
B
1
1
1
=
Integrate the above differential equation,
( )
( )
∫
=
∫
x
B
x
C
C
dx
T
k
E
eZ
C
dC
0
1
0
1
1
1
1
we have
ln
C
1
x
( )
C
1
0
( )
=
eZ
1
E
k
B
T
x
or
( )
( )
=
x
T
k
E
eZ
C
x
C
B
1
1
1
exp
0
Therefore,
( )
( )
(
)
24
.
0
1
973
10
38
.
1
01
.
0
12
10
6
.
1
exp
exp
0
23
16
1
1
1
=
×
×
×
×
−
×
×
=
=
−
−
mm
K
K
J
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 Spring '02
 CHEN
 Thermodynamics, 0 K, 0.00094 k, 0.44mm, 0.56 1mm

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