15 December 2009
Department of Materials Science and Engineering
1 of 12
Case Western Reserve University
Name:
S O L U T I O N
Final Exam — 180 minutes; 200 points; 8 questions; 12 pages; 20% of course grade
You may use a calculator, pencils or pens, eraser, and a straight edge. Handwritten notes on both sides of a
single 8.5
″×
11
″
sheet of paper are permitted.
Please turn in your note page with your test.
No other form of
stored information is permitted.
Write all answers on these pages; use the back if more space is needed.
Partial credit
will be given for correct set-ups and reasoning. Give
units
on numerical answers where ap-
propriate and use the
correct number of significant figures.
Constants
:
Wiedemann-Franz constant,
£
=
π
2
k
B
2
/(3
q
e
2
)
=
2.445
×
10
-8
W
Ω
K
-2
Boltzmann’s constant,
k
B
= 1.381
×
10
-23
J K
-1
= 8.620
×
10
-5
eV K
-1
gas constant,
R
: 8.314 J mol
-1
K
-1
Charge on an electron, |
q
e
| = 1.602
×
10
-19
C
Faraday’s constant,
F
= 96,500 C mol
–1
1)
Gaseous hydrogen at a constant pressure
P
i
will flow
through a thin-walled nickel tube. Outside the tube, the hydrogen pressure will be constant at
P
o
. If the
thickness of the tube wall is
∆
x
, the concentration gradient of hydrogen across the tube wall is given by
dC
H
dx
=
A
Δ
x
P
i
1/2
−
P
o
1/2
( )
exp
−
B
RT
⎛
⎝
⎜
⎞
⎠
⎟
(1)
where
T
is the temperature of the tube and gas in Kelvin, and
A
and
B
are materials properties of nickel
(assumed to be independent of temperature). The diffusion coefficient of hydrogen in nickel is given by
D
H
=
M
exp
−
Q
RT
⎛
⎝
⎜
⎞
⎠
⎟
(2)
where
M
and
Q
are materials properties of nickel (assumed to be independent of temperature).
How will the steady-state flux of hydrogen through the tube wall change (i.e.,
increase, decrease,
or
stay the same
) if the following changes are made and all of the other parameters defined above are held
constant? Briefly
justify
your answers in terms of Fick’s first law.
a)
(4 points)
Increase the wall thickness,
∆
x
According to eq. 1, increasing
∆
x
will decrease the concentration gradient (
2 pts
).
Per Fick’s first law, this will decrease the outward flux (
J
= -D(
dC
H
/
dx
) ) (
2 pts
).
b)
(4 points)
Increase the pressure of hydrogen in the tube, P
i
According to eq. 1, increasing
∆
will increase the concentration gradient (
2 pts
).
Per Fick’s first law, this will increase the outward flux (
= -D(
H
/
) ) (
2 pts
).
c)
(8 points)
Increase the temperature, T
Increasing the temperature will cause the exponents in both eqs. 1 and 2 to become
smaller negative numbers (
2 pts
). This will make both the concentration gradient
(
2 pts
) and the diffusion coefficient (
2 pts
) larger numbers. Per Fick’s first law, each
of these effects will increase the outward flux (
2 pts
).
continued on next page