age 1 of 5
Fall 2005

ID number: _____ _____ - _____ _____ _____ _____ (LAST SIX DIGITS ONLY!)

Question 1: Electrical Resistivity, Electrical Conductivity and Thermal Conductivity- 20 points
At room temperature the electrical conductivity and the electron mobility for aluminum (Al) are
and
(
)
1
7
3.8 10
m (
)
3
2
1.2 10
/
m
V s respectively. The Wiedemann-Franz coefficient for aluminum
is
.
8
2
2.24 10
/
W K
a)
Compute the number of free electrons per cubic meter for aluminum at room temperature.
b)
What is the number of free electrons per aluminum atom? (Assume a density of 2.7g/cm
3
).
c)
The electrical conductivity for aluminum (Al) at 500
o
C is
(
)
1
7
1.4 10
m
. Compute the electrical resistivity
due to the increase in temperature from room temperature to 500
o
C.
d)
Compute the thermal conductivity for aluminum at 500
o
C.


a)
From
(
)
1
7
3.8 10
e
n e
m
=
=
(Callister Equation 19.8) one gets the number of charge carriers per unit
volume as
(
)
[ ]
(
)
1
7
29
3
29
3
19
3
2
3.8 10
1.98 10
2.0 10
1.6 10
1.2 10
/
m
n
m
C
m
V s

m =
=
(using
[ ] [ ]
C
As
=
and
[ ] [
]
/
V A
=
, respectively).

b)
From the density
(
and the atomic weight of Al (26.982 [g/mol]), the volume per Al atom is
given by
)


3
2.7
/
g cm
= 23
3
29
26.982 / 6.022140 10
1.66 10
2.7
Al
Al
m
V
cm
3
m
=
=
=
. The average number of electrons per Al
atom is then determined as
. This value is approximately equal to the
number of valance electrons of Al (3), the additional charges may be due to electrons from impurities and/or
alloying elements.
29
3
29
3
2.0 10
1.66 10
3.3
Al
n
m
m
=


c)
The resistivity at RT and 500
o
C are
[
]
8
1/
2.63 10
RT
RT
m =
=
and is
[
]
8
500
500
1/
7.14 10
m =
=
. The
resistivity increase due to the increase in temperature is
[
]
8
8
8
7.14 10
2.63 10
4.51 10
T
m = =


d)
The Wiedemann-Franz law (Callister Equation 20.7) states
k
L
T =
. The thermal conductivity can be
calculated from the electrical conductivity at 500
o
C (=773 K) according to
(
)
[ ]
(
)
[
]
1
8
2
7
2.24 10
/
1.4 10
773
242.4
/
k L T
W K
m
K
W
m K =
=
=
ENGR 145

Exam #3 - Page 2 of 5
Fall 2005

ID number:
_____ _____ - _____ _____ _____ _____ (LAST SIX DIGITS ONLY!)

Question 2: Adiabatic Expansion - 20 points
Suppose
[
]
2.00 mol of an ideal, monatomic gas is initially at a pressure of
[ ]
3.00 atm and a temperature of
[ ]
350
T
=
K . The gas is expanded irreversibly and adiabatically
(
)
0
q
= against a constant external pressure of
[ ]
1.00
atm until the volume has doubled.
a)
Calculate the final volume.
b)
Calculate the work
( )
w for this process in joules.
c)
Calculate the energy change
( )
E
for this process in joules.
d)
Calculate the final temperature of the gas.


The First Law of Thermodynamics (OGN Equation 7.3) states E q w
= + . For an adiabatic expansion,
0
q
= , and
. Also, from the ideal gas law, a closed system containing a constant number of molecules
E w
=
( )
n one gets
1 1
2 2
1
2
.
PV
PV
nR
const
T
T
=
=
=

a)
The initial volume is then given by
[
]
(
) (
)
[
]
[ ]
[
]
[ ]
1
1
1
2.0
0.0820574
/
350
19.14
3.00
mol
L atm
mol K
K
nRT
V
L
P
atm
=
=
=
.
The problem states that the volume doubled during the expansion, so
[ ]
2
38.28
V
L
=
.

b)
The work is calculated from
(
)
[
]
[ ]
2
2
1
V= V
V = 19.14
1.94
Ext
w
P
P
atm L
kJ
= =
since the external
pressure is constant at
[
]
2
1.0
Ext
P
P
atm
=
=

(Note: The relations between the units
[
]
[ ]
1.0
101.32
atm L
J =
can be calculated from the different units given for
the gas constant;
(
)
[
]
(
) (
)
[
]
8.31447
/
0.0820574
/
R
J
mol K
L atm
mol K
= =
)

c)
For an adiabatic expansion,
[ ]
1.94
E w
kJ
= =
.

d)
The final temperature is given by
[ ]
[ ]
2
2
2
1
1
1
1
350
2 233.3
3
P
V
T
T
K
K
P
V
= = =
.
Note: This can also be calculated from
[
]
[ ]
[
]
(
) (
)
[
]
[ ]
2 2
2
1.0
38.28
233.3
2.0
0.0820574
/
atm
L
PV
T
K
nR
mol
L atm
mol K =
=
= ENGR 145

Exam #3 - Page 3 of 5
Fall 2005

ID number:
_____ _____ - _____ _____ _____ _____ (LAST SIX DIGITS ONLY!)

Question 3: Chemical Reactions - 20 points
The molar enthalpy of fusion of solid ammonia is
[
]
5.65
/
kJ mol , and the molar entropy of fusion is
(
)
[
]
28.9
/
J
K mol .
a)
Calculate the Gibbs free energy change for the melting of 1.00 mol of ammonia at 170 K.
b)
Calculate the Gibbs free energy change for the conversion of 3.60 mol of solid ammonia to liquid ammonia at
170 K.
c)
Will ammonia melt spontaneously at 170 K?
d)
At what temperature are solid and liquid ammonia in equilibrium at a pressure of 1 atm?


The molar Gibbs free energy of fusion at standard state pressure is given by
0
0
0
f
f
G
H
T S
f =

. The enthalpy of
fusion
[
]
(
5.65
/
kJ mol
)
corresponds to the heat absorbed as a mole of ammonia melts at constant pressure.

a)
The free energy change for the melting of 1.00 mol of ammonia at 170 K is thus
[
]
[ ]
(
)
[
]
[ ]
[ ]
0
5.65
/
170
28.9
/
737
740
f
G
kJ mol
K
J
K mol
J
J = =

b)
The Gibbs free energy change for the conversion of 3.60 mol of solid ammonia to liquid ammonia at 170 K is
then
{
}
[ ]
[ ]
[ ]
3.6
3.6 737
2,650
2.65
G
mol
J
J
kJ =
=



c)
Since
, this reaction will not be spontaneous. [The fact that
0
G
>
0
S
>
is not a sufficient requirement alone
to state that a reaction will be spontaneous, the change in Gibbs free energy must be smaller than zero as well, i.e.,
see OGN section 8.7 pages 253-255].
0
G
<


d)
The equilibrium temperature is determined from the condition for which
0
0
0
0
f
f
f
G
H
T S =

= , or
[
]
(
)
[
]
[ ]
[ ]
3
0
0
5.65 10
/
195.5
196
28.9
/
f
f
J mol
H
T
K
S
J
K mol
=
=
= K
ENGR 145

Exam #3 - Page 4 of 5
Fall 2005

ID number:
_____ _____ - _____ _____ _____ _____ (LAST SIX DIGITS ONLY!)

Question 4: Phase Diagrams - 20 point

The binary Ge-Si phase diagram is shown below. The density of Ge and Si are
and
3
5.32
/
g cm
3
2.33
/
g cm ,
respectively.

=30
16
44
C
0
C
L
C
S
52
32
67
Liquid
+ Solid
Liquid
Solid
[Ge-Si Solid Solution]
a)
Consider an alloy having 30 % Si by weight. What phases are present at 1200
o
C.
b)
Determine the compositions of the phases present at 1200
o
C in weight % and in atomic %, respectively.
c)
Determine the weight fraction of the phases present at 1200
o
C.
d)
Determine the molar (or atomic) fraction of the phases present at 1200
o
C.

a)
The phases present at 1200
o
C are liquid (l) and a solid Ge-Si solution (ss), respectively.

b)
The compositions of the phases present at 1200
o
C are:
Weight % Liquid: C
L
16 weight %;
Solid: C
S
44 weight %
Atomic %
Liquid: C
L
32 atom %;
Solid: C
S
67 atom %

c)
The weight fractions of the phases present at 1200
o
C are:
Liquid:
0
44 30 50%
44 16
W
S
L
S
L
C
C
f
C
C
=
=
=
Solid:
0
30 16 50%
44 16
W
L
S
S
L
C
C
f
C
C
=
=
=


d)
The molar (or atomic) fraction of the phases present at 1200
o
C is:
Liquid:
0
67 52 42.9%
67 32
A
S
L
S
L
C
C
f
C
C
=
= Solid:
0
52 32 57.1%
67 32
A
L
S
S
L
C
C
f
C
C
=
=
=
ENGR 145

Exam #3 - Page 5 of 5
Fall 2005

ID number:
_____ _____ - _____ _____ _____ _____ (LAST SIX DIGITS ONLY!)

Question 5: Definitions - 20 points
Match a statement in Column I (A through T) with a description in Column II (1 through 20) by
WRITING THE APPROPRIATE LETTER from Column I IN FRONT OF THE NUMBER in Column II.

COLUMN I
COLUMN II
A) Thermal Conductivity
______ 1) Chemical reaction giving off heat
E




B) Microstructure
______ 2) Amount of heat required to raise the temperature by 1 K
T




C) Tie Line
______ 3) Proportionality constant between the heat flux and temperature




gradient
A
D) Intrinsic Semiconductor
______ 4) Stress due to a thermal strain
H




E) Exothermic Reaction
______ 5) Concentration of electron and hole carriers are identical
D




F) Thermodynamic Process
______ 6) Concentration of electron and hole carriers modified by dopants





M
G) First Law of Thermodynamics ______ 7) A macroscopic, time-independent equilibrium condition
Q




H) Thermal Stress
______ 8) A process that leads to a change in thermodynamic state
F




I) Third Law of Thermodynamics ______ 9) The change in internal energy of a system equals the sum of the
G




heat transferred to it and the work done
J) Lever Rule
______ 10) The heat transferred at constant pressure
O




K) Solid Solution
______ 11) State function determined by the number of microstates
S

available
in
a
system
L) Phase Equilibrium Diagram
______ 12) The entropy of the system and its surrounding must increase
N


M) Extrinsic Semiconductor
______ 13) The entropy of any pure substance approaches zero at absolute
I

zero
temperature
N) Second Law of Thermodynamics ______ 14) A thermodynamic criterion for spontaneity
P




O) Enthalpy
______ 15) Graphic representation of the stable phase at a given set of
L

thermodynamic
variables
P) Gibbs Free Energy
______ 16) Homogeneous crystalline phase containing two or more

chemical
species
K
Q) Thermodynamic State
______ 17) The maximum concentration of solute that may be added




without forming a new phase
R
R) Solubility Limit
______ 18) Structural fe