1516SEM1-PC2230
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National University of Singapore *
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Course
3321
Subject
English
Date
Nov 24, 2024
Type
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6
Uploaded by Theslyjumpy08
NATIONAL
UNIVERSITY
OF
SINGAPORE
EXAMINATION
FOR
THE
DEGREE
OF
B.
ENG.
(ENGINEERING
SCIENCE)
PC2230
-
THERMODYNAMICS
AND
STATISTICAL
MECHANICS
(Semester
I:
AY
2015-16)
Time
Allowed:
2
Hours
INSTRUCTIONS
TO
CANDIDATES
1.
Please
write
your
student
number
only.
Do
not
write
your
name.
2.
This
examination
paper
contains
SIX
short
questions
in
Part
I
and
TWO
long
questions
in
Part
II.
It
comprises
SIX
printed
pages.
3.
Answer
ALL
the
questions
in
this
paper.
4.
The
answers
to
Part
I
and
II
are
to
be
written
on
separate
answer
books,
which
are
to
be
submitted
at
the
end
of
the
examination.
5.
Please
start
each
question
on
a
new
page.
6.
This
is
a
CLOSED
BOOK
examination.
7.
The
last
page
contains
a
list
of
formulae.
8.
The
total
mark
for
Part
I
is
48
and
that
for
Part
I
is
52.
Part
1
This
part
of
the
examination
paper
contains
six
short-answer
questions,
which
carry
8
marks
each.
Answer
ALL
questions.
1.
The
equation
of
state
of
a
gas
is
given
by
P
=
-1‘—’—53—7—'
-
191_,_1\5’_3.
Now
0.5
kmol
of
this
gas
is
expanded
in
a
quasi-equilibrium
manner
from
2
to
4
m>
at
a
constant
temperature
of
300
K.
(a)
Draw
the
process
on
a
P-V
diagram.
(2
marks)
(b)
Determine
the
work
done
during
this
isothermal
expansion
process.
(6
marks)
2.
A
system
has
two
non-degenerate
energy
levels
with
an
energy
gap
ot
0.1
eV
(=
1.6
x
10729
J).
Determine
the
following:
(a)
the
probability
of
the
system
being
in
the
upper
level
if
it
is
in
thermal
contact
with
a
heat
bath
at
a
temperature
of
300
K.
(4
marks)
(b)
the
temperature
if
the
probability
to
be
in
the
upper
level
state
is
0.25.
(4
marks)
3.
A
20
kg
aluminium
block
initially
at
200°C
is
brought
into
contact
with
a
20
kg
block
of
iron
at
100°C
in
an
insulated
enclosure.
Determine
(a)
the
final
equilibrium
temperature,
and
(4
marks)
(b)
the
total
entropy
change
for
this
process.
(4
marks)
(Note:
Specific
heat
of
aluminum,
C,
=
0.973
kJkg™
'K~
and
specific
heat
of
iron,
C,
=
0.45
kJkg7'K™!)
4.
Consider
a
gas
of
oxygen
at
a
temperature
of
100
K
and
a
pressure
of
1
x
107
Pa.
Determine
whether
classical
statistics
would
be
appropriate
for
its
study.
The
molecular-weight
of
oxygen
molecule
is
32
g/mol.
(8
marks)
5.
If
a
700
N
man
stands
on
ice
skates
for
which
the
area
of
contact
with
the
ice
is
10
mm?,
by
how
many
degrees
is
the
melting
point
of
the
ice
lowered?
(The
latent
heat
of
fusion
is
6.01
x
10°
J/mole.
At
the
melting
point,
the
density
of
ice
is
0.917
g/cm?,
and
that
of
liquid
water
is
0.9999
g/cm?>.
The
molecular
weight
of
water
is
18
g/mol.)
(8
marks)
6.
The
molecules
of
a
diatomic
gas
possess
rotational
energy
levels
hz
€
=
-2-7r(r+
1),
r=0,12,...,
where
[
is
a
constant
and
the
level
¢,
is
(2r
+
1)-fold
degenerate.
Obtain
an
expression
for
the
rotational
partition
function
at
Jow
temperatures.
Hence,
find
the
molar
rotational
heat
capacity
of
the
gas
at
low
temperatures.
(8
marks)
End
of
Part
1
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Part
11
This
part
of
the
examination
paper
contains
two
(2)
long-answer
questions.
An-
swer
BOTH
questions.
1.
(a)
A
reversible
cycle
device
does
work,
while
exchanging
heat
with
three
constant
temperature
reservoirs
as
illustrated
in
Figure
1.
T,=1000K
|
Reservorrl
y
REVERSIBLE
=
CYCLIC DEVICE
Figure
1:
Reversible
cycle
device
The
three
reservoirs
1,
2,
and
3
are
at
temperature
of
1000
K,
300
K,
and
500
K,
respectively.
It
is
known
that
400
kJ
is
transferred
from
reservoir
1
to
the
device,
and
the
total
work
done
by
the
cyclic
device
is
100
kJ.
Determine
i.
the
magnitude
and
direction
of
the
heat
transfer,
Q,,
with
the
reservoir
2,
and
(8
marks)
ii.
the
magnitude
and
direction
of
the
heat
transfer,
O3,
with
the
reservoir
3.
(5
marks)
(b)
The
energy
density
(E/V)
of
blackbody
radiation
is
known
to
be
(4/c)aT*,
where
c
is
the
speed
of
light
and
o
is
the
Stefan-Boltzmann
constant.
Obtain
i.
the
expression
for
the
pressure,
and
(8
marks)
[Hint:
Use
dE
=
TdS—pdV
and
Helmholtz
function:
F(T,
V)
=
E
-TS]
ii.
the
expression
for
the
specific
heat
capacity
C,.
(5
marks)
4
2.
(a)
Consider
a
system
made
up
of
two
identical
fermionic
particles
which
can
occupy
4
different
states.
i.
Enumerate
all
the
possible
choices
for
the
occupation
numbers
of
the
single
particle
states,
expressing
your
answers
in
the
form
(ny,n,,n3,
nyg),
where
the
ns
are
the
usual
occupation
numbers.
(5
marks)
ii.
Assume
that
the
4
single
particle
states
have
the
following
ener-
gies:
€1
=
€3
=
0
and
€3
=
€4
=
¢
and
that
the
system
is
in
thermal
equilibrium
at
a
temperature
7.
Compute
the
mean
occu-
pation
number
of
the
first
single-particle
state
7;
at
temperature
T'.
Calculate
also
the
average
energy
of
the
system.
(11
marks)
(b)
The
latent
heat
of
vaporization
for
hydrogen
molecules
near
its
triple
point
(T,
=
14
K)
is
1.01
kI
mol™!.
The
liquid
density
is
71
kgm™3,
the
solid
density
is
81
kgm™3,
and
the
melting
temperature
is
given
by
7,
=
13.99
+
P/3.3,
where
T,
and
P
are
measured
in
K
and
MPa
respectively.
Compute
the
latent
heat
of
sublimation
given
that
the
molecular
weight
of
hydrogen
is
2
g/mol.
(10
marks)
Exam
Formulae
Sheet
Boltzmann
constant
kp
=
1.38
x
102>
m?*kgs
2
K™
Planck
constant
#
=
6.626
x
107>*
m?*kgs™'
Gas
constant
R
=
8.31
Jmol™!
K™!
1
atm
=
1.013
x
10°
Pa,
1
mole
=
6.022
x
10?3
dE
=
TdS
—
PdV
+
udN,
dF
=
—SdT
—
PdV
+
udN
dG
=
—SdT
+
VdP
+
udN,
dQ
=
—SdT
—
PdV
—
Ndu
For
an
adiabatic
change
of
a
perfect
gas,
RdAT
Tv?~!
=
constant,
Pv?
=
constant,
Work
done
=
—
y
P
I'(n)=
/()“x,,_.le_x
dx,
I'(n+1)=nl@m),
I'A/2)=r,
TI'l)=1
V
47
p?
S:—kBZprlnpr,
f(p)dp
=
h3p
dp
3
.
e
h(l)3
dw
Planck’s
blackbody
radiation
law:
u(w,T)
dw
=
2
exp
o
K
T)
—
1]
__
4
_
w’ky
_
4
_
~8
1o
=2
—lyr—4
u(T)-aT,
amm,
I
=0T,
o0=567x10"7"Jm
“s
K
1
1
1
Stirling’s
formula:
I
N!=
NInN
—
N+2]nN+-1n(2n)+0(N)
‘
.
.
2rmk
B
T
3/2
s
.
e
»
u.
Je—
Single-particle
translational
partition
function:
Z;
=V
(
73
)
h
Agg
=
-E,
E
=
hw
=
pc
(photon)
E?
=
p*c®
+
mdc?,
—
En,
Gibbs
distribution:
py,
=
2P
(”‘Z
v
2T
V)
=
Y
explB(uN—Ew)]
F(T,V,N)
=
—kgT
W
Z(T,V,N),
S2(T,V,n)
=
—kgT
WnZ(T,V,pn),
Z
=O
[Ter
=TIt
=
Koy,
[T
20
=
[
[T
=
Ke
D,
7(T)
=
=
dP
L
Clausius-Cl
Equation:
—
=
——
ausius-Clapeyron
Equation
T
=
TAV
.
aan)
.
1(31112
o
+FD
B
B
Jvn
T
B\
&
Jroesy
0
ePEWE1
-
BE
—
END
OF
PAPER
—
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