Page 1 of 6
ChE 359 Exam #2
Solutions prepared by Professor Lo
1.
(50 points) The Haber process is commonly used to make ammonia (NH
3
)
from nitrogen gas (N
2
) and hydrogen gas (H
2
), by passing the reactants
over an iron oxide catalyst at high temperature and pressure.
We can
model all of these as ideal gases, where the molecules are coarse grained
into spherical particles.
We will use the lattice model to study the
thermodynamics of mixing when the reactants are combined; we will not
study the reaction itself; that will come later in the semester.
a.
(10 points) The reaction chamber can be modeled by a square lattice
consisting of 100 lattice sites.
There are 25 nitrogen particles and 75
hydrogen
particles.
Assume
that
the
nitrogen
particles
are
indistinguishable, and the hydrogen particles are indistinguishable.
First, write out the factorial expression for the multiplicity
W
A
of
the heterogeneous system (i.e., the chamber is segregated into a
nitrogenonly region with 25 lattice sites, and a hydrogenonly
region with 75 lattice sites), and second, write out the factorial
expression for the multiplicity
W
B
of the homogeneous system (i.e.,
all particles are allowed to mix freely within the reaction chamber).
We first calculate
W
A
, which we know is 1 because the particles
are indistinguishable.
Each of the 25 nitrogen lattice sites can
only be occupied by a nitrogen particle, and each of the 75
hydrogen lattice sites can only be occupied by a hydrogen
particle.
W
A
=
N
N
n
N
×
N
H
n
H
=
25!
25! 25
−
25
( )
!
×
75!
75! 75
−
75
( )
!
=
1
We then calculate
W
B
.
In this case, each of the 100 chamber sites
can be occupied by either a nitrogen particle or a hydrogen
particle.
There are many more possibilities!
W
A
=
N
N
+
N
H
n
N
=
100!
25! 100
−
25
( )
!
=
5.35
×
10
29
b.
(10 points) Use the ratio
W
B
/W
A
to calculate the change in entropy
[J/K] when the particles are allowed to mix freely.
We use
Boltzmann’s law:
Δ
S
=
S
B
−
S
A
=
k
ln
W
B
−
k
ln
W
A
=
k
ln
W
B
W
A
=
1.38
×
10
−
23
J
K
( )
ln 5.35
×
10
29
( )
=
9.45
×
10
−
22
J
K
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c.
(10 points) First, state the probability
p
N
that a given lattice site is
occupied by a nitrogen particle, and the probability
p
H
that a given
lattice site is occupied by a hydrogen particle in the fully mixed
scenario.
Second,
given
that
the
energy
of
any
one
of
the
indistinguishable nitrogen particles is
ε
N
and the energy of any one
of
the
indistinguishable
hydrogen
particles
is
H
,
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 Spring '10
 CynthiaLo
 Catalyst, Mole, Kinetics, partition function, methane gas, deep sea conditions

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