CBE 102B
Problem Set 5
Due: Friday, May 7 2010
1.
(15 pts.)
A system contains C, CO, CO
2
, H
2
, and H
2
O in chemical equilibrium.
Find the
number of independent reactions,
R
, and suggest stoichiometric coefficients
ν
j
.
2.
(15 pts.)
According to the Debye theory of solids
3
4
12
5
pm
D
T
C
R
π
⎛
⎞
=
⋅
⎜
⎟
Θ
⎝
⎠
at low
T
.
The constant
D
Θ
has units of temperature and is known as the Debye temperature of
the solid.
Derive an expression for the molar entropy,
, and show that
m
S
1
3
m
p
S
C
=
m
(
,
)
at low
T
.
3.
(70 pts.)
[ABET]
In this problem, we shall investigate the thermodynamics of an
incompressible Einstein crystal.
In the quadratic or harmonic approximation, a crystal of
N
atoms is treated as a collection of 3
N
independent harmonic oscillators.
Each oscillator vibrates
at its own characteristic frequency,
ν
.
Treating the harmonic vibration with quantum mechanics,
and using statistical mechanics, one finds the following result for the Helmholtz function,
A
A N T
=
N
, of a crystal of
N
atoms
 1 
(
)
1
2
(
)
( )
ln 1
e
.
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 Spring '11
 Thermodynamics, pts, lowT asymptotic forms

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