Chapter 14: Ensemble and Molecular Partition Functions
Problem numbers in italics indicate that the solution is included in the
Student’s Solutions Manual.
Questions on Concepts
Q14.1)
What is the canonical ensemble? What properties are held constant in this
ensemble?
An ensemble is a collection of identical units or replicas of the system of interest.
In the canonical ensemble,
V
,
N
, and
T
are held constant.
Q14.2)
What is the relationship between
Q
and
q
? How does this relationship differ if
the particles of interest are distinguishable versus indistinguishable?
Q
is the canonical partition function, while
q
is the molecular partition function,
or the partition function that describes an individual member of the ensemble.
The relationship between
Q
and
q
depends on if ensemble members are
distinguishable are indistinguishable.
For
N
members or units in the ensemble:
Q
=
q
N
(distinguishable)
Q
=
q
N
N
!
(indistinguishable)
Q14.3)
List the atomic and/or molecular energetic degrees of freedom discussed in this
chapter. For each energetic degree of freedom, briefly summarize the corresponding
quantum mechanical model.
Translational:
Particle in a box
Vibrational:
Harmonic oscillator
Rotational:
Rigid rotor
Electronic:
Hydrogenatom model, MO theory
Q14.4)
For which energetic degrees of freedom are the spacings between energy levels
small relative to
kT
at room temperature?
At room temperature, the spacings between translational energy levels are
extremely small relative to
kT
.
The rotational energy level spacings are also
generally small relative to
kT
, although some species involving light atoms (such
as H
2
) may have energy levels spacings that are comparable to
kT
.
Vibrational
and electronic energy levels usually have spacings that are greater than
kT
.
141
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Chapter 14/Ensemble and Molecular Partition Functions
Q14.5)
For the translational and rotational degrees of freedom, evaluation of the
partition function involved replacement of the summation by integration. Why could
integration be performed? How does this relate back to the discussion of probability
distributions of discrete variables treated as continuous?
Integration can be performed in evaluating
q
for the translational and rotational
degrees of freedom because the energy level spacings along these degrees of
freedom are small relative to
kT
.
Since there are numerous states within the
energy range of interest (given by
kT
, it is reasonable to treat the variable of
interest (i.e., the number of states) as continuous.
Q14.6)
When is the highT approximation for rotations and vibrations? For which
degree of freedom do you expect this approximation to be generally valid at room
temperature?
The hightemperature approximation is when the product
kT
is significantly
greater than the energy level spacings.
This will be true for translations at all but
lowest temperatures.
For rotations, the hightemperature limit is appropriate with
T
≥ 10
Θ
R
where
Θ
R
is the rotational temperature, equal to B/k where B is the
rotational constant.
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 Fall '09
 Abra
 Mole, Statistical Mechanics, Orders of magnitude, partition function, Molecular Partition Functions

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