Overview
Distinguishable and indistinguishable particles have been considered. Consider the
...
single-particle
...
partition
...
function
...
among
...
distinguishable
...
particles.
...
Label
...
and
...
assign
...
criteria
...
among
...
different
...
particles.
...
Apply
...
the
...
Boltzman
...
approximation
...
at
...
high
...
temperature,
...
low
...
density,
...
and
...
high
...
mass
...
to
...
indistinguishable
...
particles
...
and
...
obtain
...
the
...
relation
...
below.
...
Adjust
...
for
...
overcounting.
...
The
...
number
...
of
...
particles
...
is
...
much
...
less
...
than
...
the
...
number
...
of
...
state.
...
Apply
...
to
...
a
...
monoatomic
...
gas.
...
At
...
low
...
temperature
...
and
...
high
...
density,
...
the
...
Boltzman
...
approximation
...
is
...
not
...
good.
...
Different
...
types
...
of
...
statistics
...
have
...
been
...
mentioned.
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} \[ Q = \frac{q_{\mbox{single}}^N}{N\!} \] {latex} |
Today
...
a
...
diatomic
...
gas
...
is
...
considered.
...
In
...
a
...
solid,
...
atoms
...
are
...
fixed
...
in
...
space.
...
A
...
solid
...
solution
...
and
...
vibrations
...
in
...
a
...
solid
...
are
...
considered.
...
Degrees
...
of
...
freedom
...
in
...
a
...
diatomic
...
molecule
...
There
...
are
...
different
...
degrees
...
of
...
freedom
...
in
...
a
...
diatomic
...
molecule.
...
Below
...
is
...
a schematic.
There are various contributions to the Hamiltonian of a single particle. Contributions include translation, rotation, vibration, electric, and nuclear. The diatomic molecule is part of an ideal gas. Separate the Hamiltonians. Separate the interactions and particles. Under the rigid rotor approximation, rotations do not see the changing bond length. The contributions of the Hamiltonian due to rotation and vibration are independent. When vibrations are large, though, the bond length is changing. The first excited electric state would be considered. McQuarrie considers the first excited nuclear state. Thermodynamics cares about derivatives of the partition function.
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schematic. !Diatomic_molecule_--_translation%2C_vibration%2C_rotation.PNG! There are various contributions to the Hamiltonian of a single particle. Contributions include translation, rotation, vibration, electric, and nuclear. The diatomic molecule is part of an ideal gas. Separate the Hamiltonians. Separate the interactions and particles. Under the rigid rotor approximation, rotations do not see the changing bond length. The contributions of the Hamiltonian due to rotation and vibration are independent. When vibrations are large, though, the bond length is changing. The first excited electric state would be considered. McQuarrie considers the first excited nuclear state. Thermodynamics cares about derivatives of the partition function. {latex} \[ \hat H_{\mbox{single}} = \hat H_{\mbox{trans}} + \hat H_{\mbox{rot}} + \hat H_{\mbox{vib}} + \hat H_{\mbox{elec}} + \hat H_{\mbox{nucl}} \] {latex} |
The
...
partition
...
function
...
is
...
a
...
measure
...
of
...
the
...
number
...
of
...
thermally
...
excited
...
states.
...
Consider
...
the
...
single-particle
...
partition
...
function.
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} \[ q_{\mbox{single}} = q_{\mbox{trans}} \cdot q_{\mbox{rot}} \cdot q_{\mbox{vib}} \] {latex} |
There
...
is
...
a
...
high
...
temperature
...
approximation.
...
Assuming
...
"normal
...
conditions"
...
the
...
Boltzman
...
approximation
...
can
...
be
...
used.
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} \[ Q_{\mbox{sys}} = \frac{q_{\mbox{single}}^N}{N\!} \] {latex} h2. Translation Look at the center of mass when considering the translational component of the single-particle partition |
Translation
Look at the center of mass when considering the translational component of the single-particle partition function.
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function.
{latex} \[ q_{\mbox{trans}} = \left ( \frac{2 \pi ( m_1 + m_2 ) k_B T}{n^2} \right )^{\frac{3}{2}} \] |
Vibrations
There is interest in expressions of
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{latex}
h2. Vibrations
There is interest in expressions of {latex} \[ q_{\mbox{rot}} \] |
and
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{latex} and {latex} \[ q_{\mbox{vib}} \] {latex} |
and
...
energy
...
spectrum.
...
Diatomic
...
molecules
...
see
...
a
...
potential
...
well
...
and
...
there
...
is
...
an
...
equilibrium
...
bond
...
length.
...
The
...
deviation
...
is
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} \[ \zeta \] {latex} |
,
...
and
...
the
...
term
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|---|
} \[ \mbox{De} \] {latex} |
is
...
the
...
energy
...
required
...
to
...
dissociate.
...
Assuming
...
the
...
vibrations
...
are
...
small,
...
the
...
energy
...
associated
...
with
...
vibrating
...
bonds
...
can
...
be
...
expressed
...
with
...
a
...
Taylor
...
expansion.
...
!Potential_versus_spacing_--_Ro.PNG
...
!
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} \[ U( \zeta ) = U(0) + \left ( \frac{\partial U}{\partial \zeta} \right )_{\zeta = 0} \zeta + \frac {1}{2} \left ( \frac{\partial^2 U}{\partial \zeta^2} \right )_{\zeta = 0 } \zeta^2 + ... \] {latex} |
The
...
first
...
term,
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} \[ U(0) \] {latex} |
,
...
is
...
constant,
...
and
...
the
...
second
...
term
...
is
...
equal
...
to
...
zero
...
since
...
there
...
is
...
no
...
force.
...
The
...
second-order
...
term
...
is
...
proportional
...
to
...
the
...
strength
...
of
...
the
...
bond
...
between
...
two
...
particles.
...
There
...
is
...
a
...
force
...
constant
...
between
...
two
...
particles.
...
Write
...
the
...
Hamiltonian
...
with
...
the
...
approximation
...
of
...
small
...
vibrations.
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} \[ \hat H_{\mbox{vibr}} = \frac{\hbar^2}{2 m_r} \frac{\partial^2}{\partial \zeta^2} + \frac{1} {2} f \zeta^2 \] {latex} |
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|---|
{latex} \[ m_r = \frac{m_1 m_2}{m_1 + m_2} \] {latex} |
The
...
first
...
term
...
is
...
a
...
kinetic
...
contribution,
...
and
...
the
...
force
...
constant,f,
...
is
...
a
...
property
...
of
...
the
...
molecule.
...
Consider
...
the
...
quantum
...
mechanical
...
solutions
...
of
...
a
...
harmonic
...
oscillator.
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} \[ \epsilon_n = \left ( n + \frac{1}{2} \right ) h \nu \] {latex} |
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{html}
|
...
<P> </P>{html} |
...
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|---|
...
\[ n=0,1,2 \] |
...
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{html} |
...
<P> </P>{html} |
...
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|---|
...
\[ \nu = \frac{1}{2 \pi} \sqrt{\frac{f}{m}} \] |
...
There
...
is
...
a
...
relation
...
between
...
a
...
spring
...
constant
...
and
...
mass.
...
Light
...
masses
...
or
...
masses
...
with
...
a
...
stiff
...
spring
...
vibrate
...
rapidly.
...
Write
...
the
...
mini-partition
...
function.
...
and
...
simplify
...
with
...
the
...
relation
...
pertaining
...
to
...
a
...
geometric
...
series.
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} \[ q_{\mbox{vib}} = \sum_n^{\infty} e^{-\beta \epsilon_n} \] {latex}{ |
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{html} |
...
<P> </P>{html} |
...
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|---|
...
\[ q_{\mbox{vib}} = \sum_{n=0}^{\infty} e^{\left ( n + \frac{1}{2} \right ) h \nu} \] |
...
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{html} |
...
<P> </P>{html} |
...
| Latex |
|---|
...
\[ q_{\mbox{vib}} = e^{\beta \frac{h \nu}{2}} \sum_{n=0}^{\infty} e^{\beta h \nu n} \] |
...
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{html}
|
...
<P> </P>{html} |
...
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|---|
...
\[ \mbox{Geometric Series} \] |
...
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{html}
|
...
<P> </P>{html} |
...
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|---|
...
\[ x < 1 \] |
...
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{html} |
...
<P> </P>{html} |
...
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|---|
...
\[ \sum_{n = 0}^{\infty} x^n = \frac{1}{1-x} \] |
...
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|---|
{html}
|
...
<P> </P>{html} |
...
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|---|
...
\[ q_{\mbox{vib}} = \frac{e^{-\beta \frac{h \nu}{2}}}{1-e^{-\beta h \nu}} \] |
...
Express
...
in
...
terms
...
of
...
the
...
characteristic
...
vibrational
...
temperature,
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|---|
} \[ \theta_{\nu} \] {latex} . {latex} |
.
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|---|
\[ \theta_{nu} = \frac{h \nu}{k_B} \] {latex} |
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{html}
|
...
<P> </P>{html} |
...
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|---|
...
\[ q_{\mbox{vib}} = \frac{e^{-\frac{\theta_{\nu}}{2 T}}}{1-e^{-\frac{\theta_{\nu}}{2 T}}} \] |
...
The
...
characteristic
...
termperature,
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|---|
} \[ \theta_{\nu} \] {latex} |
,
...
is
...
typically
...
between
| Latex |
|---|
} \[ 3000 - 6000 K \] {latex} |
.
...
Calculate
...
the
...
probability
...
of
...
being
...
in
...
a
...
certain
...
state.
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|---|
} \[ P_n = \pi_n \] {latex}{ |
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{html} |
...
<P> </P>{html} |
...
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|---|
...
\[ P_n = \frac{e^{-\beta \epsilon_n}}{q_{\mbox{vib}}} \pi_{n>0} \] |
...
A
...
general
...
expression
...
of
...
being
...
in
...
any
...
state
...
other
...
than
...
the
...
ground
...
state
...
is
...
below,
...
as
...
well
...
as
...
a
...
calculation
...
at
| Latex |
|---|
} \[ 300 K \] {latex} |
and
...
a
...
characteristic
...
temperature
...
of
| Latex |
|---|
} \[ 3000 K \] {latex} . {latex} |
.
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|---|
\[ \pi_{n>0} = 1 - \pi_0 \] {latex}{ |
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{html} |
...
<P> </P>{html} |
...
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|---|
...
\[ \pi_{n>0} = e^{-10} \] |
...
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|---|
{html} |
...
<P> </P>{html} |
...
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|---|
...
\[ \pi_{n>0} \approx 10^{-5} \] |
...
The
...
probability
...
of
...
a
...
molecule
...
being
...
in
...
an
...
excited
...
state
...
is
...
very
...
small.
...
It
...
is
...
hard
...
to
...
excite
...
a
...
vibrational
...
state
...
above
...
the
...
ground
...
state.
...
Rotation
Rotational motion is associated with angular momentum. The rotation of a diatomic molecule can be considered equivalent to the rotation of one particles moving freely along a sphere with fixed radius,
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\[ R_0 \] |
, and a reduced mass. Consider the Hamiltonian. The term
| Latex |
|---|
\[ I \] |
is the moment of inertia and an expression of the eigenvalues,
| Latex |
|---|
\[ \epsilon_j \] |
, to the spherical coordinate problem is below.
!Diatomic_molecule_--_rotation_with_axis.PNG
...
!
| Latex |
|---|
} \[ \hat H = -\frac{\hbar^2}{2I} \left ( \frac{1}{\sin \theta} \frac{\partial}{\partial \theta} \left ( \sin \theta \frac{\partial}{\partial \theta} \right ) + \frac{1}{\sin^2 \theta} \frac{\partial^2}{\partial r^2} \right ) \] |
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|---|
{latex} {latex} \[ I = m_R R_0^2 \] {latex} {latex |
| Latex |
|---|
} \[ \epsilon_j = \frac{\partial ( \zeta + 1 ) h^2}{8 \pi^2 I} \] |
| Latex |
|---|
{latex} {latex} \[ j = 0, 1, 2 \] {latex} |
Identify
...
the
...
Hamiltonian,
...
solve
...
with
...
Schrodinger
...
or
...
remember
...
the
...
energy
...
spectrum.
...
In
...
an
...
unsymmetric
...
molecules,
...
the
...
two
...
masses
...
are
...
not
...
equivalent.
...
A
...
summation
...
in
...
the
...
mini-partition
...
function
...
is
...
not
...
as
...
easy
...
as
...
the
...
vibrational
...
case.
...
Consider
...
two
...
limits.
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|---|
} \[ q_{\mbox{rot}} = \sum_{\epsilon_j} = w_j e^{\beta \epsilon_j} \] {latex} {latex} |
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{html}<p>{html} |
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|---|
\[ q_{\mbox{rot}} = \sum_{\epsilon_j} = (2j + 1) e^{-j (j+1) \frac{\theta}{T} } \] |
...
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{html}<p>{html} |
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|---|
\[ \theta_r = \frac{h^2}{8 \pi^2 I k_N} \] |
...
Case 1
The first case involves high temperatures, or temperatures much greater than\theta_r.
...
If
...
the
...
temperature
...
is
...
large,
...
the
...
exponential
...
term
...
goes
...
to
...
zero.
...
States
...
are
...
dense,
...
and
...
it
...
is
...
possible
...
to
...
approximate
...
with
...
an
...
integral.
| Latex |
|---|
} \[ q_{\mbox{rot}} = \int (2j + 1) e^{-j (j+1) \frac{\theta}{T}} dj \] {latex} {latex} |
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{html}<p>{html} |
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|---|
\[ q_{\mbox{rot}} = \frac{T}{\theta_r} \] |
...
Case 2
Consider when the temperature is close to\theta_r.
...
Caclulate
...
a
...
few
...
of
...
the
...
terms,
...
and
...
see
...
that
...
the
...
terms
...
decrease
...
rapidly.
...
Assume
...
that
...
the
...
larger
...
terms
...
are
...
zero.
...
Consider
...
the
...
first
...
couple
...
terms
...
of
...
the
...
summation.
| Latex |
|---|
} \[ q_{\mbox{rot}} = 1 + 3 e^{-\frac{2 \theta_r}{T}} + 5 e^{-\frac{6 \theta_r}{T}} + ... \] {latex} |
There
...
is
...
overcounting
...
of
...
the
...
number
...
of
...
states
...
with
...
symmetric
...
molecules.
...
Rotational
...
degrees
...
of
...
freedom
...
are
...
coupled.
...
Read
...
MacQuarrie
...
6-4.
...
The
...
level
...
of
...
quantum
...
mechanics
...
is
...
above
...
what
...
is
...
required.
...
There
...
is
...
inversion
...
of
...
the
...
nucleus,
...
two
...
indistinguishable
...
configurations,
...
and
...
overcounting.
...
Regarding
...
the
...
quiz,
...
one
...
should
...
be
...
able
...
to
...
do
...
a
...
problem
...
with
...
a
...
symmetric
...
molecule.
...
Correction
...
for
...
overcounting
...
The
...
mini-partition
...
function
...
is
...
a
...
measure
...
of
...
the
...
number
...
of
...
thermally
...
accessible
...
states.
...
The
...
mini-partition
...
function
...
of
...
rotation
...
is
...
written
...
below
...
in
...
the
...
first
...
case
...
wherein
...
the
...
temperature
...
is
...
much
...
greater
...
than\theta_r.
| Latex |
|---|
} \[ q_{\mbox{rot}} = \frac{T}{2 \theta_r} \] {latex} |
The
...
case
...
where
...
the
...
temperature
...
is
...
less
...
than\theta_rrequires
...
rigorous
...
quantum
...
mechanical
...
treatment
...
(really
...
onlyH_2
...
!).
...
Calculate
...
the
...
probability
...
to
...
be
...
in
...
a
...
certain
...
state.
| Latex |
|---|
} \[ P_j = \frac{N_j}{N} \] {latex} {latex} |
| Latex |
|---|
\[ P_j = \frac{(2j + 1) e^{-j (j+1) \frac{\theta_r}{T}}}{q_{\mbox{rot}}} \] {latex}
|
Other
...
states
...
are
...
seen
...
other
...
than
...
the
...
lowest
...
one.
...
A
...
graph
...
is
...
below.
...
!Nj_n_versus_j.PNG
...
!
...
Deriving
...
Thermodynamic
...
Quantities
...
from
...
Partition
...
Function
...
From
...
the
...
single-partition
...
function,
...
build
...
the
...
entire
...
system.
| Latex |
|---|
} \[ Q_{\mbox{sys}} = \frac{\left ( q_{\mbox{trans}} \cdot q_{\mbox{rot}} \cdot q_{\mbox{vib}} \right )^N}{N\!} \] {latex} {latex} |
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{html}<p>{html} |
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|---|
\[ F = -k_B T \ln Q \] |
...
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{html}<p>{html} |
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\[ F = -k_B T \left ( \ln q_{\mbox{trans}} + \ln q_{\mbox{rot}} + \ln q_{\mbox{vib}} - \ln N\! \right ) \] |
...
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{html}<p>{html} |
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|---|
\[ F = F_{\mbox{trans}} + F_{\mbox{rot}} + F_{\mbox{vibr}} \] |
Pressure
Calculate the pressure. Only the translational component of the partition function is proportional to volume.
| Latex |
|---|
{latex}
h2. Pressure
Calculate the pressure. Only the translational component of the partition function is proportional to volume.
{latex} \[ P = - \left ( \frac{\partial F}{\partial V} \right )_{N, T} \ln q_{\mbox{trans}} = \left ( \frac{2 \pi \left (m_1 + m_2 \right ) k_B T} {n^2} \right )^{\frac{3}{2} V} \] |
| Latex |
|---|
{latex} {latex} \[ P = \frac{N k_B T}{V} \] {latex} |
This
...
expression
...
is
...
the
...
same
...
as
...
for
...
a
...
monoatomic
...
gas.
...
The
...
ideal
...
gas
...
equation
...
holds
...
for
...
both
...
monoatomic
...
and
...
diatomic
...
gases.
...
Heat
...
Capacity
...
Calculate
...
the
...
heat
...
capacity.
...
Decouple
...
energies
...
and
...
heat
...
capacities.
| Latex |
|---|
} \[ E = k_B T^2 \left (\frac{\partial \ln Q}{\partial T} \right )_{N, V} \] |
| Latex |
|---|
{latex} {latex} \[ E = E_{\mbox{trans}} + E_{\mbox{rot}} + E_{\mbox{vibr}} \] {latex} {latex} |
| Latex |
|---|
\[ C_v = C_{v, \mbox{trans}} + C_{v, \mbox{rot}} + C_{v, \mbox{vibr}} \] {latex}
h3. Translational energy
Below is an expression of the translational energy
{ |
Translational energy
Below is an expression of the translational energy
| Latex |
|---|
latex} \[ E_{\mbox{trans}} = \frac{3}{2} N k_B T \] {latex} h3. Rotational energy An expression of the rotational energy is below. Consider at normal temperature and a system that is |
Rotational energy
An expression of the rotational energy is below. Consider at normal temperature and a system that is notH_2.
...
An
...
expression
...
of
...
the
...
rotational
...
partition
...
function
...
is
...
also
...
below.
| Latex |
|---|
} \[ E_{\mbox{rot}} = N k_B T^2 \left ( \frac{ \partial \ln q_{\mbox {rot}}}{\partial T} \right )_{N, V} \] {latex} {latex} |
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\[ E_{\mbox{rot}} = N k_B T \] |
Vibrational energy
The vibrational energy is expressed below.
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{latex} h3. Vibrational energy The vibrational energy is expressed below. {latex} \[ E_{\mbox{vibr}} = N k_B T^2 \left ( \frac{ \partial \ln q_{\mbox{vibr}}}{\partial T} \right )_{N, V} \] {latex} {latex} |
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\[ E_{\mbox{rot}} = N k_B T \left ( \frac{ \frac{h \nu}{k_B T}}{e^{\frac{h \nu}{k_B T}} - 1} \right ) + \frac{N h \nu}{2} \] |
...
Analyze
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the
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vibrational
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energy
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with
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two
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limits.
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Case
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1
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In
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the
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first
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case,
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consider
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the
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vibrational
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energy
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as
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the
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temperature
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approaches
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infinity.
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The
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expression
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of
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vibrational
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energy
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can
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be
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written
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in
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terms
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ofx,
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and
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is
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written
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below.
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As
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the
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temperature
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approaches
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infinity,xapproaches
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zero.
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} \[ x = \frac{h \nu}{k_B T} \] {latex} {latex} |
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\[ \lim_{x \to 0} \frac {x}{e^x - 1} = 1 \] |
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\[ |
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E_{\mbox{vibr}} = N k_B T + \frac {N h \nu}{2} \] |
Case 2
An expression of the vibrational energy in the case of temperature much less than
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} \] {latex}
h4. Case 2
An expression of the vibrational energy in the case of temperature much less than {latex} \[ \theta_v \] |
.
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{latex}. {latex} \[ E_{\mbox{vib}} = \frac{N h \nu}{2} \] {latex} h3. Translational component of heat capacity The translational component of heat capacity is found by |
Translational component of heat capacity
The translational component of heat capacity is found by differentiating the translational energy with respect to temperature.
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differentiating the translational energy with respect to temperature. {latex} \[ C(v, \mbox{trans}) = \frac{\partial E_{\mbox{trans}} }{\partial T} \] {latex} {latex} |
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\[ C(v, \mbox{trans}) = \frac{3 N k_B}{2} \] |
Rotational component of heat capacity
The relation below of the rotational component of heat capacity is used unless the temperature is very low or for
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{latex}
h3. Rotational component of heat capacity
The relation below of the rotational component of heat capacity is used unless the temperature is very low or for {latex} \[ H_2 \] |
.
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{latex}. {latex} \[ C(v, \mbox{rot}) = \frac{\partial E_{\mbox{rot}}}{\partial T} \] {latex} {latex} |
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\[ C(v, \mbox{rot}) = N k_B \] |
Vibrational component of heat capacity
The expression of the vibrational component of heat is below and is evaluated in two cases.
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] {latex} h3. Vibrational component of heat capacity The expression of the vibrational component of heat is below and is evaluated in two cases. {latex} \[ C(v, \mbox{vibr}) = \frac{\partial E_{\mbox{vib}}}{\partial T} \] {latex} h4. Case 1 The temperature can be |
Case 1
The temperature can be evaluated in the case of temperature much lower than
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evaluated in the case of temperature much lower than {latex} \[ \theta_r \] |
.
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{latex}. {latex} \[ C(v, \mbox{vibr}) = 0 \] {latex} h4. Case 2 An express of the vibrational component of heat |
Case 2
An express of the vibrational component of heat capacity in the case of temperature greater than
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capacity in the case of temperature greater than {latex} \[ \theta_r \] {latex} |
is
...
below.
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} \[C{v, \mbox{vibr}} = N k_B \] {latex} h3. Total heat capacity The total heat capacity is considered in two cases |
Total heat capacity
The total heat capacity is considered in two cases.
Case 1
Consider when the temperature is much less than
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. h4. Case 1 Consider when the temperature is much less than {latex} \[ \theta_{\nu} \] {latex} |
and
...
greater
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than
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} \[ \theta_r \] {latex}. {latex} |
.
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\[ C(v, \mbox{tot}) = \frac {3}{2} N k_B + N k_B + 0 \] {latex}
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\[ C(v, \mbox{tot}) = \frac{5}{2} N k_B \] |
Case 2
An expression of the heat capacity is below in the case of the temperature greater than or equal to
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{latex} h4. Case 2 An expression of the heat capacity is below in the case of the temperature greater than or equal to {latex} \[ \theta_v \] {latex} |
and
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much
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greater
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than
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} \[ \theta_r \] {latex} {latex}] |
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\[ C(v, \mbox{tot}) = \frac{3}{2} N k_B + N k_B + N k_B \] {latex}
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\[ C(v, \mbox{tot}) = \frac{7}{2} N k_B \] |
Comments
Define different kinds of motion. Solve the Schrodinger equation. Evaluate as well as possible. Take limits in the rotational case. Boltzman is a system approximation. Don't need to be too occupied with the math.
Configurational Properties
A figure below is of a mixture of atoms in a solid. Consider some arrangement on a lattice of atoms and vacancies or a binary mix. In general different arrangements are associated with different energies. Consider
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{latex} h1. Comments Define different kinds of motion. Solve the Schrodinger equation. Evaluate as well as possible. Take limits in the rotational case. Boltzman is a system approximation. Don't need to be too occupied with the math. h1. Configurational Properties A figure below is of a mixture of atoms in a solid. Consider some arrangement on a lattice of atoms and vacancies or a binary mix. In general different arrangements are associated with different energies. Consider {latex} \[ M \] {latex} |
sites
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with
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} \[ N \] {latex} |
atoms
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of
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one
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kind.
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The
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term
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} \[ x \] {latex} |
is
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the
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concentration.
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!Binary_mix.PNG! {latex} \[ M \mbox{sites} \] {latex} {html}<P>< |
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\[ N \mbox{atoms} \] |
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\[ x = \frac{N}{M} \] |
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Assume
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now
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that
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the
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energy
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is
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only
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a
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function
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of
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concentration.
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Fix
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the
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concentration.
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With
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} \[ N \] {latex} |
fixed,
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there
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is
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only
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one
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energy.
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Treat
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in
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terms
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of
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a
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canonical
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or
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microcanonical
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ensemble.
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The
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number
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of
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ways
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to
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distribut
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} \[ eN \] {latex} |
distinguishable
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particles
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on
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} \[ M \] {latex} |
sites
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is
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expressed
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below
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in
...
terms
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of
| Latex |
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} \[ \Omega (E) \] {latex}. {latex} |
.
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\[ Q = \Omega (E) \cdot e^{-\beta E} \] {latex}
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\[ \Omega (E) = \frac{M!}{N!(M-N)!} \] |
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\[ F = E - TS \] |
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\[ F = E - k_B T \ln \Omega \] |
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| Latex |
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\[ S = k_B \ln \Omega \] |
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The
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term
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} \[ S \] {latex} |
is
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the
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configurational
...
entropy.
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Construct
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a
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phase
...
diagram.
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Consider
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an
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electronic
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or
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vibrational
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shift.
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Configurational
...
entropy
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is
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most
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important.
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Below
...
is
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an
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expression
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simplified
...
with
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Stirling's
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approximation.
...
The
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configurational
...
entropy
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can
...
be
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expressed
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as
...
a
...
function
...
of
...
concentration.
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} \[ S_{\mbox{config}} = k_B \ln \left ( \frac{M!}{N! (M-N)!} \right ) \] {latex} {html}<P>< |
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\[ S_{\mbox{config}} = k_B ((M - N) \ln M + N \ln M - N \ln N - (M - N) \ln (M - N) \] |
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\[ S_{\mbox{config}} = -k_B \left ( \frac{N} {M} \ln \frac{N}{M} + \frac{(N-M)} {M} \ln \frac {M-N}{M} \right ) \] |
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\[ S_{\mbox{config}} = -k_B M ( x \ln x + (1 - x) \ln (1 - x)) \] |
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The
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expression
...
above
...
is
...
of
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the
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ideal
...
solid
...
solution
...
configurational
...
entropy.
...
Calculate
...
the
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propert
...
of
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the
...
system
...
with
...
increased
| Latex |
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} \[ T \] {latex} |
.
...
Consider
...
phase
...
diagrams.
...
An
...
expression
...
of
...
Hemholtz
...
free
...
energy
...
is
...
below.
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} \[ F = E - TS \] {latex} {html}<P>< |
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...
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\[ F = E_{G, ST = 0} - T (S_{\mbox{conf}} + S_{\mbox{el}} + S_{\mbox{vibr}} ) \] |
...
The
...
largest
...
impact
...
is
...
from
...
the
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configurational
...
entropy.