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This module is registered in the
"advanced" part, despite the fact that the chemical potential belongs
to basic thermodynamics. The reason is that people with a mostly physical
background (like me) may often have learned exciting things like
Bose-Einstein condensations and the Liouville theorem in their thermodynamics
courses, but not overly much about chemical potentials and chemical
equilibrium. |
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First we will address, somewhat
whimsically, a certain problem related to the name "Chemical potential". It is, in the view
of many (including professors and students), a slightly unfortunate name for
the quantity ∂G/∂ni; meaning the partial
derivative of the free enthalpy with respect to the particle sort
i and all other variables kept constant. |
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In other words, the "chemical
potential μ" is a measure of how much the
free enthalpy (or the free
energy) of a system changes (by dGi) if you add or
remove a number dni particles of the particle species
i while keeping the number of the other particles (and the temperature
T and the pressure p) constant: |
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Since particle numbers are pure
numbers free of dimensions, the unit of the chemical
potential is that of an energy, which justifies the name
somewhat. |
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However, the particles considered in
the context of general thermodynamics do not have to be only atoms or molecules
(i.e. the objects of chemistry). They can be electrons, holes, or anything else
that can be identified and numbered. In considering e.g., the equilibrium
between electrons and holes in semiconductors, physically minded people do not
feel that this involves chemistry. Moreover, they feel since electrons and
holes are Fermions, classical thermodynamics as expressed in the chemical
potential or the mass actions law, might not be the right way to go at it. The
"chemical potential" of the electrons, however, is still a major
parameter of the system (to the annoyance of the solid state physicists - they
therefore usually call it "Fermi
energy"). |
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A better name, perhaps, would help.
How about "particle potential"? But such a name would not be too good
either. Because now there is the danger of mixing-up the thermodynamic Potential G of the particles,
and the "Particle Potential",
which is a partial derivative of G (not to mention the common
electrostatic or gravitational potential). Now,
what exactly is a potential?
Use the link to refresh your memory! |
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The Gibbs energy G, e.g., may be
viewed as a thermodynamic potential because it really is a "true"
potential. Not only does it satisfy the basic conditions that its value is
independent of the integration path (i.e. it does not matter how you got
there), but it is also measured in units of energy and its minima (i.e.
dG = 0) denote stable (or metastable) equilibrium. |
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The chemical potential meets the
first two criteria, albeit the second one only barely. This is so because if
you define it relative to the particle concentration and not the number (which
would be equally valid), you end up with an energy density and not an energy.
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The last condition, however, is not true for the chemical potential. Its minima
do not necessarily signify equilibrium; the equilibrium conditions if several
particles are involved are rather |
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Below is a
detailed derivation for this. |
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Lets try a different approach. In a
formal way, the particle numbers are general
coordinates of the free enthalpy for the system under consideration.
Since the partial derivatives of thermodynamic potentials with respect to the
generalized coordinates can be viewed as generalized forces (in direct and
meaningful analogy to the gravitational potential), the chemical potentials
could just as well be seen as chemical
forces. |
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The equilibrium conditions are
then immediately clear: The sum of the forces must be zero. If there is only
one particle in the system (e.g. vacancies in a crystal), equilibrium exists if
there is no "chemical force", i.e. μvac = ∂G/∂nV = 0. If there are more
particles that are coupled by some reaction equation, the left-hand sum of the
chemical potentials (times the number of particles involved) must be equal to
the right hand sum. An example: |
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Reaction |
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SiO2 + 2CO |
↔ |
Si + 2CO2 |
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Think of a beam balance and you get the
drift. |
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This suggests yet another name:
"Particle force" or "Particle change force".
Of course, now we would have a force being measured in terms of energy - not
too nice either, but maybe something has to give? |
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Unfortunately, there is another drawback. If we
look at currents (electrical or otherwise), i.e. at non-equilibrium conditions, the driving forces for
currents very generally can be identified with the gradients of the chemical
potentials (which still may be defined even under global non-equilibrium as long as we have local equilibrium). Now we would have a force being
the derivative of a force - and that is not too clear either. In this context a
potential would be a much better name. |
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So - forget
it! ∂G/∂ni is called, and will be
called "chemical potential of the particle sort
i". But by now, you know what it means. Still, if you feel
uncomfortable with the name "Chemical Potential" in the context of
looking at non-chemical stuff, e.g. the behavior of electrons, use your own
name while thinking about it, keep in mind what it means, but do write down
"chemical potential". |
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The good part about the chemical
potential is its simplicity - after you have dug through the usual
thermodynamical calculations. It is especially easy to obtain for (ideal)
gases. |
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An ideal gas is a system of particles of any kind whatsoever that obeys the equation
p·V = N·R·T with N
= Number of mols in the system; or p·V =
n·k·T with n = Number of particles in
the system. |
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Lets go through this quickly (haha), because we
are not really interested in gases, but only want to remember the nomenclature
and the way to go at it. |
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From regular thermodynamics we get a
lot of relations between the partial derivatives of state functions and
therefore also for the chemical potential, e.g. |
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with the proper
quantities kept constant and with care as to the use of absolute or molar
values |
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From these equations we obtain
for the chemical potential of a pure ideal gas, i.e. a system consisting only
of one kind of component - a bunch of O2 molecules in a
container, or a bunch of vacancies in a
crystal: |
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μideal gas(p,T) |
= |
μ0ideal gas +
RT · ln |
p
p0 |
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Now wait a
minute! In the case of vacancies, we seem to have two components - the vacancies and the crystal, not
to mention that considering vacancies as an ideal gas seems to be stretching
the concept a bit. |
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Well - yes, there is the crystal, but for the
real gas there is the vacuum in which the particles move. As long as the
"container" of the ideal gas particles does not do anything, we may
ignore it (if we don't, math will do it for us as as soon as we write down
equations like the mass action law or
others that tell us what happen inside the "container"). |
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So get used to the idea
of treating point defects like an ideal gas for a start! |
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What is μ0ideal gas? It is called
something like "the standard chemical potential
for the pure phase". Lets look at what it means from two points of view. |
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First, if we
stay with the vacancy example, i.e. we consider an ideal gas of vacancies, the
pressure is given by pV = n · kT with
n = number of vacancies in the crystal, or p = n
· kT/V. Likewise, p0, the
pressure at some reference state, can be written as p0 =
NkT/V0 with N = number of
vacancies at the reference state and V0 volume of the
system at the reference state. |
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Rewriting the chemical potential of our vacancies
for n gives (in 3 easy steps) |
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p
p0 |
= |
n · k · T · V0
N · k · T · V |
= exp |
μV
μV0
RT |
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Since the volume of the crystal will not change
much no matter at what state you look, we have
(V0/V) » 1.
Moreover, in equilibrium we demand μV = 0. This leaves us with |
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And this looks very familiar! If we chose the
standard state to be N = number of atoms of the crystal = number
of sites for vacancies, μV0 must be the energy of
forming one mol of vacancies which is simply the formation energy measured in
kJ/mol. If you like electron volts, simply replace R by
k. |
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In other words, the standard reference state is
very important, but also a bit trivial. You can chose whatever you like, but
there are smart choices and not so smart choices. Best to stick with the
conventions - they usually are smart choices and you can use the numbers given
in books and tables without conversion to some other system. |
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Now the
second point of view. |
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Since the chemical potential is an energy (with
many properties very similar to the better known gravitational or electrostatic
potential energy), there is no unique choice of its zero point. All hat counts
are changes, i.e. μi(state x)
μi(state 0). |
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For μi(state 0) we write μi0 and call it standard
potential. |
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So far so good. But
what about the chemical potential of some stuff (always particles) in a
mixture with other particles? To start
easy, lets take a mixture of ideal gases - O2 with
N2, vacancies and interstitials (both uncharged, so there is
negligible interaction). |
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We want the chemical potential
μimix(p,T) of the
component i in a mixture of ideal gases as a function of the temperature and
the (total) pressure. We first need the quantities "mole fraction" and "partial pressure" to describe a mixture. |
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The mole
fraction xi is simply the amount of phase
i (measured in mols or particle numbers) divided by the sum of
the amounts of all phases. |
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The partial
pressure pi of gas
number i in a mixture of gases is simply the pressure that gas number i
would have if you take all the other gases away and let it occupy the available
volume. It follows that the total pressure p = Σi pi and
pi/p = xi (for ideal
gases). |
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With that we obtain for the chemical
potential μi of the component
i in a mixture of ideal gases |
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μimix(p,T)
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= μipure(p,T) +
RT · ln |
pi
p |
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With pi = partial
pressure of component i and p = actual pressure =
Σpi |
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In words: The
chemical potential of gas number i in a mixture of gases at a
certain temperature T and pressure p is equal to
the chemical potential of this gas in the pure phase at p and
T plus RT·
lnxi. But note that
xi < 1 for all cases and thus RT
· lnxi < 0. |
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Gases like to mix! It lowers their chemical
potentials and thus their free enthalpy. |
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Now comes a big (and, to the eye of a physicist), somewhat
confusing trick: |
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We call μipure(p,T) now
the standard state and write it μi0 which is only the same thing
as our old μi0 as long
as p = p0, or, in the vacancy example above,
N = N0 = Lohschmidts number (= number of particles in a
mol). Again, you are free in your choices oft standard states - use it
wisely! |
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Considering this,
we obtain a kind of "master
equation" for the chemical potential of the component
i in some mixture of ideal gases: |
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μiid(p,T)
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= μi0 +
RT · ln |
pi
p |
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The ln term simply contains
the entropy of mixing; otherwise, when we mix two gases, we would only add up
the enthalpy/energy contained in the two pure components before the
mixing. |
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This is one way of writing down the
chemical potential for a mixture of gases.
Again note that whenever we see the
Gas constant R instead of the Boltzmann constant k, you
know that you are dealing with amounts that are taken per mol of a substance instead of per particle.
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Again, what exactly is μi0 now? Nothing but the
reference for the energy scale, but nevertheless a quantity of prime
importance, called the "standard potential of
component i" ( the superscript
"0" always refers to the "standard"
reference frame; in the case of gases mostly to atmospheric pressure and room
temperature). It is also called standard reaction enthalpy and
gives the change in the total free enthalpy at standard conditions if you wiggle the
concentration of particle i a bit via |
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In other words:
μi0 = ΔG0/Δni or μi0 = the increase in enthalpy
(or sloppily, energy) if you add a unit of the particles under consideration to
the particles already in place. |
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What do the equations mean? If
we use the unit "particle", μ
0 is exactly the amount of free enthalpy needed to add (or
subtract) one particle; usually given in [eV/particle] which is
[eV]. If we use the unit "mol", it is the free enthalpy
needed to add (or subtract) one mol, usually given in [kJ/mol]. |
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So far we have
considered rather straight-forward thermodynamics; the difficulties arise if we
use the concept of the chemical potential for non-ideal gases, for liquids and solids, for
mixtures gases liquids and solids, or, as we do, for things like vacancies
which are not usually described in those terms anyway. The first step is to
consider non-ideal gases: |
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If the gas is non-ideal, which means
that it has some kind of interaction between its particles, it will obey some
virial equation (any equation replacing
p·V = N·RT). The simplest possible virial
equation is V = R·T/p + B and for this
we obtain |
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μnon-id(p) |
= μ0 +
RT · ln |
p
p0 |
+ B · p |
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For any other virial equation we can
derive the corresponding formula for the chemical potential of that particular
non-ideal gas. It will always have some extra terms containing the
pressure. |
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However, to make things easy,
chemists like to keep the simple equation
for μid even in the case of
non-ideal gases by substituting the real pressure
p by a quantity called fugacity f chosen in such a way that the correct
value for μnon-id results. |
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Fugacity and pressure thus are
necessarily related and we define |
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The dimensionless numberφ can always be calculated from the virial equation
applicable to the situation. In our example we have |
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As long as we look at gases, there is
no problem. Fugacity is a well defined concept, even if needs getting used to.
The next step, however, is a bit more problematic. |
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Now we will turn to solids (and in
one fell swoop we also include liquids in this). The good news is that the
equation for a mix of ideal gases is equally valid for a mix of ideal
condensed phases, i.e. ideal solids. The
bad news is: An ideal solid in analogy to gases, i.e. without any interaction
between the atoms, is an
oxymoron (i.e. a contradiction
in itself). |
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What then are ideal solids supposed
to be? Since we need interactions between the atoms or molecules, we must mean
something different from gases. What is meant by "ideal" in this
cases is that the interactions between the constituents of the solid are the
same, regardless of their nature. |
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Now that is
certainly not a good approximation for most solids. So we use the same trick as
in gases, we replace the mole fraction (which is a concentration)
xi of the component i by a quantity that
contains the deviation from ideality; that quantity is called "activity" ai. |
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Again, we define the activity
ai of component i by |
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With φi now carrying the burden of
non-ideality. |
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In contrast to gases, φi is not all that easily calculated, in fact it is almost
quite hopeless. You may have to resort to an experiment and measure it. |
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In any case, if we
use activities instead of concentrations or fugacities (which we treat as
special case of activities), we are totally general and obtain for the chemical
potentials of whatever component in any mixture: |
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Now, in looking at simple vacancies
we already had the formula for the chemical
potential of a vacancy; it read (if you put the various equations given in the
link together): |
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∂G
∂nV |
= 0 = GF
kT · ln |
N
n |
= |
μV |
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with n/N =
nV, the equilibrium concentration of vacancies which we
now also may call aV, the activity of vacancies, if we
want to be totally general. |
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We have
k instead of R, so we must
be considering energies per particle and not per mol - which we did. We
therefore do not have a mol fraction but a particle number fraction; but this
is identical, anyway. All we have to do to get the activity is to reshuffle the
ln: |
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∂G
∂nV |
= μV =
GF + kT · ln |
n
N |
= GF + kT · ln
aV |
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Now this is exactly the formula for
an ideal gas or solid if we identify the formation enthalpy
GF of a vacancy with its standard chemical potential
μ0(vacancy) - and we
did that already, too. |
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Replacing the concentration
n/N of the vacancies with the activity of the vacancies is
fine - but fortunately, for vacancy concentrations in elemental crystals, there
is no difference between concentration and activity, because vacancy
concentrations are always small (below 104) - the
vacancies are far apart and therefore do not interact very much -
they do behave like an ideal gas! |
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The situation, however, may be
completely different for point defects in large
concentrations, e.g. impurity atoms or vacancies and interstitials
in ionic crystals. |
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The latter case is special because the
concentration of intrinsic point defects may depend on
the stoichiometry and on impurities: If there is e.g. a trace of
Ca++ in a NaCl crystal, there must be a corresponding
concentration of Na - vacancies to maintain charge neutrality and this
concentration can not only be much larger than the maximum concentration in
thermal equilibrium for "perfect" crystals, it will also be constant,
i.e. independent of the temperature! |
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How to use the chemical potentials and activities
in this context is described in a series of modules in the "backbone II" section of
chapter 2. Here we will only give one example - equilibrium between
phases. |
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Consider some
substance at constant pressure and temperature, but with two possible phases. |
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An everyday example is water in
contact with ice, or any binary substance with a given composition (e.g.
Pb and Sn - solder) at some point at its phase diagram where two
phases coexist (consult the module "phase diagrams"), for that matter. |
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How many particles will be contained
in phase 1 and how many in phase 2? Given N
particles altogether, we will have N1 particles in
phase 1 and N2 = N
N1 in phase 2. How large is
N1? |
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Lets look at the free enthalpy of the
substance, or better yet, at its change with the particle numbers. In full
generality, we have two equations: |
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1. |
dG(p, T, N1, N1)
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= |
∂G
∂T |
· dT + |
∂G
∂p |
· dp + |
∂G
∂N1 |
· dN1 + |
∂G
∂N2 |
· dN2 |
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Since we look at a
situation with constant pressure and
temperature, ∂G/∂T =
∂G/∂p = 0. |
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For equilibrium, we demand
dG = 0. Moreover, from equ. (2) we get |
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Substituting that in equ. (1)
yields |
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∂G
∂N1 |
· dN1 |
∂G
∂N2 |
· dN1 |
= |
dG = 0 |
∂G
∂N1 |
· dN1 |
= |
∂G
∂N2 |
· dN1 |
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q.e.d. |
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What happens if μ(N1) > μ(N2); i.e. if we have
non-equilibrium conditions with μ(N1), the
chemical potential of the particles in phase 1 being larger
than in phase 2? |
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We now must change the particle
numbers in the phases until equilibrium is achieved. |
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So do we have to increase
N1 (at the same time decreasing
N2) or should it go the other way around? |
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Well, whatever we do,
it must decrease G, so
dG must be negative if we change the particle numbers the right
way. For dG we had (a few lines above) |
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dG |
= |
∂G
∂N1 |
· dN1 |
∂G
∂N2 |
· dN1 |
dG |
= |
μ(N1) |
· dN1 |
μ(N2) |
· dN1 |
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For positive
dN1, we will have dG > 0 since
μ(N1) > μ(N2). This necessarily leads to the
general conclusion: |
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dN1 must
be < 0 if the system is to move towards equilibrium. |
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In
words this means: The phase with the larger chemical potential will have to to
shrink and the phase with the smaller chemical potential will grow until
equilibrium is achieved and μ(N1) = μ(N2). |
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This is a very general
truth. Electrons, e.g., move from the phase with the higher chemical
potential (than called Fermi energy)
to the phase with the lower one. |
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We can also turn it around: Vacancies in
supersaturation will tend to move to vacancy agglomerates and increase their
size. It follows that the chemical potential of supersaturated single vacancies
must be larger than that of vacancies in an agglomerate. |
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Following up this line of thought
leads straight to the law of mass
action, which will be dealt with in another module. |
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© H. Föll