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| | Names and Meanings
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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. |
| | 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/¶n
i; meaning the partial derivative of the free enthalpy with respect to the
particle sort i and all other variables kept constant (See a pure thermodynamic script as well). |
|  |
In other words, the "chemical potential m"
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"). |
| 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. |
| | 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 |
| |
|
| | Belowis a detailed derivation for this. |
| 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. mvac=¶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: |
|
| | Reaction | | |
| | | SiO
2 + 2CO | Û
| Si + 2CO2 |
| |
| | Think of a beam balance and you get the drift. |
|
| 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? |
| | 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. |
| 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. |
|
| 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. |
| | 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. |
| 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. |
| | ¶m
i
¶p | =
| Vi | | |
| | ¶mi
¶T | = |
– Si | | |
| |
with the proper quantities kept constant and with care as to the use
of absolute or molar values |
| | 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: |
| |
| mideal gas(p,T) | =
| m0ideal gas
+ RT· ln | p
p0
| | |
| 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. |
| | 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"). |
|
| So get used to the idea of treating point defects like
an ideal gas for a start! |
| What is m0
ideal 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. |
| | 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 p
0=NkT/V0 with N= number of vacancies
at the reference state and V0 volume of the system at the reference
state. |
| | Rewriting the chemical potential of our vacancies for n
gives (in 3 easy steps) |
|
| |
p
p
0 | = | n ·
k · T · V0
N · k · T ·
V | = exp | m
V – mV0
RT | | |
| | Since
the volume of the crystal will not change much no matter at what state you look, we have (V
0/V) » 1. Moreover, in equilibrium we demand mV=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,
mV0 must be the energy of forming one
mol of vacancies and that is simply the formation energy measured in kJ/mol. If you
like electron volts, simply replace R by k. |
| | 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. |
| Now the second point of view. |
| | 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. mi(state x) – mi
(state 0). |
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|
For mi(state 0)
we write mi0 and call it standard potential. |
| 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). |
| | We want the chemical potential mimix
(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. |
|
|
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. |
| | 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=Si pi
and pi/p =xi (for ideal gases). |
| With
that we obtain for the chemical potential mi of
the component i in a mixture of ideal gases |
| | | mi mix(p,T) | =
m ipure(p,T) + RT
· ln | pi
p
| | |
| | With pi=partial pressure of component
i and p= actual pressure=Spi
|
| | 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. |
|
| Gases like to mix! It lowers their chemical potentials and thus their free enthalpy. |
| Now comes a big (and, to the eye of a physicist), somewhat
confusing trick: |
| | We call mipure
(p,T) now the standard state and write
it mi0 which is only the same thing
as our old mi0 as long as p=p
0, 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! |
| Considering this, we obtain
a kind of "master equation" for the chemical
potential of the component i in some mixture of ideal gases: |
| |
| miid(p,T)
| = mi0
+ RT· ln | pi
p |
| |
|
| 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. |
| | 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. |
| Again, what exactly
is mi0 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 |
|
| |
| In other words: mi0=
DG0/Dni or mi0= the increase in enthalpy (or sloppily,
energy) if you add a unit of the particles under consideration to the particles already in
place. |
| | What do the equations mean? If we use the unit "particle", m 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]. |
| 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 |
| |
| mnon-id
(p) | = m0
+ RT · ln | p
p0
| + B· p | |
|
| | 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. |
| However,
to make things easy, chemists like to keep the simple equation
for mid 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
mnon-id results. |
| | Fugacity and pressure thus are necessarily
related and we define |
|
| |
| | The dimensionless numberj can always be calculated
from the virial equation applicable to the situation. In our example we have |
| | |
|
| 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. |
| |
|
| 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). |
| | 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. |
| | 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
" a
i. |
| | Again, we define the activity ai
of component i by |
| | |
| | With ji
now carrying the burden of non-ideality. |
|
|
In contrast to gases, j
i is not all that easily calculated, in fact
it is almost quite hopeless. You may have to resort to an experiment and measure it. |
|
| 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: |
| | |
| 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): |
| | ¶G
¶nV
| = 0 = GF –
kT · ln | N
n
| = | mV | | |
| | 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. |
| | Wehave 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: |
| |
¶G
¶
nV | = mV
= GF + kT · ln |
n
N | = GF
+ kT· ln aV |
|
|
| 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 m0(vacancy)
- and we did that already, too. |
| | 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 10–4) - the vacancies
are far apart and therefore do not interact very much - they do behave
like an ideal gas! |
| | The situation, however, may be completely different for point defects in large concentrations, e.g. impurity atoms or vacancies and interstitials
in ionic crystals. |
|
| 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! |
| | 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. |
|
| |
| | Consider some substance at constant pressure and temperature, but with two possible phases. |
|
|
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. |
| | 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 N
2 = N – N1 in phase 2. How large
is N1? |
| 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: |
| |
| 1. |
dG(p, T, N1, N1)
| = | ¶G
¶T | ·
dT + | ¶G
¶p | · dp
+ | ¶G
¶N1 | · dN1
+ | ¶G
¶N2 | · dN2
|
| |
|
|
Since we look at a situation
with constant pressure and temperature, we have that dT = 0 = dp. |
| | For
equilibrium, we demand dG = 0. From equ. (2) we get |
| | |
| | Substituting that in equ. (1) yields |
| |
¶G
¶N1 | · dN1 –
| ¶G
¶N2 | · dN
1 | = | dG =
0 |
¶G
¶N
1 | · dN1 | = |
¶G
¶N2 | · dN1 |
| |
| |
q.e.d. |
|
| What happens if m(N
1) > m(N2); i.e.
if we have non-equilibrium conditions with m(N1),
the chemical potential of the particles in phase 1being
larger than in phase 2? |
| | We now must change the particle numbers in the phases
until equilibrium is achieved. |
| | So do we have to increase N1
(at the same time decreasing N2) or should it go the other way around? |
| |
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) |
| |
| dG | =
| ¶G
¶N1 | · dN1
– | ¶G
¶N2 |
· dN1 |
| dG | = | m(N1) | · dN1
– | m(N2) |
· dN1 | |
|
| |
For positive dN1, we will have dG> 0 since m(N1) > m(N2)
. This necessarily leads to the general conclusion: |
|
|
dN1 must be < 0
if the system is to move towards equilibrium. |
| 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 m(N1)=m(N2). |
|
|
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. |
| | 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. |
| Following up this line of thought leads straight to the law of mass action, which will be dealt with in another module. |
| |
© H. Föll (Defects - Script)
2.1.1
Simple Vacancies and Interstitials 
With frame