# The chemical potential

## Names and Meanings

 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 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 dG/dni; meaning the partial derivative of the free enthalpy with respect to the particle sort i and all other variables kept constant. In other words, the "chemical potential m" is a measure of how much the free enthalpy 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 Volume V) constant (dGi = (dG/dni)dni). 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. The particles, however, 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, physically minded people do not feel that this involves chemistry. The "chemical potential", however, is still the 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 be "particle potential" or "particle change energy". But such a name would not be too good either. Because we still have two potentials with the danger of some mix-up: the thermodynamic Potential G, and the "Particle Potential", which is a partial derivative of G. Now, what exactly is a potential? The Gibbs energy G, e.g., may be viewed as a thermodynamic potential because it is a "true" potential. Not only does it satisfy the basic condition that its value is independent of the integration path (i.e. it does not matter how you got there), it is measured in units of energy, and its minima (i.e. dG = 0) denote stable 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. The last condition, however, is not true for the chemical potential. Its minima do not necessarily signify equilibrium; the equilibrium conditions are rather m = 0 or mi = mj = const. if several particles are involved. Let´s try an other 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. 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 = 0. If there are more particles that are coupled by some reaction equation, the left-hand sum oft he chemical potentials (times the number of particles involved) must be equal to the right hand sum. An example: Reaction: SiO2 + 2CO Si + 2CO2 Equilibrium condition: mSiO2 + 2mCO = mSi + 2mCO2 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! dG/dni 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, and write down "chemical potential".

## The Burden of History: Gases and Fugacity

 Now, the good part of the chemical potential is its simplicity after you have dug through the usual thermodynamical calculations. It is especially easy to obtain for (ideal) gases. (Ideal gases obey the reaction p·V = R·T). Let´s go through this quickly, 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. dmi/dp = Vi, or dmi/dT = Si with the proper quantities kept constant and with care as to the use of absolute or molar values. From these equations we obtain mid(p) = m0 + RT · ln(p/p0) with p = pressure. Whenever we see the Gas constant R instead of the Boltzmann constant k, we know that we are dealing with amounts that are taken per mol of a substance instead of per atom. The superscript "0" always refers to the "standard" reference frame, in the case of gases to atmospheric pressure. m0 thus is the chemical potential of a gas at atmospheric pressure. If the gas is non-ideal, i.e. has some kind of interaction between its particles, it will obey some virial equation (any equation replacing p·V = R·T), we obtain for the simplest possible virial equation V = R·T/p + B mnon-id(p) = m0 + RT · ln(p/p0) + B · p 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 by a quantity called fugacity f that gives the right value for mnon-id. Fugacity and pressure are related by the definition f = jp, and j, a dimensionless number, can always be calculated from the virial equation applicable to the situation. In our example we have lnj = (B·p)/(R·T). Now let´s look at a mixture of gases. We 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 moles or particles) 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 = Sipi and pi/p = xi (for ideal gases) The chemical potential of the gas number i with mole fraction xi in a mixture of gases then is mi(mix) = mi(pure) + RT · lnxi 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. Mixing gases increases their chemical potential - it is nothing but the entropy of mixing that enters here.

## Solids and Activities

 Now to solids. 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: There are no ideal solids an analogy to gases, i.e. solids without any interaction between the atoms. 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) by a quantity that contains the deviation from ideality; that quantity is called "activity" a Again, we define the activity ai of component i by ai = fi·xi with fi carrying the burden of non-ideality. In contrast to gases, fi is not all that easily calculated, in fact it is quite hopeless. You may have to resort to an experiment and measure it. Now, in looking at simple vacancies we (unknowingly) already derived a formula for the chemical potential of a vacancy; it read dG/dnV = gF - k·T·ln(N/n) 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 is to reshuffle the ln and we obtain dG/dnV = gF + k·T·ln(n/N). Now this is exactly the formula for mixes of ideal gases or solids if we identify the formation enthalpy gF of a vacancy with its standard chemical potential mV(pure) (see above). Since it is a bit hard to envision a substance consisting exclusively of vacancies, lets look at m(pure) in a slightly different way. (to be continued)

2.4 Point Defects in Ionic Crystals

2.4.2 Notations and their Use

Potential