183 The drive of substance flows – particle number density or chemical potential?

Subject In physics and chemistry, we are confronted with “currents” or “flows” of physical quantities of all kinds: electric currents (= currents of the electric charge), mass and volume currents, and also sub- stance currents, or better: currents of the amount of substance, be- cause the flowing quantity here is the amount of substance. Each current can be hindered by a “resistance”. It is then said to be dissi- pative. In this case it needs a “drive” or “driving force”: in the electri- cal case it is an electric potential gradient, the mass flow needs a gravitational potential gradient, a heat flow needs a temperature gradient. For the flow of the amount of substance, the gradient of the particle number density is usually introduced as the drive quanti- ty. The transport itself is then called diffusion. It is said that a sub- stance diffuses from places with a higher particle density to places with a lower particle density.

Deficiencies First of all, a detail: The physical quantity, the density of which is at issue here, is the amount of substance. It is a basic quantity of the SI unit system. If one uses the particle number density instead, it is like using the elementary charge number density instead of the elec- tric charge density. Just as it can sometimes be interesting to look at the swarming electrons, in the case of diffusion it may sometimes be practical to look at the swarming of particles. For most practical questions, however, it is a good idea to operate with the charge density or with the density of the amount of substance, respectively. The sentence one would like to pronounce in connection with diffu- sion would then rather be: The substance diffuses from the higher to the lower density of the amount of substance. Now our actual topic. The quantitative formulation of the statement is Fick’s first law; in modern spelling: ! jn = –D · gradρn (1)

ρn is the molar density (density of the amount of substance n) and ! jn the flow density of the amount of substance. The factor D in front of the gradient is the diffusion constant. For ideal gases, it is inde- pendent of the molar density. In this description of the diffusion, the gradient of the molar density appears as the cause or as the drive of the substance flow. One can see that the equation belongs to a series of several other equations which all play an important role in the thermodynamics of irreversible processes. They describe flows or currents of extensive physical quantities where a resistance has to be overcome, so- called dissipative currents, i.e. currents with entropy production. A well-known example is the expression for the electric current den- ! sity jQ : ! jQ = –σ · gradϕ (2) Here � is the electric potential, and σ the electric conductivity. Equation (1) tells us that the substance current flows from high to low molar density, but equation (2) does not tell us that the electric current flows from high to low charge density. This may sometimes be the case, but only sometimes. As far as the molar current is concerned, in certain cases the molar density can be considered as the driving force, namely whenever the system in which the current flows is homogeneous (apart from the inhomogeneity of the molar density) and when the diffusing sub- stance follows the ideal gas equation. In general, however, the ap- propriate measure for the drive is the chemical potential μ, which is also formally analogous to the other cases. Instead of equation (1) one has then: ! jn = –K · gradµ (3) In this formulation the equation applies always, i.e. not only for ideal gases and homogenous systems (provided of course that there is no other drive, so we are not dealing with coupled flows). In the case of the ideal gas

ρn µ = µ0 + RT ln ρn0 and the factor K in equation (3) is proportional to the mass density: Dρ K = n RT But if D is independent of the mass density, is not equation (1), at least for ideal gases, the simpler, the more beautiful equation? The simpler yes, the more beautiful not. Because if one interprets the equation as it is reasonable, namely that the gradient represents the driving force for the current, then equation (1) makes a statement that does not fit into the picture: For a given drive one would expect that the current is proportional to the density of the “flowing quantity”. This is true in the electrical case (and also in the thermal case). The electric conductivity in equation (2) is known to be proportional to the charge density of the moving charge carriers.

Origin Fick’s first law, Equation (1), was published in 1855, i. e. before Gibbs (1873) introduced the chemical potential. One can see here, as well as in many other places of the of physics syllabus: Once in- troduced, nothing can be changed.

Disposal Introduce the chemical potential, an easy to understand, benign and universally usable quantity. Then Fick’s law can be written in the form of equation (3) and its resemblance with the corresponding electrical law becomes visible. By the way, it very nicely says in Wikipedia: “At a fixed pressure p and a fixed temperature T, the gra- dient of the chemical potential μ is the driving force of the substance flow from the point of view of thermodynamics”.