# How to calculate the potential of dissolved substances

The chemical potentialμ characterizes the possibilities of a substance,

A reaction, transformation or redistribution can only take place voluntarily if the chemical potential in the initial state is greater than in the final state.

A reaction runs by itself if the sum of the μ-Values ​​of the starting materials is greater than those of the end materials; A substance changes its crystal structure, its state of aggregation, its degree of association etc. only when the μ-Value in the initial form is greater than in the final form; a substance only moves in one direction in which the μValue decreases (provided there is no other drive).

The chemical potential is therefore suitable for describing all types of material conversions, including reactions in which photons, phonons, electrons or defect electrons are involved.

It is defined by the fundamental Gibbs equation of internal energyU: T = absolute temperature

S. = Entropy

p = Pressure

V. = Volume

ni = Amount of substance of the system component i

The size μ, which we call chemical potential today, was introduced by Josiah Willard Gibbs (1839-1903). He called it the potential of a substance; occasionally also, in order to avoid confusion with other potentials, intrinsic potential.

### Alternative formulations

From Gibb's fundamental equation it follows that μ from the function U(V., S., ni) can be calculated: Alternatively, μ also from Legendre-transformed by U(V., S., ni) to calculate:   Here is H = U + pV the enthalpy, F. = UTS the free energy and G = U + pVTS the free enthalpy.

The index indicates the quantities to be kept constant. The nj are the substance quantities of all system components except ni

### Equilibrium conditions

In chemical equilibrium, the chemical potential of the educts is the same as that of the products, similar to the thermal equilibrium between two systems whose temperatures are the same or in electrical equilibrium the electrical potentials.

With a suitable process management, the condition for chemical equilibrium can also be formulated as an extremal principle.

Keep constant of Size whose value becomes minimal
V. and S.U
p and S.H
V. and TF.
p and TG

Since pressure and temperature are often kept constant in chemical reactions, the minimum principle is free enthalpy G the most commonly used criterion. The more general equilibrium criterion, however, is the disappearance of the chemical potential difference, because it applies regardless of which other quantities are kept constant; it still holds true if none of the other variables remain constant during the process of equilibrium adjustment.

### interpretation

μi gives the dependence of the thermodynamic energy quantities U, H, F. and G from the composition of the system.

Each potential difference describes the system's ability to do work. Chemical reactions therefore have a driving force ("voluntariness of running") until an equilibrium is established in which the chemical potentials of all substances equalize (the sum of the chemical potentials multiplied by the stoichiometric coefficients results in zero, or in other words: the affinity becomes Zero).

For pure substances (one-component systems) the chemical potential is constant and equal to the molar Gibbs energy. The molar Gibbs energies for the formation of many substances under standard conditions are tabulated and allow the calculation of chemical potential differences in reaction mixtures. Given p, T, V., S. and nj = 1 k This enables the calculation of the heat of reaction and the direction of the reaction.

For crystalline solids, the chemical potential of the electrons corresponds to T = 0 K of the Fermi energy.

For quasiparticles, μ = 0, since the particle number is not obtained for these.

### Chemical potential values

The values ​​of the chemical potential are for standard conditions (T = 298.15 K; p = 101.325 kPa), see below under "Weblinks". If the chemical potential for a certain state is known (e.g. under standard conditions), it can be calculated in a linear approximation for pressures and temperatures in the vicinity of this state:
μ(T) = μ(T0) + α(TT0)
or.
μ(p) = μ(p0) + β(pp0)
Here is the temperature coefficient and the pressure coefficient.
With the help of the Maxwell relations and it follows that the temperature coefficient is equal to the negative molar entropy and the pressure coefficient is equal to the molar volume.

### Concentration dependence

For the practical definition of the concentration dependency, the chemical potential is split into a concentration-independent and a concentration-dependent summand. The concentration-dependent term is defined using the thermodynamic activity of component i: μi° chemical potential under standard pressure, which is still dependent on the temperature. (Standard conditions) Activity of the substance in relation to the standard activity

(The curly brackets are to be understood here as a mathematical operator that means: "Divide ai by the standard activity. ")

The activity can often be calculated with sufficient accuracy through the concentration ci (for dissolved substances) or the partial pressure pi (for gases). If the system under consideration is an electrolyte, the Debye-Hückel theory provides an activity coefficient with the help of which the concentration can be converted into the activity.

### literature

• The Scientific Papers of J. Willard Gibbs, I (1873-1878), Dover Publications, New York 1961
• G. Job and F. Herrmann, Chemical Potential - a quantity in search of recognition, Eur. J. Phys. 27 (2006) 353 - 371