Chemical Potential

The chemical potential of salt solutions is the basis for phase equilibrium calculations for solutions with salts. This includes vapor-liquid equilibrium, solid-liquid equilibrium, and liquid-liquid equilibrium.

The chemical potential of a substance i is the partial molar derivative of the free energy G, but can also be derived from the enthalpy H, the Helmholtz energy A, or the internal energy U of substance i:

Matter flows spontaneously from a region of high chemical potential to a region of low chemical potential just like electric current flows from a region of high electric potential to a region of low electric potential and mass flows from a position of high gravitational potential to a position of low gravitational potential. The chemical potential can therefore be used to determine whether or not a system is in equilibrium. When the system is in equilibrium, the chemical potential of each substance will be the same in all the phases appearing in the system.

The ideal solution

The ideal solution can be defined as a solution in which the chemical potential of each species is given by the expression:

In this expression, μ_i⁰(T,P) is the chemical potential of pure species i in the same state of aggregation as the solution i.e. in a liquid mixture, μ_i⁰(T,P) is the chemical potential of pure liquid i at temperature T and pressure P. μ_i⁰(T,P) is referred to as the standard state chemical potential. From the expression above, it is seen that the chemical potential μ_i of a species in an ideal solution is lower than the chemical potential of the pure component: the mole fraction is less than one and the second term is therefore negative. Alternative expressions are used for the chemical potential of salt solutions. Instead of using the chemical potential of pure species i, the chemical potential of species i in the aqueous state is used.

Colligative properties

Colligative properties are solution properties that have their origin in the fact that the chemical potential of a species in an ideal solution is lower than the chemical potential of the pure component. In ideal solutions all solutes cause identical colligative properties. No solution is completely ideal. Colligative properties are therefore dependent on the identity and the concentration of the solute. Many text books on Physical Chemistry claim mistakenly that colligative properties per definition are independent of the identity of the solute. This is only true for very dilute solutions. The colligative properties are

the vapor pressure lowering/boiling point elevation
the freezing point depression
the osmotic pressure

In an aqueous solution of salt or of another solute, the chemical potential of water is lower than the chemical potential of pure water. The chemical potentials of pure ice and pure water vapor however, are independent of the composition of the solution from which it was formed. At equilibrium water will be in the state that has the lowest chemical potential. Therefore, an aqueous solution has a lower freezing point and a higher boiling point (lower vapor pressure) than pure water.

The two graphs to the right show the freezing points and boiling points of various salt solutions. It is clear that the freezing points as well as the boiling points are dependent on the type of salt and should therefore not be labeled as “colligative properties”. From the graphs it can also be seen that the two sulfates shown have the opposite deviation from ideality as the chlorides. Notice also that the graph with freezing points extends to much higher concentrations than the diagram with bubble points. “Moles of solutes” is here identical to “moles of ions”. The calculation of these two graphs is explained in “Electrolyte Solutions: Thermodynamics, Crystallization, Separation methods”: https://doi.org/10.11581/dtu:00000073 by Kaj Thomsen.

Non-ideal solutions

Usually, only very dilute solutions can be considered ideal. When calculating the chemical potential of species i, a term taking the deviation from ideality into account is therefore added. This term is called an excess term, and can be either positive or negative. The term is usually written RTlnγ_i, γ_i is called the activity coefficient of component i. The activity coefficients are calculated with an activity coefficient model such as the Extended UNIQUAC model.

The complete expression for the chemical potential of species i in a non-ideal solution is:

As mentioned above, in this expression, μ_i⁰(T,P) is the chemical potential of pure species i in the same state of aggregation as the solution. For pure species i, x_i is one, and γ_i consequently must be one too.
For aqueous solutions of salts μ_i⁰(T,P) represents the chemical potential of pure ions. This chemical potential can not be measured experimentally. Instead of using this hypothetical standard state, the activity coefficients of the ions are often normalized by introducing the unsymmetric activity coefficient:

γ_i^∞ is the activity coefficient of species i at infinite dilution. It therefore follows that at infinite dilution the unsymmetric activity coefficient, γ_i*, is equal to one. If the chemical potential of species i is expressed in terms of the unsymmetric activity coefficient, we obtain the expression:

The standard state chemical potential μ_i* =μ_i⁰ +RTlnγ_i^∞ has the advantage that it can be measured experimentally.

The molality concentration scale

“The molality m_i of solute i is the amount of solute i pr kg solvent. If the solvent is water (index w), the following relation between mole fraction and molality of solute i can be derived:”.

Using this relation, the chemical potential of ion i can be expressed as a function of the molality and the molal activity coefficient γ_i^m

M_w is the molar mass of water (kg/mol). m⁰=1 mol/kg has been included in order to make the expressions dimensionless. The standard state chemical potential is μ_i^m = μ_i⁰ +RTlnM_wm⁰γ_i^∞ when the molality concentration scale is used. The molal activity coefficient is related to the unsymmetrical mole fraction activity coefficient by: γ_i^m = γ_i* x_w, where x_w is the mole fraction of water.

Further reading about chemical potentials and other properties of aqueous electrolytes: “Electrolyte Solutions: Thermodynamics, Crystallization, Separation methods”: https://doi.org/10.11581/dtu:00000073

Chemical Potential