# Chemical Potential

### 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:

### 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:

*μ*

_{i}

^{0}(

*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

_{i}

^{0}(T,P) is the chemical potential of pure liquid

*i*at temperature

*T*and pressure

*P*.

*μ*

_{i}

^{0}(

*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

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

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The two graphs above 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 RTln*γ*_{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*_{i}^{0}(*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*_{i}^{0}(*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*_{i}^{0} +*RT*lnγ_{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).

*m*=1 mol/kg has been included in order to make the expressions dimensionless. The standard state chemical potential is

^{0}*μ*

_{i}

^{m}=

*μ*

_{i}

^{0}+

*RT*ln

*M*

_{w}m

^{0}

*γ*

_{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