The chemical potential of a substance i is the partial molar derivative of the free energy G, 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 can be defined as a solution in which the chemical potential of each species is given by the expression:
In this expression, μi0(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 μi0(T,P) is the chemical potential of pure liquid i at temperature T and pressure P.μi0(T,P) is referred to as the standard state chemical potential. From the expression above, it is seen that the chemical potential 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.
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. The colligative properties are
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 figures 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 figures it can also be seen that the two sulfates shown have the opposite deviation from ideality as the chlorides. Notice also that the diagram with freezing points extends to much higher concentrations than the diagram with bubble points. "Moles of solutes" is here identical to "moles of ions"
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 RTlnγi, γi is called the activity coefficient of component i. The complete expression for the chemical potential of species i is:
As mentioned above, in this expression, μi0(T,P)
is the chemical potential of pure species i in the same state of
aggregation as the solution. For pure species i, xi is one, and γi
consequently must be one too.
For aqueous solutions of salts μi0(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:
γi∞ is the activity coefficient of species i at infinite dilution. It therefore follows that at infinite dilution the unsymmetric activity coefficient, γi*, 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 μi* =μi0 +RTlnγi∞ has the advantage that it can be measured experimentally.
The molality mi 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 γim
Mw 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 μim = μi0 +RTlnMwm0γi∞ when the molality concentration scale is used. The molal activity coefficient is related to the unsymmetrical mole fraction activity coefficient by: γim = γi* xw, where xw is the mole fraction of water.