Ideal solution

In chemistry, an ideal solution or ideal mixture is a solution in which the gas phase exhibits thermodynamic properties analogous to those of a mixture of ideal gases. The enthalpy of mixing is zero as is the volume change on mixing by definition; the closer to zero the enthalpy of mixing is, the more "ideal" the behaviour of the solution becomes. The vapor pressure of the solution obeys either Raoult's law or Henry's law (or both), and the activity coefficient of each component (which measures deviation from ideality) is equal to one.
The concept of an ideal solution is fundamental to chemical thermodynamics and its applications, such as the use of colligative properties.
Physical origin.
Ideality of solutions is analogous to ideality for gases, with the important difference that intermolecular interactions in liquids are strong and cannot simply be neglected as they can for ideal gases. Instead we assume that the mean strength of the interactions are the same between all the molecules of the solution.
More formally, for a mix of molecules of A and B, then the interactions between unlike neighbors (UAB) and like neighbors UAA and UBB must be of the same average strength, i.e., 2 UAB = UAA + UBB and the longer-range interactions must be nil (or at least indistinguishable). If the molecular forces are the same between AA, AB and BB, i.e., UAB = UAA = UBB, then the solution is automatically ideal.
If the molecules are almost identical chemically, e.g., 1-butanol and 2-butanol, then the solution will be almost ideal. Since the interaction energies between A and B are almost equal, it follows that there is a very small overall energy (enthalpy) change when the substances are mixed. The more dissimilar the nature of A and B, the more strongly the solution is expected to deviate from ideality.
Formal definition.
Different related definitions of an ideal solution have been proposed. The simplest definition is that an ideal solution is a solution for which each component (i) obeys Raoult's law formula_1 for all compositions. Here formula_2 is the vapor pressure of component i above the solution, formula_3 is its mole fraction and formula_4 is the vapor pressure of the pure substance i at the same temperature.
This definition depends on vapor pressures which are a directly measurable property, at least for volatile components. The thermodynamic properties may then be obtained from the chemical potential μ (or partial molar Gibbs energy g) of each component, which is assumed to be given by the ideal gas formula 
The reference pressure formula_6 may be taken as formula_7 = 1 bar, or as the pressure of the mix to ease operations.
On substituting the value of formula_2 from Raoult's law, 
This equation for the chemical potential can be used as an alternate definition for an ideal solution.
However, the vapor above the solution may not actually behave as a mixture of ideal gases. Some authors therefore define an ideal solution as one for which each component obeys the fugacity analogue of Raoult's law formula_10, 
Here formula_11 is the fugacity of component formula_12 in solution and formula_13 is the fugacity of formula_12 as a pure substance. Since the fugacity is defined by the equation 
this definition leads to ideal values of the chemical potential and other thermodynamic properties even when the component vapors above the solution are not ideal gases. An equivalent statement uses thermodynamic activity instead of fugacity.
Thermodynamic properties.
Volume.
If we differentiate this last equation with respect to formula_16 at formula_17 constant we get:
but we know from the Gibbs potential equation that:
These last two equations put together give:
Since all this, done as a pure substance is valid in a mix just adding the subscript formula_12 to all the intensive variables and 
changing formula_22 to formula_23, standing for Partial molar volume.
Applying the first equation of this section to this last equation we get
which means that in an ideal mix the volume is the addition of the volumes of its components:
Enthalpy and heat capacity.
Proceeding in a similar way but derivative with respect of formula_17 we get to a similar result with enthalpies
derivative with respect to T and remembering that formula_29 we get:
which in turn is formula_31.
Meaning that the enthalpy of the mix is equal to the sum of its components. 
Since formula_32 and formula_33:
It is also easily verifiable that 
Entropy of mixing.
Finally since 
Which means that 
and since the Gibbs free energy per mole of the mixture formula_38 is
formula_39 
then 
At last we can calculate the molar entropy of mixing since 
formula_41 and formula_42
Consequences.
Solvent-solute interactions are similar to solute-solute and solvent-solvent interactions
Since the enthalpy of mixing (solution) is zero, the change in Gibbs free energy on mixing is determined solely by the entropy of mixing. Hence the molar Gibbs free energy of mixing is
or for a two component solution 
where m denotes molar, i.e., change in Gibbs free energy per mole of solution, and formula_3 is the mole fraction of component formula_12.
Note that this free energy of mixing is always negative (since each formula_49, each formula_50 or its limit for formula_51 must be negative (infinite)), i.e., "ideal solutions are always completely miscible". 
The equation above can be expressed in terms of chemical potentials of the individual components
where formula_53 is the change in chemical potential of formula_12 on mixing.
If the chemical potential of pure liquid formula_12 is denoted formula_56, then the chemical potential of formula_12 in an ideal solution is
Any component formula_12 of an ideal solution obeys Raoult's Law over the entire composition range:
where
It can also be shown that volumes are strictly additive for ideal solutions.
Non-ideality.
Deviations from ideality can be described by the use of Margules functions or activity coefficients. A single Margules parameter may be sufficient to describe the properties of the solution if the deviations from ideality are modest; such solutions are termed "regular". 
In contrast to ideal solutions, where volumes are strictly additive and mixing is always complete, the volume of a non-ideal solution is not, in general, the simple sum of the volumes of the component pure liquids and solubility is not guaranteed over the whole composition range. By measurement of densities thermodynamic activity of components can be determined.