An ideal solution or ideal mixture is a
solution that exhibits thermodynamic properties analogous to those of a mixture of
ideal gas
An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is ...
es.
[ The enthalpy of mixing is zero as is the volume change on mixing.][ The ]vapor pressure
Vapor pressure or equilibrium vapor pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases (solid or liquid) at a given temperature in a closed system. The equilibrium vapor pressure is an indicat ...
s of all components obey Raoult's law
Raoult's law ( law) is a relation of physical chemistry, with implications in thermodynamics. Proposed by French chemist François-Marie Raoult in 1887, it states that the partial pressure of each component of an ideal mixture of ''liquids'' is ...
across the entire range of concentrations,[ and the activity coefficient (which measures deviation from ideality) is equal to one for each component.
The concept of an ideal solution is fundamental to both ]thermodynamics
Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
and chemical thermodynamics
Chemical thermodynamics is the study of the interrelation of heat and work with chemical reactions or with physical changes of state within the confines of the laws of thermodynamics. Chemical thermodynamics involves not only laboratory measure ...
and their applications, such as the explanation 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 (''U''AB) and like neighbors ''U''AA and ''U''BB must be of the same average strength, i.e., 2 ''U''AB = ''U''AA + 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., ''U''AB = ''U''AA = ''U''BB, 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 only 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 obeys Raoult's law
Raoult's law ( law) is a relation of physical chemistry, with implications in thermodynamics. Proposed by French chemist François-Marie Raoult in 1887, it states that the partial pressure of each component of an ideal mixture of ''liquids'' is ...
for all compositions. Here is the vapor pressure
Vapor pressure or equilibrium vapor pressure is the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phases (solid or liquid) at a given temperature in a closed system. The equilibrium vapor pressure is an indicat ...
of component above the solution, is its mole fraction
In chemistry, the mole fraction or molar fraction, also called mole proportion or molar proportion, is a quantity defined as the ratio between the amount of a constituent substance, ''ni'' (expressed in unit of moles, symbol mol), and the to ...
and is the vapor pressure of the pure substance at the same temperature.[P. Atkins and J. de Paula, ''Atkins’ Physical Chemistry'' (8th edn, W.H.Freeman 2006), p.144]
This definition depends on vapor pressure, which is a directly measurable property, at least for volatile components. The thermodynamic properties may then be obtained from the chemical potential
In thermodynamics, the chemical potential of a Chemical specie, species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potent ...
μ (which is the partial molar Gibbs energy ''g'') of each component. If the vapor is an ideal gas,
:
The reference pressure may be taken as = 1 bar, or as the pressure of the mix, whichever is simpler.
On substituting the value of 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 . Here is the fugacity of component in solution and is the fugacity of 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.[P.A. Rock, ''Chemical Thermodynamics: Principles and Applications'' (Macmillan 1969), p.261]
Thermodynamic properties
Volume
If we differentiate this last equation with respect to at constant we get:
:
Since we know from the Gibbs potential equation that:
:
with the molar volume , these last two equations put together give:
:
Since all this, done as a pure substance, is valid in an ideal mix just adding the subscript to all the intensive variables and changing to , with optional overbar, standing for partial molar volume:
:
Applying the first equation of this section to this last equation we find:
:
which means that the partial molar volumes in an ideal mix are independent of composition. Consequently, the total volume is the sum of the volumes of the components in their pure forms:
:
Enthalpy and heat capacity
Proceeding in a similar way but taking the derivative with respect to we get a similar result for molar enthalpies:
:
Remembering that we get:
:
which in turn means that and that the enthalpy of the mix is equal to the sum of its component enthalpies.
Since and , similarly
:
It is also easily verifiable that
:
Entropy of mixing
Finally since
:
we find that
:
Since the Gibbs free energy per mole of the mixture is
then
:
At last we can calculate the molar entropy of mixing since
and
:
:
Consequences
Solvent–solute interactions are the same as solute–solute and solvent–solvent interactions, on average. Consequently, the enthalpy of mixing (solution) is zero and the change in Gibbs free energy
In thermodynamics, the Gibbs free energy (or Gibbs energy as the recommended name; symbol is a thermodynamic potential that can be used to calculate the maximum amount of Work (thermodynamics), work, other than Work (thermodynamics)#Pressure–v ...
on mixing is determined solely by the entropy of mixing. Hence the molar Gibbs free energy of mixing is
:
or for a two-component ideal solution
:
where m denotes molar, i.e., change in Gibbs free energy per mole of solution, and is the mole fraction of component . Note that this free energy of mixing is always negative (since each