In
chemistry, an ideal solution or ideal mixture is a
solution
Solution may refer to:
* Solution (chemistry), a mixture where one substance is dissolved in another
* Solution (equation), in mathematics
** Numerical solution, in numerical analysis, approximate solutions within specified error bounds
* Soluti ...
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 a ...
es. The
enthalpy of mixing
In thermodynamics, the enthalpy of mixing (also heat of mixing and excess enthalpy) is the enthalpy liberated or absorbed from a substance upon mixing. When a substance or compound is combined with any other substance or compound, the enthalpy o ...
is zero as is the volume change on mixing by definition; the closer to zero the enthalpy of mixing is, the more "ideal" the behavior of the solution becomes. The
vapor pressure
Vapor pressure (or vapour pressure in English-speaking countries other than the US; see spelling differences) or equilibrium vapor pressure is defined as the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phas ...
s of the solvent and solute obey
Raoult's law and
Henry's law
In physical chemistry, Henry's law is a gas law that states that the amount of dissolved gas in a liquid is directly proportional to its partial pressure above the liquid. The proportionality factor is called Henry's law constant. It was formulate ...
, respectively, and the
activity coefficient
In thermodynamics, an activity coefficient is a factor used to account for deviation of a mixture of chemical substances from ideal behaviour. In an ideal mixture, the microscopic interactions between each pair of chemical species are the same ( ...
(which measures deviation from ideality) is equal to one for each component.
The concept of an ideal solution is fundamental to
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 measurem ...
and its applications, such as the explanation of
colligative properties
In chemistry, colligative properties are those properties of solutions that depend on the ratio of the number of solute particles to the number of solvent particles in a solution, and not on the nature of the chemical species present. The number ...
.
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
Interaction is action that occurs between two or more objects, with broad use in philosophy and the sciences. It may refer to:
Science
* Interaction hypothesis, a theory of second language acquisition
* Interaction (statistics)
* Interactions o ...
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 + U
BB 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
1-Butanol, also known as butan-1-ol or ''n''-butanol, is a primary alcohol with the chemical formula C4H9OH and a linear structure. Isomers of 1-butanol are isobutanol, butan-2-ol and ''tert''-butanol. The unmodified term butanol usually refers ...
and
2-butanol
2-Butanol, or ''sec''-butanol, is an organic compound with formula C H3CH( OH)CH2CH3. Its structural isomers are 1-butanol. isobutanol, and ''tert''-butanol. 2-Butanol is chiral and thus can be obtained as either of two stereoisomers designat ...
, 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 for all compositions. Here
is the
vapor pressure
Vapor pressure (or vapour pressure in English-speaking countries other than the US; see spelling differences) or equilibrium vapor pressure is defined as the pressure exerted by a vapor in thermodynamic equilibrium with its condensed phas ...
of component
above the solution,
is its
mole fraction
In chemistry, the mole fraction or molar fraction (''xi'' or ) is defined as unit of the amount of a constituent (expressed in moles), ''ni'', divided by the total amount of all constituents in a mixture (also expressed in moles), ''n''tot. This ex ...
and
is the vapor pressure of the pure substance
at the same temperature.
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 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 potential of a species ...
μ (which is the
partial molar Gibbs energy
In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature and pre ...
''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
In chemical thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the mechanical partial pressure in an accurate computation of the chemical equilibrium constant. It is equal to the pressure of an ideal gas whic ...
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 variable
Physical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes. According to IUPAC, an intensive quantity is one ...
s and changing
to
, with optional overbar, standing for
partial molar volume
In thermodynamics, a partial molar property is a quantity which describes the variation of an extensive property of a solution or mixture with changes in the molar composition of the mixture at constant temperature and pressure. It is the p ...
:
:
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
In thermodynamics, the entropy of mixing is the increase in the total entropy when several initially separate systems of different composition, each in a thermodynamic state of internal equilibrium, are mixed without chemical reaction by the therm ...
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; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature and ...
on mixing is determined solely by the
entropy of mixing
In thermodynamics, the entropy of mixing is the increase in the total entropy when several initially separate systems of different composition, each in a thermodynamic state of internal equilibrium, are mixed without chemical reaction by the therm ...
. 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