Ideal solution
Encyclopedia
In chemistry, an ideal solution or ideal mixture is a solution
with thermodynamic properties analogous to those of a mixture of ideal gas
es. The enthalpy of solution (or "enthalpy of mixing") is zero as is the volume change on mixing; the closer to zero the enthalpy of solution is, the more "ideal" the behavior of the solution becomes. The vapour pressure of the solution obeys Raoult's law, and the activity coefficient
s (which measure deviation from ideality) are equal to one.
The concept of an ideal solution is fundamental to chemical thermodynamics
and its applications, such as the use of colligative properties
.
, with the important difference that intermolecular interactions in liquids are strong and can not 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, 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.
for all compositions. Here 
is the vapor pressure
of component i above the solution,
is its mole fraction and 
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
may be taken as 
= 1 bar, or as the pressure of the mix to ease operations.
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.
at 
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
to all the intensive variables and
changing
to 
, 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.
we get to a similar result with enthalpies
derivative with respect to T and remembering that
we get:
which in turn is
.
Meaning that the enthalpy of the mix is equal to the sum of its components.
Since
and 
:
It is also easily verifiable that
Which means that
and since
then
At last we can calculate the entropy of mixing
since
and 
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
is the mole fraction of component 
.
Note that this free energy of mixing is always negative (since each
is positive and each 
must be negative) i.e. ideal solutions are always completely miscible.
The equation above can be expressed in terms of chemical potential
s of the individual components
where
is the change in chemical potential of 
on mixing.
If the chemical potential of pure liquid
is denoted 
, then the chemical potential of 
in an ideal solution is
Any component
of an ideal solution obeys Raoult's Law over the entire composition range:
where
is the equilibrium vapor pressure
of the pure component
is the mole fraction of the component in solution
It can also be shown that volumes are strictly additive for ideal solutions.
s or activity coefficient
s. 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.
Solution
In chemistry, a solution is a homogeneous mixture composed of only one phase. In such a mixture, a solute is dissolved in another substance, known as a solvent. The solvent does the dissolving.- Types of solutions :...
with thermodynamic properties analogous to those of a mixture of ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...
es. The enthalpy of solution (or "enthalpy of mixing") is zero as is the volume change on mixing; the closer to zero the enthalpy of solution is, the more "ideal" the behavior of the solution becomes. The vapour pressure of the solution obeys Raoult's law, and the activity coefficient
Activity coefficient
An activity coefficient is a factor used in thermodynamics to account for deviations from ideal behaviour in a mixture of chemical substances. In an ideal mixture, the interactions between each pair of chemical species are the same and, as a result, properties of the mixtures can be expressed...
s (which measure deviation from ideality) are equal to one.
The concept of an ideal solution is fundamental to chemical thermodynamics
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...
and its applications, such as the use of colligative properties
Colligative properties
Colligative properties are properties of solutions that depend on the number of molecules in a given volume of solvent and not on the properties/identity of the molecules. Colligative properties include: relative lowering of vapor pressure; elevation of boiling point; depression of freezing point...
.
Physical origin
Ideality of solutions is analogous to ideality for gasesIdeal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...
, with the important difference that intermolecular interactions in liquids are strong and can not simply be neglected as they can for ideal gases. Instead we assume that the mean strength of the interactions
Intermolecular force
Intermolecular forces are forces of attraction or repulsion which act between neighboring particles: atoms, molecules or ions. They are weak compared to the intramolecular forces, the forces which keep a molecule together...
are the same between all the molecules of the solution.
More formally, for a mix of molecules of A and B, 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
N-Butanol
n-Butanol or n-butyl alcohol or normal butanol is a primary alcohol with a 4-carbon structure and the molecular formula C4H9OH. Its isomers include isobutanol, 2-butanol, and tert-butanol...
and 2-butanol
2-Butanol
2-Butanol, or sec-butanol, is an organic compound with formula CH3CHCH2CH3. This secondary alcohol is a flammable, colorless liquid that is soluble in 12 parts water and completely miscible with polar organic solvents such as ethers and other alcohols. It is produced on a large scale, primarily as...
, 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 for all compositions. Here is the vapor pressureVapor pressure
Vapor pressure or equilibrium vapor pressure is the pressure of a vapor in thermodynamic equilibrium with its condensed phases in a closed system. All liquids have a tendency to evaporate, and some solids can sublimate into a gaseous form...
of component i above the solution,
is its mole fraction and 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
Chemical potential
Chemical potential, symbolized by μ, is a measure first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. It is the potential that a substance has to produce in order to alter a system...
μ (or partial molar Gibbs energy g) of each component, which is assumed to be given by the ideal gas formula.
,The reference pressure
may be taken as = 1 bar, or as the pressure of the mix to ease operations.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 fugacityFugacity
In chemical thermodynamics, the fugacity of a real gas is an effective pressure which replaces the true mechanical pressure in accurate chemical equilibrium calculations. It is equal to the pressure of an ideal gas which has the same chemical potential as the real gas. For example, nitrogen gas ...
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
Activity (chemistry)
In chemical thermodynamics, activity is a measure of the “effective concentration” of a species in a mixture, meaning that the species' chemical potential depends on the activity of a real solution in the same way that it would depend on concentration for an ideal solution.By convention, activity...
instead of fugacity.
Volume
If we differentiate this last equation with respect to at 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
to all the intensive variables andchanging
to , 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 we get to a similar result with enthalpiesEnthalpy
Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...
derivative with respect to T and remembering that
we get:which in turn is
.Meaning that the enthalpy of the mix is equal to the sum of its components.
Since
and :It is also easily verifiable that
Entropy of mixing
Finally sinceWhich means that
and since
then
At last we can calculate the entropy of mixing
Entropy of mixing
In thermodynamics the entropy of mixing is the increase in the total entropy of a compound system, when different and chemically non-reacting chemical substances or material components are mixed by removing partition between the system's initially separate volumes...
since
and Consequences
Solvent-Solute interactions are similar to solute-solute and solvent-solvent interactionsSince the enthalpy of mixing (solution) is zero, the change in Gibbs free energy
Gibbs free energy
In thermodynamics, the Gibbs free energy is a thermodynamic potential that measures the "useful" or process-initiating work obtainable from a thermodynamic system at a constant temperature and pressure...
on mixing is determined solely by the entropy of mixing
Entropy of mixing
In thermodynamics the entropy of mixing is the increase in the total entropy of a compound system, when different and chemically non-reacting chemical substances or material components are mixed by removing partition between the system's initially separate volumes...
. 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
is the mole fraction of component .Note that this free energy of mixing is always negative (since each
is positive and each must be negative) i.e. ideal solutions are always completely miscible.The equation above can be expressed in terms of chemical potential
Chemical potential
Chemical potential, symbolized by μ, is a measure first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. It is the potential that a substance has to produce in order to alter a system...
s of the individual components
where
is the change in chemical potential of on mixing.If the chemical potential of pure liquid
is denoted , then the chemical potential of in an ideal solution isAny component
of an ideal solution obeys Raoult's Law over the entire composition range:where
is the equilibrium vapor pressureVapor pressure
Vapor pressure or equilibrium vapor pressure is the pressure of a vapor in thermodynamic equilibrium with its condensed phases in a closed system. All liquids have a tendency to evaporate, and some solids can sublimate into a gaseous form...
of the pure component
is the mole fraction of the component in solutionIt 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 functionMargules function
A Margules function is a function added to the Raoult's law description of a liquid solution to account for deviations from ideality. See also Margules activity model.The amended Raoult's law description of the vapor pressure above the solution becomes:...
s or activity coefficient
Activity coefficient
An activity coefficient is a factor used in thermodynamics to account for deviations from ideal behaviour in a mixture of chemical substances. In an ideal mixture, the interactions between each pair of chemical species are the same and, as a result, properties of the mixtures can be expressed...
s. 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
Regular solution
In chemistry, a regular solution is a solution that diverges from the behavior of an ideal solution only moderately. Its entropy of mixing is equal to that of an ideal solution with the same composition, due to random mixing without strong specific interactions...
.
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.
See also
- Activity coefficientActivity coefficientAn activity coefficient is a factor used in thermodynamics to account for deviations from ideal behaviour in a mixture of chemical substances. In an ideal mixture, the interactions between each pair of chemical species are the same and, as a result, properties of the mixtures can be expressed...
- Entropy of mixingEntropy of mixingIn thermodynamics the entropy of mixing is the increase in the total entropy of a compound system, when different and chemically non-reacting chemical substances or material components are mixed by removing partition between the system's initially separate volumes...
- Margules functionMargules functionA Margules function is a function added to the Raoult's law description of a liquid solution to account for deviations from ideality. See also Margules activity model.The amended Raoult's law description of the vapor pressure above the solution becomes:...
- Regular solutionRegular solutionIn chemistry, a regular solution is a solution that diverges from the behavior of an ideal solution only moderately. Its entropy of mixing is equal to that of an ideal solution with the same composition, due to random mixing without strong specific interactions...
- Coil-globule transitionCoil-globule transitionIn polymer physics, the Coil-globule transition is the collapse of a macromolecule from an expanded coil state through an ideal coil state to a collapsed globule state, or vice-versa. The coil-globule transition is of importance in biology due to the presence of coil-globule transitions in...
- Apparent molar property
- Dilution equation