In thermodynamics, an apparent molar property of a solution component in a mixture or solution is a quantity defined with the purpose of isolating the contribution of each component to the non-ideality of the mixture. It shows the change in the corresponding solution property (for example, volume) per mole of that component added, when all of that component is added to the solution. It is described as apparent because it appears to represent the molar property of that component in solution, provided that the properties of the other solution components are assumed to remain constant during the addition. However this assumption is often not justified, since the values of apparent molar properties of a component may be quite different from its molar properties in the pure state.
For instance, the volume of a solution containing two components identified as solvent and solute is given by
where is the volume of the pure solvent before adding the solute and its molar volume (at the same temperature and pressure as the solution), is the number of moles of solvent, is the apparent molar volume of the solute, and is the number of moles of the solute in the solution. By dividing this relation to the molar amount of one component a relation between the apparent molar property of a component and the mixing ratio of components can be obtained.
This equation serves as the definition of . The first term is equal to the volume of the same quantity of solvent with no solute, and the second term is the change of volume on addition of the solute. may then be considered as the molar volume of the solute if it is assumed that the molar volume of the solvent is unchanged by the addition of solute. However this assumption must often be considered unrealistic as shown in the examples below, so that is described only as an apparent value.
An apparent molar quantity can be similarly defined for the component identified as solvent . Some authors have reported apparent molar volumes of both (liquid) components of the same solution. This procedure can be extended to ternary and multicomponent mixtures.
Apparent quantities can also be expressed using mass instead of number of moles. This expression produces apparent specific quantities, like the apparent specific volume.
where the specific quantities are denoted with small letters.
Apparent (molar) properties are not constants (even at a given temperature), but are functions of the composition. At infinite dilution, an apparent molar property and the corresponding partial molar property become equal.
Some apparent molar properties that are commonly used are apparent molar enthalpy, apparent molar heat capacity, and apparent molar volume.
Relation to molality
The apparent (molal) volume of a solute can be expressed as a function of the molality b of that solute (and of the densities of the solution and solvent). The volume of solution per mole of solute is
Subtracting the volume of pure solvent per mole of solute gives the apparent molal volume:
For more solutes the above equality is modified with the mean molar mass of the solutes as if they were a single solute with molality bT:
,
The sum of products molalities – apparent molar volumes of solutes in their binary solutions equals the product between the sum of molalities of solutes and apparent molar volume in ternary of multicomponent solution mentioned above.
,
Relation to mixing ratio
A relation between the apparent molar of a component of a mixture and molar mixing ratio can be obtained by dividing the definition relation
to the number of moles of one component. This gives the following relation:
Relation to partial (molar) quantities
Note the contrasting definitions between partial molar quantity and apparent molar quantity: in the case of partial molar volumes , defined by partial derivatives
, one can write , and so always holds. In contrast, in the definition of apparent molar volume, the molar volume of the pure solvent, , is used instead, which can be written as
, for comparison. In other words, we assume that the volume of the solvent does not change, and we use the partial molar volume where the number of moles of the solute is exactly zero ("the molar volume"). Thus, in the defining expression for apparent molar volume ,
, the term is attributed to the pure solvent, while the "leftover" excess volume, , is considered to originate from the solute. At high dilution with , we have , and so the apparent molar volume and partial molar volume of the solute also converge: .