Partial molar property
A partial molar property is a thermodynamic 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 partial derivative of the extensive property with respect to the amount (number of moles) of the component of interest. Every extensive property of a mixture has a corresponding partial molar property. Definition. The partial molar volume is broadly understood as the contribution that a component of a mixture makes to the overall volume of the solution. However, there is  more to it than this: When one mole of water is added to a large volume of water at 25 °C, the volume increases by 18 cm. The molar volume of pure water would thus be reported as 18 cm mol. However, addition of one mole of water to a large volume of pure ethanol results in an increase in volume of only 14 cm. The reason that the increase is different is that the volume occupied by a given number of water molecules depends upon the identity of the surrounding molecules. The value 14 cm is said to  be the partial molar volume of water in ethanol. In general, the partial molar volume of a substance X in a mixture is the change in volume per mole of X added to the mixture. The partial molar volumes of the components of a mixture vary with the composition of the mixture, because the environment of the molecules in the mixture changes with the composition. It is the changing molecular environment (and the consequent alteration of the interactions between molecules) that results in the thermodynamic properties of a mixture changing as its composition is altered If, by formula_1, one denotes  a generic extensive property of a mixture, it will always be true that it depends on the pressure (formula_2), temperature (formula_3), and the amount of each component of the mixture (measured in moles, "n"). For a mixture with "q" components, this is expressed as Now if temperature "T" and pressure "P" are held constant, formula_5 is a homogeneous function of degree 1, since doubling the quantities of each component in the mixture will double formula_1. More generally, for any formula_7: By Euler's first theorem for homogeneous functions, this implies where formula_10 is the partial molar formula_1 of component formula_12  defined as: By Euler's second theorem for homogeneous functions, formula_10 is a homogeneous function of degree 0 which means that for any formula_15: In particular, taking formula_17 where formula_18, one has where formula_20 is the concentration expressed as the mole fraction of component formula_12. Since the molar fractions satisfy the relation the "x" are not independent, and the partial molar property is a function of only formula_23 mole fractions: The partial molar property is thus an intensive property - it does not depend on the size of the system. The partial volume is not the partial molar volume. Applications. Partial  molar properties are useful because chemical mixtures are often maintained at constant temperature and pressure and under these conditions, the value of any extensive property can be obtained from its partial molar property. They are especially useful when considering specific properties of pure substances (that is, properties of one mole of pure substance) and properties of mixing (such as the heat of mixing or entropy of mixing). By definition, properties of mixing are related to those of the pure substances by: Here formula_26 denotes a pure substance, formula_27 the mixing property, and formula_28 corresponds to the specific property under consideration . From the definition of partial molar properties, substitution yields: So from knowledge of the partial molar properties, deviation of properties of mixing from single components can be calculated. Relationship to thermodynamic potentials. Partial molar properties satisfy relations analogous to those of the extensive properties. For the internal energy "U", enthalpy "H", Helmholtz free energy "A", and Gibbs free energy "G", the following hold: where formula_2 is the pressure, formula_35 the volume, formula_3 the temperature, and formula_37 the entropy. Differential form of the thermodynamic potentials. The thermodynamic potentials also satisfy where formula_42 is the chemical potential defined as (for constant n  with j≠i): This last partial derivative is the same as formula_44, the partial molar Gibbs free energy. This means that the partial molar Gibbs free energy and the chemical potential, one of the most important properties in thermodynamics and chemistry, are the same quantity. Under isobaric (constant "P") and isothermal (constant "T ") conditions, knowledge of the chemical potentials, formula_45, yields every property of the mixture as they completely determine the Gibbs free energy. Measuring partial molar properties. To measure the partial molar property formula_46 of a binary solution, one begins with the pure component denoted as formula_47 and, keeping the  temperature and pressure constant during the entire process, add small quantities of component formula_48; measuring formula_1 after each addition. After sampling the compositions of interest one can fit a curve to the experimental data. This function will be formula_50. Differentiating with respect to formula_51 will give formula_46. formula_53 is then obtained from the relation: Relation to apparent molar quantities. The relation between partial molar properties and the apparent ones can be derived from the definition of the apparent quantities and of the molality. The relation holds also for multicomponent mixtures, just that in this case subscript i is required. Partial