text: 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 , one denotes a generic extensive property of a mixture, it will always be true that it depends on the pressure (), temperature (), 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,  is a homogeneous function of degree 1, since doubling the quantities of each component in the mixture will double . More generally, for any :
  By Euler's first theorem for homogeneous functions, this implies
  where  is the partial molar  of component  defined as:
  By Euler's second theorem for homogeneous functions,  is a homogeneous function of degree 0 (i.e., is an intensive property) which means that for any :
  In particular, taking  where , one has
  where  is the concentration expressed as the mole fraction of component .
  Since the molar fractions satisfy the relation
  the x are not independent, and the partial molar property is a function of only  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.