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: .
Quantitatively, the relation between partial molar properties and the apparent ones can be derived from the definition of the apparent quantities and of the molality.  For volume,