In thermodynamics, a partial molar property is a 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 cm3. The molar volume of pure water would thus be reported as 18 cm3 mol−1. However, addition of one mole of water to a large volume of pure ethanol results in an increase in volume of only 14 cm3. 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 cm3 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 Z {\displaystyle Z} , one denotes a generic extensive property of a mixture, it will always be true that it depends on the pressure ( P {\displaystyle P} ), temperature ( T {\displaystyle T} ), and the amount of each component of the mixture (measured in moles, n). For a mixture with q components, this is expressed as
Z = Z ( T , P , n 1 , n 2 , ⋯ , n q ) . {\displaystyle Z=Z(T,P,n_{1},n_{2},\cdots ,n_{q}).}
Now if temperature T and pressure P are held constant, Z = Z ( n 1 , n 2 , ⋯ ) {\displaystyle Z=Z(n_{1},n_{2},\cdots )} is a homogeneous function of degree 1, since doubling the quantities of each component in the mixture will double Z {\displaystyle Z} . More generally, for any λ {\displaystyle \lambda } :
Z ( λ n 1 , λ n 2 , ⋯ , λ n q ) = λ Z ( n 1 , n 2 , ⋯ , n q ) . {\displaystyle Z(\lambda n_{1},\lambda n_{2},\cdots ,\lambda n_{q})=\lambda Z(n_{1},n_{2},\cdots ,n_{q}).}
By Euler's first theorem for homogeneous functions, this implies
Z = ∑ i = 1 q n i Z i ¯ , {\displaystyle Z=\sum _{i=1}^{q}n_{i}{\bar {Z_{i}}},}
where Z i ¯ {\displaystyle {\bar {Z_{i}}}} is the partial molar Z {\displaystyle Z} of component i {\displaystyle i} defined as:
Z i ¯ = ( ∂ Z ∂ n i ) T , P , n j ≠ i . {\displaystyle {\bar {Z_{i}}}=\left({\frac {\partial Z}{\partial n_{i}}}\right)_{T,P,n_{j\neq i}}.}
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