We know from experience that:

In other words, since the interactions between molecules of A-A and B-B differs from the interaction between A-B, we get:

$V = V_{mixture} \ne V_A + V_B$

but we know that

$n_{mixture} = n_A + n_B$

Combining these, we can show that the *intensive*
property $v$ gives

$v = v_{mixture} \ne x_Av_A + x_Bv_B$

This is actually quite general. That is, typically the mixture
properties (either extensive or intensive) are **not**
simply the sum (or weighted average) of the individual pure
component properties.

Recall that, by the state postulate, we can write (note that we will now drop the subscript "mixture"):

$v = F(P, T)$

We know that this works for a single phase pure component (we have a common example, $Pv=RT$). We can also write

$V = F (P, T, n)$

which we also have experience with ($PV = nRT$). It turns out
that for multiple components, the **Duhem relation**
tells us that we can write

$V = F (P, T, n_1, n_2, n_3, ... , n_m)$

Therefore, following the same procedure that we used for fundamental property relations, we can write

$dV = \left (\frac{\partial V}{\partial P} \right )_{T, n_{i}} dP + \left (\frac{\partial V}{\partial T} \right )_{P, n_{i}} dT + \sum_j \left (\frac{\partial V}{\partial n_j} \right )_{P, T, n_{i\ne j}} dn_j$

Write exact differentials for extensive properties in terms of m+2 independent variables for mixtures of m species

since at equilibrium we have $dP = dT = 0$, we can reduce this to

$dV = \sum_j \left (\frac{\partial V}{\partial n_j} \right )_{P, T, n_{i\ne j}} dn_j$

We define the **partial molar properties** as
being

$\bar V_j = \left (\frac{\partial V}{\partial n_j} \right
)_{P, T, n_{i\ne j}}$

which we can think of as the contribution of species $j$ to the
mixture property.

We can rewrite our expression for $dV$ then as

$dV = \sum_j \bar V_j dn_j$

Integrating this (and noting that the constant of integration is zero -- we will simply accept this for now):

$V = \sum_j \bar V_j n_j$

dividing by $n_{tot}$ we can also write:

$v = \sum_j \bar V_j x_j$

Just to be sure that we have our nomenclature right ...

We define the **total solution properties** as
being the mixture properties which we will denote $v_{mixture}$ or
simple $v$ (with specific volume as an example).

We define the **pure component properties** as
being the properties of the individual components when they are
*not* in a mixture and will denote it as $v_i$.

We define the **partial molar properties** as
being

$\bar V_j = \left (\frac{\partial V}{\partial n_j} \right
)_{P, T, n_{i\ne j}}$

which we can think of as the contribution of species $j$ to the
mixture property.

Define and explain the difference between the terms: pure species property, total solution property, and partial molar property