CfE: Mixture Properties

Mixing: One Phase

We know from experience that:

mixing

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

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

Note:

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

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

Outcome:

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

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

Definition:

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 ...

Definition:

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).

Definition:

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 $v_i$.

Definition:

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.

Outcome:

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