# EoS: Equations of State for Mixtures

### Equations of State for Mixtures

When dealing with mixtures the material-dependent constants require mixing rules due to the fact that the potential number of combinations is infinite.

#### Virial Equation for Mixtures

The most straight-forward, and theoretically sound, mixing rules apply to the virial equation.

• $B$ depends on two-body interactions
• two-body interactions ($i-j$) depend on $y_i$ and $y_j$
• $B_{mix} = \sum_i\sum_j y_iy_jB_{ij}$
• $C$ depends on three body interactions
• three-body interactions ($i-j-k$) depend on $y_i$, $y_j$, and $y_k$
• $C_{mix} = \sum_i\sum_j\sum_k y_iy_jy_kC_{ijk}$

#### Cubic EOS for Mixtures

While not as theoretically sound the terms in the cubic equations of state have the following rough meanings:

• $b$ relates to the excluded volume due to molecule size
• $a$ relates to intermolecular interactions (assumed to be two-body interactions, for simplicity)

therefore:

$a_{mix} = \sum_i\sum_j y_iy_ja_{ij}$

or

$a_{mix} = \sum_i\sum_j y_iy_j[a\alpha(T)]_{ij}$

where

$a_{ij} = \sqrt{a_ia_j}$

or (for some EOS)

$a_{ij} = \sqrt{a_ia_j}(1-k_{12})$

The excluded volume is simpler (and more "correct"), given by:

$b_{mix} = \sum_i y_ib_i$

#### Corresponding States for Mixtures

There is no theoretically sound way to adapt the concept of corresponding states. The most commonly used is Kay's Rules, which defines "pseudo-critical" properties:

$T_{r_{mix}} = \frac{T}{T_{pc}}$, for example, where

$T_{pc} = \sum_i y_iT_{c,i}$

$P_{pc} = \sum_i y_iP_{c,i}$

$\omega_{p} = \sum_i y_i\omega_i$

##### Outcome

Write the van der Waals mixing rules and explain their functionality in terms of molecular interactions

##### Outcome

Write the mixing rules for the virial coefficients and for pseudo-critical properties using Kay's rule

##### Outcome

Using mixing rules to solve for P, v, and T of mixtures

##### Test Yourself

A gas mixture contains 20.0 mole % CH4, 30.0% C2H6, and the balance C2H4. Ten kilograms of this gas is compressed to 200 bar at 90C. What is the volume?