### Intermolecular Potentials and Equations of State

We are now ready to build EoS based on our understanding on
intermolecular interactions.

##### Note

Intermolecular interactions are *at least* pairwise,
therefore, as $v$ decreases binary interactions are the first ones
to become important

The number of pairwise interactions scales with the number of
molecules as follows:

# of pairwise interactions **per** molecule =
$\left ( N-1 \right )$

total # of pairwise interactions = # of pairwise interactions
**per** molecule X # of molecules

total # of pairwise interactions = $\left ( N-1 \right )xN
\approx N^2$

##### Note

Since the number of pairwise interactions *increases*
with decreasing $v$ -- total # of pairwise interactions $\propto
\frac{1}{v^2}$

Revisiting our friend the van der Waals equation, we can then
write:

$P = \frac{RT}{v-b} - \frac{a}{v^2}$

##### Question?

Why do we "correct" P? Why is that "correction" negative?

We could also write this as:

$v^3 - \left (\frac{RT+Pb}{P}\right )v^2 + \left
(\frac{a}{P}\right )v - \frac{ab}{P} = 0$

##### Outcome

Explain how the terms in the van der Waals equation relax these
assumptions

##### Outcome

Describe how cubic equations of state account for attractive and
repulsive interactions