In this video we'll describe the properties of the equation of state that's used to relate
the pressure, temperature, and volume properties of real gas.
Instead of an ideal gas, now we account for the fact that the interaction between molecules
and at higher pressures, molecules occupy some volume.
So the simplest is the Van der Waals equation of state.
This is the case where now we can write this Z. Z is the compressibility factor and relate
it to the pressure, volume, gas constant, and temperature.
And so I've written this terms now of density.
So this is volume per mole, this is density, so the inverse of volume per mole.
The other way of writing the equation, and we have two parameters, a and b here, the
other is to write it in terms of volume where the parameters a and b are both greater than
zero.
So this accounts for the fact that molecules occupy some volume.
So ideal gas law assumes that individual molecules have no volume, but as we get to high pressures,
this assumption is not as good and so this accounts for the molecules having volume.
Potentially the volume if we were to close pack the molecules.
And this accounts for attractive interactions between molecules.
So as molecules get closer together, so this can be higher pressures, lower temperatures,
but as molecules get closer together, the attractive interactions are more important
as the molecules are closer together and the end result is that the pressure will be lower
than predicted by the ideal gas law.
So it's easiest to look at an example.
Let's look at carbon dioxide and we're going to compress it to two-tenths of a liter.
So volume is 0.2 L and pressure is 112 bar to get to that volume if we're ideal gas.
And the pressure from the Van der Waals equation is only 53 bar says Van der Waals equation
of state.
So a rather dramatic difference.
The pressure is smaller because of attractive interactions between molecules.
However, if we continue to decrease the volume, so again carbon dioxide now volume of 0.05
liters per mole, again pressure is 448 bar if it were an ideal gas.
The pressure is actually now much higher from the Van der Waals equation because now, we're
getting the volume small enough that this b term up here becomes important.
The molecules occupy volume and to compress it this much takes a much higher pressure.
So instead of the attractive interactions here, it's the attractive interactions a term,
if we like, and here it's the volume of the molecule, so the b term.
And these a and b terms depend on the molecule and there are correlations to calculate these
from reduced temperatures and reduced pressures using critical pressures and temperatures.
Now this equation of state is cubic, so the Van der Waals equation of state is cubic which
means, and this is cubic in volume, at any given pressure, we have three solutions for
volume at a given pressure.
Well, it turns out that in a number of cases, only one is a real solution, and others, all
three are real solutions.
So we look at on a pressure-volume diagram, this is just a dashed line to roughly indicate
the phase envelope where saturated liquid here and saturated vapor here.
The shape is just an approximate one.
Well the actual behavior would be the high pressures and low volumes where liquid type
behavior for an isotherm.
So we'd have phase change at constant pressure and then we'd have the lower pressure gas
out here.
As we lower the pressure, the volume increases.
Well we're trying to model this isotherm with this cubic equation of state, so we can fit
reasonably well, this area, and we can also reasonably well fit out here.
But to have just one equation, we can't model that horizontal line.
The cubic equation of state would look something like this.
Not drawn very well.
And probably more realistic.
And so we see we have three solutions.
Well this would be the liquid volume, the vapor volume in equilibrium, and this would
be one that does not physically mean anything, so this is not a physical solution, the one
in between.
So regions up here above the critical point, we're only going to have one real solution.
Here, we'd have three real solutions from the isotherm.
So now instead of ideal gas law, by using cubic equations of state, we can model liquid
phase and vapor phase regions and indeed vapor-liquid equilibrium.
And then there are more complicated equations of state.
And so I've just listed the Peng-Robinson equation of state in the density form and
the volume form.
Again, you can see there are two parameters if we look at the volume form a and b and
the Peng-Robinson equation is also a cubic equation of state so it's gonna have three
roots, similar behavior to the Van der Waals equation of state.