I'm going to describe an interactive simulation that is available on the Wolfram demonstration
project site.
And it uses the van der Waals equation of state to generate isotherms on a PV diagram
and generate isobars on a TV diagram.
On the left we plot the log of the pressure versus the log of the volume.
On the right we are plotting temperature versus the log of the volume.
So the blue line on the left and on the right corresponds to the van der Waals equation
of state, including the dashed blue line.
The dashed blue line, however, does not correspond to physical reality and instead, if we look
in the left side, we can see I've circled the purple circles, the point where we draw
a horizontal line at constant pressure, constant temperature, that connects the liquid phase
and the vapor phase that are in equilibrium at this temperature.
And so the physical representation in this two phase region the way it corresponds to
any point along this dashed line, this horizontal dashed line, is some fraction of liquid and
some fraction of vapor.
The location of this horizontal line is found by setting the fugacities of the liquid and
the vapor phases equal.
It could also be determined on a linear plot by setting this green area above the line
equal to the green area below the line.
But notice they are not equal because this is a log plot but on a linear plot those areas
would be equal.
Now the right side shows the corresponding temperature volume diagram at an isobar.
Now we can't say areas are going to be equal, but again the fugacities will be equal to
determine the horizontal line.
And on the right side we can change the pressure and change the location of this isobar.
And on the left side we can change the temperature and therefore look at a different isotherm.
So we will go ahead and look at the interactive simulation to get a little better understanding
of these pressure volume and temperature volume diagrams.
Remember this is for a single component that we are modeling by a cubic equation of state.
So here's the interactive simulation and looking at the pressure volume diagram.
And as I change the temperature, of course the isotherm moves up.
Vapor liquid equilibrium moves to higher pressure.
Again these green areas on a linear plot of pressure versus volume would be the same size
area.
And if I increase the temperature enough now that's at the critical temperature and then
we move higher up we are above the critical temperature we are in a super-critical region
up at the higher pressures and we are in a vapor region at the lower.
And so now all the points on the blue line correspond to physical meaningful values.
Whereas, of course, in the two phase region we draw this horizontal line that connects.
And as we lower the temperature we lower the saturation pressure, equilibrium pressure,
where the horizontal line is located.
And then on a temperature volume diagram now we are looking at an isobar and as we raise
the pressure of course we go to higher temperature, higher saturation temperature and sufficiently
high temperature again we have a graph that represents all the points corresponds to physical
realities.
So you can use this to get a little better understanding of how the cubic equation of
states in this case the van der Waals equation of state can model the real states of a single
component.