In this video we will look at how the Gibbs Free Energy depends on pressure and temperature
because that also gives us insight on how the fugacity depends on these variables.
Start with the definition of the Gibbs Free Energy.
And then we are interested in the differential form.
So dH- d(TS).
And so then we will go back to the first law for a closed system, and reversible, so the
change in internal energy is the heat added, write that reversibly, and the work added,
where work is positive if we add energy to the system.
If we add energy we do work on the system.
And then we can write the reversible heat transfer in terms of entropy change and the
reversible work in terms of pressure times the volume change.
Well to get to Gibbs Free Energy we are going to use, so let's start with the definition
of H is U+ PV.
So dH= dU +d(PV).
So that becomes TdS- PdV form this equation.
And then we'll use the product rule and we notice that this term cancels with this term.
So we have dH as TdS+VdP.
But with our definition of Gibbs Free Energy, this is dH-d(TS), so this becomes, by substituting
in for dH and then multiplying this out, TdS-SdT.
And these terms cancel so we end up with the equation that we'll use to calculate Gibbs
Free Energy changes, temperature and pressure changes.
And then first what we're going to look at is a single component system at constant pressure.
This term goes to zero, there's no pressure change so we're looking at the case, -SdT.
So let's get an idea of how we expect the Gibbs Free Energy to change.
So this is the plot.
We have Gibbs Free Energy, that's this line here.
The entropy change, so as the temperature increase, entropy as temperature increases.
These data were calculated from a program where we stepped along, we didn't pick a fine
enough step in the sense that this is a vertical change.
This is the entropy change, this is the delta S for vaporization.
This is delta H for vaporization.
Liquid at low temperature, vapor at high temperature.
Enthalpy increases dramatically, entropy increases and then the Gibbs Free Energy decreases.
So notice that Gibbs Free Energy doesn't change very much with temperature for the liquid.
So this is the equation for constant pressure.
So this says, as I raise the temperature, entropy is a positive term, the Gibbs Free
Energy will decrease.
Since entropy is a small term for the liquid then this change is small.
Entropy is a much larger term for the vapor so notice the Gibbs Free Energy changes much
more for the vapor.
And also very important to notice is when we have the phase change, so right here we
go from liquid to vapor at constant T and P, there is no change in the Gibbs Free Energy.
And that ends up becoming one of our equilibrium criteria, that Gibbs Free Energy is the same
in both phases.
Now we want to look at this same starting equation, for single component.
Now we are going to look at constant temperature.
Which means this term goes to zero and the change in Gibbs Free Energy is VdP.
This says as I raise the pressure, the volume is positive, the Gibbs Free Energy increases.
Let's look at a gas, and for now let's look at an ideal gas, then volume, RT/P, where
that's a volume per mole, using specific volume here.
dG is RTdlnP.
And for a liquid dG is the volume of the liquid dP.
Remember the volume of the liquid is much smaller than the volume of the vapor.
And so we were looking at water, and we will continue water and steam.
So let's look at 400K the volume of the liquid so I've just from the steam tables listed
the volume of the liquid and volume of the vapor.
And you can see it's a factor of 700, more than 700 between the two.
And so when you look at this term you can see that the Gibbs Free Energy is not going
change very much for the liquid when we change the pressure, but it does for the vapor.
So here's the data for Gibbs Free Energy, and in this case also enthalpy as a function
of pressure.
Enthalpy goes through a step function change, this is where we have our phase change, so
this is saturation temperature at 400 K. Of course the entropy also decreases.
So the volume is going to exhibit also a decrease.
And the volume will exhibit some sort of behavior, of course on a different scale, until we get
to the point where we have the phase change and then the volume will hardly change as
we change the pressure.
So here's liquid, here's vapor.
This is just Volume on a different scale.
So the Gibbs Free Energy here, dG, is volume of liquid, dP.
Very small change, looks practically constant.
But here the volume of the vapor, so much larger numbers and then we change the pressure,
if we're an ideal gas, RTdlnP, right.
So here dG=RTdlnP.
So Gibbs Free Energy increases as we increase the pressure significantly for a vapor right
because this is a vapor region.
Over here in the liquid region the Gibbs Free Energy doesn't change very much.
It's because of this form for ideal gas at constant T.
This is isothermal, dG=RTdlnP, that's ideal gas.
It's a real gas then we introduce another variable, that tells how Gibbs Free Energy
changes for a gas and that variable is fugacity instead of pressure for real gas.
And so we can relate fugacity to Gibbs Free Energy.
So if we were to plot fugacity versus pressure, if we were an ideal gas that would be a straight
line, and then when we became a liquid at high pressures- so liquid, vapor, gas phase
change here, saturation pressure- for the liquid just like Gibbs Free Energy doesn't
change, since Gibbs Free Energy is related to fugacity means that fugacity doesn't change
very much as we change the pressure.
And then for a real gas this would of course not be a straight line, and it would start
down here close to ideal gas and it might deviate something like so.
And the fugacity of the liquid equals fugacity of vapor, the gibbs free energy of the liquid
equals the gibbs free energy of the vapor at saturated conditions, at vapor liquid equilibrium.