This video, we will briefly describe chemical potential and thermodynamic activity, where
these terms come from since they're used extensively.
We use thermodynamic activity quite a bit and chemical equilibrium, chemical potential,
and its variations in phase equilibrium.
And so I've written down the notations used, we'll now define them.
Fugacity and the little hat indicates it's one component, component i, in a mixture.
This is the activity of that same component, thermodynamic activity since activity is used
in other fields.
The partial molar Gibbs free energy, so the line over the top indicates it's a partial
molar quantity.
And then the fugacity of just a pure component, no mixture, the Gibbs free energy of pure
component, and then the total Gibbs free energy of a mixture.
So this variable is extensive, all these variables are intensive.
These variables are independent of the amount, they are molar quantities for example.
This depends on the amount, it's the total.
And so we're going to look at, then, how these relate and where the term activity comes from.
So what we're going to see is chemical potential is defied in terms of Gibbs free energy.
Then fugacity is used typically instead of chemical potential because it's an easier
variable to work with and maybe it's a little more physically meaningful.
And then activity is sort of a reduced fugacity.
So this relation then will be shown during the rest of this video.
So the relation that is in thermodynamics textbooks and basically results from writing
down the first law and then using entropy to replace heat in a reversible process, and
then using the definitions of H and G, we'll end up with this relation.
The first two terms for a single component.
And now, for a mixture, we've added this other term and we'll try and understand what this
results from.
So the exact differential means that this volume is the derivative of this total Gibbs
free energy with respect to pressure, keeping temperature constant and keeping all of the
other species, namely the number of moles of each of these components, constant.
There's a similar relationship here.
And then this means here that the change of the Gibbs free energy total, respect to i,
ni because this is a sum over all i, constant pressure, constant temperature and constant
nj.
And what nj means is that j does not equal i.
So all the other species, moles are kept constant.
We change the moles of say, component 3, we'd have the chemical potential of component 3
here.
And so this chemical potential is used instead of writing down this derivative every time.
And then we also have another term for this, it's a partial molar quantity and the derivative
like this with respect to number of moles, with pressure and temperature and the other
moles constant is the partial molar quantity.
So this is the partial molar Gibbs free energy.
This means how much does Gibbs free energy change for a mixture when I add a very small
amount of component i.
So the partial molar quantity is the chemical potential, so mu is the chemical potential.
If we have a pure material, this is one component in the mixture.
If we were to have a pure component, only component i, its chemical potential would
be its Gibbs free energy, an intensive variable but it's not a partial molar quantity because
it's only one species.
So, Gibbs free energy, chemical potential have the same units, Joules per mole.
Well, we will end up not using chemical potential as much as other variables.
Keep in mind, Gibbs free energy is like enthalpy in that there's no absolute scale.
We pick some reference condition and say enthalpy is zero here and then same for entropy.
We can calculate Gibbs free energy.
And so the units for chemical potential, are this, and as we show, for example if we had
vapor and liquid equilibrium, component i in the mixture in the liquid phase has the
same chemical potential as the vapor phase, and this is our equilibrium criteria.
However, what we end up doing is creating another variable that is easier to work with,
and that's fugacity.
It has units of pressure.
And, if it's an ideal gas, then fugacity is equal to pressure.
So for a component in a mixture, it's equal to partial pressure.
But we define fugacity as differential in the chemical potential gas constant, absolute
temperature, differential the log of the fugacity of that component.
So again we're talking about one component in a mixture, and we can write this for each
of the components.
If we do an integration, so let's say chemical potential in the alpha phase minus the pure
chemical potential, that's RT the log of the fugacity in that alpha phase, for example,
the vapor phase, divided by the fugacity of the pure component, which is f sub i without
the small hat on it.
Or the other way of looking at it is that the fugacity of one component in the alpha
phase divided by its pure component fugacity is the exponential of this difference in chemical
potential.
Chemical potential in the phase, the mixture, pure chemical potential, divided by the gas
constant.
So we define the thermodynamic activity of component i as its fugacity in the mixture
over its pure component fugacity, and the notation is at a standard state.
So the standard state is typically pure component, the same temperature and pressure of the system,
and this is our standard state.
And so, if we had an ideal gas, then the activity would just be the mole fraction of that component.
If we have an ideal liquid, ideal mixture, the activity would again be the mole fraction.
If we were to have just a pure material, then the activity would be one, for the same conditions
as our standard state, temperature and pressure.
And then for a real system, the activity is going to vary from these values, very non-ideal
can vary quite a bit, for example, liquid phase from just the mole fraction.
But we use this activity and our thermodynamic equilibrium calculations instead of mole fractions,
for example.
That's what shows up in relation with the equilibrium constant.