We're going to look at what are the changes and some state variables, mainly enthalpy,
entropy and Gibbs free energy, when we mix ideal gases, or as it turns out, liquids that
form ideal solutions.
So we're going to just look at a binary mixture to make it easy to see the behavior and then
it can be extended to multi-component mixtures.
So this is going to be volume per mole of the mixture.
And if it's ideal solution or ideal gas, then it's going to just be the weighted averages
of the molar volumes of the pure material.
So this is the molar volume of component two, V1 the molar volume of component one and this
of course is the mole fraction.
What this says is if I took and mixed one liter of component one and one liter of component
two, the mixture is going to be two liters in volume, if ideal solution.
So we're looking at a liquid solution in this case.
So the volumes are additives.
And a similar relation exists for the enthalpies.
Enthalpy of the mixture, this is enthalpy now per mole of the mixture, and it's X1 times
enthalpy per mole of pure component one and mole fraction of component two, enthalpy per
mole of component two.
So this says that there's not enthalpy of mixing, there's no heat of mixing.
And again, we're talking about ideal solution and ideal gas is a subset of that.
So this means I could write enthalpy for the mixture as X1 H1 plus substituting for the
mole fraction of component two in terms of component one would have H2 plus X1, H1 minus
H2.
So this is a straight line.
This says enthalpy of the mixture versus the mole fraction of component one from zero to
one.
Here is pure component two enthalpy, here's pure component one.
Ideal solution is just a straight line because no heat of mixing, no enthalpy of mixing.
So ideal solution is essentially saying there are interactions between molecules in the
liquid phase but interactions between the unlike molecules, and this is unlike molecules,
is just the average of interaction between the like molecules.
And so delta H of mixing is zero.
If it were an ideal gas of course this is true because there are no interactions between
the molecules.
And likewise delta U is zero.
But, important difference, delta entropy of mixing is greater than zero for ideal solution.
So the entropy for the mixture is the sum of the pure component entropies times their
mole fraction minus this term for mixing.
So if we write out all of the terms, minus R X1 log of X1 minus R, X2, log X2.
And these are both positive numbers because X1 is less than one so therefore the log of
X1 is less than zero.
So the entropy increases for a mixture.
So here's the behavior we expect.
This is the entropy of the mixture as a function of mole fraction of component one.
It's going to go through a maximum as this is a positive term.
And then the last variable, Gibbs free energy, which is H minus TS, the change in Gibbs free
energy of mixing is the change of enthalpy of mixing, change of entropy of mixing.
Well this is zero for ideal and what we're looking at everywhere here is ideal, so for
ideal solution or ideal gas.
And so delta G of mixing is going to be minus RT(X1 lnX1 + X2 lnX2).
And then there's a minus sign here and of course a minus sign here, which makes this
a plus sign.
Now this says delta G of mixing is less than zero.
Again, looking at an ideal solution.
And so we're saying the Gibbs free energy for the mixture is X1G1 plus X2G2, the pure
component values, plus RT(X1 lnX1 +X2 lnX2).
And so the Gibbs free energy goes through a minimum when we mix.
So when we plot Gibbs free energy versus X1 from zero to one, G2, G1, then we're going
to see a minimum.
And so just to emphasize all of these values that we're looking at are ideal solutions,
which means they also apply for mixtures of ideal gases.