Lecture Notes for Chapter 5

Lecture Notes for Chapter 5

In this chapter we will take the ideas of the second law and apply them further. We first consider simple transitions like phase transitions that only involve one thing. The book simply gives you an expression for the change in free energy of a substance as a function of temperature and pressure change. But this is a bit confusing. After all, the change in free energy is not the change in free energy of a substance, it is the change in free energy of the universe. So, how can we talk about things like molar free energies of a substance (also called the chemical potentials) if the free energy corresponds to the whole universe? We can’t have moles of the universe?? True, but we can calculate the change in free energy of the universe when a mole of something changes in some way. We are going to go through the derivation of these ideas.

Chapter 5.1 – 5.3

The first law says:

The second law says:

Our definition of expansion-type work tells us:

putting all of this together we have:

OK, now a little slight of hand from partial differential equations (no, this won’t be on the exam). What this equation implies is that we can think of the total change in U in terms of two components, one the depends on the change with respect to S alone and one that depends on the change with respect to V alone (in each case holding all else constant):

By comparison, we can see that:

If it is not obvious to you why we can do this, that is not going to be a big deal in this course, but this kind of manipulation is very common in thermodynamics and other areas where one want to consider functions that depend on multiple variables.

Now, in fact we do not care too much about internal energy. That was sort of a warm up. Let’s now try this instead with Gibbs energy. Consider two phases of water, ice and liquid. The point is that as we change temperature (or pressure) the phase of water can change. How do we know which is more stable? We look at something called the molar Gibbs energy of water in the two forms and determine which form has the lowest molar Gibbs energy. To determine this, we need expressions like the ones above that tell us how the Gibbs energy depends on P and T.

So, let’s develop relationships for the Gibbs energy in analogy to those derived above for the internal energy. Recall that G=H-TS. Thus we can write down in general for dG:

but recall that

This comes from the definition of H, H=U+PV. We found a little bit ago that:

If we combine all of this together:

This again suggests that G should be a function of P and T and we can write:

comparing this to the equation above, we see that

With these two relationships, we can determine what will happen to the Gibbs free energy when either the pressure is adjusted or the temperature is changed. This is very important when we try and work through how the phase of a substance changes with pressure and temperature. At any given temperature and pressure, which phase has the lowest Gibbs free energy? These equations tell us how that energy should change with temperature and pressure and thus allow us to predict which phase (gas, liquid or solid) will have the lowest free energy. We can then generalize this to chemical reactions as well.

At this point, the book brings up the concept of molar Gibbs energy which they call G_m. This is really very similar to something the book does not bring up until next chapter called the chemical potential. I will use that here as well, because it is easier to learn the term now rather than later (see below). The book avoids the partial differential equations I used above and sticks with the earlier relationship we derived:

dG = VdP – SdT

It then converts the Gibbs energy into the molar Gibbs energy (the Gibbs energy change per mole of the substance of interest). The equation must also be put in terms of the molar volume (the volume of a mole of the substance of interest) and the molar entropy (which we have seen before):

dG_m = V_mdP – S_mdT

Instead of explicitly using partial differential equations as we have done above, the book just states that we will hold either P or T constant and then consider the dependence of Gibbs energy on the other. For the dependence on pressure at constant temperature we have:

dG_m = V_mdP (const. T)

Thinking about this in terms of phase transitions, we can see that it makes sense. As the pressure increases, the phase with the smallest molar volume is going to be favored (because that will have the lowest increase in Gibbs energy). Generally speaking, a solid has a lower molar volume than a liquid (water being a weird exception to this rule) and liquid has a lower molar volume than gas. Thus as you increase pressure, you generally go from gas to liquid to solid, as you would expect.

Water, unlike almost all other pure substances, expands when it freezes. Arguably, this is one of the properties of water that allows for the persistence of many different lifeforms on earth because it means that ice floats. Thus lakes do not freeze solid in the winter time because the ice actually forms an insulating layer.

This property of water also makes it possible to ice skate. The blade generates pressure on the ice which results in the ice right under the blade melting.

We can also hold P constant instead of T and get:

dG_m = -S_mdT

In other words, the phase with the greatest molar entropy will decrease the most when the temperature increases. Gas obviously has the greatest entropy per mole and indeed at high temperature, it has the most negative Gibbs energy (it is the most stable phase at high temperature). However, this equation says that as you drop the temperature, gas destabilizes faster than liquid or solid, so at some point the liquid (which has an intermediate molar entropy) will dominate and as the temperature is decreased farther, the solid (with the lowest molar entropy) will dominate.

So in the next chapter, the book tells you that the chemical potential is just the partial derivative of G with respect to n, the number of moles of the stuff in question. But for a pure substance, this just comes down to the Gibbs free energy per mole, so we will go ahead and call G_m a chemical potential. The symbol used for chemical potential is the greek letter m. So below, I will use G_m and m interchangably. We can now see that our partial differential equations can be put in terms of molar quantities:

This means that for a constant molar volume, the chemical potential is linearly related to the pressure with a slope of the molar volume. It is also linearly related to temperature and in this case the slope of the line is the negative molar entropy. Since both molar entropy and molar volume change from one phase to another, the slopes of these lines will be different depending on which phase is under consideration. For these reasons, at any particular temperature and pressure, one of the phases will have the lowest chemical potential and that one will dominate:

Here we see the effects of temperature on the chemical potential. The slope of the chemical potential vs. Temperature line is the negative of the entropy. Thus, gases, which have large entropies, will have the steepest slopes and solids, which have the least entropies, will have the smallest slopes. Because they have different slopes, the lines cross. At the crossing points, one phase becomes more stable than another. We can see that at high temperature, the Gas has the lowest chemical potential, but at low temperature, the solid is lowest. In between, the liquid is lowest.

The effect of pressure is sort of opposite that of temperature. Here the slope of the dependence of chemical potential on pressure is the molar volume. The molar volume of solids is low, but that of gases is high. Again the slope is greatest for the gas and least for the solid and thus the various lines cross. Where they cross, the phase changes. At low pressures, gases are favored. At high pressures, solids are favored.

It is worth going through a short aside here. What can ask what the functional form of the G_m or chemical potential is on pressure and temperature. Obviously, if the molar volume and molar entropy are constant, then changing the pressure at constant temperature or the other way around gives a linear plot (like the ones shown above). For example:

dG_m = V_mdP

when integrated for a constant V_m gives:

DG_m = V_mDP

Which we can see is a linear equation. This makes sense for solids and liquids which do not compress much (V_m is pretty constant with pressure), but obviously not for a gas. In that case, we can use the idea gas law for V_m remembering that this is for one mole so n = 1:

V_m = RT/P

dG_m = RT dP/P

Integrating

DG = RT ln(P_f/P_i)

The resemblance of this equation to the familiar

DG = RT ln(K_eq)

Is not a coincidence, but more on this at a later time.

Chapter 5.4 – 5.6

This is a phase diagram. It takes the information from the last two diagrams and combines it showing where the phase boundaries are between the solid, liquid and gas phases. Notice that there is a place on the diagram where all three phases coexist. This is a special point called the triple point. Finally, there is something called the critical temperature above which there is no difference between liquids and gases (they have the same density at higher temperatures).

I am not going to worry much about the derivation of phase boundries.

Now let's consider several topics by way of tutorials:

Chapter 5.7

In this we focus one equation: F = C - P + 2. This is the Gibbs phase rule. It applies to phases of a system which are in equilibrium with one another. Now all we have to do is to figure out what F, C, and P are.

P is the number of phases present (solid, liquid, gas). Note that there can be more than one phase of a particular type in multicomponent systems (such as oil and water which form two liquid phases).

C is the number of components present. This definition is kind of strange. The number of components in a system is the minimum number of independent species necessary to define the composition of all phases present in the system. The important word here is independent. If I have a chemical reaction, A à B +C, and if I know that I added a certain amount of A and formed a certain amount of B, then by stochiometry, I know the amount of C as well. C is not independent of A and B, so there are only two components. However, if I added an arbitrary amount of both A and B to begin with, then C would not be defined (because we messed up the stochiometry between A, B and C by adding extra B). In this case there would be three components. If there are no reactions, then the number of components is just equal to the number of chemical species present.

F is the number of degrees of freedom or variance in the system. This is the number of intensive variables like temperature, pressure or composition that can be changed independently without changing the number of phases. For example, if we have water at 25 C, we can change both the temperature and the pressure independently without changing the number of phases (which is one). Thus, there are 2 degrees of freedom. This is not true with an ice/water mixture. If we want to change the temperature and still maintain two phases, we will also have to change the pressure. Thus, there is only one degree of freedom. If we have a mixture of two miscible liquids (two liquids that freely mix) we have 3 degrees of freedom: the temperature the pressure and the composition of the mixture (mole fractions of the two components).

So the Gibbs phase rule allows us to predict how many degrees of freedom we have for a certain number of phases and a certain number of components. For the three cases above (water, ice/water and two miscible liquids) we see that it works. Water at 25 C has one phase and one component, so F = 2. Ice/water has 2 phases and one component, so F = 1. The two miscible liquids have 1 phase and 2 components, so F = 3. You should try and run through different examples for yourself and see if the equation makes sense.

It turns out that this is not an easy equation to derive precisely, but the concept is straightforward. Gibbs energy is a function of temperature and pressure, as we have discussed before. It can have only one value for any given temperature and pressure. At a phase boundary, the Gibbs energies of all phases present are equal. Now if I tell you that

G_liquid(T, P) = G_solid(T,P) you would realize that as long as this condition holds (as long as there is both ice and liquid water at equilibrium in your container), the system is constrained – if I change P, I must change T accordingly. There is one equation (one constraint) and two variables (P and T). The system is not uniquely determined, but there is a relationship that must exist between P and T. In other words, T becomes a simple valued function of P (or the converse). For any value of T there is a single corresponding value of P.

OK, but what happens if we have three phases in equilibrium (this happens at the triple point of water, for example). Now we can say that G_liquid(T, P) = G_solid(T,P) and G_liquid(T, P) = G_gas(T,P). There is one more constraint. We now have two equations constraining a system with two variables. Two equations, two variables, now the solution is unique. There can only be one value of T and P which will satisfy these equations.

F = C-P+2 is just a formal way of counting the number of degrees of freedom (related to the number of constraints (equations) and the number of variables which is always 2 – the reason a 2 appears in the equation). For a one component system:

The number of constraints (equations) = P-1.

This is because the number equilibrium relationships is always one less than the number of phases (for ice and water, there are two phases but only one relationship – that the gibbs energy of the two phases must be equal when they are both present).

The number of degrees of freedom = number of variables - the number of constraints = 2 – (P-1)

F = 2 – P + 1

Which is exactly the equation above for one component (C=1). The fact that F cannot be less than zero means that for a single component system, you can only have 3 phases in equilibrium at any given time. Never any more. The point where all three are in equilibrium is called the triplet point.

Chapter 5.8

In this chapter, I only want you to look at the phase diagram for water below 2 atm and compare that with other phase diagrams. Water is weird stuff as described above, the phase boundry between liquid and solid angles backward with a negative slope in this pressure region. This is because ice is less dense than liquid water.