Thermodynamics

# Thermodynamics

Granted, thermodynamics is a pretty broad subject area as you already know from your introductory thermodynamics courses. We will limit our discussion here to primarily condensed phases, which means we don't have to be concerned about fugacities and virial coefficients, and we will be describing mostly classical thermodynamics, which means you don't need to use Sterling's approximation or solve Schroedinger's equation (though it would be nice if you remembered what these things were!) Because of these limitations, this chapter and its material-specific counterparts will primarily use the classical thermodynamic quantities of free energy, enthalpy and entropy to describe the thermodynamic state and associated properties of liquids and solids. A general outline of the chapter is shown below.

Phase Equlibria
Phase Stability
Gibbs Phase Rule
Types of Phase Diagrams
Unary Phase Diagram

As we progress to multi-component systems, it will be useful to discuss binary- and ternary systems in the context of specific materials. The Fe-C (metallic) phase diagram is a classic example of a binary system, and the Al2O3-CaO-SiO2 ternary phase diagram is probably the most widely-utilized ceramic phase diagram known. Thermodynamics of polymeric systems encompasses much more than simply phase equilibria. Aspects of solution thermodynamics, chain conformations and phase separation are dealt with in a separate chapter on polymers. Finally, thermodynamic properties of composite systems are discussed.

## Phase Equilibria

It is useful to review some basic concepts of phases before introducing the thermodynamics. A phase is a homogeneous region of matter. This region need not be continuous. The bubbles in a liquid/gas system represent one phase, even though the bubbles are separated from each other. But don't confuse a phase with a substance. An oil/water mixture is a good example of a two phase system (with two substances), but ice in water (one substance) is an equally valid two-phase system. Both phases of the same substance simply coexist at a specific temperature and pressure. This brings us to the concept of phase transformations, which we should all be familiar with. We will discuss phase transformations in more detail later, but it is helpful for our review to look at how a substance can transform from one phase to another, usually through the addition of heat. The diagram below shows the typical phase changes a pure substance will go through as heat is applied:

Here, Java Equation is the latent heat of fusion (or melting, or crystallization, as the case may be) and Java Equation the latent heat of vaporization. We will return to phase transformations later, but let's now see why these phases form in the first place and why each one is stable over such a wide range of temperatures.

## Phase Stability

The stability of a phase is determined by the Gibbs free energy, Java Equation:
Java Equation

Java Equation = enthalpy
Java Equation = entropy
T = absolute temperature

Any system is said to be in equilibrium if there are no unbalanced forces within the system, i.e., Java Equation. If the free energy is less than zero, a phase can spontaneously transform to another phase; e.g., solid to liquid. This type of transformation is said to be "thermodynamically favorable." If the free energy is greater than zero, a phase transformation will not occur spontaneously. Keep in mind that the free energy condition is a thermodynamic one only; it does not describe the rate at which transformations from one phase to another may take place, even if they are thermodynamically favorable. Phase transformations are rate-dependent, and as such are discussed in the appropriate chapters on kinetics.

## Gibb's Phase Rule

For a system in equilibrium, the phase rule relates the number of components (substances), variables (temperature, pressure) and phases to something called the degrees of freedom:
F = C + N - P

F = degrees of freedom; the number of independent variables that must be arbitrarily fixed to establish the intensive state of a system
C = number of components
P = number of phases
N = number of noncompositional variables

This is a very important relationship. It describes the intensive state of a system of components. Recall that an intensive variable is one that is independent of the size of the system. (Recall, also, that an extensive variable depends on the system size.) The temperature at which pure water boils at one atmosphere is 100°C, regardless of whether there is 10 ml or 10,000 gallons. The Gibbs Phase Rule will be used extensively in the coming sections, so it is important that you understand it completely and utilize it effectively. Let's look at each of these variables in a little more detail.

The number of components, C, is fairly straightforward. If I mix oil and water, I have two components. If I put some tin into molten lead, I have two components. Water is an example of a one-component system. If I put salt in water, I once again have two components. You might argue that when I dissolve salt in water I have only one component: salt water. What I have is one phase; there are still two components. What if I put in too much salt? It begins to precipitate out and I get two phases, but in both instances I have two components: the salt and the water.

As you can see in the saltwater example, the number of phases, P, is something that is not always known. Sometimes this is the quantity we desire, but more often than not, we can determine the number of phases in a system by inspecting the system, either visually or with a microscope. If we keep in mind that a phase is a homogeneous region of matter, we can usually identify how many of these regions are present. Remember that these regions need not be continuous. Bubbles in a liquid represent only two phases, even though the bubbles are separated from each other by the liquid.

The number of noncompositional variables, N, is simply the process variables we wish to consider other than composition. Usually, these are temperature and pressure, so that N=2, but for condensed systems; e.g., two solids, the effect of pressure is often times negligible, so that N can also be 1.

The degrees of freedom, F, are difficult to conceptualize. They don't have any physical significance, that is, they don't represent a certain phase, or a process variable. Just as with the other variable in the Gibbs Phase Rule, though, they count the number of something. You can think of the degrees of freedom as counting whatever is left from adding and subtracting the other variables (C, N and P). If F=0, we say the system is invariant. Similarly, we can have monovariant, divariant or trivariant systems for F=1, 2 or 3, respectively.

Often times, we wish to determine the degrees of freedom. We know how many components we have, we know how many process variables (noncompositional variables) we wish to vary, and we sometimes know how many phases we have. By determining F, we find out how many of the process variables can be changed independently without affecting the number of phases present. The best way to strengthen your grasp on this concept is to forge ahead with phase diagrams. As the complexity of the diagrams increases, you should see why it is important to know the degrees of freedom for a given system.

## Types of Phase Diagrams

Phase diagrams are a graphical representation of three types of information:
1. number of phases present
2. composition of each phase
3. quantity of each phase
Phase diagrams can be grouped according to how the information is presented; i.e., what types of variables are being graphically represented:
Java Equation = intensive variable; P, T, µ (chemical potential), electrical potential, etc.
q = extensive variable; V, S (entropy), ni (number of moles), Q (charge), etc.

### Type 1 Phase Diagram

Java Equation vs Java Equation
e.g., P vs. T; T vs. µ1

### Type 2 Phase Diagram

Java Equation vs. q2/q3 or q2/(q2+q3)
e.g., T vs. n1/n1+ n2 (where n1/n1+ n2 is X1, mole fraction of component 1)

### Type 3 Phase Diagram

q1/q3 vs. q2/q3
e.g., nO /(nCo + nNi ) vs. nNi/(nCo+ nNi ) for NiO-CoO system

## Unary Phase Diagram

The unary phase diagram is usually a Type 1 phase diagram. It has one component, hence the name "unary". Applying Gibbs Phase Rule shows us that C=1 (one component), and N=2 (temperature and pressure), so that F + P = 3. This relationship must hold true for any point on the diagram. So, in a single phase field (P=1) say, the liquid region, the number of degrees of freedom must be two. This tells us that both temperature and pressure can be changed independently within this region, up to any phase boundaries.

As in any phase diagram, lines indicate phase boundaries, which are transition regions where both phases coexist, that is, both phases are in thermodynamic equilibrium. The critical point is where two phases have exactly the same density and are indistinguishable. The triple point is where all three phases coexist at equilibrium (P=3, F=0). At the critical point, you can't change any variable without losing a phase. The region above the critical point is called the "supercritical" region. Supercritical fluids like CO2 (it's nearly impossible to distinquish between liquid and vapor at this point, so we call it a "fluid") are technologically important for separation and extraction purposes.