In this chapter, we quickly review some basic definitions and concepts from thermodynamics. We then provide a brief description of the first and second laws of thermodynamics. Next, we discuss the mathematical consequences of these laws and cover some relevant theorems in multivariate calculus. Finally, free energies and their importance are introduced.
A state function is a function that depends only on the current properties of the system and not on the history of the system. Examples of state functions include density, temperature, and pressure.
A path function is a function that depends on the history of the system. Examples of path functions include work and heat.
An extensive property is a characteristic of a system that is proportional to the size of the system. That is, if we double the size of the system, then the value of an extensive property would also double. Examples of extensive properties include total volume, total mass, total internal energy, etc. extensive properties will be underlined. For example, the total entropy of the system, which is an extensive property, will be denoted as S.
An intensive property is a characteristic of a system that does not depend on the size of the system. That is, doubling the size of the system leave the value of an intensive property unchanged. Examples of intensive properties are pressure, temperature, density, molar volume, etc. By definition, an intensive property can only be a function of other intensive properties. It cannot be a function of properties that are extensive because it would then depend on the size of the system.
The first law of thermodynamics is simply a statement of the conservation of energy. Energy can take on a variety of forms, for example kinetic energy, chemical energy, or thermal energy. These different forms of energy can transform from one to another, however, the sum total of all the types of energy must remain constant.
The second law of thermodynamics formalizes the observation that heat is spontaneously transferred only from higher temperatures to lower temperatures. From this observation, one can deduce the existence of a state function of a system.
Note that the second law of thermodynamics is unique among the various laws of nature in that it is not symmetric in time. It sets a direction in time, and consequently there is a distinction between running forward in time and running backwards in time. We can notice that a film is being palyed in reverse because we observe events that seem to violate the second law.
Note that in an isolated system, every spontaneous event that occurs always increases the total entropy. Therefore, at equilibrium, where the properties of a system no longer change, the entropy of the system will be maximized.
The Gibbs free energy is minimized for a system for a system at constant temperature, pressure, and total number of moles. The Gibbs free energy is important because in most experiments, the temperature and pressure are variables that we control. This will become useful to us later when we consider phase equilibria.
As we have seen, free energies such as the internal energy and Gibbs free energy are useful in that they tell us whether a process will occur spontaneously or not. A process in which the requisite free energy decreases will occur spontaneously. A process in which the free energy increases will not occur spontaneously. This does not mean that the process cannot happen; we can force the process to occur by performing work on the system. Therefore, we see that free energies are useful to us, qualitatively, in that they tell us the direction in which things will naturally happen.
Free energies also provide us with quantitative information about processes. The change in the free energy is equal to the maximum work that can be extracted from a spontaneous process, or in the case of a non-spontaneous processes, the minimum amount of work that is required to cause the process to occur.
Figure 2.1 shows the pressure-temperature projection of the phase diagram for a general one-component system. Depending on the temperature and pressure, the system can exist in either a solid, liquid or vapor phase. Lines separate the various phases. On the lines, two phases coexist. The line separating the vapor and liquid phases is known as the vapor pressure curve. On crossing this curve, the system will transform discontinuously from a liquid to a vapor (or vice-versa). At high temperatures, the vapor pressure curve ends at a critical point. Beyond this point, there is no real distinction between vapor and liquid phases. By going around the critical point, a liquid can be continuously transformed into vapor.
The line separating the solid and liquid phases is known as the melting or freezing curve. The line separating the solid and vapor phases is known as the sublimation curve. The point where the vapor pressure curve, the melting curve, and the sublimation curves meet is the triple point. At these conditions, the solid, liquid, and vapor phases can simultaneously coexist.
In figure 2.2, we show the temperature-density phase diagram for a general pure substance. As with the pressure-temperature diagram, the temperature-density phase diagram is divided by various curves into vapor, liquid, and solid phases. Outside these curves, the system exists as a single phase.
The dashed-line represents the triple point. Anywhere along the dash-line, the vapor, liquid and solid phases can simultaneously exist.
Now let's derive the mathematical conditions for equilibrium between two coexisting phases. We consider an isolated system that is separated into two phases, which we label A and B. The volume occupied by each phase can change, in addition, the both phases can freely exchange energy and material with each other. Because the system is isolated, the total energy U, the total volume V, and the total number of moles N in the system must remain constant.
In this section, we derive the Clapeyron equation. This equation relates changes in the pressure to changes in the temperature along a two-phase coexistence curve. This is one form of the Clapeyron equation. It relates the slope of coexistence curve to the entropy change and volume change of the phase transition.
Entropy is not directly measureable, and, therefore, the Clapeyron equation as written above is not in a convenient form. However, we can relate entropy changes to enthalpy changes, which can be directly measured.
Thus, the entropy change of a phase transition, which is not directly measureable, can be determined from the enthalpy change of the phase transition, which is directly measurable.
The conditions for phase equilibria can also be extended to multicomponent systems.
How many variables need to be specified in order to fix the state of a system? In order to fix the state of a one-phase system, the composition of the phase must be specified as well as two additional intensive variables.
The maximum number of degrees of freedom that a system can have is given when there is only one phase present. For a binary mixture, we find that there are at most three degrees of freedom. This means that we can represent the state of binary mixture using a three dimensional diagram. An example of such a diagram is given in Fig. 3.1, which is for mixtures of methane and ethane.
The key feature of this phase diagram is a solid body in the center of the figure. Within this solid body, the system exists as a two-phase mixture, with a coexisting liquid and vapor phase. Above this body, the system exists as a single liquid phase; below this body, the system is a single vapor phase. The upper surface that bounds the body is the locus of bubble points (i.e., the points at which bubbles begin to appear in a liquid). The lower surface (marked by green points) is the locus of dew points (i.e., the points at which droplets begin to appear in a vapor).
The points C1 and C2 are the critical points of pure methane and ethane, respectively. The line connecting these two points, which is the intersection of the bubble point and dew point surface, is the critical locus. This is the set of critical points for the various mixtures of methane and ethane. The black curve connecting points A and C1 is the vapor pressure curve of pure methane, and the violet curve connecting points B and is the vapor pressure curve of pure ethane.
For a one component system, the bubble point and the dew point are the same and lie along the vapor pressure curve, however, this is not necessarily the case for a mixture. Within envelopes contained between the vapor pressure curves of the pure components, a mixture consists of a coexisting vapor and liquid phases. The upper part of the envelope (the solid curve with filled symbols) is the bubble point curve; the lower part of the envelope (the dashed curve with open symbols) is the dew point curve. Different envelopes correspond to different mixture compositions.
In the systems that we have examined so far, the bubble point and the dew point of the mixture vary monotonically with the composition. This is the case for ideal systems. However, for very non-ideal systems, there may be a maximum or a minimum in the bubble and dew point curves. This is the case for azeotropic systems. An example of a system that exhibits a low-boiling azeotrope is a mixture of n-heptane and ethanol, which is shown in Figure 3.5. For this type of system, both the bubble and the dew point temperature curves have a local minimum at the same composition. At this composition, these two curves meet. This point is known as the azeotrope. At the azeotrope, the composition of the coexisting liquid and vapor phases are identical. In this case at the azeotrope, the boiling temperature of the liquid is lower than the boiling temperature of either pure components. The corresponding bubble and dew point pressure curves have a maximum at the azeotrope.
When two liquids are mixed together, they do not always form a single, homogenous liquid phase. In many cases, two liquid phases are formed, with one phase richer in the first component and the other phase richer in the second component. The classic example of a system that exhibits this behavior is a mixture of oil and water.
The maximum of the liquid-liquid phase envelope and is known as the critical point of the mixture. Above the critical temperature (i.e., the temperature at the critical point), the system exists as a single liquid phase. Below the critical temperature, the system can split into two coexisting liquid phases, depending on the overall composition.
The basic reason why liquid-liquid phase separation occurs is that the attractive interactions between different molecules are weaker than the attractive interactions between similar molecules. As a result, similar molecules prefer to be near to each other and than to dissimilar molecules.
As the pressure of the system decreases, the boiling temperature decreases, in general. Therefore, we expect the vapor-liquid coexistence envelope to drop to lower temperatures as the pressures decrease. Changes in pressure, however, do not have a strong influence on the phase behavior of liquids. As a result, we do not expect the liquid-liquid phase envelope to change much with pressure.
In many situations, we need to predict the properties of a mixture, given that we already know the properties of the pure species. To do this requires a model that can describe how various components mix. In mathematical terms, this means that we need to relate the Gibbs free energy of a mixture to the Gibbs free energy of the various pure components. One of the simplest models that achieves this is the ideal solution mode. In this lecture, we present the ideal solution model. Then we apply this model to describe vapor-liquid equilibria, and as a result, derive Raoult's law.
In a ideal solution, we see that the chemical potential of a species depends on its mole fraction and not directly on the composition of the other components in the system. Also, we see that mixing causes the chemical potential of each component to decrease.