In previous parts of this book we have developed rigorous, generalized, thermodynamic descriptions of phenomena. With this chapter we begin to convert those descriptions into specific forms that can be applied to phase and reaction equilibrium calculations. Such calculations always requires us to make decisions—to select from among alternative computational strategies. For example, a common decision to be faced is this: Which of the five famous fugacity formulae should I use? If, in a given situation, our models are all reliable and their parameters are all known, and if we can solve all appropriate thermodynamic relations, then our choices are relatively simple: our decisions are dictated by the process, the substances involved, and their states. Unfortunately, most situations are not so simple: we usually have limited information about the process, all necessary properties of the substances may not be known, some models may be of limited reliability, and rigorous computational routes may be inaccessible. Such constraints complicate the selection process, forcing us to balance thermodynamic rigor, model reliability, and computational simplicity.

In § 10.1 we present the basic thermodynamic relations that are used to start phase-equilibrium calculations: we discuss vapor-liquid, liquid-liquid, and liquid-solid calculations. We have seen that the most interesting phase behavior occurs in nonideal solutions, but when we describe nonidealities using an ideal solution as a basis, we must select an appropriate standard state. Common options for standard states are discussed in § 10.2; they include pure-component standard states and dilute-solution standard states.

Thermodynamics is fundamental and applicable to all technical endeavors. Its two brief laws provide a complete basis for establishing the states of pure substances and their mixtures. It shows us the directions in which those states tend to change when systems are prodded by external forces. It provides a secure foundation for scientific investigations into all forms of matter. It reveals constraints on interconversions of heat and work, on separations of components from solutions, and on ultimate extents of chemical reactions. It can guide screening for feasibility of alternative processes, and when a design has been selected, it can contribute to the optimization of that design.

Although thermodynamics describes natural phenomena, those descriptions are in fact products of creative, systematic, human minds. Nature unfolds without any explicit reference to energy, entropy, or fugacity; these are unnatural concepts created by humans. Nevertheless, the complexities observed in Nature can be organized by appealing to thermodynamic methodology. With proper understanding, generalized thermodynamic techniques can be used to deal effectively with many aspects of reality. But to gain that understanding, thermodynamics must be studied in a systematic way that uncovers its structure and economy.

Thermodynamic ideas originated almost 200 years ago, but the subject continues to evolve. Although some claim that “there is nothing new in thermodynamics,” scholars still find challenges in its abstractness, rigor, and universality. They debate the “best” ways to phrase its basic principles and to identify the limits of its application.

When we apply thermodynamics to industrial and research problems, we should draw fundamental ideas from Parts I and II, devise an appropriate solution strategy, as in Chapter 10, and combine those with a computational technique, as in Chapter 11. Such a procedure provides values for measurables that can be used to interpret novel phenomena, to design new processes, and to improve existing processes. The procedure is illustrated in this chapter for several well-developed situations. They include conventional phase-equilibrium calculations for vapor-liquid, liquid-liquid, and solid-solid equilibria (§ 12.1); solubility calculations for gases in liquids, solids in liquids, and solutes in near-critical solvents (§ 12.2); independent variables in steady-flow processes (§ 12.3); heat effects for flash separators, absorbers, and chemical rectors (§ 12.4); and effects of changes of state on selected properties (§ 12.5).

PHASE EQUILIBRIA

When two or more bulk phases are in contact and at equilibrium, the measurables of interest are usually temperature, pressure, and the compositions of the phases. Of these measurables, the most important are often the compositions; for example, in the design and operation of separation processes, we routinely need the composition of a particular phase, or when the temperature and pressure change, we need to know the extent to which the compositions also change. When engineering applications involve fluid-fluid equilibria, we often find that, besides absolute compositions, relative compositions can be informative and important.

In the previous chapter we accomplished our first objective: we showed how the process variables heat and work are related to changes in system properties, the internal energy U and the entropy S. Those relations are provided by the first and second laws. Now our problem is to learn how to compute changes in U and S. Since U and S cannot be obtained directly from experiment, we must first relate ΔU and ΔS to measurable state functions, particularly temperature, pressure, volume, composition, and heat capacities. When we can establish such relations, our strategy in a process analysis can take the path on the left branch of the diagram shown in Figure 3.1.

Unfortunately, ΔU and ΔS are not always simply related to measurables, nor are ΔU and ΔS always directly related to convenient changes of state. So to ease conceptual and computational difficulties, we create additional state functions. Then we must establish how ΔU and ΔS are related to these new state functions and, in turn, how changes in the new functions are related to measurables. In these situations, our strategy follows the right branch of the diagram in Figure 3.1. In this chapter we develop relations that allow us to follow both strategies represented in the figure.

Our long-term goal is to be able to analyze processes, and since processes cause changes in system states, we begin by discussing the conditions that must be satisfied to characterize a state (§ 3.1).

When two or more homogeneous systems are brought into contact to form a single heterogeneous system, any of several actions may occur before equilibrium is reestablished. The possibilities include mass and energy transfers, chemical reactions, and the appearance or disappearance of phases. In this chapter we provide thermodynamic criteria for determining whether and to what extent such phenomena actually occur. Surprisingly, these criteria invoke no new thermodynamics—we need only combine familiar thermodynamic quantities in new ways and, in some cases, apply to those quantities mathematical operations that we have not used heretofore.

The heterogeneities of most concern to us are those that involve the presence of more than one phase. The analysis of multiphase systems can be important to the design and operation of many industrial processes, especially those in which multiple phases influence chemical reactions, heat transfer, or mixing. For example, phase-equilibrium calculations form the bases for many separation processes, including stagewise operations, such as distillation, solvent extraction, crystallization, and supercritical extraction, and rate-limited operations, such as membrane separations.

Analysis of multiphase systems is a principal theme of chemistry and chemical engineering; another is analysis of chemical reactions—processes in which chemical bonds are rearranged among species. Rearranging chemical bonds is the most efficient way to store and release energy, it drives many natural processes, and it is used industrially to make substitutes for, and concentrated forms of, natural products.

You are a member of a group assigned to experimentally determine the behavior of certain mixtures that are to be used in a new process. Your first task is to make a 1000-ml mixture that is roughly equimolar in isopropanol and water; then you will determine the exact composition to within ±0.002 mole fraction. Your equipment consists of a 1000-ml volumetric flask, assorted pipettes and graduated cylinders, a thermometer, a barometer, a library, and a brain. You measure 300 ml of water and stir it into 700 ml of alcohol—Oops!—the meniscus falls below the 1000-ml line. Must have been careless. You repeat the procedure: same result. Something doesn't seem right.

At the daily meeting it quickly becomes clear that other members of the group are also perplexed. For example, Leia reports that she's getting peculiar results with the isopropanol-methyl(ethyl)ketone mixtures: her volumes are greater than the sum of the pure component volumes. Meanwhile, Luke has been measuring the freezing points of water in ethylene glycol and he claims that the freezing point of the 50% mixture is well below the freezing points of both pure water and pure glycol. Then Han interrupts to say that 50:50 mixtures of benzene and hexafluorobenzene freeze at temperatures higher than either pure component.

During the design and operation of chemical processes, we routinely propose a state for a system by specifying a temperature, pressure, composition, and phase. Then the question is, Can the system be brought to that state? This is a question of observability. In many situations, particularly those involving multicomponent mixtures, the answer is not at all obvious. For example, at certain values for T and P, mixtures of phenol and water can undergo drastic phase changes in response to slight changes in composition: a mixture of phenol in water might be a one-phase vapor, or a one-phase water-rich liquid, or a phenol-rich liquid in equilibrium with a water-rich liquid, or it might be in three-phase vapor-liquid-liquid equilibrium.

In the previous chapter we derived criteria for identifying equilibrium states; for example, in a closed system at fixed T and P, the equilibrium state is the one that minimizes the Gibbs energy. That minimization is equivalent to satisfying the equality of component fugacities. More generally, we derived criteria for thermal, mechanical, and diffusional equilibrium in open systems. But although those criteria can be used to identify equilibrium states, they are not always sufficient to answer the question of observability. Observability requires stability. Thermodynamic states can be stable, metastable, or unstable; a stable equilibrium state is always observable, a metastable state may sometimes be observed, and an unstable state is never observed.

Introduction

Throughout history there has been a never-ending effort to develop materials with higher yield strength. However, a higher yield strength is generally accompanied by a lower ductility and a lower toughness. Toughness is the energy absorbed in fracturing. A high-strength material has low toughness because it can be subjected to higher stresses. The stress necessary to cause fracture may be reached before there has been much plastic deformation to absorb energy. Ductility and toughness are lowered by factors that inhibit plastic flow. As schematically indicated in Figure 13.1, these factors include decreased temperatures, increased strain rates, and the presence of notches. Developments that increase yield strength usually result in lower toughness.

In many ways the fracture behavior of steel is like that of taffy candy. It is difficult to break a warm bar of taffy candy to share with a friend. Even children know that warm taffy tends to bend rather than break. However, there are three ways to promote its fracture. A knife may be used to notch the candy bar, producing a stress concentration. The candy may be refrigerated to raise its resistance to deformation. Finally, rapping it against a hard surface raises the loading rate, increasing the likelihood of fracture. Notches, low temperatures, and high rates of loading also embrittle steel.

There are two important reasons for engineers to be interested in ductility and fracture. The first is that a reasonable amount of ductility is required to form metals into useful parts by forging, rolling, extrusion, or other plastic working processes.