Gibbs Energy and Chemical Potential

We started the last chapter with the question, Will a process occur spontaneously We found that it is the total entropy change of the universe—the system and the surroundings—that determines spontaneity. If AS niv is greater than zero, the process is spontaneous. However, it may not always be convenient to determine the entropy change of both the system and the environment. To have a spontaneity condition that depends on the system would be more convenient for assessing chemical reactions. It would also be convenient if this spontaneity condition were useful under conditions that are common for chemical reactions, mainly fixed temperature and fixed pressure. In this chapter, we will determine such a spontaneity condition and apply it to physical and chemical systems. 

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Exercise 3.6 Partial molar Gibbs energy and chemical potential... 

FIGURE 4.12 Calculated Gibbs energy and chemical potentials for an equilibrium mixture of normal and isobutane. (This plot assumes ideal solution behavior as discussed in Chapter 7, and pure species Gibbs energies as discussed in Chapter 12 and Appendix F.)... 

OSMOTIC PRESSURE THERMODYNAMIC FOUNDATIONS 3.2a Gibbs Free Energy and Chemical Potential... 

These equations allow calculation of the effect of temperature and pressure on the partial Gibbs energy (or chemical potential). They are the partial-property analogs of two equations that follow by inspection from Eq. (10.2) ... 

Of the three quantities (temperature, energy, and entropy) that appear in the laws of thermodynamics, it seems on the surface that only energy has a clear definition, which arises from mechanics. In our study of thermodynamics a number of additional quantities will be introduced. Some of these quantities (for example, pressure, volume, and mass) may be defined from anon-statistical (non-thermodynamic) perspective. Others (for example Gibbs free energy and chemical potential) will require invoking a statistical view of matter, in terms of atoms and molecules, to define them. Our goals here are to see clearly how all of these quantities are defined thermodynamically and to make use of relationships between these quantities in understanding how biochemical systems behave. 

To apply the preceding concepts of chemical thermodynamics to chemical reaction systems (and to understand how thermodynamic variables such as free energy vary with concentrations of species), we have to develop a formalism for the dependence of free energies and chemical potential on the number of particles in a system. We develop expressions for the change in Helmholtz and Gibbs free energies in chemical reactions based on the definition of A and G in terms of Q and Z. The quantities Q and Z are called the partition functions for the NVT and NPT systems, respectively. 

It should be evident from the examples in Chapters 10, 11, and 12 that the evaluation of species fugacities or partial molar Gibbs energies (or chemical potentials) is central to any phase equilibrium calculation. Two different fugacity descriptions have been used, equations of. state and activity coefficient models. Both have adjustable parameters. If the values of these adjustable parameters are known or can be estimated, the phase equilibrium state may be predicted. Equally important, however, is the observation that measured phase equilibria can be used to obtain these parameters. For example, in Sec. 10.2 we demonstrated how activity coefficients could be computed directly from P-T-x-y data and how activity coefficient models could be fit to such data. Similarly, in Sec. 10.3 we pointed out how fitting equation-of-state predictions to experimental high-pressure phase equilibrium data could be used to obtain a best-fit value of the binary interaction parameter.. /"... 

Note that the chemical potential G , defined by (3.2.23), has the structure of (3.4.5) that is, the chemical potential is the partial molar Gibbs energy. This is why we use the partial-molar notation for the chemical potential the notation reminds us that the chemical potential has mathematical and physical characteristics in common with other partial molar properties. For example, the integrated form of dG in (3.2.32) is consistent with the mole-fraction average (3.4.4) and the pure-fluid chemical potential (3.2.24) is consistent with (3.4.6) for the molar Gibbs energy. The chemical potential plays a central role in phase equilibria and chemical reaction equilibria therefore, we will need to know how G,- responds to changes of state. 

With m , and s" determined, we can use the defining Legendre transforms to relate the residual Gibbs energy, Helmholtz energy, and chemical potential. The results are... 

Since CO2 gas can be used as an acidifying agent in chemical or biochemical processes, it can also be used to dissolve or precipitate metal carbonate salts through these common-ion effects. The chemical state of a system is described by means of Gibbs energy G, chemical potential /ij of species i, and mole amount [11-14]. 

A second-order phase transition or continuous change is characterized by a change in the heat capacity of a substance without the evolution of heat. While the first derivatives of the Gibbs energies (or chemical potentials) are continuous, the second derivatives with respect to temperature and pressure, that is, heat capacity, thermal expansion, and compressibility are discontinuous, for example, the transition... 

Next, for a mixture of perfect gases, we interpret p as the partial pressure of the gas, and is the partial molar Gibbs energy, the chemical potential. Therefore, for a mixture of perfect gases, for each component J present at a partial pressure pj,... 

For equilibrium calculations we almost always use the partial molar Gibbs energy or chemical potential. But the partial volumes and enthalpies appear most often in the form of partial mass properties, and are used for calculations other than equilibrium. Wherever a dnt appears in this chapter (or elsewhere in this book) it could be replaced by... 

The chemical potential, plays a vital role in both phase and chemical reaction equiUbria. However, the chemical potential exhibits certain unfortunate characteristics which discourage its use in the solution of practical problems. The Gibbs energy, and hence is defined in relation to the internal energy and entropy, both primitive quantities for which absolute values are unknown. Moreover, p approaches negative infinity when either P or x approaches 2ero. While these characteristics do not preclude the use of chemical potentials, the appHcation of equiUbrium criteria is faciUtated by the introduction of a new quantity to take the place of p but which does not exhibit its less desirable characteristics. 

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