Pure substance, chemical potential

But a pure-substance chemical potential is merely the molar Gibbs energy, so (7.2.26) can also be written as... 

Numbers for and in the standard state are available for many substances (for instance Stull et al. [14], Reid et al. [8], Frenkel et al. [15, 16]). Numbers for and at other conditions can be obtained from the standard state data using information on heat capacities and enthalpies of vaporization, which are also available in many cases, for instance in the sources cited above. It should be noted that the accuracy of chemical equilibrium constants obtained by this way is limited and may not be sufficient for a given application. This is mainly caused by the fact that due to the summation of the chemical potentials in equations such as equation (4.13) or (4.15), even small errors in the numbers for the pure component chemical potentials may become very important in the calculation of the equilibrium constant from equations such as equations (4.13) and (4.15). Therefore, the chemical equilibrium constants generally have to be determined from direct experimental investigations. A comparison of some chemical equilibrium constants of esterifications and transesterifications as obtained from direct measurements and from estimated numbers for is given in Table 4.1, underlining the need for accurate experimental data on chemical equilibrium constants. 

The energy of a system can be changed by means of thermal energy or work energy, but a further possibility is to add or subtract moles of various substances to or from the system. The free energy of a pure substance depends upon its chemical nature, its quantity (AG is an extensive property), its state (solid, liquid or gas), and temperature and pressure. Gibbs called the partial molar free heat content (free energy) of the component of a system its chemical potential... 

The intensive function Gm for a pure substance is known as the chemical potential. We will see that it is the potential that drives the flow of mass during a chemical process or a phase change. 

We can derive very simply an important interpretation of the chemical potential for a pure substance, for which we can write... 

Hence, for a pure substance, the chemical potential is a measure of its molar Gibbs free energy. We next want to describe the chemical potential for a component in a mixture, but to do so, we first need to define and describe a quantity known as a partial molar property. 

We now have the foundation for applying thermodynamics to chemical processes. We have defined the potential that moves mass in a chemical process and have developed the criteria for spontaneity and for equilibrium in terms of this chemical potential. We have defined fugacity and activity in terms of the chemical potential and have derived the equations for determining the effect of pressure and temperature on the fugacity and activity. Finally, we have introduced the concept of a standard state, have described the usual choices of standard states for pure substances (solids, liquids, or gases) and for components in solution, and have seen how these choices of standard states reduce the activity to pressure in gaseous systems in the limits of low pressure, to concentration (mole fraction or molality) in solutions in the limit of low concentration of solute, and to a value near unity for pure solids or pure liquids at pressures near ambient. 

In equilibrium the chemical potential must be equal in coexisting phases. The assumption is that the solid phase must consist of one component, water, whereas the liquid phase will be a mixture of water and salt. So the chemical potential for water in the solid phase fis is the chemical potential of the pure substance. However, in the liquid phase the water is diluted with the salt. Therefore the chemical potential of the water in liquid state must be corrected. X refers to the mole fraction of the solute, that is, salt or an organic substance. The equation is valid for small amounts of salt or additives in general ... 

At equilibrium, all components of a mixture have the same molar free energy, i.e., the same chemical potential, in any phase in which they are present, and they have the same chemical potential as all other components. However it is not always convenient to use the same standard state for all components or even for the same component in all phases. Just as Equation 6 defines fugacity, Equation 7 or 8 defines activity. Furthermore, Equations 6-8 define / and a for all substances, not just gases. However we should keep in mind that we do not use the same standard state for a substance in all the phases, mixtures, or pure states in which it may occur or for all components of a mixture. 

When Xa is unity, the left-hand side becomes the chemical potential of the pure substance ... 

For a pure substance, the chemical potential n is merely another name for the molar Gibbs function. 

For the purposes of this investigation-rather than adopting any single definition of a reactive chemicaT-CSB focuses on the broadest range of practices to identify reactive hazards and to manage the risk of reactive incidents. A reactive chemical may include any pure substance or mixture that has the capability to create a reactive incident. CSB defines a reactive incident as a sudden event involving an uncontrolled chemical reaction-with significant increases in temperature, pressure, or gas evolution-that has caused, or has the potential to cause, serious harm to people, property, or the environment. 

NFPA developed Standard 704 as a tool for identification and evaluation of potential hazards during emergency response, not for application to chemical process safety. The instability rating is a part of this standard. It was not intended to be used to measure reactivity, but rather to measure the inherent instability of a pure substance or product under conditions expected for product storage. The instability rating does not measure the tendency of a substance or compound to react with other substances or any other process-specific factors, such as operating temperature, pressure, quantity handled, chemical concentration, impurities with catalytic effects, and compatibility with other chemicals onsite. 

Solvent in Solution. We shall use the pure substance at the same temperature as the solution and at its equilibrium vapor pressure as the reference state for the component of a solution designated as the solvent. This choice of standard state is consistent with the limiting law for the activity of solvent given in Equation (16.2), where the limiting process leads to the solvent at its equilibrium vapor pressure. To relate the standard chemical potential of solvent in solution to the state that we defined for the pure liquid solvent, we need to use the relationship... 

As the mole fraction of A in the mixture iucreases toward uuity, the secoud term iu Eq. (2.17) teuds toward zero, aud the chemical poteutial of A teuds toward the standard chemical potential, Pa = G, the molar Gibbs euergy (or chemical poteutial) of A iu the realizable standard state of pure A, in the sense that pure liquid A is a known chemical substance. A similar equation holds for the component B. 

Equation (13) reminds us that the chemical potential has its greatest value, p,, for a pure substance. Any value of a, less than unity will cause n, to be altered from by an amount RT In a which will be negative for a, < 1. Second, any pressure on a liquid that exceeds p°, increases n above This is seen from the combination of Equations (13) and (18). Thus consideration of the chemical potential of the solvent makes it clear how osmotic equilibrium comes about. The presence of a solute lowers the chemical potential of the solvent. This is offset by a positive pressure on the solution, the osmotic pressure 7r, so that the net chemical potential on the solution side of the membrane equals that of the pure solvent on the other side of the membrane. This is summarized by the expression... 

Let us now continue with our discussion of how to relate the chemical potential to measurable quantities. We have already seen that the chemical potential of a gaseous compound can be related to pressure. Since substances in both the liquid and solid phases also exert vapor pressures, Lewis reasoned that these pressures likewise reflected the escaping tendencies of these materials from their condensed phases (Fig. 3.9). He thereby extended this logic by defining the fugacities of pure liquids (including subcooled and superheated liquids, hence the subscript L ) and solids (subscript s ) as a function of their vapor pressures, pil ... 

Equation 9.20 gives the pressure dependence of the Gibbs free energy of a pure substance. More generally, for a mixture one should consider the chemical potential /r, which is defined as the partial molar free energy of species k ... 

Figure 15-2B gives the corresponding chemical potentials calculated as in Equation 15-1. A loop also appears on this figure. The loop is nonexistent physically but can be used analytically. The point of intersection, e, meets the requirements of equilibria for the gas and liquid of a pure substance. At point e, the pressure of the gas equals the pressure of the liquid, and the chemical potentials of the two phases are equal. Point f has the same pressure as points e but is not an equilibrium point because its chemical potential is higher than that of points e. 

Remember that chemical potential for the liquid must equal chemical potential for the gas at equilibrium. For a pure substance this means that at any point along the vapor pressure line, the chemical potential of the liquid must equal the chemical potential of the gas. Thus Equation 15-3 shows that the fugacity of the liquid must equal the fugacity of the gas at equiHbliuffl ffn"thre Vaporf "pressure" line. So gas-liquid Equilibria can be calculated under the condition that... 

Chemical Potential—The Fugacity—Fugacity Coffi-cient—Example of State Calculation for a Pure Substance Mixtures 425... 

For a pure substance, the molar Gibbs free energy is also known as the chemical potential8 p. In a solution, the partial molar free energy is the chemical potential Pi. Hence,... 

The quantity gk T) in Equation (7.67) is again a molar quantity, characteristic of the individual gas, and a function of the temperature. It can be related to the molar Gibbs energy of the fcth substance by the use of Equation (7.67). The first two terms on the right-hand side of this equation are zero when the gas is pure and ideal and the pressure is 1 bar. Then gk(T) is the chemical potential or molar Gibbs energy for the pure fcth substance in the ideal gas state at 1 bar pressure. We define this state to be the standard state of the fcth substance and use the symbol 1 bar, yk = 1] for the... 

When the concentration of a multicomponent system is expressed in terms of the molalities of the solutes, the expression for the chemical potential of the individual solutes and for the solvent are somewhat different. For dilute solutions the molality of a solute is approximately proportional to its mole fraction. (The molality, m, is the number of moles of solute per kilogram of solvent. When two or more substances, pure or mixed, may be considered as solvents, a choice of solvent must be clearly stated.) In conformity with Equation (8.68), we then express the chemical potential of a solute in a solution at a given temperature and pressure as... 

B) As a second example, suppose that the original reference state of the /cth component, considered as a solute, is the pure substance and that mole fractions are used as the composition variable. It is then desired to make the infinitely dilute solution the reference state and to use the molality for the composition variable. Here, again, we express the chemical potential of the fcth component in the two equivalent ways ... 

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