Chemical potential of a pure substance

This last equation is the Gibbs-Duhem equation for the system, and it shows that only two of the three intensive properties (T, P, and fi) are independent for a system containing one substance. Because of the Gibbs-Duhem equation, we can say that the chemical potential of a pure substance substance is a function of temperature and pressure. The number F of independent intensive variables is T=l — 1+2 = 2, and so D = T + p = 2 + l = 3. Each of these fundamental equations yields D(D — l)/2 = 3 Maxwell equations, and there are 24 Maxwell equations for the system. The integrated forms of the eight fundamental equations for this system are ... 

This equation shows that, because entropy is always positive, the chemical potential of a pure substance decreases as die temperature is increased. As S(g) > Sflj > S(s), die slope of the plot of p versus T is steeper for the vapour than for the liquid, and steeper for the liquid than for the solid. 

The chemical potential of a pure substance i indicates the thermodynamic energy level of the substance relative to the energy level of the chemical elements that make up the substance i. In chemical thermodynamics the chemical potentials of elements are conventionally all set zero in the stable state of them at the standard temperature 298 K and pressure 101.3 kPa. The chemical potential of a substance (a chemical compound) / at the standard state, as a result, is equal to the free enthalpy (Gibbs energy) required to form one mole of the substance i from its constituent elements in their stable standard state. 

The chemical potential of a pure substance depends on its state of aggregation, its crystal structure, etc. For example, liquid water and water vapour or diamond and graphite have different potentials at the same temperature and pressure. In order that the p values are unambiguously given the aggregation state of the substance concerned is added to the formula by a vertical line and the abbrevations s for solid, I for liquid and g for gaseous modifications can be characterised, for example, by their names. In the case of a solute, the solvent used can be labelled in the same manner for water the abbreviation w is chosen. 

The relation between the chemical potential of a pure substance and its molar entropy is given by Eq. 7.8.3 ... 

Just as the chemical potential of a pure substance at a given elevation is defined in this book as the molar Gibbs energy at that elevation (page 196), the chemical potential of substance i in a mixtme at elevation h is the partial molar Gibbs energy at that elevation. 

Since the chemical potential of a pure substance p = Gm, the molar Gibbs free energy, we have... 

Equations (6.1-1) and (6.1-3) contain p (T, P), the chemical potential of pure substance i at temperature T and pressure P. The chemical potential of a pure substance is equal to its molar Gibbs energy. It was shown in Chapter 4 that the molar Gibbs energy of a pure solid or liquid is nearly pressure-independent. We will assume that p. is independent of pressure. 

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

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 ... 

The condition of equilibrium is also applicable to changes of state that involve heterogenous reactions, and the same methods used for homogenous reactions to obtain expressions of the equilibrium constant are used for heterogenous reactions. One difference is that in many heterogenous reactions one or more of the substances taking part in the change of state is a pure phase at equilibrium. In such cases the standard state of the substance is chosen as the pure phase at the experimental temperature and pressure. The chemical potential of the pure substance in its standard state still appears in Y.k vkPk but the activity of the substance is unity and its activity does not appear in the expression for the equilibrium constant. 

Generally, the chemical potential of a constituent substance i in a mixture consists of a unitary part, which is inherent to the pure substance i and independent of its concentration, and a communal part, which depends on the concentration of constituent i [Ref. 3.]. The communal part of the chemical potential of a constituent i in a mixture arises from the entropy of mixing of i For an ideal mixture the molar entropy of mixing of i, s,M, is given from Eq. 3.51 by = -j ln x, and hence the communal part of the chemical potential is expressed by p 4 = -TsM = RT nx, at constant temperature, where x, is the molar fraction of... 

Gibbs-Duhem Relationship The partial molar properties of a multicomponent phase cannot be varied independently (the mole fractions, jc, = ,/E of the components total unity). For example, for the chemical potentials, /i, the Gibbs-Duhem relationship is En, dni = 0 (for details, see e.g., Atkirs, 1990 Blandamer, 1992 Denbigh, 1971). Similar constraints apply to the partial molar volumes, enthalpies, entropies, and heat capacities. For pure substances, the partial molar property is equal to the molar property. For example, the chemical potential of a pure solid or liquid is its energy per mole. For gaseous, liquid, or solid solutions, X, = X,(ny), that is, the chemical potentials and partial molar volumes of the species depend on the mole fractions. 

A mixture is called ideal if this relation is not only valid for low mole fractions but in the whole range 0 < x < 1 and in particular for xq = 1. If we write again for the chemical potential of the pure substance, the equation above simplifies to (cf. Sect. 12.3) ... 

The constant Ki is called the Henry constant. Some values of the Henry constant are given in Table 8.1. In the region where Henry s law is valid, Ki is not equal to the vapor pressure of the pure substance. The graphical significance of the Henry constant is shown in Fig. 8.3. (Also, where Henry s law is valid, in general the chemical potential of the reference state p9 is not the same as the chemical potential of the pure substance.) Only for a perfect solution do we have Ki = p when jc,- 1, but such solutions are very rare. Many dilute solutions obey Raoult s law and Henry s law. 

The values of thermodynamic properties are usually specified with reference to a standard state. We have already defined the standard state of gases and pure liquids and solids to correspond to a pressure of P° (exactly 1 bar). We now choose the standard state for a component of an ideal liquid solution to be the pure liquid substance at pressure P°. For a component of an ideal solid solution the standard state is the pure solid at pressure P°. We will define all of our standard states to correspond to a pressure P°. Since we assume that the chemical potential of a pure liquid or solid substance is nearly pressure-independent, Eq. (6.1-1) becomes, to a good approximation,... 

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... 

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. 

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. 

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... 

These equations are used whenever we need an expression for the chemical potential of a strong electrolyte in solution. We have based the development only on a binary system. The equations are exactly the same when several strong electrolytes are present as solutes. In such cases the chemical potential of a given solute is a function of the molalities of all solutes through the mean activity coefficients. In general the reference state is defined as the solution in which the molality of all solutes is infinitesimally small. In special cases a mixed solvent consisting of the pure solvent and one or more solutes at a fixed molality may be used. The reference state in such cases is the infinitely dilute solution of all solutes except those whose concentrations are kept constant. Again, when two or more substances, pure or mixed, may be considered as solvents, a choice of solvent must be made and clearly stated. 

We consider only binary solutions in this discussion. The standard states of the two components are defined as the pure components, and the chemical potentials of the components are based on the molecular mass of the monomeric species. We designate the components by the subscripts 1 and 2 and the monomeric species by the subscripts A1 and B1, respectively. From the discussion given in Section 8.15 we know that the chemical potential of a substance considered in terms of the species present in a solution must be... 

The chemical potential of a substance i in a homogeneous mixture depends on the temperature, pressure, and concentrations of constituent substances, p, = p,(T,p,xl , ) whereas, that of a pure substance is a function of temperature and pressure only. As mentioned in the foregoing chapters, the mixing of substances causes an increase in entropy of the system and hence changes the chemical potentials of the substances... 

Real gases are usually non-ideal. Thermodynamics describes both ideal and non-ideal gases with the same type of formulas, except that for non-ideal gas mixtures the fugacity f is substituted in place of the pressure pi and that the activity at is substituted in place of the molar fraction xi or concentration c, of constituent substance i. We have already seen that in the ideal gas of a pure substance the chemical potential is expressed by Eq. 7.5. By analogy, we write Eq. 7.9 for the non-ideal gas of a pure substance i ... 

It readily follows that for a total of 1 mole of a pure substance, G = p, i.e., free energy is identical with chemical potential. 

Since any partial molar property of a pure substance is simply the corresponding molar property, the chemical potential of a component i in pure form, denoted by p°, is evidently equal to the molar Gibbs free energy G° of pure component i at the same temperature and pressure. 

We can represent the regions of stability of gases, liquids and solids under various temperature and pressure conditions using a phase diagram showing at which phase each substance is the most stable. As we know from thermodynamics, the most stable phase of a pure substance at a particular temperature and pressure is the one with the lowest chemical potential. A phase transition is the spontaneous conversion of one phase into another phase, which occurs at a characteristic temperature at a given pressure. For example, as seen in Figure 4.1, under latm external pressure, above 0°C, the chemical potential of... 

Primarily, this approach was based on the formal analogy between a first order phase transition and the micellisation. When a new phase of a pure substance is formed the chemical potential of this substance and its concentration in the initial phase do not change with the total content of this substance in the system. A similar situation is observed above the CMC, where the adsorption and the surface tension become approximately constant. In reality variations of these properties are relatively small to be observed by conventional experimental methods. The application of the Gibbs adsorption equation shows that the constancy of the surfactant activity above the CMC follows from the constancy of the surfactant adsorption T2 [13]... 

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