Unitary chemical potential

Fig. 3-1. Ionic dissociation of a gaseous molecule XY and X ion level in molecule XY t = energy, unitary X ion level (unitary chemical potential) Dxt = ionic dissociation energy of XY.

Two cases then arise with respect to the ideality of mixtures One is the case in which the mixture is ideal for all values of x and for all constituent substances. This type of mixture is thermodynamically called the perfect mixture, for which the Raoult s law (a linear relation between pt and In x. in the whole range of concentrations) holds valid and in which the unitary chemical potential pi (T,p) of i equals the chemical potential pi, T, p) of pure substance i for all the substances in the system as shown in Eq. 5.24 ... 

It follows, in general, that the standard chemical potential p) of a chemical compound i corresponds to the free enthalpy of formation for one mole of the compound substance i at the standard state, the value of which is tabulated in chemical handbooks as shown for a few compounds in Table 5.1. For ions in electrolytic solutions the chemical potential in their pure state can not be defined, but we may use the standard state of an ion in which the ionic activity is equal to unity (a, = 1) to define the unitary chemical potential of the ion as will be discussed in chapter 9. 

Since the unitary chemical potential p is a function of T and p only, the partial molar enthalpy hf of each constituent i of an ideal system is independent of the composition of the... 

In the summations, the stoichiometrical coefficient v, is negative for the reactants and positive for the products. In Eq. 6.2 the first term on the right hand side is the unitary affinity A, which comprises of the stoichiometrical sum of the unitary chemical potentials of the reactants and products, and the second term is the affinity of mixing AM, which comprises of the stoichiometrical sum of the chemical potentials of mixing for the reactants and products. By substituting the unitary affinity A for the first term on the right hand side of Eq. 6.2 and defining this to be equal to RT In K(T, p we obtain Eq. 6.3 ... 

The unitary affinity of a reaction can be obtained, as mentioned in the foregoing chapter 5, from the unitary chemical potentials of the reactants and products. 

From Eq. 7.32 we have the unitary chemical potential fi (T,p) as follows ... 

We bear in mind however that the values of p (T,p) and y, depend upon the choice of the ideal reference system. If we choose for the solvent a reference system in which y, becomes unity as xt approaches unity, the unitary chemical potential p (T,p) is given by the chemical potential p (T,p) of the pure solvent i ut(T,p) = p (T,p). On the other hand, if we choose for the solute substances a reference system in which y, becomes unity as xt approaches zero, the unitary chemical potential ju (r,p) is given by the chemical potential p (T,p) of the solute i at infinite dilution p ( T,p) = p (T,p). 

A solution is called perfect, if Eq. 8.1 is valid over the whole range of concentration for all constituent substances. The perfect solution is realized if the molecules of the solvent and the solutes are similar to one another in their nature. In perfect solutions the unitary chemical potential p (T,p) of a constituent substance i equals the chemical potential p°(T,p) of the pure substance i for all the constituent substances Raoult s law. 

In contrast to a perfect solution, a solution is called an ideal solution, if Eq. 8.1 is valid for solute substances in the range of dilute concentrations only. Moreover, the unitary chemical potential p2(T,p) of solute substance 2 is not the same as the chemical potential p2( T,p) of solute 2 in the pure substance p2(T,p) p2(T,p) Henry s law. For the main constituent solvent, on the other hand, the unitary chemical potential p[( T,p) is normally set to be equal to f l p) in the ideal dilute solution p"(T,p) = p°(l p). The free enthalpy per mole of an ideal binary solution of solvent 1 and solute 2 is thus given by Eq. 8.10 ... 

As mentioned in section 8.1, the value of the unitary chemical potential pi depends on the choice of the reference system. There are two reference systems which are commonly used one is unsymmetrical and the other is symmetrical. In discussing the reference systems we shall for convenience limit ourselves to a binary solution. 

The other choice is to define each unitary chemical potential if as being equal to the chemical potential p° in the pure state for both solvent 1 and solute 2 n (T, p) = tf(T, p). We then obtain Eqs. 8.15 and 8.16 for the chemical potentials of solvent 1 and solute 2 in both an ideal and a non-ideal solution ... 

The unitary real potential, ay., of the surface metal ion consists of the chemical potential, Py, and the electrostatic surface term e x as shown in Eqn. 3-7 ... 

Fig. S-14. Energy change in hydration of A ions p = outer potential of aqueous solution x = surface potential Pa (.,) il A <.q)) = unitary electrochemic (chemical) potential of hydrated A ions Oa ( i) unitary real potential of hydrated A ions = Pa ( )-zex( ...

The activity of a perfect gas, as for any substance, is unitary, by definition, at standard state. Moreover, for a perfect gas, activity is (numerically) equivalent to pressure, at all pressures. Let us consider the relationship existing, with T held constant, between the chemical potential of component i in gaseous phase g at 1 bar (/a, 17 ) and at pressure P... 

The chemical potential of species B in the unitary gas phase above a pure solution of B is... 

The chemical potential is defined as an intensive energy function to represent the energy level of a chemical substance in terms of the partial molar quantity of free enthalpy of the substance. For open systems permeable to heat, work, and chemical substances, the chemical potential can be used more conveniently to describe the state of the systems than the usual extensive energy functions. This chapter discusses the characteristics of the chemical potential of substances in relation with various thermodynamic energy functions. In a mixture of substances the chemical potential of an individual constituent can be expressed in its unitary part and mixing part. 

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... 

If this linear relation between the chemical potential and the logarithm of the molar fraction of i holds valid in the whole concentration range extending from x, = 0 to xf = 1, the unitary part of the chemical potential (r,p) is identical with the chemical potential Li (T,p) of the pure substance i. The linear relation of Eq. 5.22, however, is not necessarily valid over the whole range of concentrations but in the range of dilute concentrations only. In such a case the unitary part of pi (T,p) is usually set at the value estimated by extrapolation from the dilute concentration range to the mole fraction of xt - 1. 

In this case the unitary value of the chemical potential of solute substance i can be estimated, as mentioned above, by extrapolating the chemical potential of dilute constituent i to xt = 1 from the dilute concentration range in which the linear relation of Eq. 5.22 holds. 

We call this quantity A the unitary affinity of the reaction. Since the chemical potentials of solid carbon C and of gaseous molecular oxygen 02 are set zero in the standard state... 

The concept of the unitary and mixing terms described above can apply, in general, not only to the chemical potential of substances in a mixture but also to the other thermodynamic... 

The modified unitary approach of Philip (1977a), supplemented by adsorption terms presented by Iwamatsu and Horii (1996), provides a means of calculating equilibrium liquid-vapor interfaces for various chemical potentials during drainage and imbibition. Four major steps are discerned during the transition from adsorption to capillary-dominated imbibition (Fig. 1-6). At low matric potentials,... 

In other words, the chemical potential may be equivalently characterized by a unitary uni-parameter group which appears as a rest of the cocycles from above ... 

One of us has used molecular polarization potentials (MPP) to study the interaction of aromatic molecules, including furan, thiophene, and pyridine with a positive unitary charge, these maps being powerful tools for the study of intermolecular interactions and chemical reactivity [129,130], This kind of study leads us to examine theoretically the problem of the interaction between cations and anions with aromatic rings. We were pioneers in proposing that, in parallel with cation-7i-systems (for instance, benzene), there should exists anion-perfhiorinated-7i-systems (for instance, hexafluorobenzene) [131]. These studies include tetrafluorofuran and tetrafluorothiophene (128, 129). Simultaneously, Mascal et al. [132] described the same phenomenon but with 1,3,5-triazine (130) and 2,4,6-trifluoro-l,3,5-triazine (131) as acid 7i-systems. The group of the University of Palma de Mallorca has published a large number of papers on this topic [133] that are well summarized in a two recent reviews [134,135],... 

The invariance of the first-order density matrix with respect to unitary transformations ensures the invariance of all one-electron properties, like electrostatic potentials. Thus the transformation to localized orbitals does not alter the value of the potential at any point r of the space, but permits a chemically meaningful partition of this quantity. In fact, the lone pair , bond and core localized orbitals resulting from the Boys transformation are particularly suitable for our attempt a) to give a rational basis to the additivity rules for group contributions, and b) to find some criteria by which to measure the degree of conservation of group properties. 

---