Chemical potential pressure variation

Note that the response of the chemical potential to variations in the density is different for each set of thermodynamic variables. The three derivatives in (4.123)-(4.125) correspond to three different processes. The first corresponds to a process in which the chemical potential of the solvent is kept constant (the temperature being constant in all three cases) and therefore is useful in the study of osmotic experiments. This is the simplest expression of the three, and it should be noted that if we simply drop the condition of constant, we get the appropriate derivative for the pure A component system. This is not an accidental result in fact, this is the case where strong resemblance exists between the behavior of the solute 4 in a solvent B under constant and a system 4 in a vacuum which replaces the solvent. We return to this analogy later and compare the virial expansion of the pressure with the corresponding virial expansion of the osmotic pressure. 

Notwithstanding their very low vapor pressure, their good thermal stability (for thermal decomposition temperatures of several ionic liquids, see [11, 12]) and their wide operating range, the key property of ionic liquids is the potential to tune their physical and chemical properties by variation of the nature of the anions and cations. An illustration of their versatility is given by their exceptional solubility characteristics, which make them good candidates for multiphasic reactions (see Section 5.3.4). Their miscibility with water, for example, depends not only on the hydrophobicity of the cation, but also on the nature of the anion and on the temperature. 

Thus, for each component, its chemical potential is the same in all phases that are in equilibrium. We will see below that the relationships involving the pressure and temperature variations of the chemical potential that we have developed earlier will be helpful in explaining the effect of these variables on phase equilibria. 

The local conditions of temperature and pressure, as well as the new energy source in the form of the electrochemical gradient, can all be incorporated into the Gibbs free energy by adding new terms to the chemical potential. Variation of AG and AH with temperature are all standard thermodynamics, although we will resist the temptation to explore them here. 

For condensed phases (liquids and solids) the molar volume is much smaller than for gases and also varies much less with pressure. Consequently the effect of pressure on the chemical potential of a condensed phase is much smaller than for a gas and often negligible. This implies that while for gases more attention is given to the volumetric properties than to the variation of the standard chemical potential with temperature, the opposite is the case for condensed phases. 

Many solvent properties are related to density and vary with pressure in a SCF. These include the dielectric constant (er), the Hildebrand parameter (S) and n [5], The amount a parameter varies with pressure is different for each substance. So, for example, for scC02, which is very nonpolar, there is very little variation in the dielectric constant with pressure. However, the dielectric constants of both water and fluoroform vary considerably with pressure (Figure 6.3). This variation leads to the concept of tunable solvent parameters. If a property shows a strong pressure dependence, then it is possible to tune the parameter to that required for a particular process simply by altering the pressure [6], This may be useful in selectively extracting natural products or even in varying the chemical potential of reactants and catalysts in a reaction to alter the rate or product distributions of the reaction. 

An alternative theory, first proposed by Rettori and Villain (1988). takes the point of view which corresponds to the variation depicted in Fig. 1 (a), in which case the excess chemical potential of the top terrace is much lower than Eq. (6) suggests. In fact, in their treatment, the shrinkage of the top terrace is drivrii by the pressure from the step trains on either side, which has its origin in the repulsive interaction (of elastic or entropic origin) of like steps. This yields an effective chemical potential for the top terrace,... 

This equation gives the variation of the chemical potential of a constituent in a mixture with respect to temperature at constant pressure and composition of the system. S i is the partial molal entropy of zth component of the mixture. 

Shown variation of chemical potential with pressure... 

It is important to note that the concept of osmotic pressure is more general than suggested by the above experiment. In particular, one does not have to invoke the presence of a membrane (or even a concentration difference) to define osmotic pressure. The osmotic pressure, being a property of a solution, always exists and serves to counteract the tendency of the chemical potentials to equalize. It is not important how the differences in the chemical potential come about. The differences may arise due to other factors such as an electric field or gravity. For example, we see in Chapter 11 (Section 11.7a) how osmotic pressure plays a major role in giving rise to repulsion between electrical double layers here, the variation of the concentration in the electrical double layers arises from the electrostatic interaction between a charged surface and the ions in the solution. In Chapter 13 (Section 13.6b.3), we provide another example of the role of differences in osmotic pressures of a polymer solution in giving rise to an effective attractive force between colloidal particles suspended in the solution. 

Because variations in solvent chemical potential are generally much easier to determine experimentally (e.g., by osmotic pressure measurements, as described in Section 7.3.6), (6.37) gives the recipe for determining the more difficult solute from its Gibbs-Duhem dependence on other easily measured thermodynamic intensities. Equations such as (6.35)-(6.37) are sometimes referred to as Gibbs-Duhem equation(s), but they are really only special cases of (and thus less general than) the Gibbs-Duhem equation (6.34). 

We see that the ratio of the activities is always constant at constant temperature and pressure the ratio of the mole fractions is not constant over a range of concentrations of x 3, and the variation of the value of the ratio depends upon the difference of the excess chemical potentials. 

Determination of pure component parameters. In order to use the EOS to model real substances one needs to obtain pure component below its critical point, a technique suggested by Joffe et al. (18) was used. This involves the matching of chemical potentials of each component in the liquid and the vapour phases at the vapour pressure of the substance. Also, the actual and predicted saturated liquid densities were matched. The set of equations so obtained was solved by the use of a standard Newton s method to yield the pure component parameters. Values of exl and v for ethanol and water at several temperatures are shown in Table 1. In this calculation vH and z were set to 9.75 x 10"6 m3 mole"1 and 10, respectively (1 ). The capability of the lattice EOS to fit pure component VLE was found to be quite insensitive to variations in z (6[Pg.90]

In a binary solution, the Gibbs-Duhem relation [Eq. (15)] determines the variation of a partial molar property of one component in terms of the variation of the partial molar quantity of the other component. This relation is useful for obtaining chemical potentials in binary solutions when only one of the components has a measurable vapor pressure. Applying Eq. (15) to chemical potentials in a binary solution,... 

These results are important as they help us in deriving expressions for the variation of chemical potential with temperature and pressure. 

The variation of chemical potential of any constituent i of the system with pressure may be derived by differentiating Eq. 1.17 with respect to pressure and Eq. 1.26 with respect to A/,. The results are ... 

Eq. 1.32 gives the variation of chemical potential (p ) of any constituent of the system with pressure. 

The variation of the solvation free energies with pressure is the partial molar volume and gives direct information on hydration structure. Consider a solute species such as the ion M above, diluted in a solvent denoted by W, for example, water. Recalling the chemical potential expression of Eq. (3.3), p. 33, show that the partial molar volume is... 

Here the point of reference is the chemical potential of i in pure form, at a pressure of one bar /l/(T, 1, x ) = jU,(T, l,c ) = fXi(T, 1, m ) = ixf(T, 1), known as the standard chemical potential for the pure material at temperature T. This is adopted regardless of the pressure under whieh the actual experiments are performed. The above expressions are self consistent. As usual, use of mole fractions for compositional variations offers the simplest formulation for the chemical potential. Also, in all three cases there are no problems with regard to units and dimensions. ... 

Equation (2.4.15) relates the variation of chemical potential of an ideal gas to / r In P, in accordance with (i) this suggests that ju.,- should be specified by an analogous expression, RTlnf. The quantity f is known as efugacity of the /th component of the gas. In accord with (ii) this quantity must approach the pressure P, at ideality. Since /x, is specified only to within an arbitrary constant we can determine uniquely only the difference in chemical potential of the nonideal gas in two states, 1 and 2, given by... 

The function fJ (T) which we have introduced is analogous to that introduced in the theory of perfect gases (c/. 10.11), except that here the standard pressure is zero instead of unity. The way in which the chemical potential of a condensed phase varies with pressure is, however, quite different from the logarithmic variation of p, with p in the case of a perfect gas. Here we see that, since (1 - kp) 1 the variation of jjL with p is linear and is given by the term pv (T, 0). As we have seen this term is quite negligible under ordinary conditions so that the chemical potential is independent of pressure and... 

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