Chemical potential variation with pressure

Fig. 3.21 The variation with partial pressure of the chemical potential of a perfect gas. Note that the chemical potential increases with pressure.

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. 

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... 

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... 

The equilibrium isotherms depend on the parameters that characterize the physical state of the system and the chemical potential of the compounds studied in this system selected, i.e., its temperature and pressure, the chemical compositions of the mobile and the solid phases. The last of these parameters is far more difficult to investigate than that of the other three. Suffice it to say that important variations of the isotherm parameters have been observed for different brands of Cig-bonded silica. However, the best isotherm model accormting for the adsorption data measured on these different brands remained the same [61] while the colmnn to column reproducibility of the data was excellent for at least one brand and probably for several others [11,121,122], with relative standard deviations for the parameters being of the order of a few percent [123]. 

Variation of with atomic structure. Some materials are polymorphic examples are a- and y-iron, - and / -quartz, and the graphite and diamond forms of carbon. The chemical potential has a different value in each polymorph for example, at 298 K and 1 bar pressure, the values for graphite and diamond are ... 

Figure 2.2 Variation of chemical potential of carbon with pressure.

Figure 2.4 Variation of chemical potential of carbon with pressure. Figure 2.2 has been revised to distinguish metastable states from stable states.

The variation of Gibbs free energy with temperature and pressure in a closed system was given in Equation (126), and similarly, we can write [dp = -SmdT + VmdP] for each phase. Since the chemical potentials are equal for two phases at equilibrium, it follows that... 

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