Mole fraction, chemical potential solution

For this expression to be valid, in cell A components 1 and 2 must be identical in all respects, so it is a rather special case of an ideal mixture. They are however, allowed to interact differently with the membrane, as described above, xa is the mole fraction of the solute in cell A, while p and p are the number densities of cells A and B respectively. The method was extensively tested against both Monte Carlo and equations of state for LJ particles, and the values of the chemical potential were found to be satisfactory. The method can also be extended to mixtures [29] by making... 

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 ... 

Consider now two practically immiscible solvents that form two phases, designated by and ". Let the solute B form a dilute ideal solution in each, so that Eq. (2.19) applies in each phase. When these two hquid phases are brought into contact, the concentrations (mole fractions) of the solute adjust by mass transfer between the phases until equilibrium is established and the chemical potential of the solute is the same in the two phases ... 

Figure 11-4. Comparison of exact chemical potentials with Flory-Huggins theory (FH), mole-fraction-based classical solution theory (CST), and density-based (ln(/o)) approximations. The figure is taken from Krukowski et al. [24] with permission...

Variation in solvent density corresponds to changing the amount of CO2 in a reactor of constant volume, and hence the chemical potential and the mole fraction of a solute can be varied at constant molar (mole per volume) concentration. Obviously, such changes may have a strong impact on chemical equilibria and reaction rates, which in turn determine yields and selectivities of synthetic processes [11]. In addition, a number of solvent properties of the fluid phase are directly related or change in parallel with the density. Accordingly, such properties can be tuned in SCCO2. Variation of the so-called solvent power is the most obvious application and discussed in more detail below. 

Equation (13.8) relates the chemical potential of the solute to the mole fraction of the solute in the solute. This expression is analogous to Eq. (13.5), and the symbols have corresponding significances. Since the p for the solute has the same form as the p for the solvent, the solute behaves ideally. This implies that in the vapor over the solution the partial pressure of the solute is given by Raoult s law ... 

Schnabel et al. [252] calculated the Henry s law constant of CO2 in ethanol. They evaluated the chemical potential with Widom s test molecule method [207] cf. (30). In this approach, by simulating the pure solvent, the mole fraction of the solute in the solvent is exactly zero, as required for infinite dilution, because the test molecules are instantly removed after the potential energy calculation. 

The validity of (3.76) actually goes far beyond mixtures of perfect gases. Systems in which the chemical potentials of the components can be expressed by (3.76) are called ideal systems. A special case of ideal systems are dilute solutions. A statistical derivation of (3.76) for dilute solutions may be found in LANDAU-LIFSHITZ, Theoretical Physics, Vol. V (1968). In dilute solutions, the mole fraction of a solute is approximately given as x. = N. /N where is the number of moles of the solvent. This enables us to rewrite the chemical potential of a solute approximately in terms of its concentration c. as... 

Experiments conducted by Frangois-Marie Raoult (1830-1901) in the 1870s showed that if the mole fraction of the solvent is nearly equal to 1, i.e. for dilute solutions, equation (8.1.10) is valid. For this reason (8.1.10) is called the Raoult s law. The chemical potential of the vapor phase of the solvent Ps,g = Ts,g(Po, T) + RTIn (ps/po) can now be related to its mole fraction in the solution by using the Raoult s law and setting po= P -... 

A problem of obvious importance is the determination of the chemical potential of constituents that form a liquid or solid solution. We proceed by analogy to Eq. (2.4.15) for the ideal gas mixture. This objective is sensible, at least for ideal solutions defined below, because the different constituents in an ideal condensed phase do not interact, so they form an analog to the ideal gas mixture for which the partial pressure F,- constimtes the independent variable. The corresponding composition variable is the mole fraction x,. The solutions must be sufficiently dilute for the ideal Uquid model, discussed below, to apply. 

In such a binary solution, the chemical potential of the solute and that of the solvent A/xg are related to the integral free energy of formation of the solution, AG per mole, containing a mole fraction Xp, of component A, and for component B, by the expression... 

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. 

These simple expressions may also be obtained from the chemical potentials according to Eqs. (XII-26) and (XII-32) by appropriately changing subscripts and recalling that x in these equations represents the ratio of the molar volumes, which in the present case is unity. Owing to the identity of volume fractions with mole fractions in this case, Eqs. (18) and (19) are none other than the chemical potentials for a regular binary solution in which the heat of dilution can be expressed in the van Laar form. The critical conditions (see Eqs. 2)... 

In real mixtures and solutions, the chemical potential ceases to be a linear function of the logarithm of the partial pressure or mole fraction. Consequently, a different approach is usually adopted. The simple form of the equations derived for ideal systems is retained for real systems, but a different quantity a, called the activity (or fugacity for real gases), is... 

An ideal solution is defined as one for which the chemical potential of every component (/ , ) is related to its mole fraction by... 

Equation 29 implies that is the chemical potential of a hypothetical solution in which XA = 1, but the vapor pressure over the solution still obeys Henry s law as extrapolated from infinite dilution. Thus the standard state is a hypothetical Henry s law solution of unit mole fraction. 

The difference between the chemical potential of a pure and diluted ideal gas is simply given in terms of the logarithm of the mole fraction of the gas component. As we will see in the following sections this relationship between the chemical potential and composition is also valid for ideal solid and liquid solutions. 

The flame temperature calculation is essentially the solution to a chemical equilibrium problem. Reynolds [8] has developed a more versatile approach to the solution. This method uses theory to relate mole fractions of each species to quantities called element potentials ... 

No heat is evolved when pure components that form an ideal solution are mixed. The validity of this statement can be shown from consideration of the temperature coefficient of the chemical potential. Again, from Equation (14.6) at fixed mole fraction. 

In this equation, X2 represents the mole fraction of naphthalene in the saturated solution in benzene. It is determined only by the chemical potential of solid naphthalene and of pure, supercooled liquid naphthalene. No property of the solvent (benzene) appears in Equation (14.45). Thus, we arrive at the conclusion that the solubility of naphthalene (in terms of mole fraction) is the same in all solvents with which it forms an ideal solution. Furthermore, nothing in the derivation of Equation (14.45) restricts its application to naphthalene. Hence, the solubility (in terms of mole fraction) of any specified solid is the same in all solvents with which it forms an ideal solution. 

As at constant pressure the chemical potential of the pure sohd is a function only of the temperature, and the chemical potential of the solute is a function of the temperature and mole fraction, we can express Equation (14.48) as... 

In the preceding chapters we considered Raoult s law and Henry s law, which are laws that describe the thermodynamic behavior of dilute solutions of nonelectrolytes these laws are strictly valid only in the limit of infinite dilution. They led to a simple linear dependence of the chemical potential on the logarithm of the mole fraction of solvent and solute, as in Equations (14.6) (Raoult s law) and (15.5) (Heiuy s law) or on the logarithm of the molality of the solute, as in Equation (15.11) (Hemy s law). These equations are of the same form as the equation derived for the dependence of the chemical potential of an ideal gas on the pressure [Equation (10.15)]. 

For solvents, 1, is equal to V because the standard state is the pure solvent, if we neglect the small effect of the difference between the vapor pressure of pure solvent and 1 bar. As the standard state for the solute is the hypothetical unit mole fraction state (Fig. 16.2) or the hypothetical 1-molal solution (Fig. 16.4), the chemical potential of the solute that follows Henry s law is given either by Equation (15.5) or Equation (15.11). In either case, because mole fraction and molality are not pressure dependent. 

Consider a dilute solution of a dye D in equilibrium with a black box at temperature Tg as shown in Fig. 3a. The black-body radiation will cause a very small but finite fraction of the dye molecules to be in the excited state D. Let Xg be the mole fraction of D molecules at equilibrium. Since the system is completely at equilibrium, the chemical potentials of groxind and excited states must be equal, that is... 

For ideal multicomponent systems, a simple linear relationship exists between the chemical potential fii) and the logarithm of the mole fraction of solvent and solute, respectively. 

This expression can be formally identified with the chemical potential MiIS of the central ion in the ideal solution, where c is the ionic concentration on the mole fraction scale ... 

Grunwald and Bacarella (16) have shown that the rate of change of the standard chemical potential Go of a solute with water mole fraction Z in a binary solvent mixture can be expressed by the relation ... 

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