Chemical potential of dilute solution

The chemical potentials of dilute solutions may be expressed in terms of molality (moles of solute per kilogram of solvent) or molarities Ck (moles of solute per liter of solution) instead of mole fractions X/t. In electrochemistry it is more common to use molality ntk. For dilute solutions, since Xk = (A jfc/A soivent)> we have the following conversion formulas for the different units... 

G. D. J. Phillies, /. Chem. Phys., 60, 2721 (1974). Excess Chemical Potential of Dilute Solutions of Spherical Polyelectrolytes. 

As seen from Eq. (130) an activity coefficient may deviate significantly from unity at higher salt concentrations. The activity coefficient can therefore also be used as a measure of the deviation of the salt solution from a thermodynamically ideal solution. If the chemical potential of a solute in a (pressure-dependent) standard state of infinite dilution is /x°, we find the standard partial molar volume from... 

We can show that if the solute obeys Henry s law in very dilute solutions, the solvent follows Raoult s law in the same solutions. Let us start from the Gibbs-Duhem Equation (9.34), which relates changes in the chemical potential of the solute to changes in the chemical potential of the solvent that is, for a two-component system... 

Although Pb tends to - °o as Xb tends to 0 (and In Xb also tends to - °o), the difference on the right-hand side of Eq. (2.18) tends to the finite quantity pi, the standard chemical potential of B. At infinite dilution (practically, at high dilution) of B in the solvent A, particles (molecules, ions) of B have in their surroundings only molecules of A, but not other particles of B, with which to interact. Their surroundings are thus a constant environment of A, independent of the actual concentration of B or of the eventual presence of other solutes, C, D, all at high dilution. The standard chemical potential of the solute in an ideal dilute solution thus describes the solute-solvent interactions exclusively. 

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 ... 

So far, we have used the pure liquid compound as reference state for describing the thermodynamics of transfer processes between different media (Chapter 3). When treating reactions of several different chemical species in one medium (e.g., water) it is, however, much more convenient to use the infinite dilution state in that medium as the reference state for the solutes. Hence, for acid-base reactions in aqueous solutions, in analogy to Eq. 3-34, we may express the chemical potential of the solute i as ... 

Thus, u.°A for the solute in Eq. (II) is the chemical potential of the solute in a hypothetical standard state in which the solute at unit concentration has the properties which it has at infinite dilution. 

The molar Gibbs energy of micelle formation is the Gibbs energy difference between a mole of monomers in micelles and the standard chemical potential in dilute solution ... 

Although I did not know about the concept of the combinatorial contribution, I recognized the need for such a correction even in the initial version of COSMO-RS [C9]. Since at that time I only had in mind the calculation of infinite-dilution partition coefficients and of vapor pressures, I only cared about a solvent-size correction in pure solvents. I thought of two different solvent-size effects influencing the chemical potential of a solute X in a solvent S. The first is quite obvious—in 1 mol of a homogeneous liquid S,... 

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... 

When the excess chemical potential of the solute in the liquid phase is required as a function of the mole fraction at the constant temperature T0 and pressure P, an integration of the Gibbs-Duhem equation must be used. For this the infinitely dilute solution of the solute in the solvent must be... 

As shown in Example 1, the chemical potential of a solute can be of a more complicated form than given by Eqs. (43)-(45), even though the solution shows ideally dilute behavior. This can result from a transformation of the substance when it dissolves in the solution. 

Liquid-phase extraction is a procedure by which some fraction of a solute is taken out of solution by shaking the solution with a different solvent (in which the solute usually has greater solubility). The analysis of this process assumes that the shaking is sufficient so that equilibrium is established for the solute, i, between the two solutions. At equilibrium, the chemical potentials of the solute in the two solutions are equal. Assuming ideally dilute solutions, we can write... 

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. 

Where the component of the solution is a nonelectrolyte, its chemical potential in dilute solution at a molality m, can be calculated from... 

In dilute solutions of nonvolatile solutes, Raoult s law (see section 2.3.1) can usually be assumed to be obeyed and the chemical potential of the solute is given by equation (3.50) ... 

The conventional thermodynamic approach always applies to the limit of a very dilute solution of s, where the chemical potential of the solute s can be written as... 

The concept of the ideal dilute solution i s extended to include nonvolatile solutes by requiring that the chemical potential of such solutes also have the form given by Eq. (14.18). 

For any form of (gas or liquid) chromatography, one can define the distribution of solute between the stationary and mobile phases by an equilibrium (2). At equilibrium the chemical potentials of each solute component in the two phases must be equal. The driving force for solute migration from one phase to the other is the instantaneous concentration gradient between the two phases. Despite the movement of the mobile phase in the system, the equilibrium exists because the solute diffusion into and out of the stationary phase is fast compared with the flow rate. Under dilute solution conditions, the equilibrium constant (the ratio of solute concentrations in the stationary to the mobile phases) can be related to the standard Gibbs free-energy difference between the phases at constant temperature and pressure ... 

Hence U2 can be interpreted as the chemical potential of pure solute in a hypothetical liquid state corresponding to extrapolation from infinite dilution (which serves as reference state) to X2 = 1 along a line where Y2 = U that is, along the Henry s law line. In physical terms, it might be regarded as a hypothetical state in which the mole fraction of solute is unity (pure solute), but some thermodynamic properties are those of the solute 2 in the reference state of infinite dilution in solvent 1 (e.g., partial molar heat capacity). Since from the context it should always be clear whether the superscript circle denotes standard state" or "pure substance", no further distinction is introduced. 

Fluctuation Solution Theory (FST) At infinite dilution the solubility expression contains no hypothetical chemical potential of the solute [4, 45], For dilute solutions, the Henry s law standard state can be more reliable than the pure component standard state since the unsymmetric convention activity coefficients, designated by y., are often very close to unity, y is related to y. by... 

For solutions, the chemical potentials of the solutes (Y2,..., Yk), when referred to infinite dilution instead... 

The same can be said for all the expressions for jx- ix° in Equations (8.30). They all express the difference between the chemical potential of a solute species in a real system, and the same potential in an ideal system under the same conditions. The term residual function is strictly speaking applied only when the ideal system is an ideal gas, so differences from other states such as infinitely dilute solutions or pure phases are called deviation functions (Ewing and Peters, 2000). 

The Henry s law constant can be obtained from molecular simulation using several approaches [214, 215]. It is related to the residual chemical potential of the solute i at infinite dilution by [216] ... 

For convenience, we consider a system in the T, V, N ensemble. (A similar treatment can be carried out in the T, P, N ensemble see, for example, Appendix 9-F.) The chemical potential of the solute S at infinite dilution in water is given by... 

The solute concentration C, expressed as a fraction of the host atoms, is fairly dilute, so the chemical potential of the solute can be expressed as p. = p.0 + kr In C, where k is the Boltzmann constant and T is the absolute temperature. 

---