Ideal dilute solution Chemical potentials

As with most other chromatographic separation methods, gel permeation chromatography-size-exclusion chromatography (GPC-SEC) is based on partitioning of analyte polymer molecules between the stationary phase (which, in the case of GPC-SEC, is contained within the pores of a porous stationary-phase support) and the mobile phase. Ideally, the mobile phase establishes a concentration equilibrium with the stationary phase at each plate in the column before being transferred to the next plate. The concentration equihb-rium is dictated by an equal chemical potential of the polymer chain between the two phases [1,2]. In normal conditions of GPC-SEC, the polymer concentration Cm in the mobile phase is sufficiently low, and the solution behaves ideally dilute. Its chemical potential per molecule in the mobile phase is then given as... 

Equation 13 has an important implication a clathrate behaves as an ideally dilute solution insofar as the chemical potential of the solvent is independent of the nature of the solutes and is uniquely determined by the total solute concentrations 2K yK1.. . 2x yKn in the different types of cavities. For a clathrate with one type of cavity the reverse is also true for a given value of fjiq (e.g. given concentration of Q in a liquid solution from which the clathrate is being crystallized) the fraction of cavities occupied 2kVk s uniquely determined by Eq. 13. When there are several types of cavities, however, this is no longer so since the individual occupation numbers 2k2/ki . ..,2k yKn, and hence the total solute concentration... 

In what follows we shall always write A, = fC. We assume that the ligand is provided from either an ideal gas phase or an ideal dilute solution. Hence, A, is related to the standard chemical potential and is independent of the concentration C. On the other hand, for the nonideal phase, A will in general depend on concentration C. A first-order dependence on C is discussed in Appendix D. Note also that A, is a dimensionless quantity. Therefore, any units used for concentration C must be the same as for (Aq) . 

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. 

Since ideal conditions simplify calculations, an ideal gas at 1 atm pressure in the gas phase which is infinitely dilute in solution will be utilized. Then the total standard partial molar Gibbs free energy of solution (chemical potential), AG, can be directly related to KD, the distribution constant, by the expression... 

Comparison with the standard form for the chemical potential, p = p° + RT In a [Eq. 47 of Chapter 6], shows that in the ideally dilute solution activities are equal to mole fractions for both solvent and solute. In order to find the standard state of the solvent in the ideally dilute solution, we note that at xA = 1 (infinite dilution, within the range of applicability of the model), we have p = p. The standard state of the solvent in the ideally dilute solution is pure solvent, just like the standard states of all components in an ideal solution. The solvent in the ideally dilute solution behaves just like a component of the ideal solution. Although it is also true that p° becomes p at x, = 1, this is clearly outside the realm of applicability of Eq. (43). In order to avoid this difficulty, in determining p° we make measurements at very low values ofx, and extrapolate to x, = 1 using p = p, — RT In x as if the high dilution behavior held to x, = 1. In other words, our standard state for a solute in the ideally dilute solution is the hypothetical state of pure solute with the behavior of the solute in the infinitely dilute solution. 

In order to find the vapor pressures above the ideally dilute solution, we equate the chemical potentials of the components in the solution with those in the vapor. Because the solvent in the ideally dilute solution behaves just like a component of an ideal solution, its vapor pressure follows Raoult s law. For the solute in an ideally dilute solution, we obtain... 

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 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 ... 

Chemical Potentials for Ideal Dilute Solutions at Equilibrium... 

Chemical Potential Representing Deviation from an Ideal Dilute Solution... 

Corresponding to each chemical potential there is an activity coefficient defined in terms of equation (20.4). By convention, the activity coefficients of electrolytes are always expressed in terms of the ideal dilute solution as standard reference state, cf. chap. XXI, 3. Thus in the case of an aqueous NaCl solution we may write... 

Just as was done previously, one develops the thermodynamic description of an ideally dilute solution by considering the equilibrium between the dilute solute component in the liquid solution and in the vapor phase. If the minority component is designated B, then one may write its chemical potential as... 

It should be noted that gives the standard chemical potential for the ideally dilute solute in a hypothetical system in which the mole fraction of B is unity. This is obviously a fictitious state which is impossible in reality but whose properties are obtained by extrapolating the Henry s law line to Xb = 1 (see fig. 1.12). When Henry s law is not obeyed, an activity coefficient 73 introduced so that the product Yb h b is equal to the vapor pressure Pb- The activity of the dilute component Ub is defined to be 73 3- Thus, the general expression for the concentration dependence of pb becomes... 

Thus, the standard Gibbs energy of solvation A G°p is equal to the solvation Gibbs energy. It is also determined experimentally in the same way as in (7.31). To see that, we write the chemical potential of s in the two phases in the traditional convention, valid only in the limit of ideal gas and ideal dilute solutions ... 

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). 

If a dilute solution of iodine in water is shaken with carbon tetrachloride, the iodine is distributed between the two immiscible solvents. If and are the chemical potentials of iodine in water and carbon tetrachloride, respectively, then at equilibrium n = /t. If both solutions are ideal dilute solutions, then, choosing Eq. (14.18) to express jj, and fx, the equilibrium condition becomes fx + RT nx = n + RT In x, which can be rearranged to... 

The fundamentals of liquid/liquid extraction are provided by the thermodynamic theory of equilibrium. Two immiscible liquid partial systems 1 and 2 are in equilibrium when all mass-, energy-, and impulse-transfer processes have come to a stop, that is, when the chemical potential, temperature, and pressure are the same in both phases. If Equation (2.3.4-1) is set up for a component E in phase 1 or 2, then the chemical potential describes the state of the pure component E with the properties of the ideal dilute solution ... 

Equations of state for surface layers, adsorption isotherms and surface tension isotherms can be derived by equating the expressions for the chemical potentials at the surface, Eq. (6), to those in the solution bulk. As shown for example in (Fainerman et al. 2001) for ideally dilute solutions this finally yields... 

Here/<2 is not the chemical potential of pnre 2 at Tandp. The chemical potential of the solvent in sneh a solntion is given by Equation 1.68. Assume that the surface is an ideal dilute solution in your derivation. 

By a simple rearrangement of eq. (4-39), the desired expression for the chemical potential p, of the electrons in an ideal dilute solution is obtained in the well-known form p > /i ... 

In an ideal dilute solution, that is, one that obeys Hemy s Law, each solute particle A is fully solvated, and there is no aggregation occurring that could otherwise influence the behavior of the solution. In such cases, the chemical potential of solute A is given by (3) ... 

Similar considerations apply to the extreme right-hand side of the diagram where we are concerned with an ideal dilute solution of A in B, In equation (8 14) fi will now stand for the real chemical potential of pure B and fi will stand for the chemical potential of A in a non-realizable state—and, of course, is not the same as the value of fi for the ideal solution on the left of the diagram. 

Within the composition range that a solution effectively behaves as an ideal-dilute solution, then, the fugacity of solute B in a gas phase equilibrated with the solution is proportional to its mole fraction xb in the solution. The chemical potential of B in the gas phase, which is equal to that of B in the liqvtid, is related to the fugacity by //.b = + RTlnifs/p°) (Eq. 9.3.12). Substituting /b = A h,bxb (Henry s law) into this equation, we obtain... 

By similar steps, combining Henry s law based on concentration or molality (Eqs. 9.4.17 and 9.4.18) with the relation/Lb = /t-B(8) + / 7 ln( //>°), we obtain for the solute chemical potential in the ideal-dilute range the equations... 

The quantities g and are the chemical potentials of the solute in hypothetical reference states that are solutions of standard concentration and standard molality, respectively, in which B behaves as in an ideal-dilute solution. Section 9.7.1 will show that when the pressure is the standard pressure, these reference states are solute standard states. 

We now use the Gibbs-Duhem equation to investigate the behavior of the solvent in an ideal-dilute solution of one or more nonelectrolyte solutes. The Gibbs-Duhem equation applied to chemical potentials at constant T and p can be written xt dp,i = 0 (Eq. 9.2.43). We use subscript A for the solvent, rewrite the equation as xa d/iA + = 0,... 

In an ideal-dilute solution, the chemical potential of each solute is given by /l/ = +... 

Consider a process in an open system in which we start with a fixed amount of pure solvent and continuously add the solute or solutes at constant T and p. The solvent mole fraction decreases from unity to a value x, and the solvent chemical potential changes from jji to We assume the solution formed in this process is in the ideal-dilute solution range, and integrate Eq. 9.4.33 over the path of the process ... 

Next consider a solute, B, of a binary ideal-dilute solution. The solute obeys Henry s law, and its chemical potential is given by /Tb = 3 + 7 In xb (Eq- 9.4.24) where 3... 

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