Chemical potential derivatives

We now turn in some detail to the chemical potential derivatives. At constant temperature it is found upon application of Equation 16 to Equation 14 that... 

The corresponding regional (resultant) softnesses are similarly defined by the mixed resolution chemical potential derivatives, e.g.,... 

Limiting rate constants for loss of pyrazine or of piperidine from their respective pentacyanoferrate(II) derivatives are affected to only a very small extent by the acetone content of binary aqueous mixtures.Probably here, as in the earlier case of the 4-cyanopyridine complex in aqueous alcohols, " this minor effect on rate constants conceals large but almost equal effects on the initial and transition states. This proved impossible to assess, due to the authors failure to find salts of appropriate solubility for the requisite measurements and transfer chemical potential derivations. It may be that the recently characterized transition metal(II) salts could lead to an answer. 

Derive an expression for the surface tension for a system where the molecules adsorbed in a single monolayer on the surface are described by a lattice gas with the free energy of Eq. (2.50), but where the molecules in the bulk solvent interact with either attractive or repulsive forces described by a dilute, nonideal gas (see Chapter 1). From the requirement that these two systems (surface and bulk) must be in equilibrium and thus have the same chemical potential, derive an expression for the volume fraction adsorbed on the surface as a function of the volume fraction of molecules in the bulk. 

In the original Kirkwood and Buff paper (Kirkwood and Buff 1951), the starting point for their derivation is the A matrix (see also the Prolegomenon) which has elements of the form y(3(3 j, /3A )ry, Ar. When used in combination with the B matrix, with elements B j given by Equation 1.38, one finds the matrix relationship I = A B, which is obtained directly from Equation 1.43 by taking derivatives with respect to particle numbers with volume and T constant. Others have adopted similar approaches (Ben-Naim 2006). This is usually followed by a series of thermodynamic transformations to convert the isochoric chemical potential derivatives to provide the more common and useful isobaric expressions. Here, we wish to eliminate the majority of these transformations. In fact, the results for a small number of components can be obtained directly from Equation 1.43, as we shall see in the next section. 

Hence, the derivative of the solute chemical potential (or activity) with respect to solute concentration can be expressed in terms of a combination of number densities and particle number fluctuations or KBIs. The ability to express thermodynamic properties in terms of KBIs is the major strength of FST. This has been achieved without approximation and the relationship holds for any stable binary solution at any composition involving any type of components. Derivatives of other chemical potentials can be obtained by application of the GD equation, or by a simple interchange of indices. The same approach can be applied to the second expression in Equation 1.48, with a subsequent application of Equation 1.27, to provide chemical potential derivatives with respect to other concentration scales. 

After expressions for the chemical potential derivatives have been obtained, one can use them to determine corresponding expressions for the partial molar volumes and isothermal compressibility. Using Equation 1.50 in Equation 1.47 with i = k = 2 followed by some rearrangement using the GD expression provides... 

Alternatively, one can simply use the matrix relation between the A and B matrices, and a thermodynamic relation between the isochoric and isobaric chemical potential derivatives to provide the elements of the A matrix. The relevant expressions are... 

Other equivalent expressions involving a single chemical potential derivative (usually 1122) have also appeared. The most common operative form of the inversion equations for binary mixtures is... 

The explicit expressions for the chemical potential derivatives, partial molar volumes, and isothermal compressibility become rather cumbersome for ternary systems. Experimental data are also much less common. However, there are many interesting effects that involve ternary systems (see Chapter 4). Also, we shall see that considerable simplification is obtained when one of the components is at infinite dilution (see Chapters 10 and 11). If one requires specific expressions for the various properties, it will prove convenient to define the following set of variables (Smith 2006a),... 

We do not provide expressions for the KBIs in a general form. This is a result of the many equivalent forms for the expressions that can be obtained when the chemical potential derivatives are interchanged using the expressions provided by the GD relationship. However, a useful set of expressions is (Smith 2008)... 

Both expressions are valid for systems containing any nnmber of components at any composition. The chemical potential derivatives are provided by the expressions for an component system. For low solnte concentrations, the former equation also provides an expression for the standard volume change for the process. A particularly common sitnation involves the effect of a single cosolvent on the conformational eqnilibrinm (n= 1, D N) of an infinitely dilute solute. In this case, Equation 1.98 then rednces to... 

The subsequent step is the calculation of the quantities that appear in the matrix elements A-. By means of Equation 4.15 through Equation 4.18, all chemical potential derivatives are calculated. The partial molar volumes are obtained by adding to the molar volume of the pure components the excess partial molar volumes, obtained through Equation 4.21 with = V, by applying the same procedure used to calculate In y. The mixture molar volume is then obtained as = x V + X2V2 + 3 3 and Kj. from Equation 4.5. The values of are finally obtained from Equation 4.12, for which the calculations of the concentrations, the determinant, and the cofactors are all straightforward. 

Smith, P. E. 2006a. Chemical potential derivatives and preferential interaction parameters in biological systems from Kirkwood-Buff theory. Biophysical Journal. 91, 849. 

Thermal de Broglie wavelength of species i Absolute activity of i Chemical potential of component i Chemical potential derivative (liquation 1.1)... 

In particular, variance can be expressed in terms of the chemical potential derivative for a gas composed of hard spheres over the sphere number concentration. 

The preferential hydration parameter is nearly independent of concentration, and for most small molecular weight excluded solutes is in the range 0.2-0.6 [6,74-77]. Thus, the chemical potential derivative, in terms of mass-based concentrations, becomes... 

Our starting point is the general expression for the composition dependence of the chemical potential derived in Section 3.8 ... 

The physical signiflcance of this result may be clearer if we return to the case of a regular solution defined by Eq. 6.3-6. Using the chemical potential derivative in Eq. 6.3-7, we find from Eq. 6.3-19 that... 

T—= chemical potential (derivate of free energy P, T wrt moles)... 

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