Chemical potential change

If we change the (p. T,, V ) conditions of the mixture such that the chemical potential changes by d/q, then /q 0 would need to change by an equivalent amount to maintain equilibrium. That is... 

If we are interested in the excess chemical potential change for mutating mass mA into mass mB, we obtain... 

That is, the magnitude of the chemical potential changes by 1.89 kcal/mol on going from one standard state to the other. 

In a foregoing section, we mentioned that field forces (e,g., of the electric or elastic field) can cause an interface to move. If they are large enough so that inherent counterforces (such as interface tension or friction) do not bring the boundary to a stop, the interface motion would continue and eventually become uniform. In this section, however, we are primarily concerned with boundary motions caused by chemical potential changes. From irreversible thermodynamics, we know that the dissipated Gibbs energy of the discontinuous system is T-ab, where crb here is the entropy production (see Section 4.2). Since dG/dV = dG/dV = crb- T/ A < ), we have with Eqn. (4.8) at the boundary b... 

Under the assumption of local equilibrium, the Gibbs-Duhem relation applies, which places an additional constraint on chemical potential changes in Eq. 6.7 and implies that only IV — 1 of the m can vary independently ... 

Since chemical potential changes are related to relative activities by Mb = MeT + RT In aB... 

The seminal work of Flory and Rehner [1-3] set the framework for analysis of swollen polymer gels. In their approach it is assumed that in a crosslinked rubber at equilibrium the free energy of mixing and the change in elastic component are separable and additive. The chemical potential change on dissolution at equilibrium is written as ... 

Moreover, a partial proof was provided for the HSAB principle. Consider the formation of a diatomic molecule AB. Upon neglect of the external potential perturbation, the chemical potential change for the atoms A and B will be ... 

Tanford, C. (1982a). Simple model for the chemical potential change of a transported ion in active transport Proc. Natl. Acad. Sci. USA 79,2882-2884. 

The numerator of this expression is simply the negative of the total chemical potential change -A/x gained over displacement interval X. (This is... 

The maximum energy (or chemical potential) change possible for any solute occurs if the force acts over the entire separation path L, which gives... 

The infinitesimal chemical potential change of the surfactant is given by ... 

Grunwaldt et al. (2000) XAS Cu/ZnO Wetting transition during chemical potential change, structural dynamics + + + Methanol synthesis... 

An approximate thermodynamic evaluationbased on liquid water and crystalline sodium octanoate as reference states has recently evaluated the energy conditions in the premicellar aggregate. The calculations essentially were concerned with the free energy of a gaseous soap/water complex. No attempt was made to evaluate the chemical potential changes in any of the components when dissolved in the pentanol. 

Figure 1.4 A schematic diagram of chemical potential changes at the stationary occurrence of a stepwise reaction R Yq Y2 P, where R and P are the initial reactant and final product of the reaction, while Yq and Y2 are thermalized Intermediates. The minimums in the traditional potential energy profile relate to the standard chemical potentials of thermalized external reactants and intermediates. However, actual chemical transformations of the intermediates occur at stationary values Pyi and pvz (bold lines), the rates of these transformations being dependent on the difference of the corresponding thermodynamic rushes and the values of truncated rate constants e-,j (the latter are functions of standard chemical potentials of the transition states only).

The essence of the task therefore in compnting the chemical-potential change due to the interactions of the ionic species i with the ionic solution is the calculation of the electrostatic potential produced at a reference ion by the rest of the ions in solution. Theory must aim at this quantity. 

Thus, the problem of calculating the chemical-potential change due to the interactions between one ionic species and the assembly of all the other ions has been reduced to the following problem On a time average, how are the ions distributed around any specified ion If that distribution became known, it would then be easy to calculate the electrostatic potential of the specified ion due to the other ions and then, by Eq. (3.3), the energy of that interaction. Thus, the task is to develop a model that describes the equilibrium spatial distribution of ions inside an electrolytic solution and then to describe that model mathematically. 

The genius of Debye and Hiiekel lay in their formulation of a very simple but powerful model for the time-averaged distribution of ions in very dilute solutions of electrolytes. From this distribution they were able to obtain the electrostatic potential contributed by the surrounding ions to the total electrostatic potential at the reference ion and hence the chemical-potential change arising from ion-ion interactions [Eq.(3.3)]. Attention will now be focused on their approach. 

The ionic Cloud and the Chemical-Potential Change Arising from Ion-Ion Interactions... 

The Debye-Htickel ionic-cloud model for the distribution of ions in an electrolytic solution has permitted the theoretical calculation of the chemical-potential change arising from ion-ion interactions. How is this theoretical expression to be checked, i.e., connected with a measured quantity It is to this testing of the Debye-Huckel theory that attention will now be turned. 

The existence of ions in solution, of interactions between these ions, and of a chemical-potential change arising from ion-ion interactions have all been taken to be self-evident in the treatment hitherto presented here. This, however, is a modem point of view. The thinking about electrolytic solutions actually developed along a different path. 

It was argued that, in nonideal solutions, it was not just the analytical concentration jCj of species i, but its effective concentration xJi which determined the chemical-potential change - // . This effective concentration x,/- was also known as the activity a,- of the species i, i.e.. 

Thus, the chemical-potential change in going from the standard state to the final state can be written as... 

Hence, to analyze the physical significance of the activity coefficient term in Eq. (3.57), it is necessary to compare this equation with Eq. (3.52). It is obvious that when Eq. (3 52) is subtracted from Eq. (3.57), the difference [i.e., /r,- (real) - fij (ideal)] is the chemical-potential change arising from interactions between the solute particles (ions in the case of electrolyte solutions). That is. 

Thus, the activity coefficient is a measure of the chemical-potential change arising from ion-ion interactions. There are several well-established methods of experimentally determining activity coefficients, and these methods are treated in adequate detail in standard treatises (see Further Reading at the end of this section). 

Now, according to the Debye-Hiickel theory, the chemical-potential change arising from ion-ion interactions has been shown to be given by... 

In order to test these predictions, attention was drawn to an empirical treatment of ionic solutions. For solutions of noninteracting particles, the chemical-potential change in going from a solution of unit concentration to one of concentration X/ is described by the equation... 

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