Activity or chemical potential

Since consideration of thermodynamics demand that the activity (or chemical potential) of a solute should be equal in the two phases at equilibrium, a distribution coefficient of other than unity implies that the solute must have different activity coefficients in the two phases. The origin of such a difference usually resides in the degree of interaction between the solute and the two solvents. 

Wagner factor — or thermodynamic factor, denotes usually the - concentration derivative of -> activity or - chemical potential of a component of an electrochemical system. This factor is necessary to describe the - diffusion in nonideal systems, where the - activity coefficients are not equal to unity, via Fick s laws. In such cases, the thermodynamic factor is understood as the proportionality coefficient between the selfdiffusion coefficient D of species B and the real - diffusion coefficient, equal to the ratio of the flux and concentration gradient of these species (chemical diffusion coefficient DB) ... 

Concentration Cells with a Single Electrolyte Amalgam Concentration Cells.—In the concentration cells already described the e.m.p. is a result of the difference of activity or chemical potential, i.e., partial molal free energy, of the electrolyte in the two solutions it is possible, however, to obtain concentration cells with only one solution, but the activities of the element with respect to which the ions in the solution are reversible are different in the two electrodes. A simple method of realizing such a cell is to employ two amalgams of a base metal at different concentrations as electrodes and a solution of a salt of the metal as electrolyte thus... 

The numerical value of kt in (2-3) depends on how activity is defined and on the units in which concentration is expressed (molarity, mole fraction, partial pressure). Measurement of the absolute activity, or chemical potential, of an Individual ion is one of the classical unsolved problems. Since we cannot measure absolute ion activity, we are then necessarily interested in the next best—comparative changes in activities with changing conditions. To obtain comparative values numerically, we measure activity with respect to an arbitrarily chosen standard state under a given set of conditions of temperature and pressure, where the substance is assigned unit activity. The value of ki in (2-3) thus depends on the arbitrary standard state chosen accordingly, the value of the equilibrium constant also depends on the choice of standard states. 

Lithium ion conductors are very much desired in commercial applications because of the relatively high open circuit voltages (up to 4 V) that can be achieved in electrochemical devices employing lithium-based anodes with high chemical activities (or chemical potentials). Many of the polycrystalline lithium-based solid electrolytes, that have been studied to date have ionic resistivities at 300°C in the range between 20 and 200 fl cm. While thin-film applications for these materials are possible, the biggest drawback associated with lithium ion conductors is their chemical and electrochemical instability over time at temperatures of interest in environments very high in lithium chemical activity. 

The thermodynamic term of widest use in soil chemistry is the free energy, or more explicidy, the Gibbs free energy. This is the energy of a substance or a reaction that, at constant temperature and pressure, is available for subsequent use. Energy drives chemical reactions and AG is the most widely useful. It is directiy related to (1) the activity or chemical potential, (2) the energy of formation of compounds, (3) the equilibrium constant of a reaction, and (4) the electrode potential. The first three are discussed here the electrode potential is discussed in Chapter 4. 

A distinction has to be made here between chemical and virtual potentials. As discussed at some length in Chap. 5, since the activity or chemical potential of an individual ion or charged defect cannot be defined, it follows that its chemical potential is also undefined. The distinction, however, is purely academic, because defect reactions are always written so as to preserve site ratios and electroneutrality, in which case it is legitimate to discuss their chemical potentials. 

In all of the situations shown in Figure 7.5 the principal question is where and under what conditions will the sulphide phase form. It is, therefore, important to define the general conditions for sulphide formation. This will occur whenever the local sulphur activity, or chemical potential, is greater than that required to form sulphide with the metal at the activity that obtains at that site. This condition is stated in Equation (7.21). 

Here D is known as the component diffusion coefQcient. The importance of this dehnition lies in the fact that Nemst-Einstein proportionality between a diffusion coefficient and a mobility has been retained, even though the condition of ideality has been relaxed. This is important since the apparent violation of the Nemst-Einstein equation in nonideal solutions is not a failure of the proportionality between mobility and mean displacement it is a weakness in the method of formulating the driving force for diffusion in terms of a concentration gradient (Pick s law) rather than in terms of an activity or chemical potential gradient. 

It is not straightforward to calculate phase equilibria or chemical potentials in a simulation, and special techniques must be used. This has been an active research area over the past few years [15]. Several methods have been proposed, and these can be divided into direct methods, in which the coexistence properties of the phases are calculated directly, and indirect methods, where the chemical potential is first calculated and then used to determine the phase equilibrium conditions (Table 4). 

Since the adsorption of a protein to a surface is basically a reversible process, changes of pH, ionic strength, substrate concentration, temperature, etc. may detach the biomolecule from the carrier (Carr and Bowers, 1980). In addition to the simplicity of the procedure, the advantage of adsorptive immobilization is that it does not need nonphysiologi-cal coupling conditions or chemicals potentially impairing enzyme or cell functions. An activity loss is therefore seldom observed. 

It should be evident from the examples in Chapters 10, 11, and 12 that the evaluation of species fugacities or partial molar Gibbs energies (or chemical potentials) is central to any phase equilibrium calculation. Two different fugacity descriptions have been used, equations of. state and activity coefficient models. Both have adjustable parameters. If the values of these adjustable parameters are known or can be estimated, the phase equilibrium state may be predicted. Equally important, however, is the observation that measured phase equilibria can be used to obtain these parameters. For example, in Sec. 10.2 we demonstrated how activity coefficients could be computed directly from P-T-x-y data and how activity coefficient models could be fit to such data. Similarly, in Sec. 10.3 we pointed out how fitting equation-of-state predictions to experimental high-pressure phase equilibrium data could be used to obtain a best-fit value of the binary interaction parameter.. /"... 

Next we can finally see how the activity coefficient relates to the Margules equations for this case. Recall from Chapter 9 that the partial molar quantity of one component in a binary solution can be obtained graphically from the tangent (as with the chemical potentials /j,b and /ta in the coexisting solutions of Figure 15.3a). From equation (9.6), the partial molar free energy or chemical potential of component A in a solution of A and B is given by... 

K being a constant. When the vapoui-s are not ideal, vapour pressures are replaced by thermodynamic activities. Substitution of vapour pressure or activity with chemical potential yields the relation between a and thermodvnamic ftmctions ... 

Let Pi (l)u be the volume fraction of inactive material and let P2(N)v denote the dimensionless concentration of activated Ai-mers (u is the monomer volume). The grand potential of the inactive monomers isgivenby= pi(l)[ln pi(1)l> - 1 - lnZi(l) - /Sp,], while that of the active ones, Q.2, is given by Eq. (1), where as before we replace p N) by P2 N). The equilibrium distribution of material over the active and inactive states is determined by the set of equations 5f2/5pi(l) = 0 and SQ/Sp2 N) = where = f2i - - Here, i/ acts as a Lagrange multiplier (or chemical potential) that keeps the number of active chains constant. 

The criteria for co-crystal thermodynamic stability were presented in Section 11.3. An important feature of co-crystals is that they coexist in equilibrium with solution. This occurs when their molar free energy or chemical potential is equal to the sum of the chemical potentials of each co-crystal component in solution. Thus, the individual component chemical potentials in a solution saturated with co-crystal can vary as long as their sum is constant. In terms of activities, it is the activity product that is constant. 

Many biochemical reactions can be induced by temperature increase in foods Maillard reactions, vitamin degradation, fat oxidation, denaturation of thermally unstable proteins (resulting in variation of solubility or of the germinating power of grains, for example), enzyme reactions (which can either be promoted or inhibited), and so on. Some of these biochemical reactions generate components suitable, for example, for their sensory properties (flavor development) others may be more or less undesirable for nutritional or potential toxicity reasons (vitamin losses, changes in color, taste or aroma, formation of toxic compounds). All the reactions are linked to the simultaneous evolution of product composition, temperature and water content (or chemical potential, or water activity), these factors varying diflferently from one point to another, from the center to the surface of the products. 

At finite concentrations this formula needs modifying in two ways. In the first place, diffusion is governed by the osmotic pressure, or chemical potential, gradient (not, strictly, by the concentration gradient), so that the mean activity coefficient of the electrolyte must be taken into account. In the second place, ionic atmosphere effects must be allowed for. In diffusion, unlike conductance, the two ions are moving in the same direction, and the motion causes no disturbance of the symmetries of the ionic atmospheres there is therefore no relaxation effect. There is a small electrophoretic effect, however, the magnitude of which for dilute solutions has been worked out by Onsager, and the most accurate measurements support the extended formula based on these corrections. 

To proceed fiirther, to evaluate the standard free energy AG , we need infonnation (experimental or theoretical) about the particular reaction. One source of infonnation is the equilibrium constant for a chemical reaction involving gases. Previous sections have shown how the chemical potential for a species in a gaseous mixture or in a dilute solution (and the corresponding activities) can be defined and measured. Thus, if one can detennine (by some kind of analysis)... 

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