Concentration Dependence of Chemical Potential

6 Mass Action and Concentration Dependence of Chemical Potential 

the tendency to transform and therefore the chemical potential of substances increase according to how strongly concentrated they are. Conversely, the chemical potential goes down when the concentration of a substance decreases. We will use an example from everyday life to illustrate this. According to the values of the chemical potentials, pure water vapor must condense at room conditions  

However, if the vapor is diluted by air, the value of its potential decreases below that of liquid water. It can then undergo a phase transition to the gaseous state. It evaporates. /(H20 g) //(HaOll) is required for wet laundry (Fig. 6.1), wet dishes, and wet streets to dry (if no other causes such as direct sunlight play a role). 

Concentration Coefficient The influence of concentration c uptm the tendency p of a substance to transform can basically be described by a linear relation like it was 

Mass action is an effect which superposes with other less important influences which we will address later and all of which contribute to the concentration coefficient y- T = f +/ + / + . The symbol x above a term will be used here and in the following to denote the quantities dependent upon mass action when it is desirable to distinguish them from similar quantities of different origins. Mass action appears most noticeably at small concentrations where the other influences recede more and more until they can be totally neglected, f y If one wishes to investigate this effect as directly as possible, experiments should be carried out with strongly diluted solutions c c (= 1 kmol m ). 

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For snfficiently dilute solutions the concentration dependence of chemical potential is given similarly by... 

Solntions in which the concentration dependence of chemical potential obeys Eq. (3.6), as in the case of ideal gases, have been called ideal solutions. In nonideal solntions (or in other systems of variable composition) the concentration dependence of chemical potential is more complicated. In phases of variable composition, the valnes of the Gibbs energy are determined by the eqnation... 

The departure of a system from the ideal state is due to interaction forces between the individual particles contained in the system. The dependence of chemical potential of a species on its concentration can be written as... 

Further we looked at galvanic cells where it was possible to extract electrical energy from chemical reactions. We looked into cell potentials and standard reduction potentials which are both central and necessary for the electrochemical calculations. We also looked at concentration dependence of cell potentials and introduced the Nemst-equation stating the combination of the reaction fraction and cell potentials. The use of the Nemst equation was presented through examples where er also saw how the equation may be used to determine equilibrium constants. 

Light Scattering. Light scattering methods are a long established and reliable source of thermodynamic data on both dilute and concentrated polymer solutions. Scholte showed in detail how to obtain concentration and temperature dependences of chemical potential from the intensity (/) of scattered light, and also how the extrapolation (/ -> 0) furnishes spinodal temperatures even in... 

The other term [i T ln(a,)] accounts for the influence of concentration on chemical potential. It is intuitively imderstandable that the concentration of a component, as well as its molecular structure, should influence the chemical potential of that component (and the free energy of the mixture). A very important feature of this term is the notion that chemical potentials vary linearly with the logarithm of concentration, in ideal or nearly ideal solutions. Although many solutions are nonideal and phase separation may dramatically alter the dependence of chemical potential on composition (eliminating its influence altogether under some circumstances), it is, nevertheless, useful to become very familiar with this important equation. 

In regime IE, above the CMC the surface tension is nearly constant. This is due to the very weak dependence of chemical potential on concentration, and not because of saturation of the adsorbed layer, since this already occurs in regime II. To see this, we note that the chemical potential of the surfactant, in dilute solution, is given by Eq. (4.26). Below the CMC, the total surfactant concentration is equal to the unimer concentration, c = Cs, so chemical potential increases logarithmically with concentration as for any dilute solution. However, above the CMC, Eq. (4.37) in the limit gives... 

When a diffusion layer is present in a solution, the concentration of the latter at the double-layer interface is c rather than Cg in the bulk. Because of the concentration dependence of chemical affinity, a potential difference is developed across the diffusion layer. This potential difference is called concentration or diffusion overpotential and is given by,... 

Since the infinite dilution values D°g and Dba. re generally unequal, even a thermodynamically ideal solution hke Ya = Ys = 1 will exhibit concentration dependence of the diffusivity. In addition, nonideal solutions require a thermodynamic correction factor to retain the true driving force for molecular diffusion, or the gradient of the chemical potential rather than the composition gradient. That correction factor is ... 

In that way, the thermodynamic approach with the use of conformational term of chemical potential of macromolecules permitted to obtain the expressions for osmotic pressure of semi-diluted and concentrated solutions in more general form than proposed ones in the methods of self-consistent field and scaling. It was shown, that only the osmotic pressure of semi-diluted solutions does not depend on free energy of the macromolecules conformation whereas the contribution of the last one into the osmotic pressure of semi-diluted and concentrated solutions is prelevant. 

Besides the dependence of peak potential with solution pH, there is other evidence of the acid-base and redox coupling, namely the prediction of amine deprotonation during film oxidation. Deprotonation is a response to the creation of Os(III) sites that increment the concentration of positive charges in the film. This is an example of charge regulation a chemical equilibrium at the interface is displaced as the system tries to reduce its electrostatic charge. 

In reverse osmosis both solvent and solute diffuse because of gradients in their chemical potentials. For the solvent there is no gradient of chemical potential at an osmotic pressure of x at applied pressures p greater than 7r, there is such a gradient that is proportional to the difference p — ir. To a first approximation, the gradient of the solute chemical potential is independent of p and depends on the difference between concentrations on opposite sides of the membrane. This leads to the result that the fraction of solute retained varies as [1 + const./(p — 7r)] 1. Verify that the following data for a reverse osmosis experiment with 0.1 M NaCl and a cellulose acetate membrane follow this relationship ... 

The driving force for transport within the zeolite crystals appears to be the gradient of chemical potential rather than the concentration gradient, and, for systems with a nonlinear equilibrium isotherm, the diffusivity is therefore concentration dependent (6-8). 

Let us conclude with a short remark on the concentration dependence of the phenomenological potentials p, and, in particular, when point defects are involved. It is common and convenient to split the chemical potentials into two parts 1) r (P, T), which does not depend on the composition variables TV,-, and 2) the composition dependent term R T- In , which for ideal solutions (a, = N,) is simply R T- In TV,. For non-ideal solutions, one introduces the excess term R T- In ft = R T-In a —R T-Iri TV,-. Let us write In f, as a power series of the form... 

Let us now discuss some details of practical relevance. From the Gibbs phase rule, it is evident that crystals consisting of only one component (A) become nonvariant by the predetermination of two thermodynamic variables, which for practical reasons are chosen to be Pand T. In these one-component systems, it is easy to recognize the (isobanc) concentration dependence of the point defects on temperature. From the definition of the vacancy chemical potential for sufficiently small vacancy mole fractions Nv, namely //v = /A (P, T) + RT- In Vv, together with the condition of equilibrium with the crystal s inerL surroundings (gas, vacuum), one directly finds... 

The primary difference between D and D is a thermodynamic factor involving the concentration dependence of the activity coefficient of component 1. The thermodynamic factor arises because mass diffusion has a chemical potential gradient as a driving force, but the diffusivity is measured proportional to a concentration gradient and is thus influenced by the nonideality of the solution. This effect is absent in self-diffusion. 

The concentration-dependent part of the gradient of chemical potential is... 

The excess light scattered is related to the concentration dependence of the chemical potential by... 

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