Nonideal solutions chemical potential

Perhaps the most significant of the partial molar properties, because of its appHcation to equiHbrium thermodynamics, is the chemical potential, ]1. This fundamental property, and related properties such as fugacity and activity, are essential to mathematical solutions of phase equihbrium problems. The natural logarithm of the Hquid-phase activity coefficient, Iny, is also defined as a partial molar quantity. For Hquid mixtures, the activity coefficient, y, describes nonideal Hquid-phase behavior. 

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 ... 

These expressions comprise the nonideal terms in the previous equations for the chemical potential, Eqs. (30) and (31 ). They may therefore be regarded as the excess relative partial molar free energy, or chemical potential, frequently used in the treatment of solutions of nonelectrolytesi.e, the chemical potential in excess (algebraically) of the ideal contribution, which is —RTV2/M in dilute solutions. 

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 behaviour of most metallurgically important solutions could be described by certain simple laws. These laws and several other pertinent aspects of solution behaviour are described in this section. The laws of Raoult, Henry and Sievert are presented first. Next, certain parameters such as activity, activity coefficient, chemical potential, and relative partial and integral molar free energies, which are essential for thermodynamic detailing of solution behaviour, are defined. This is followed by a discussion on the Gibbs-Duhem equation and ideal and nonideal solutions. The special case of nonideal solutions, termed as a regular solution, is then presented wherein the concept of excess thermodynamic functions has been used. 

The chemical potential of the solute in a nonideal solution is given by... 

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) . 

With respect to an enzyme, the rate of substrate-to-product conversion catalyzed by an enzyme under a given set of conditions, either measured by the amount of substance (e.g., micromoles) converted per unit time or by concentration change (e.g., millimolarity) per unit time. See Specific Activity Turnover Number. 2. Referring to the measure of a property of a biomolecule, pharmaceutical, procedure, eta, with respect to the response that substance or procedure produces. 3. See Optical Activity. 4. The amount of radioactive substance (or number of atoms) that disintegrates per unit time. See Specific Activity. 5. A unitless thermodynamic parameter which is used in place of concentration to correct for nonideality of gases or of solutions. The absolute activity of a substance B, symbolized by Ab, is related to the chemical potential of B (symbolized by /jlb) by the relationship yu,B = RTln Ab where R is the universal gas constant and Tis the absolute temperature. The ratio of the absolute activity of some substance B to some absolute activity for some reference state, A , is referred to as the relative activity (usually simply called activity ). The relative activity is symbolized by a and is defined by the relationship b = Ab/A = If... 

The extreme nonidealities characteristic of electrolyte solutions warn of the dangers inherent in approximations commonly employed in general chemistry. Except in the crossover region of intermediate m where y + 1, blithe replacement of activity by molarity is seldom justified for strong electrolytes. Elementary treatments of acid dissociation, solubility products, and the like may therefore be subject to considerable error unless the realistic variations of chemical potential with concentration are properly considered. 

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. 

In the case of nonideal solutions, we can relate the activity aA of any component A of the solution to the chemical potential /ia of that component hy the equation... 

Determination of T y. In the formulation of the phase equilibrium problem presented earlier, component chemical potentials were separated into three terms (1) 0, which expresses the primary temperature dependence, (2) solution mole fractions, which represent the primary composition dependence (ideal entropic contribution), and (3) 1, which accounts for relative mixture nonidealities. Because little data about the experimental properties of solutions exist, Tg is usually evaluated by imposing a model to describe the behavior of the liquid and solid mixtures and estimating model parameters by semiempirical methods or fitting limited segments of the phase diagram. Various solution models used to describe the liquid and solid mixtures are discussed in the following sections, and the behavior of T % is presented. 

For a nonideal solution, the gradient of the chemical potential is given by [9]... 

For concentrations sufficiently high that the infinite dilution approximation breaks down, Eq. 2.17 must be modified by the incorporation of an activity coefficient y, which compensates for additions to the chemical potential from such nonidealities as solute-solute interactions. We have... 

In earlier sections we have dealt with a variety of methods for determining the chemical potential of species in nonideal solutions. This now provides the groundwork for the study of equilibrium constants. As might be expected, the large variety of ways in which these chemical potentials may be specified is reflected in many different ways for defining equilibrium constants. As usual, care will have to be taken to ensure that these different specifications will actually lead to identically the same characterization of a given physical system. 

For nonideal liquid and solid solutions, two identical equations of chemical potential can be used ... 

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.. 

If the activity of a solute or ion were ideal, it could be taken as equivalent to the molal concentration, of the i ion or solute. However, interactions with other ions and with the solvent water are strong enough to cause nonideal behavior and the characteristic property relating concentration to chemical potential is the activity coefficient, y, ... 

Also, it is customary to refer all thermodynamic properties to chemical potentials of all species, whether in the pure state or in solution, to their values under standard conditions. In that case the equilibrium constant will be designated, as before, by fCx and the pressure in the above equations is set at P = bar. Finally, it is possible to specify compositions in terms of molarity c, or molality m, leading to the specification of Kc and Km or Kc and Km - The resulting analysis becomes somewhat involved and will not be taken up here interested readers should read Section 3.7 for a full scale analysis of the treatment of nonideal solutions. 

Much of the material covered in Chapter 2 will now be repeated in a form applicable to nonideal solutions we concentrate particularly on the proper characterization of the chemical potentials of the constituents. Once this quantity is known all thermodynamic properties of the system may be determined. Particular emphasis is placed on the many alternative concentration units that may be adopted. Pains must be taken to ensure that the various final mathematical formulations uniquely describe a given experimental situation. 

We begin our specification of the chemical potential of nonideal solutions in the condensed state that is based on the canonical formulation of ideal solutions, introduced in Section 2.5,... 

In Section 3.4 we have displayed many different modes for characterizing the chemical potential of a given species in a nonideal solution. While these various descriptions all look different, surely all physical predictions must be independent of the particular standard or reference state which has been chosen, and surely they cannot be allowed to depend on the choice of concentration units. We now adopt restrictions that guarantee that the chemical potential of any species i in any solution relative to the standard potential shall indeed be unique, i.e., invariant under any change in choice of concentration units. 

For a nonideal electrolyte solution, for example, NaCl, the chemical potential on the molal scale must then be expressed as... 

The present paper is concerned with mixtures composed of a highly nonideal solute and a multicomponent ideal solvent. A model-free methodology, based on the Kirkwood—Buff (KB) theory of solutions, was employed. The quaternary mixture was considered as an example, and the full set of expressions for the derivatives of the chemical potentials with respect to the number of particles, the partial molar volumes, and the isothermal compressibility were derived on the basis of the KB theory of solutions. Further, the expressions for the derivatives of the activity coefficients were applied to quaternary mixtures composed of a solute and an ideal ternary solvent. It was shown that the activity coefBcient of a solute at infinite dilution in an ideal ternary solvent can be predicted in terms of the activity coefBcients of the solute at infinite dilution in subsystems (solute + the individual three solvents, or solute + two binaries among the solvent species). The methodology could be extended to a system formed of a solute + a multicomponent ideal mixed solvent. The obtained equations were used to predict the gas solubilities and the solubilities of crystalline nonelectrolytes in multicomponent ideal mixed solvents. Good agreement between the predicted and experimental solubilities was obtained. 

Starting from equations (20.4) for the chemical potentials of a nonideal solution, we have instead of (21.3)... 

The term RTInyt accounts for the nonideality of the solution (it is also referred to as the partial molar energy). As it can be seen from Equation (5), the terms on the right side, except Tlrvyt, are calculated from the pure properties. However, in order to have an accurate prediction of the chemical potential of any species in the solution, RT/ny, should be known as well. In general, the activity coefficient is a function of temperature and composition and to a much less extent, the pressure. Because the activity coefficient is defined for a liquid solution, the pressure has very little effect on it. However, temperature and mole fractions of the species have significant effects on the activity coefficient of each species in a solution. 

Here D. is the diffusion coefficient of component i, fi. is its chemical potential and X. its mole fraction. Bj is a mobility termj which is always positive. Introducing the chemical potential of a nonideal solution, which is given by ... 

Thermodynamics is used in the analysis of electrochemical cells (1) to predict which electrode reactions occur spontaneously in the anodic and cathodic directions if the two electrodes are in equilibrium with their respective adjacent solutions and are connected to one another via an external wire, and (2) to quantify chemical potentials and activity coefficients in nonideal electrolytic solutions. 

A substance in solution has a chemical potential, which is the partial molar free energy of the substance, which determines its reactivity. At constant pressure and temperature, reactivity is given by the thermodynamic activity of the substance for a so-called ideal system, this equals the mole fraction. Most food systems are nonideal, and then activity equals mole fraction times an activity coefficient, which may markedly deviate from unity. In many dilute solutions, the solute behaves as if the system were ideal. For such ideally dilute systems, simple relations exist for the solubility of substances, partitioning over phases, and the so-called colligative properties (lowering of vapor pressure, boiling point elevation, freezing point depression, osmotic pressure). 

The interactions of ions with water molecules and other ions affect the concentration-dependent (colligative) properties of solutions. Colligative properties include osmotic pressure, boiling point elevation, freezing point depression, and the chemical potential, or activity, of the water and the ions. The activity is the driving force of reactions. Colligative properties and activities of solutions vary nonlinearly with concentration in the real world of nonideal solutions. 

Question. What is the lowering of the chemical potential in the nonideal solution with y = 0.93 Pa Pa... 

Nonideal Solutions Atmospheric aerosols are usually concentrated aqueous solutions that deviate significantly from ideality. This deviation from ideality is usually described by introducing the activity coefficient, y and the chemical potential is given by... 

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