The chemical potential of a solute

Raoult s law provides a good description of the vapor pressure of the solvent in a very dilute solution, when the solvent A is almost pure. However, we cannot in 

Here K, which is called Henry s law constant, is characteristic of the solute and chosen so that the straight line predicted by eqn 3.13 is tangent to the experimental curve at Xb = 0 (Fig. 3.28). Henry s law is usually obeyed only at low concentrations of the solute (close to Xg = 0). Solutions that are dilute enough for the solute to obey Henry s law are called ideal-dilute solutions. 

The Henry s law constants of some gases are Ksted in Table 3.2. The values given there are for the law rewritten to show how the molar concentration depends on the partial pressure, rather than vice versa  

The Henry s law constant, Kh, is commonly reported in moles per cubic metre per kilopascal (mol m kPa )- This form of the law and these units make it very easy to calculate the molar concentration of the dissolved gas, simply by multiplying the partial pressure of the gas (in kilopascals) by the appropriate constant. Equation 3.14 is used, for instance, to estimate the concentration of Oj in natural waters or the concentration of carbon dioxide in blood plasma. 

The number of significant figures in the result of a calculation should not exceed the number in the data. 

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This is done by starting with equation (6.84), which relates the chemical potential of a solute in solution with activity to the standard state chemical potential... 

With the help of the o-profile the surface integral can be elegantly transformed into a o-integral (right side in Eq. 11), but we should keep in mind that the chemical potential of a solute in a solvent is essentially a surface integral of a solvent specific function over the surface of the solute. This fact is important for the analysis of the problem of solubiUty prediction. 

Partitioning In Octanol-Water Systems. At equilibrium, the chemical potential of a solute (defined as pj = p + RT In y x ) Is equal In the octanol and the water phase. Hence, we may write... 

As seen from Eq. (130) an activity coefficient may deviate significantly from unity at higher salt concentrations. The activity coefficient can therefore also be used as a measure of the deviation of the salt solution from a thermodynamically ideal solution. If the chemical potential of a solute in a (pressure-dependent) standard state of infinite dilution is /x°, we find the standard partial molar volume from... 

At equilibrium, the chemical potential of a solute (naphthalene, etc.) will be given as... 

The chemical potential of a solute in equilibrium with its solid or liquid phase is equal to the chemical potential of the neat compound. It is therefore hardly surprising that some papers claiming to report on on water chemistry merely report reactivity comparable to reactivity of the neat reactants. In the author s opinion, the term on water chemistry would preferably be reserved for those processes for which additional rate-enhancing effects are found. 

Although I did not know about the concept of the combinatorial contribution, I recognized the need for such a correction even in the initial version of COSMO-RS [C9]. Since at that time I only had in mind the calculation of infinite-dilution partition coefficients and of vapor pressures, I only cared about a solvent-size correction in pure solvents. I thought of two different solvent-size effects influencing the chemical potential of a solute X in a solvent S. The first is quite obvious—in 1 mol of a homogeneous liquid S,... 

When the concentration of a multicomponent system is expressed in terms of the molalities of the solutes, the expression for the chemical potential of the individual solutes and for the solvent are somewhat different. For dilute solutions the molality of a solute is approximately proportional to its mole fraction. (The molality, m, is the number of moles of solute per kilogram of solvent. When two or more substances, pure or mixed, may be considered as solvents, a choice of solvent must be clearly stated.) In conformity with Equation (8.68), we then express the chemical potential of a solute in a solution at a given temperature and pressure as... 

As an approximation, he considers the chemical potential of a solute as being the sum of the potential differences of the various groups of which the molecule is composed. In consequence, all isomers having the same groups would be expected to have the same partition coefficient. 

As shown in Example 1, the chemical potential of a solute can be of a more complicated form than given by Eqs. (43)-(45), even though the solution shows ideally dilute behavior. This can result from a transformation of the substance when it dissolves in the solution. 

For any imaginary ideal solution of an electrolyte, at any given T and P, in which all activity and osmotic coefficients are unity, we can write for the chemical potential of a solute s. 

The chemical potential of a solute can be expressed in terms of molality... 

Consider the boundary between an aqueous solution of a nonionic surfactant and a hydrophobic phase, air or oil. The dividing surface is usually chosen to be the equimolecular surface with respect to water, that is F , = 0. Then Equation 5.1 reduces to da = -F,r/pj, where the subscript 1 denotes the surfactant. Because the bulk surfactant concentration is usually not too high, we can use the expression for the chemical potential of a solute in an ideal solution Pj = + A rinc, where k... 

In order to understand the equilibrium properties of gas and liquid solutions we should explain how the chemical potential of a solution varies with its composition. For gases in a closed system, dG = VdP - SdT, as we know from Equation (126). When the temperature is constant, dT = 0, we can calculate the Gibbs free energy at one pressure in terms of its value at another pressure... 

For liquid solutions, we need to express how the chemical potential of a solution va ries with its composition. In order to derive a useful expression, we should remember that, in equilibrium, the chemical potential of a substance in the liquid phase must be equal... 

If a solute has a partial vapour pressure proportional to its mole fraction it is said to follow Henry s Law. Raoult s Law may be regarded as a special case of Henry s Law. The chemical potential of a solute that follows Henry s Law is given by the expression... 

There is no explicit thermodynamic derivation of the expression for the chemical potential of a solute in solution. However, experimentally it is found that the expression for the emf of a reversible cell has an nRT loge term in it. Since the emf is directly related to AG for the cell reaction, AG = —zFE, then the expression for p. for a solute is given by analogy as ... 

It follows that Uj = nij as nij = 0. Thus, for the chemical potential of a solute in the practical system, we have... 

The solubility method of determining medium effects is more generally applicable. The chemical potentials of a solute MX in saturated solution in solvent S and in water are equal since each is in equilibrium with the same solid phase, provided no crystal solvates are formed (sect. 2.3.1), thus... 

The superscript ( ) in Eqs. (89) and (90) indicates that the solute Y and the solution are considered in equal states of aggregation, which is not necessarily the stable state of the pure compound Y, with chemical potential /xf The general definition of the chemical potential of a solute Y in solution (Yi is the solvent S) is based on the activity coefficient of the type given by Eq. (90). 

The same can be said for all the expressions for jx- ix° in Equations (8.30). They all express the difference between the chemical potential of a solute species in a real system, and the same potential in an ideal system under the same conditions. The term residual function is strictly speaking applied only when the ideal system is an ideal gas, so differences from other states such as infinitely dilute solutions or pure phases are called deviation functions (Ewing and Peters, 2000). 

It is also possible to derive the Nernst distribution law from thermodynamic considerations using the concept of free energy (Lewis and Randall 1923). At equilibrium, the chemical potential of a solute X has to be the same in the aqueous and the organic phase, i.e.. 

The validity of (3.76) actually goes far beyond mixtures of perfect gases. Systems in which the chemical potentials of the components can be expressed by (3.76) are called ideal systems. A special case of ideal systems are dilute solutions. A statistical derivation of (3.76) for dilute solutions may be found in LANDAU-LIFSHITZ, Theoretical Physics, Vol. V (1968). In dilute solutions, the mole fraction of a solute is approximately given as x. = N. /N where is the number of moles of the solvent. This enables us to rewrite the chemical potential of a solute approximately in terms of its concentration c. as... 

Again in each of these equations, we replace the expression in brackets, which depends on T and p but not on composition, with the chemical potential of a solute reference state ... 

We begin by obtaining the chemical potential of a solution. The general expression for the chemical potential of a substance is T) = p°(po5 T)+... 

The key to linking the properties of a solution to those of a gas and setting up an expression for the chemical potential of a solute is the work done by the French chemist Fran( ois Raoult (1830-1901), who spent most of his fife mccisuring the vapor pressures of solutions. He measured the partial vapor pressure, pj, of each component in the mixture, the partial pressure of the vapor of each component in... 

Henry s law lets us write an expression for the chemical potential of a solute in a solution. We show in the following Justification that the chemical potential of the solute when it is present at a mole fraction is... 

It is useful to summarise the assumptions considered for the thermodynamic analysis discussed above, and the main results obtained. Based on the stress strain relationship for the glassy system in the form of equation 3, the chemical potential of a solute component in the polymeric mixture may be calculated as the derivative of the specific non-equilibrium Helmholtz free energy with respect to the moles of solute per polymer mass Yi at constant temperature, pressure and specific volume, as expressed in eqaution 12. On the other side, under the same assumption, the nonequilibrium Helmholtz free energy has a unique value at given temperature, specific volume and composition, whatever is the pressure of the system, as stated in equation 13. 

According to the above three theories, the chemical potentials of a solute are composed of an intrinsic free energy of a pure component, an interaction energy between solute and solvent, and a concentration term. The quantities measured experimentally do not represent the chemcal potentials of the solute but rather their differences, and depend strongly on the interaction with solvent. The following section discusses how the solubility is determined. 

Equation (21) is of enormous practical value. The knowledge of the chemical potential of a solute X in almost arbitrary solvents allows for the calculation of almost any equilibrium of solutes between different solvents and between solvents and vapors. Thus it allows for the calculation of vapor pressures, partial pressures of components over mixed fluids, and partition coefficients of all kinds. A few examples are given below. Beyond the calculation of free energies and hence of chemical potentials equation (20) also is the key to the calculation of heats of solution, and even of surface tensions. For the sake of brevity we do without details here. 

Derivation of a partial molar Helmholtz free energy equation for an ideal solution will provide a tool by which ideal and real solution behavior can be differentiated. Specifically, we will make use of the fact that the partial molar enthalpy of a real solution will depend on the type and eoncentration of solutes in a solution while for an ideal solution, the partial molar enthalpy for a solute is independent of the solution composition [18]. As a brief proof of this ideal solution property, consider the defining Eq. (12) for the chemical potential of a solute, Y y, in an ideal solution ... 

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