In two phases, at equilibrium

Solubility Equilibria The Solubility Product Principle.—It was seen on page 133 that the chemical potential of a solid is constant at a definite temperature and pressure consequently, when a solution is saturated with a given salt Mv A, the chemical potential of the latter in the solution must also be constant, since the chemical potential of any substance present in two phases at equilibrium must be the same in each phase. It is immaterial whether this conclusion is applied to the undissociated molecules of the salt or to the ions, for the chemical potential is given by... 

This assumption is supported by many experimental studies. The concentrations of a component in two phases at equilibrium are related by equilibrium ratios, which remain approximately constant over a substantial concentration range. 

According to equation (3.49) the chemical potentials in two phases at equilibrium are equal. Thus, from equation (3.52),... 

Equality of the chemical potential of a component/ species in two contiguous immiscible phases constituting two regions rarely implies equality of concentration. In fact, the relation between the different concentrations of different species in two phases at equilibrium in a separator will be developed for various types of separation phenomena in Section 4.1 using these relations. 

In Section 1.5, we learned that P and T are not independent for a pure species that exists in two phases at equilibrium. We now wish to come up with an expression relating the pressure at which two phases can coexist to the temperature of the system. This expression will allow us to calculate, for example, how the saturation pressure changes with temperature. Recall that the saturation pressure, is defined as the unique pressure at which pure species boils at a given temperature. 

For mixtures, the calculation is more complex because it is necessary to determine the bubble point pressure by calculating the partial fugacities of the components in the two phases at equilibrium. 

An interesting extension of the original methodology was proposed by Lopes and Tildesley to allow the study of more than two phases at equilibrium [21], The extension is based on setting up a simulation with as many boxes as the maximum number of phases expected to be present. Kristof and Liszi [22, 23] have proposed an implementation of the Gibbs ensemble in which the total enthalpy, pressure and number of particles in the total system are kept constant. Molecular dynamics versions of the Gibbs ensemble algorithm are also available [24-26]. 

In many practical situations solute A may dissociate, polymerize or form complexes with some other component of the sample or interact with one of the solvents. In these circumstances the value of KD does not reflect the overall distribution of the solute between the two phases as it refers only to the distributing species. Analytically, the total amount of solute present in each phase at equilibrium is of prime importance, and the extraction process is therefore better discussed in terms of the distribution ratio D where... 

The temperature-composition diagram can be used to calculate the composition of the two-phase system according to the amount of each solvent present. For example, at temperature T, the composition of the most abundant phase, which consists of liquid A saturated with liquid B, is represented by the point a and the composition of the minor phase, consisting of liquid B saturated with liquid A, is represented by point a. The horizontal line connecting these two points is known as a tie line as it links two phases that are in equilibrium with each other. From this line the relative amounts of the two phases at equilibrium can be calculated, using the lever rule, under the conditions described by the diagram. The lever rule gets its name from a similar rule that is used to relate two masses on a lever with their distances from a pivot, i.e. ... 

The temperature also affects the composition of the two phases at equilibrium, but the effect is not equivalent in all systems. In the example shown in Figure 2.6, raising the temperature increases the solubility of the two phases and this is what is usually observed. The diagram shows that by heating the system, more of A dissolves in B and vice versa. However, other solvent pairs become less miscible with raised temperature, for example, water and ethylamine. In the case of these liquid pairs, the temperature-composition diagram is essentially reversed, as shown in Figure 2.7. 

The first term in parentheses on the right side of equation 5.213 is the distribution coefficient (K ), and the second groups activity coefficients related to the mixing behavior of components in the two phases. The equilibrium constant is thus related to the interaction parameters of the two phases at equilibrium. For example, the equilibrium between two regular mixtures is defined as... 

Let us call the melt phase a and the solid phase with complete immiscibility of components y. P is constant and fluids are absent. The Gibbs free energy relationships at the various T for the two phases at equilibrium are those shown in figure 7.2, with T decreasing downward from Ty to Tg. The G-X relationships observed at the various T are then translated into a T-X stability diagram in the lower part of the figure. 

In an attempt to minimize the compositional effects of phases on trace partitioning, Henderson and Kracek (1927) also introduced the concept of normalized partition coefficient D, which compares the relative trace/carrier (Tr/Cr) mass distributions in the two phases at equilibrium—i.e.. 

Liquid-Liquid Extraction Principle. If a liquid solvent which is either immiscible or only partially miscible is mixed with a solution containing solute A, the solute will distribute between the two liquids until equilibrium is established. The solute s concentration in the two phases at equilibrium will depend on its relative affinity for the two solvents. Although... 

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. 

Partition coefficients relate the concentrations in each of the two phases at equilibrium ... 

The equations derived for calculating the fractions of total i present in each phase at equilibrium in a two-phase system (Eqs. 3-62 and 3-63) can be easily extended to a multiphase system containing n phases (e.g., to a unit world ). If we pick one phase (denoted as phase 1) as the reference phase and if we use the partition constants of i between this phase and all other phases present in the system ... 

The wide varity of the properties of chemical compounds does not enable the use of a universal apparatus for the measurement of thermodynamic properties for pure components and mixtures at high pressure. In the case of two-phase equilibria like vapour-liquid equilibria, the typical set of data to be determined is the pressure, the temperature, and the composition of the two phases at equilibrium. Some experimental apparatus also allows the... 

Since the fugacity and activity coefficients are mathematically complex functions of the compositions, finding corresponding compositions of the two phases at equilibrium when the equations are known requires solutions by trial. Suitable procedures for making flash calculations are presented in the next section, and in greater detail in some books on thermodynamics, for instance, the one by Walas (1985). In making such calculations, it is usual to start by assuming ideal behavior, that is,... 

Waals forces are acting. Since the attractive energy term is smaller in this case, the equilibrium particle concentrations in the two phases at equilibrium do not differ as much as in the presence of free polymer. 

When it is placed at a temperature below the melting point of the solute (i.e., the solute becomes a solid), the solubility of this solid in a given liquid solvent is therefore limited. Thus, the two phases at equilibrium are the saturated solution containing the solute and the solid solute. At equilibrium, the chemical potentials of the solute in the saturated solution and the solid are equal to ... 

On cooling dilute solutions, the solvent usually separates as the solid phase. There are two phases at equilibrium solid solvent and liquid solution with a solute. Assume that the solute does not dissolve in the solid solvent. The thermodynamic approach to this equilibrium is identical to the one for saturated solutions as described in Section 3.1.1. Following the same reasoning as in Section 3.1.1, Equation (3.1) to Equation (3.6) can be applied to the solvent (component 1), and the freezing point of an ideal solution becomes ... 

The Ostwald partition coefficient, L, is a widely used and physically intuitive measure of gas solubilities and oil-water partition coefficients. It is defined as the ratio of concentrations of a solute between two phases at equilibrium. These two phases can be the ideal gas and a liquid phase, in which case the Ostwald partition coefficient gives the gas solubility, or two immiscible liquids - e.g., oil and water - in which case L is an oil-water partition coefficient. For the gas solubility of component 2 in liquid 1,... 

Consider a polymeric species with degree of polymerization i in solution. The homogeneous solution can be caused to separate into two phases by decreasing the affinity of the solvent for the polymer by lowering the temperature or adding some poorer solvent, for example. If this is done carefully, a small quantity of polymer-rich phase will separate and will be in equilibrium with a larger volume of a solvent-rich phase. Tlie chemical potential of the i-mer will be the same in both phases at equilibrium, and the relevant Flory-Huggins expression is... 

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