## Theory and Equations of Inorganic Solvent Extraction

The solvent extraction process to separate and/or preconcentrate an analyte of interest is performed by using two immiscible solvents. A complex (typically neutral in charge) is formed with the element of interest, typically in an aqueous solution and will partition into a mutually insoluble (organic solvent) phase. The Nernst partition (or distribution law) states that at equilibrium a given solute will be distributed between two essentially immiscible liquids according to the following equation:

Kd = 7o[A]o/yaq[A]aq where KD is the distribution coefficient (also called the partition coefficient) and [A] is the concentration of the analyte, y are activity coefficients, subscript 'o' denotes organic phase and the 'aq' subscript denotes aqueous phase. The above equation holds true in only the most rigorously well-defined thermodynamic systems. For simplicity the relationship assumes that no side reactions occur in either the aqueous or organic phase and that no stable intermediates are formed with the analyte (e.g. metal) of interest. From a practitioner's standpoint, the total amount of analyte (e.g. metal) transferred from one phase to the other is of most interest. An empirical distribution ratio, D, is defined by the simplified relationship given below:

where [AT]o includes all (T = total) complexes of the analyte of interest in the organic phase and activity coefficients are assumed unity. The assumption of y = 1 for chelating extractions is reasonable. However, for ion-pair extractions where the electrolyte concentration is high, to assume unity for the activity coefficients is a poor assumption. The simplifying relationship is still often employed, however, with the assumption that the (yo/yaq) ratio will remain constant.

The extraction efficiency, %E, which defines the amount of analyte transferred from the aqueous phase to the organic phase, is defined as follows:

where Vaq is the volume of the aqueous phase and Vo is the volume of the organic phase. An important property of the above relationship is that the extraction efficiency is independent of the initial analyte concentration. High extraction efficiencies can be achieved when the Vaq/Vo ratio is small (that is, small aqueous volumes used with large organic volumes). There is of course a practical limit to this approach. Multiple extractions with reasonable volumes perform better than a single extraction with one large volume. Large values of D, distribution ratio, correspond to high extraction efficiencies (e.g. D = 100 then %E = 99%, D = 0.1 then %E = 10%, for a 1 : 1 volume ratio).

An extraction reaction may be described by the general chemical equation given below:

where A is the analyte of interest (e.g. metal ion) with charge n #, HX is the extracting agent (e.g. chelating agent). Note, that extracting agents are often acidic. From the following extraction reaction the equilibrium constant, Kex is:

By substituting the distribution ratio, D, the equation simplifies to:

hence:

The logarithmic form for the distribution coefficient is then:

From this model, for a given system, the degree of extraction increases as the concentration of the chelate [HX]o increases. Extraction increases with increasing pH (decreasing hydrogen concentration) in the aqueous phase. A one unit increase in pH results in a factor of 10 increase in the distribution coefficient for n = 1; for n = 2, the distribution coefficient increases by a factor of 100. Hydrolysis of the metal ion and decreased solubility of the chelate occur at high pH limiting this general approach.

Plots of log D versus pH (or %E versus pH or versus pH1/2) are often used to define extraction systems. These types of plots produce sigmoidal curves, with the overall position relative to the pH axis dependent on Kex with the slope = n. For purposes of comparison, if D = 1 (i.e. E = 50%) and [HX]o = 1, the pH is constant and equal to log Kex/n. This term is referred to as pH1/2, and is characteristic of the extraction process. Analyte/chelate agent values of pH1/2 are often cited and are used as a measure of the feasibility of separating two analytes.

Further theoretical discussion is beyond the scope of this chapter but includes topics on solvent properties, such as the solvent Hildebrand parameter, solvent dielectric constant, and complex properties such as the complex size, polarity and polarizability, as well as pH, temperature and reaction kinetics.

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