FUNCIONES QUE DEBERÁN REALIZAR LOS RECURSOS PREVENTIVOS EN ESTA OBRA

The composition of a mixture need not be given in terms of the mole fractions of its components. Other scales of concentration are frequently used, in particu- lar, when one of the components, say, A, can be designated as the solvent and the other (or others), B, (C, . . . ) as the solute (or solutes). When the solute is an electrolyte capable of dissociation into ions (but not only for such cases), the

molal scale is often employed. Here, the composition is stated in terms of the

number of moles of the solute, m, per unit mass (1 kg) of the solvent. The symbol m is used to represent the molal scale (e.g., 5 m= 5 mol solute/1 kg solvent). The conversion between the molal and the rational scale (i.e., the mole fraction scale, which is related to ratios of numbers of moles [see Eq. (2.2)] proceeds according to Eqs. (2.32a) or (2.32b) (cf. Fig. 2.4):

m x M x x M x m m M m M B B A B B A B B B A B A a b = − ≈ = + ≈ / ( ) / ( . ) /( / ) ( . ) 1 2 32 1 2 32

where MA is the molar mass of the solvent A, expressed in kilograms per

mole (kg mol−1). For aqueous solutions, for instance, MA= 0.018 kg mol−1and

1/MA= 55.5 mol kg−1. The approximate equalities on the right-hand sides of

Eqs. (2.32a) and (2.32b) pertain to very dilute solutions. Activity coefficients on the molal scale are designated byγ, so that Eq. (2.29) becomes

µB=µB (m)+ B+ lnγB

∞ _{R R}

Tlnm T ( .2 33)

where, again,µB∞(m)= lim(mB → 0)(µB− RT ln mB) is the standard chemical potential

of B on the molal scale, and lim(mB → 0)γB= 1.

Another widely used concentration scale is the molar scale, that describes the number of moles of the solute, c, per unit volume of the solution [i.e., per 1 dm3_{= 1 L (liter)]. The symbol M = mol/L is used to represent this scale [e.g.,}

0.5 M= 0.5 mol solute/1 liter (L) of solution]. Conversion between the molar and molal scales is made according to Eqs. (2.34a) and (2.34b) by means of the density of the solution,ρ, in kg L−1(= g mL−1= g cm−3; seeFig. 2.5):

c m m M m m M m c M c c M B B B B B A B B B B cB B B A B B b = + ≈ + = − ≈ − ρ ρ ρ ρ / ( ) / ( ) ( . ) / ( ) / ( ) ( . ) 1 1 2 34 2 34 a

where MBis the molar mass of the solute (in kg mol−1). The approximate equali-

Fig. 2.5 Concentration scales. The molality mB[in mol/(kg solvent)] and molarity cB [in mol (L solution)−1] of solute B, which has a molar mass MB0.100 kg mol−1and a molar volume VB= 0.050 L mol

−1

, in the solvent A, which has MA= 0.018 kg mol −1

and

VA= 0.018 L mol−1(water), are shown as a function of the mole fraction xBof the solute. Note that the molarity tends toward a maximal value (1/VB), whereas the molality tends toward infinity as xBincreases toward unity.

solutions. For aqueous solutions at 25°C, ρA= 0.997 kg L−1, and a generally

negligible error is made ifρAis replaced by 1 for very dilute aqueous solutions.

The activity coefficient on the molar scale is designated by y, so that Eq. (2.29) becomes

µB=µB (c)+ B+ B

∞ _{R R}

where, as before,µ∞B(c)= lim(cB→0)(µB− RT ln cB) is the standard chemical poten-

tial of B on the molar scale, lim(cB→0)yB= 1.

As an example for such concentration scale conversions, consider 20 wt% tri-n-butyl phosphate (TBP, B) in toluene (A). The mole fraction of the TBP is xB= (20/MB)/[(20/MB)+ (80/MA)]= 0.080, its molality is mB= (0.200/MB)/

0.800= 0.94 m, and its molarity is cB= 0.94 ρA/(1+ 0.94 MB)= 0.65 M (MBis

taken in kg mol−1).

The quantity µB is independent of the concentration scale used, being a

true property of the solution, but the three standard chemical potentials µB∞(x),

µB∞(m), and µB∞(c) are not equal. Consequently, differences between the standard

chemical potentials of a solute in the two liquid phases employed in solvent extraction also depend on the concentration scale used. Thus, PB defined in

Eq. (2.23) is specific for the rational concentration scale, and does not equal corresponding quantities pertaining to the other scales. Therefore, Eq. (2.30) might be rewritten with a subscript (x), to designate the rational scale, [i.e., with

DB(x)and PB(x)]. Similar expressions would then be

DB(m)=mB′/mB′′ =PB(m)( B / B )γ γ′′ ′ ( .2 36a)

for the molal scale, and

DB( )c =cB′/cB′′ =PB( )c( B / B )y ′′y ′ ( .2 36b)

for the molar scale. The subscripts are often omitted, if the context unequivo- cally defines the concentration scale employed.

2.5

AQUEOUS ELECTROLYTE SOLUTIONS

In many practical solvent extraction systems, one of the two liquids between which the solute distributes is an aqueous solution that contains one or more electrolytes. The distributing solute itself may be an electrolyte. An electrolyte is a substance that is capable of ionic dissociation, and does dissociate at least partly to ions in solution. These ions are likely to be solvated by the solvent (or, in water, to be hydrated) [5]. In addition to ion–solvent interactions, the ions will also interact with one another: repulsively, if of the same charge sign, attractively, if of the opposite sign. However, ion–ion interactions may be negli- gible if the solution is extremely dilute. The electrolyte Cν+Aν−, is made up of

ν+positive ions, or cations, Cz+, and ν−− negative ions, or anions, Az−, or of

altogether ν = ν++ ν− ions, and is designated in the following by the symbol

CA. A general principle that is always obeyed in electrolyte solutions is that of the electroneutrality of the solution as a whole. This means that the number of cations weighted by their charge numbers z₊(the number of protonic charges each has) equals that of the anions similarly weighted with z₋. Per unit volume of the solution this can be written as:

∑c z+ ++ ∑c z− − =0 ( .2 37) (Note that z−is a negative number, so thatν+z++ ν−z−= 0.) The electroneutral-

ity principle is equivalent to stating that it is impossible to produce a solution that contains, for example, only cations or an excess of positive charge (i.e., that ions cannot be considered as independent components of solutions). Only entire electrolytes are components that can be added to a solution.

It is expedient to consider the mean ionic properties of the electrolytes. An electrolyte that dissociates into ν ions has a mean molality m_±= (ν₊ m₊+

ν−m−)/ν and a mean molarity c±= (ν+c++ ν−c−)/ν.

A useful concept that is used when the activities of electrolytes are calcu- lated is that of the ionic strength of the solution. This is defined (on the molar scale) as:

I = ∑1 c z

2 i i 2 38

2

( . )

where the summation extends over all the cations and anions present in the solution. Since the charges appear to the second power in the expression for the ionic strength, all the terms are positive. An electrolyte that is completely disso- ciated to univalent ions is designated as a 1:1 electrolyte, examples being aque- ous HCl and LiNO3. If only a single 1:1 electrolyte is present at a molar concen-

tration cCA, then I= cCA. According to this notation, MgCl2, and (NH4)2SO4 are

2:1 and 1:2 electrolytes, respectively, for which I= 3 cCA. For 1:1, 2:2, . . . (i.e.,

symmetric) electrolytes, m±= mCA, and I= cCA, 4 cCA, . . . The relationship be-

tween mCAand cCAis given by Eq. (2.34.)