Distribution of the Third Component between Two Immiscible Liquids
If the third component is added to a system consisting of two immiscible liquids then it will be distributed in them in a certain proportion corresponding to the established dynamic equilibrium between the I and II phases. The equality of chemical potentials of the third (i-th) component in both the phases is a condition of phase equilibrium at a constant temperature, as follows:
µiI = µiII . |
(3.14) |
If µi for each phase is expressed through the activity, аi, of the i-th
components and the standard chemical potential, µi°, then the following equation is obtained:
µi0,I + RT ln aiI |
= µi0,II + RT ln aiII , |
(3.15) |
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from which |
aII |
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µ0,I −µ0,II |
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ln |
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(3.16) |
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aiI |
RT |
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Since the standard chemical potentials, |
µi0,I and |
µi0,II are constant, the |
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activity ratio of the third component in two phases is also constant: |
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aII |
= K 0 . |
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i |
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(3.17) |
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aiI |
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The obtained equation expresses the distribution law, which establishes that the equilibrium activity ratio of the third component in two immiscible liquids is a constant value at a constant temperature that is called the thermodynamic distribution (partition) coefficient or constant.
The partition coefficient, K°, depends on temperature and the nature of all of 3 substances, but does not depend on the concentration of the dissolved substance.
By expressing the activity of the dissolved substance in both solvents via the activity coefficient, γi, and the concentration, Сi, the following equation is obtained:
К0 = |
aII |
= |
CII γ II |
= K |
γ II |
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i |
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(3.18) |
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aI |
CI γ I |
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K = |
CII |
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(3.19) |
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Thus, we obtain one more expression for the distribution law, i.e. the equilibrium concentration ratio of the third component (the dissolved substance) in two immiscible liquids at a constant temperature is a constant called the distribution coefficient, К.
In contrast to K°, the distribution coefficient, К, depends also on the concentration of the dissolved substance in both solvents. For diluted
solutions Сi → 0, γi → 1, and K → K°, i.e. the distribution coefficient, K almost does not depend on the concentration.
Nikolai Shilov and Walther Nernst have discovered that the distribution law in its simplest form is applicable when a size of the dissolved substance particles is the same in both phases, i.e. there are no dissociation or association of molecules of the dissolved substance. Upon dissociation or association of the dissolved substance, complex equilibrium is established between the simple and associated molecules or ions in each phase (Fig. 3.3.). Let us apply a condition of the equilibrium distribution of КА in two phases to the case shown in Fig. 3.3а without consideration of the activity coefficients, since it is valid for dilute solutions, as follows:
Fig. 3.3. The examples of distribution of the dissolved component in two immiscible liquids in case of dissociation or association
µKA0,I + RT ln cKAI |
= µKA0,II + RT ln cKAII |
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(3.20) |
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where cKAI is the equilibrium concentration of КА, |
which, |
in turn, is |
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determined by the dissociation constant: |
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KD = |
cKI + cAI − |
= |
(CKAI αKAI )2 |
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CKAI (1−αKAI ) |
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(3.21) |
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where сi is the equilibrium concentration of particles, αKAI is the degree of
dissociation of КА that is determined by the ratio of the number of dissociated molecules to the total number of molecules before dissociation,
CKAI is the analytical concentration of КА in the I phase. According to Eq.
(20) and taking into consideration Eq. (3.21), the thermodynamic distribution constant is determined by the following expression:
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µ0,I −µ0,II |
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cKAII |
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CKAII |
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K |
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KA KA |
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(3.22) |
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cI |
(C I |
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/ K |
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KA |
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KA KA |
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D |
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where CKAII is the analytical concentration of КА in the II phase. Under the
condition that the degree of dissociation of the component, КА, in the I phase does not depend on its concentration (in particular that is valid for the
case of highly dilute solutions atαKAI ≈1), the law of distribution of the
third component between two phases can be expressed via the analytical concentration, as follows:
K = (αI |
)2 (K 0 / K |
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CII |
≈ const . |
(3.23) |
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KA |
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For the case shown in Fig. 3b, the equilibrium concentration in the I phase is determined by the degree of association and via the corresponding association constant:
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c(IKA)2 |
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CKAI βKAI |
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KAssoc = |
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(3.24) |
(cKAI )2 |
(CKAI )2 (1−βKAI )2 |
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where c is the equilibrium concentration, βKAI is the degree of association
of the dissolved substance, KA, in the I phase. The equilibrium concentration of КА in the I phase is expressed as follows:
cKAI = CKAI (1− βKAI )= |
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CKAI βKAI |
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(3.25) |
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KAssoc |
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Taking into account Eqs. (20) and (24), the distribution law under the condition that the degree of association does not depend on the concentration can be expressed in this case in the following way:
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K 0 |
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C II |
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K = |
β I |
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≈ const. |
(3.26) |
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KAsooc |
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CKA |
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It can be shown in the same way that the distribution law for the case given in Fig. 3c is expressed by the formula:
K = |
C II |
≈ const. |
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(3.27) |
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For the practical purposes, the distribution law, which considers possible association or dissociation of the dissolved substance in both phases, is more convenient to give in the following form:
СII |
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К = (CiIi )n , |
(3.28) |
where n is the index, which does not depend on the concentration and considers possible dissociation or association of the dissolved substance:
n = |
M II |
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i |
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(3.29) |
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where МiII is the average molecular weight of the dissolved substance in the II phase, and МiI is the average molecular weight of the dissolved substance in the I phase.
Eq. (3.28) unites into one common form all the considered cases expressed in Eqs. (3.19), (3.23), (3.26), and (3.27). If n > 1, i.e. МiII > МiI, then the association in the II phase or dissociation in the I phase is possible. If n < 1, i.e. МiII < МiI, then, on the contrary, dissociation in the II phase or association in the I phase is possible. If n = 1, then МiII = МiI, i.e. the dissolved substance in both phases is present in the same molecular form, that means there is no dissociation or association.
For example, acetic acid, as well as other carboxylic acids, in nonpolar organic solvents exists mainly in the form of coupled molecules. Such dimeric molecules are formed due to the presence of hydrogen bonds. Acetic acid in aqueous solutions is not dimerized, since water breaks up the dimeric molecules, and itself forms hydrogen bonds with acetic acid. A molecule of acetic acid in water partially dissociates into ions, since acetic acid is a weak electrolyte.
If both parts of Eq. (3.28) for the distribution law that takes into account possible association or dissociation of the dissolved substance are given in the logarithmic form, then the following expression is obtained:
lgCiII = lgK + nlgCiI, |
(3.30) |
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which forms a linear dependence in the coordinates, lgCiII ~ f (lg CiI) (see Fig. 3.4.).
A segment intercepted by the linear dependence on the ordinate axis, lgCiII, allows one to determine the lgК value and, hence, the average value of the distribution constant, К. The tangent of the slope angle gives the value of the index, n.
The method of extraction is based on the distribution of the dissolved substance between two immiscible solvents.
The separation of the dissolved substance from a solution by means of another solvent (extracting agent), which is almost immiscible with the first one and better dissolves the separating substance, is called extraction.
Diethyl ether is such a solvent for many organic substances, and as water for inorganic substances. The extractable substance for complete extraction is transferred into the molecular form, which is
contained in both phases. Thus, for example, for the extraction of weak organic acid it is beneficial to lower its degree of dissociation by adding mineral acid. Undissociated molecules of organic acid in this case are extracted more fully. The solubility of organic substances in water significantly decreases in the presence of salts due to the effect of salting out.
The extraction is one of the most widespread processes in chemical, pharmaceutical, food, and other industries. The extraction is widely used for separation of essential oils from the vegetative raw materials and for their purification, and also for separation of the components from complex natural and technological solutions. The extraction process in winemaking is used for the preparation of crude materials for producing vermouths, processing of grape millcakes, etc.
The extraction of the dissolved substance can be performed several times by small portions of the extracting agent (multiple extractions) or once by the same total amount of the extracting agent (single extraction).
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The weight, mq, and the concentration, Сfin, of the extracted substance remaining in the initial solution after q extractions with equal volumes of extracting agent are quantitative characteristics of the extraction procedure. The other characteristics are the degree of extraction, Rq, of a substance after q extractions that is determined by the ratio of the weight (quantity) of the substance, which have transferred into the extracting agent, to the weight (quantity) of a substance in the starting solution, and also the number of extractions, n, necessary for reaching the desired degree of extraction or reaching the specified concentration of a substance remained in the initial solution.
Let us determine the weight, mq, which remains in solution (I) after q extractions by equal volumes of extracting agent. If the initial weight of the i-th component in the initial solution, (I), is m0, and after the first extraction a weight of m1 is remained, then the following equation for the equilibrium constant (partition coefficient) is obtained upon establishing the equilibrium between two solvents:
K = |
CII |
= |
(m |
−m ) V |
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1 1 . |
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The following equation can be derived from Eq. (31):
m1 = m0 |
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where V1 is the volume of the extracted solution, and V2 is the volume of extracting agent.
After the second extraction
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where m2 |
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is the weight of a substance remained in Solution (I) after the |
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second extraction. |
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The weight of a substance remained in the initial solution after n
extractions can be calculated as follows: |
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V1 |
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(3.35) |
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Upon the extraction with one portion of the extracting agent with a total volume of qV2, the weight of a substance remained in the extracted solution equals:
m′ = m0 |
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(3.36) |
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Since К > 1 in any case (the extracted substance is dissolved better in the extracting agent), the fractions in Eqs. (3.35) and (3.36) are proper and,
hence, mq << m′.
As followed from the above, q extractions (multiple extractions) are more effective than one extraction with the same volume of extracting agent. A similar expression can be obtained for the molar concentration, Сfin, of the dissolved substance remaining in Solution (I) after q extractions:
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(3.37) |
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+ KV |
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init V |
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where Сfin = |
mq |
, Mi is the molecular weight of the dissolved substance, |
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MiV1 |
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Сinit is the initial molar concentration of the dissolved substance in solution
(I), and С |
init |
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m0 |
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MiV1 |
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The degree of extraction of the dissolved substance after q extractions is determined by the formula:
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m0 − mq |
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mq |
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V |
q |
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(3.38) |
Rq = m |
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− V |
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To determine the number of extractions, q, required for reaching the specified degree of extraction, the logarithmic form of Eq. (37) for Сfin should be obtained, and then the q value can be expressed as follows:
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lgCinit −lgC fin |
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q = |
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lg(V + KV |
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