## Theory

The separation of chemical compounds by partitioning between two liquid phases, so-called liquid-liquid extraction, can be made more effective by using it as a cascade process. One way in which this can be carried out is by multiplicative partitioning, also called countercurrent distribution (CCD). This process, in which complete partition equilibrium is achieved in each step, is presented schematically in Figure 1. The principle is that two sets of liquid phases, the upper and lower phase, come into contact with each other stepwise. The bottom phases are numbered 0, 1, 2 and so on. The sample to be analysed (fractionated) is included in the first system (containing bottom phase number 0). Before each transfer of the upper phases (to the right in Figure 1) the two-phase systems are equilibrated by mixing and the sample components are distributed between each pair of phases (each full two-phase system). The partitioning of a pure substance between the phases of a two-phase system can be expressed either by a partition coefficient, K, defined as the ratio of the concentrations (C) of the component in the phases:

or by a partition ratio, G, defined as the ratio of the masses (m) of the components in the phase:

Volume (phase I)

Volume (phase II)

In the following the upper phase is chosen as phase I. A convenient way of analysing the CCD process is to calculate the amounts (in fractions) of a pure

component in the various phases based on its G value. In the initial two-phase system (number 0) the component is distributed with the fractions p in the upper phase (phase I) and q in the lower phase (phase II). By definition p + q = 1. When the upper phase of system number 0 is combined with the pure lower phase number 1 equilibration of this system will make the transferred fraction p distribute as p/q ( = G) giving the fraction p2 in the upper phase and the fraction pq in the lower phase. In the new system number 0, with a new upper phase, the remaining fraction q will partition likewise and the equilibration results in the fraction pq in the upper phase and q2 in the lower phase. The resulting distribution after 10 transfers in the 11 tubes 0-10 is shown in Figure 2. In the same figure the material in each tube (upper plus lower phase) has been calculated. These values are the same as the terms obtained when (q + p)10 is written as a polynomial. More generally, the amount of material Tnii (in fractions), in tube number i after a CCD with n transfers is given by eqn [4]:

iP'qT

The volume ratio is kept constant during the process. By using the relations Gi = p'/q' and (1 + G)n =

pq |
10 PV |
i i 45p q |
120 pq |
210pV |
252pV |
210pV |
45p q |
10 P% |
p | |

q |
1(W° |
ir ! ! 45p q |
120 pV |
210pV |
252pq |
210pV |
120pV |
„ a i 45p q |
10 pV |
p q |

Amount in upper phase

Amount in lower phase

Amount in upper phase

Amount in lower phase

q° 10pq9 45pV 120pV 210pV 252pV 210pV 120pV 45pV 10p9q p10 ^ach tube

Figure 2 Distribution of a component which partitions in the ratio p/q between the upper and lower phase - after a CCD with 10 transfers. The amounts are given in fractions, i.e. p + q= 1.

This relationship has been used to calculate the distribution profiles in Figure 3 for three cases: i.e. with the [5] number of transfers (n) equal to 10, 100 and 1000. In each case the distributions of components with

G values of 0.05, 0.1, 0.25, 0.5, 1, 2, 4, 10 and 20 are shown. The resolution of components with various G values increases with the increase in the number of transfers. This is due to the fact that the difference in the position (tube number) of the peaks (with given G values) is proportional to the number of transfers, while the peak width only increases with the square root of the number of transfers.

The width of a peak Ar, covering 99.7% of the compound, can be approximately calculated using eqn [6]:

7KK2

n |
Ar (no. oftubes) |
Arrel (%) |

20 |
13 |
67 |

50 |
21 |
42 |

100 |
30 |
30 |

200 |
42 |
21 |

500 |
67 |
13 |

1000 |
95 |
9.5 |

2000 |
134 |
6.7 |

5000 |
212 |
4.2 |

Table 2 Number of transfers, n, in a CCD necessary to separate two substances, 1 and 2, with a given separation factor ß = K1/K2. The volume ratio is chosen in such a way (eqn [7]) that gixg2 = 1

The relative width of a peak, Arrel = Ar/n, is given in Table 1 for G = 1 and a number of n values.

The most effective separation of two components, with partition coefficients K1 and K2, is achieved when their distribution peaks are oriented symmetrically around the middle of the tube train, i.e. tube i = n/2. This, as can be seen in Figure 3, is equal to the relation G1 = 1/G2 which can be written as GjxG2= 1. By combining this relation with eqn [3] the volume ratio, V, to be used for optimal separation can be calculated as shown in eqn [7]:

Table 2 Number of transfers, n, in a CCD necessary to separate two substances, 1 and 2, with a given separation factor ß = K1/K2. The volume ratio is chosen in such a way (eqn [7]) that gixg2 = 1

ß |
G1 |
G2 |
n |

2 |
0.709 |
1.414 |
264 |

3 |
0.577 |
1.732 |
110 |

4 |
0.500 |
2.00 |
70 |

9 |
0.333 |
3.00 |
30 |

16 |
0.250 |
4.00 |
20 |

25 |
0.200 |
5.00 |
16 |

36 |
0.167 |
6.00 |
14 |

49 |
0.143 |
7.00 |
12 |

81 |
0.111 |
9.00 |
10 |

400 |
0.050 |
20.0 |
6 |

2500 |
0.020 |
50.0 |
4 |

The G value can also be obtained by comparing the amount of a substance in two consecutive tubes, numbers i and i + 1. By combining the expressions for Tnii and Tn,i +1 obtained from eqn [5] the G value is obtained as in eqn [9]:

The ratio between the K values of two substances 1 and 2 is called the separation factor, ft, and is a measure of the separability of the substances. In Table 2 the number of transfers necessary for virtually complete (more than 99.5%) separation has been calculated for various ft values.

The G value of a component which has its maximum peak value in tube number nL after n transfers is given by the approximate eqn [8]:

By following the apparent G values over a distribution peak the presence of several substances differing slightly in their G values can be detected.

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