IDENTIFICATION OF SMALL AMOUNTS OF ORGANIC COMPOUNDS BY DISTRIBUTION STUDIES

V. CALCULATION OF THEORETICAL CURVES*

BY BYRON WILLIAMSON AND LYMAN C. CRAIG

(From th,e Laboratories of The Rockefeller Institute for Medical Research, New York)

(Received for publication, December 13, 1946)

The method of counter-current distribution for the purpose of fraction- ation, detection of inhomogeneity, or the characterization of an unknown compound has been described in several previous publications (l-3). Be- cause of the nature of the process and the attainment of essential equilib- rium at each step, the results are particularly adaptable t,o exact mathe- matical interpretation or analysis for any particular procedure chosen. Such mathematical interpretation is useful from the standpoint of regular practice as well as in understanding the underlying nature of such a proc- ess, and therefore is deserving of considerable attention. It is the purpose of the present treatment to deal with one phase of the interpretation, namely that of a machine distribution when nothing is withdrawn from the machine until the desired distribution is finished (Procedure 1 (l), or with the procedures given in the two later papers (2,3)). These represent per- haps the simplest of all counter-current, procedures and arc therefore the most easy to interpret mathematically, since they require only the direct applica.tion of the binomial expansion (4) (X+Y)“, where Y is considered the fraction being transferred in the upper layer or K/(K+ 1) in terms of the partition coefficient and for equal volumes of the two phases. X is then the fraction remaining in the lower layer, l--K/(X+1) or l/(X+1). The expansion is therefore

1 K 7L __ - K+l+K+l 1

Once such an actual distribution with a solute, homogeneous or otherwise, has been reached, the results can best be followed or interpreted by plotting a distribution curve. This curve is drawn by plot.ting the total amount of substance in each tube or cell of the machine VRKSUS the consecutive numbers on the tubes. Highly useful deductions can then be drawn from the general shape of the curve and the number of maxima obtained. For the proper interpretation of the curve, the theoretical distribution of a single

* The work described in this paper was done under a contract, recommended by the Committee on Medical Research, between the Office of Scientific Research and Development and The Rockefeller Institute for Medical Research.

687

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688 DISTRIBUTION STUDIES. V

pure substance having the same partition coefficient as the component in question is necessary and can be calculated directly from the binomial theorem. In this paper several simple and rapid short cuts are presented for the calculation of such theoretical distributions when the partition isot,herm is a linear one.

The calculat,ion of theoretical curves by a method of approximation has been referred to previously in t,he papers mentioned above. However, a direct application of the binomial expansion is necessary, when applied to relatively few transfers such as in eight transfer distributions, and for the higher distributions in which Tube 0 (or the highest tube) contains ap- preciable material. As the calculation depends only on t,he knowledge of the value of the partition coefficient of the pure solute in the system em- ployed, methods of finding this value will be discussed later in the paper.

The problem presented is the calculation of t,he fraction of the pure ideal solute present in each of the tubes (both layers combined) after t,he dis- tribution has been effected. From a step by step analysis of the binomial theorem, one is able to establish a t.able of general terms, Table I, giving the fraction of the original solute present in each tube at every stage of the analysis as a function of the partition coefficient, i7.l The terms of Table I are those from the binomial expansion

1 K n - ___ K+l+K+l 1

Any single term, T,,,, may be calculated directly from the mathematics of the binomial theorem, as given in Formula 1.

n U-0 (1)

T tt,c is, therefore, the fraction of the original substance present in the Tube r in a distribution of n transfers or plates. K is the partition coeffi- cient, or distribution constant, and is always defined as t,he concentration of solute in the upper phase divided by the concentration of solute in the lower phase. Thus formula (1) may be used to calculate a theoretical distribution for any given partition coefficient. For an eight, transfer distribution, n = 8 and r = 0, 1,2; . . 8, nine values must be calculated; for a twenty-four transfer distribution, n = 24, T = 0,1,2; . .24, twenty-five values must be calculated.

1 All formulae derived in this paper will apply only to distributions in which the upper phase migrates, as in a machine distribution. They will apply to distributions in which the lower phase migrates if l/K is substituted for K. Also the volumes of the two phases are assumed to be equal. When they are not equal, the calculation may be done by substituting the product K X T wherever K is used. r in this case is the ratio of the upper and lower volumes.

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B. WILLI.4bISON AND L. C. CRSIG 689

h

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T T

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m co m 3

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690 DISTRIBUTION STUDIES. V

Rather than calculate each value independently from equation (I), more rapid methods are possible which in principle determine only one term directly and calculate all other terms from this one. For a specific ex- pansion, i.e. for a given value of n, adjacent terms of the expansion are very simply related to one another. This is evident upon examination of Table T, e.g.

(n = 8)

T, = 8KTo

7 TZ = - KTI , etc.

2 (2)

In general terms, the rth term is related to the (T- 1)th and the (r+ 1)th term by formulae (3) and (4) respectively.

T, = FKTrmI (3)

T, = F’ 0

1 T,+I K

(4)

where F = (n+l-r)/r and F’ = (r+l)/(n - r). For eight and twenty-four transfer distributions the factors F and F’ are

given in Table II. A direct application of the use of these factors is dis- cussed later in this paper. For eight transfer distributions in which the lower phase migrates, a further convenience results if a series of nine graphs are plotted from equation (I), (l/K substituted for K) with T, as the ordinate and K as the common abscissa, for values of r = 0,1,2;. .8. From these graphs one is able to read all nine values, one from each curve, for any value of K, and hence obtain the theoretical concentrations in a very few minutes. These curves are given in Fig. 1 for values of K from 0.40 to 2.50.

As T, is the fraction of the original solute in Tube r, it is numerically equal to the amount of solute in gm. only provided the experiment was begun with 1 gm. In general, experiments are not begun with a unit weight, so that to determine the actual amount present in Tube r the value T, must be multiplied by the total weight of solute used.

In applying these data to the interpretation of an actual distribution, i.e. the fitting of a curve, it is well to emphasize the basic point, namely that for a pure solute and a given number of transfers the relative amount of substance in each tube is fixed (assuming a linear isotherm). Thus in an inhomogeneity analysis, if the values of I’, are multiplied by the weight of the inhomogeneous sample, the experimental concentration values for the tubes at the extremities of the distribution will exceed the theoretical ones by nearly the actual amount of the impurity present, while the theoretical

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B. WILLIAMSON AND L. C. CRAIG 691

Factor

20 5/20 21 4/21 22 3/22 23 2/23 24 1/a

TABLE II

Eight and Twenty-Four Transfer Distributions

8 transfer distribution 24 transfer distribution

3/6 7/2 217 8

l/8

P I F’

24 23/2 22/3 21/4 20/5 19/6 18/7 1718 16/9

l/N 2/23 3/22 4/21

I

5/20 6/19 7/18 8/17 9/16

10/15

21/4 22/3 23/2 24

0.40 38 %

36 2

.34 z

z .32 2

0.30 2

.2a

.26

p” 5 .24 3

C .22 1 .; 0.20 .j .I8

z 2 16

.14

.12 4

0.10

.06 0

.04 5

.oe 0.00 %I

0.1 0.2 0.3 0.4 Q50 0.6 O.? 0.8 0.9 100 1.1 1.2 1.3 14 1.50 1.6 1.7 1.8 1.9 200 2.1 2.2 2.3 2.4 2YJ’

Fadition coefficient

FIG. 1. Graph of eight transfer distribution for different partition coefficients

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692 DISTRIBUTION STUDIES. V

values for the central tubes will exceed the experimental ones by a pro- portional amount.

The pure solute from the inhomogeneous sample’ will usually be con- centrated in the tubes comprising the peak of the distribution, and it is thus valid to att,empt to fit the maximum of the theoretical curve to that of t.he experimental one. Impurities are then quantitatively indicated by the deviations at the extremities of the two curves, the experimental values always exceeding the theoretical ones by the amount of impurity present. This fit is accomplished by multiplying all T,. values by the factor T’,,,J- T mnx., dwre T’,,,. is the experimental value for the tube of ma.x- imum conrentration and T,,,. is the theoretical value for this same tube on the basis of unity, i.e. as calculated from formula (l), or as read from Fig. 1. In the event the means of analysis does not result in f;he direct determination of the actual weight of substance in each of the tubes (e.g. spectroscopic analysis), the figures proportional to weight may be used.

In pract,ice, however, especially with distributions of more than eight transfers matching the theoretical curve to the experimental one is best accomplished directly by the procedure now to be outlined. After dcter- mining the experimental distribution, a value is chosen near the peak of the distribution curve, preferably the maximum one, and, assuming this to lie on the theoretical curve, the other values composing the theoretical distribution are calculated by use of the factors F and F’. The use of the point near the maximum of the distribution curve involves the previously stated assumption that the solution represented by this point is that of a pure compound.

If a dist,ribution is attempted with a mixture of two substances whose partition coefficients are close together, for example a mixture of isomers, a clear separation may not result, and instead of obtaining a curve of two distinct peaks a rather broad, single peak may occur. In attempting to fit a theoretical curve to such a distribution, one must assume the existence of at least two substances and find two curves which, when added together, will give the experimental one. The application of this method to sterio- isomers, diasterioisomers, and other very closely related compounds is being undertaken and appears to offer promise. Methods for a more com- plete separation of compounds with nearly identical partition coefficients may also be found in variations of the standard distribution procedures. Such possible variations are being further investigated. Two typical curves are given in Figs. 2 and 3. Fig. 2 is that of an eight transfer distribution of a mixture of two compounds consisting of 90 per cent of Compound A (K = 1) and 10 per cent of Compound B (K = 10). Fig. 3 is a twenty-four transfer distribution of the same mixture, showing the increased separation rendered by the use of a greater number of transfers. In all cases in which

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B. WILLIAMSON AND L. C. CRAIG 693

an applied theoretical curve will not coincide with the experimental one, it is indicated by a dotted line.

By use of the above methods, the theoretical distribution of a pure sub- stance can be determined exactly for any given partition coefficient.

012345670

0 ExpesGnentol x Theoretical

Tube numbso

FIG. 2. Pattern of eight transfer distribution

.20

.10 - p” 2 .16

0 ExprAtnenral * Theoretical

I I

$ .12 3

a, Jo

's .00

.4 .c6 s d .04

& .02

Tube numbea

FIG. 3. Pattern of twenty-four transfer distribution

Therefore in contrasting theoretical concentrations with those obtained experimentally in an inhomogeneity analysis, the partition coefficient of the pure substance must be determined. The ways in which this may be achieved are several.

Direct Measurement-The partition coefficient as measured before the distribution is that of t,he inhomogeneous solute, and is consequently not

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694 DISTRIBUTION STUDIES. V

usable in the theoretical determination. The material in the tubes near the peak of the experimental distribution, however, should be the most, nearly homogeneous, except in the case of an inhomogeneity with an iden- t,ical or nearly identical partition coefficient. These solutions may thus be used to determine experimentally the partition coefficient of the pure solute, unless otherwise indicated by the shape of the curve. Direct measurement. is always used in this laboratory to serve as a check on experimental pro- cedure or calculations, as well as on possible deviations from a linear partition isotherm; but for use in the calculation of a theoretical curve to be fitted K may be derived as given below, without the separate experi- mental determination.

Indirect CaMation-When the tubes near the peak of the curve contain homogeneolls material, this part of the experimental curve is representative of the true partition coefficient of the pure solute. Further, the form and position of t,his part of the curve are quite sensitive to the value of the partition coefficient, and herein lies the basis for several methods of de- termining t,his coefficient. Two are mentioned here.

A previously reported application (1) of this principle is in the use of the formula

(5)

which defines the relationship between the position N of the peak of the curve and the partition coefficient, K, where n is the number of t’ransfers in the distribution. This formula assumes a continuous function and is exact only if K is equal to unity or if there are an infinite number of transfers in the distribution. For distributions involving a number of transfers greater than twenty, when the pnrt(ition coefficient is near unity, the above formula is saGsfactory in ordinary pract,ice. Its disadvantages are 2-fold; namely, the abscissa of the peak of the curve, N, can only be estimated rather than determined accurately, and secondly, it cannot be applied to eight transfer distributions.

A second method of determining the partition coefficient from the region near the peak or from any other part of the experimental distribution curve utilizes the condition that the ratio of the concentrations in any two tubes is a specific function of the partition coefficient. The concentration in any one tube is a function of both the initial concentration and the partition coefficient. The ratio of the concentrations in any two tubes, however, is a function of the part,ition coefficient only, and is independent of the initial concentration. For example, in an eight transfer distribution, Td/Tc = 5/4(K). Hence

K = f5/4WT3 (6)

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33. WILLIAMSON AND L. C. CRAIG 695

Tn general, from equations (3) and (4),

Ratio of tern(Y): term(v’+l)

FIG. 4. Graph for determining partition coefficient

(7)

(8)

If the various ratios, !Z’,/T,+,, are plotted against K, straight lines will be obtained (Fig. 4) and from such graphs the part,ition coefficient can be read quickly and accurately. Thus through the use of the graphs of Figs. 1 and 4 an eight transfer tjheoret’ical distribution may be quickly calculated and applied.

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696 DISTRIBUTION STUDIES. V

The methods given in this paper for the calculation of theoretical curves

(9)

will supersede in many instances the formula earlier derived for the calcu- lation of theoretical curves. Equation (1) is exact, while equation (9) is an approximation. Nevertheless, equation (9) is of interest because it permits easy calculation of curves for numbers of transfers higher than twenty-five. Further, it is apparent that very close approximations may be made with this type of mathematical calculation. In Fig. 5 curves for K = 0.707

FIG. 5. Agreement of calculation with the exact calculation and the approximation

Tube number

are given for twenty-four and forty-eight plate distributions calculated exactly by formula (1) and by the approximation of formula (9). The constant a, which was derived empirically, approaches 2 for either large or small coefficients, since it is equal to 2(l/K+l). Here the recipro- cal of K, l/K, must be used when K is greater than 1. Thus equation (9) becomes equation (10)

where y is the ordinate or the fraction of the substance in the tube in question, K is the partition coefficient, n is the number of transfers, and x is the number of tubes between the one in question and the maxima.

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B. WILLIAMSON AND L. C. CRAIG 697

Thuswhen x: is 0, y is the maximum. Equation (9) is in accord with previous equations (5) for this type of calculation and coincides with the curve of error.

Our theoretical considerations have thus far been restricted to the sim- plest possible type of manipulation of the machine. In this no attempt is

made to prevent the band from spreading or otherwise to’modify it, such as periodic withdrawal of certain fractions, evaporation to dryness, and return to the preceding tube would achieve. This latter procedure offers the possibility of one way of introducing a principle with the same effect in our machine that the principle of reflux has, as used in fractional distillation. Also the periodic withdrawal of certain fractions offers itself certain in- teresting possibilities. It is our intention to discuss these approaches in a forthcoming paper when more experimental data are at hand.

SUMMARY

Methods for rapid and accurate calculation of theoretical curves for use in the “counter-current distribution” method have been presented.

BIBLIOGRAPHY

1. Craig, L. C., J. Bid. Chem., 156, 519 (1944). 2. Craig, L. C., Golumbic, C., Mighton, H., and Titus, E., J. Biol. Chem., 161, 321

(1945). 3. Craig, L. C., Golumbic, C., Mighton, H., and Titus, E., Science, 103, 587 (1946). 4. Martin, A. J. P., and Synge, R. L. M., Biochem. J., 35, 1358 (1941). 5. Rietz, H. L., Mathematical statistics, Chicago, 34 (1927).

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Byron Williamson and Lyman C. CraigCURVES

CALCULATION OF THEORETICAL BY DISTRIBUTION STUDIES: V.

AMOUNTS OF ORGANIC COMPOUNDS IDENTIFICATION OF SMALL

1947, 168:687-697.J. Biol. Chem. 

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