Anal. Chem. 1997, 69, 4495-4497
Technical Notes
Matrix Representation of Solution Mixing by Aliquot Exchange Matthew R. Linford* and Helmuth Mo 1 hwald
Max-Planck-Institut fu¨ r Kolloid und Grenzfla¨ chenforschung, Rudower Chaussee 5, D12489 Berlin, Germany
We present a mathematical description of mixing two solutions by exchanging aliquots back and forth between them. We propose that this method of aliquot exchange can be used to automate calibration curve preparation in a way that produces less solvent waste than conventional serial dilution methods. We also show its use in quickly mixing solutions. The process of aliquot exchange is represented mathematically by a 2 × 2 symmetric matrix, A, that is a function of the volume or percentage of liquid, p, that is exchanged. Each cycle of aliquot exchange is represented by operating the matrix, A, on the previous concentrations. That is, after n mixing cycles, the final concentrations (Cin) are given by A(pn)A(pn-1)...A(p1) operating on the initial concentrations (Ci0), or (A(p))nCi0 if the same amount of liquid is exchanged in each step. We observe close agreement between theory and experiment. For solutions that have equal initial volumes, both the matrix A(p) and the product of any number of such matrices that may have the same or different values of p have equal diagonal and equal off-diagonal elements (they are symmetric), the sum of the elements in any row or column sums to unity, and the operation of any of these matrices on a set of concentrations produces two new concentrations that sum to the same value as the sum of the initial concentrations. We follow the mixing process for two solutions of equal volume by plotting the matrix element A12n of An, which approaches 0.5 as n increases. As expected, the larger the exchanged aliquot, the more quickly the solutions mix. By varying the fraction, p, that is exchanged, we show that it should be possible to produce a calibration curve with values that vary in concentration over at least 3 orders of magnitude from just two solutions. The automated preparation of a set of solutions to make a calibration curve can be performed by serial dilution,1 where a robot arm may remove and discard a certain amount of a solution in a measurement cell and then replace that which was removed by an aliquot of clean solvent.2 We suggest that exchanging aliquots of solution back and forth between two solutions may be an equally effective way of preparing a series of solutions over a range of concentrations that uses less solvent and thus creates less waste. We present a mathematical (matrix) description of (1) Hamilton, M. A.; Rinaldi, M. G. Stat. Med. 1988, 7, 535-544. (2) Lauda Operating Instructions Tensiometer TE!C/3 + SAE/KM 5, 48a. S0003-2700(97)00191-1 CCC: $14.00
© 1997 American Chemical Society
aliquot exchange. We finally show that aliquot exchange can be used to quickly mix two solutions. We demonstrate that aliquot exchange can be represented mathematically as a 2 × 2 matrix, A, that operates on two initial concentrations, and that each repeated exchange of aliquots between the same two solutions is equivalent to an additional operation of this matrix A. That is, the concentrations, after n aliquot exchange cycles with equal exchange volumes, are given by operating the matrix An on the initial concentrations. For equal initial solution volumes, A and An are symmetric matrices that have equal diagonal elements, and the sum of the elements in any row or column of these matrices is unity. The theoretical predictions agree well with experiment. As a safety consideration, we note that mixing a highly concentrated solution with a dilute solution may be strongly exothermic. THEORY We first present a mathematical representation of aliquot exchange between two solutions that have the same volume but different initial concentrations (CI0 and CII0). The result of the more complete treatment of unequal volumes is footnoted.3 The matrix obtained in the general (footnoted) case of unequal volumes can be substituted into any of the eqs 5-7 given below. Equation 1 gives the concentration in the second solution (CII1) after a certain percentage, p, of the first solution is injected into it. The initial solution volumes are taken to be 100, so that p also represents the volume of the first solution that is exchanged.
CII1 )
pCI0 + 100CII0 100 + p
(1)
We assume here and throughout this work complete mixing of the solutions, that the solutions are made with the same solvent (to avoid problems with possible changes in optical absorbances), (3) Consider two solutions with initial concentrations and volumes of CI0, CII0, VI, and VII, where a volume fVI (f is a fraction of VI) is exchanged between the solutions. After carrying out the procedure described in the text, we again find a matrix that operates on the initial concentrations to yield the final concentrations:
[ ] CI1 CII1
)
[
fVI + VII - fVII VII + fVI fVI VII + fVI
fVII VII + fVI VII VII + fVI
][
CI0 CII0
]
This general result reduces to the special case of equal volumes (eq 4) if the conditions considered in the text (VI ) VII ) 100, and fVI ) p) are substituted into this matrix.
Analytical Chemistry, Vol. 69, No. 21, November 1, 1997 4495
and that the solution volumes do not change upon dilution or concentration. Now removing the same volume from the second solution and adding it to the first solution changes the concentration of the first solution to CI1:
CI1 )
(100 - p)CI0 + pCII1 100
(2)
Substituting CII1 into the equation for CI1 and simplifying allows us to express CI1 as a function of the initial concentrations:
CI1 )
100CI0 + pCII0 100 + p
(3)
Equations 1 and 3 can now be combined in matrix form:
[
100 CI1 ) 100 + p p CII1 100 + p
[ ]
][
p 100 + p CI0 100 CII0 100 + p
]
(4)
where we define A to be the 2 × 2 matrix shown in eq 4. When p ) 0, the final concentrations equal the initial concentrations, as they should. When p ) 100, complete mixing is obtained (CI1 ) CII1 ) 1/2CI0 + 1/2CII0), as expected. A can be viewed as an operator that represents the mixing of any two solutions of equal volume by aliquot exchange where a percentage, p, of the initial solution is exchanged back and forth. This matrix has equal diagonal elements (A11 ) A22) and off-diagonal elements (A21 ) A12) (it is symmetric), and its rows and columns sum to unity. We can represent eq 4 more compactly as
Ci1 ) ACi0
(5)
where i runs over I and II. From the derivation of eq 4, it is clear that eq 5 is not limited to operating on the initial concentrations but should be valid for any set of solutions in the mixing process such that Cin ) ACin-1. Thus, after n mixes with the same p value for each mix, the final solution concentrations are given in terms of the initial concentrations by
Cin ) AnCi0
(6)
It is easy to verify (see footnote 4) that when a 2 × 2 matrix that has equal diagonal and off-diagonal elements, where the rows and columns sum to unity, is multiplied by another matrix with these same properties, the product is also a matrix with equal diagonal and off-diagonal elements that has rows and columns that sum to unity. Also, when such a matrix operates on two concentrations, the sum of the output concentrations is the same as the sum of (4) Consider multiplication of matrices A and B with elements A11 ) E, A12 ) F, A21 ) F, A22 ) E, B11 ) G, B12 ) H, B21 ) H, B22 ) G, where E + F ) 1 and G + H ) 1. Their product (AB) is a symmetric matrix with diagonal elements equal to EG + FH and off-diagonal elements equal to EH + FG. The sum of the elements in any row or column is given by EG + FH + EH + FG, which equals unity. Thus, if A is a matrix with equal diagonal and equal off-diagonal elements, where the sum of the elements in any row or column sum to unity, An also has these properties.
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Figure 1. Demonstration of a possible series of concentrations that would make up a calibration curve, as predicted by eqs 4 and 7. Concentrations of two solutions of equal volume were initially taken at 0 (9) and 2 M (b). The x-axis is the sum of the volumes, or percentage of the first solution, that were exchanged, e.g., the first point gives the concentration after aliquot exchange of 0.05% of the liquid, the second point after a second aliquot exchange of 0.05% (0.1% ) 0.05% + 0.05%), etc.
the initial concentrations (see footnote 5). Numerical values for the elements of An are easily obtained with any number of readily available software routines. We used both Mathematica and the student version of Matlab for matrix multiplications. RESULTS AND DISCUSSION Aliquot exchange may provide a simple method for automating the preparation of calibration curves. For example, a robot arm with a syringe attached to it could exchange liquid between two cuvettes, slowly increasing the concentration in one of the cuvettes while its concentration is monitored by a technique such as ultraviolet-visible spectroscopy. Different values of p (different amounts of liquid) could be used for each exchange. For variable p values, the concentration after n steps is given by
Cin ) A(pn)A(pn-1)...A(p1)Ci0
(7)
Figure 1 shows how aliquot exchange between two solutions with initial concentrations of 0 and 2 M could be used to incrementally vary the concentration of the solution that initially had no solute in it over 3 orders of magnitude (from 1 mM to 1 M). (If we consider 1 mL solutions, a typical cuvette volume, the smallest amount of liquid that must be exchanged is 0.5 µL.) It is clear that, if the concentration of the second solution is lowered to 2 mM and this same set of exchanges is repeated, we obtain a set of solutions with concentrations that vary from 1 µM to 1 mM. For precise work, it may be advantageous to use the robot arm to mechanically stir the solutions after injecting liquid into them. An additional possible use for aliquot exchange is to quickly mix two solutions. We have often needed to mix solutions that almost fill the vials they are in. We have found that mixing by aliquot exchange is often much quicker than cleaning a new piece of glassware and that it provides sufficient accuracy for many (5) Consider operation of the matrix A (A11 ) C, A12 ) D, A21 ) D, A22 ) C) on the concentrations Ci0 (CI0 ) E, CII0 ) F), where C + D ) 1 and E + F ) k. The sum of the new concentrations (Ci1 ) ACi0) is CE + FD + ED + FC, which equals k.
Figure 2. Element A12n of matrix An (A12n ) A21n, A11n ) A22n, A11n + A12n ) 1) as a function of the number of aliquot exchange cycles (n) shown for different percentages of the volume that are exchanged: 5% (9), 10% (b), 15% (2), 20% ([), and 25% (1). These predictions are for solutions with equal volumes. Complete mixing is obtained at A12n ) 0.5.
applications. From eq 6, we see that equal concentrations are obtained after aliquot exchange when the matrix elements A11n ) A12n ) 0.5 or, equivalently, when A21n ) A22n ) 0.5. Figure 2 is a plot of the matrix element A12n as a function of n for different percentages, p, of the initial volume that are removed from the first solution. As the percentage of the solution, p, that is exchanged is increased, A12 approaches 0.5 more quickly; that is, mixing takes place more rapidly, as expected. For example, if two solutions are initially prepared that are 105 and 95% ((5%) of a desired concentration after five aliquot exchanges, which takes less than 30 s, the compositions of these solutions will be (3.0, (1.8, (1.1, and (0.7% of the desired concentration if we exchange 5, 10, 15, and 20% of the initial solution volume, respectively. After 10 aliquot exchanges, which takes less than 1 min, the compositions of these solutions will be (1.8, (0.7, (0.2, and less than (0.1% of the desired concentration if we again exchange 5, 10, 15, and 20%, of the initial solution volume, respectively. A typical (long) Pasteur pipet has an internal volume of 1.5-2.0 mL, which is 15-20% of the 10 mL solution volume we typically work with. In the worst case scenario for mixing two solutions by aliquot exchange (0 and 100% of a given solute), Figure 1 shows that 10, 15, and 20 cycles lead to nearly total mixing (within 2% of complete
Figure 3. Absorbances at 586 nm of an aqueous dye solution (10 mL of ∼6.25 mM Direct Blue 71, Aldrich, Catalog No. 21,240-7) and water (10 mL) after aliquot exchange. The dye solution had an initial absorbance of approximately 0.50 au. 9’s are paired with b’s, 2’s are paired with 1’s, and each pair of points represents a unique experiment. Theoretical values for the process, as predicted by eq 6, are given by the solid lines. Approximately 15% of the solution was exchanged during the process (p ) 15). Spectra were taken with a Cary Model 4E UV-visible spectrophotomer.
mixing) when 20, 15, and 10%, respectively, of the solution is exchanged. These theoretical predictions are confirmed by experiment. Figure 3 shows the optical absorbance of pairs of solutions of a water-soluble organic dye (Direct Blue 71) and water after aliquot exchange between them. Two separate experiments (four points) were performed for 1, 2, 5, 10, 15, and 20 aliquot exchanges. The upper points represent the dye solutions after exchange, the lower points represent the initially pure water after exchange, and the solid lines are the theoretical predictions. ACKNOWLEDGMENT M.R.L. acknowledges the Max Planck Society for a stipend. Received for review February 17, 1997. Accepted July 11, 1997.X AC970191X X
Abstract published in Advance ACS Abstracts, September 15, 1997.
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