MAY, 1968 THE ANALYST Vol. 93, No. I106 A General Graphical Method for Evaluating Experimental Results that Should Fit a Linear Equation BY H. M. N. H. IRVING (Department of Inorganic and Structural Chemistry, Th.e University, Leeds 2) When the results of an experiment should be representable by a linear equation of the general form y = fllx + c the errors inevitably associated with the values of xj and yj ( j = 1 to n >2) give rise to a set of n simultaneous equations that will be inconsistent. As an alternative to a least-squares computation, a graphical procedure is described for finding the best values of the constants m and c. The method is quite general and it is illustrated by worked examples of problems encountered in analytical chemistry, e.g., the potentiometric titration of a dibasic acid, the spectrophotometric deter- mination of the stability of a weak complex, the evaluation of redox potentials and the liquid -liquid extraction of a weak monobasic acid.WITH the increasing availability and use of computers, graphical methods of handling data are tending to drop out of favour. There are experiments, however, in which graphical procedures have advantages, even if only as a preliminary to more rigorous computational methods; sometimes the precision of the experimental data is such that conclusions from graphical methods are adequate. Many chemical problems yield experimental data in terms of sets of two variable quantities xj and yj that can be related by a linear equation which can be presented graphically as a straight line passing through the points (XI, yj).The problem is to determine the values of the constants m and c. Provided only two sets of observa- tions have been made ( j = 0 and 1) there will be a unique solution to the resulting pair of simultaneous equations, irrespective of the experimental errors made in the measurements. When, as in practice, there may be more than two sets of observations, each of which will be subject to experimental error, the corresponding linear equations will probably not be consistent, and the problem is now to find the “best” values for the constants m and c. This can be done graphically by drawing the “best straight line” through the sets of points (xj, yj) or by computing the slope and intercept of this line by the “least squares method.” It should be pointed out that in this procedure it is common to assume that all the random errors are associated with one of the observations (e.g., with measurements of xj), while measurements of the other variable, yj, are free from error.The object of the present paper is to present an alternative graphical procedure for the solution of sets of (possibly inconsistent) linear equations. No novelty is claimed for this procedure, which appears to have been suggested by Professor Finsler and has been used extensively by Schwarzenbach, Willi and Bach.l However, the method is not widely known and experience has shown that many research workers have not found the description in the German text easy to follow. I have therefore taken the opportunity of generalising the procedure and showing, by worked examples, how it can be applied in several different situations.BASIC PRINCIPLES Instead of plotting the data (xj, yl) as a series of points through which the “best straight line” is to be drawn, the basic principle is to use a co-ordinate system such that each associated pair of observations (xj andyj) is used to define a straight line; the various straight lines for all the observations are then to intersect in a common point whose co-ordinates are a measure of the desired constants m and c. yj = mxj + c .. .. .. .. * ’ (I), 0 SAC and the author. 273274 IRVING: A GENERAL GRAPHICAL METHOD FOR EVALUATING [AvtalySt, VOl. 93 Fig. 1. Plots of lines conforming to the general equation (i)K + ($)w = 1 Suppose a straight line is drawn to pass through the points (A,O) and (0,B) situated along the x-axis and y-axis, respectively, of a rectilinear co-ordinate system (Fig.1 ) . If this line also passes through the point (K,K') it is easy to show that its equation must be . . (2). ( ; ) K + ( ; ) K ' = . .. .. .. The graphical procedure now involves several stages- (i) The exact linear equation representing the process under investigation is written down in its appropriate symbols. (ii) The form of this equation is adjusted to conform exactly with equation (2), i.e., with the desired constants on the left-hand side, and only unity on the right. (iii) By direct comparison with equation (2), A is expressed in terms of the experi- mentally determinable quantities and the same procedure is carried out for B.The relationship of K and K' to the two desired constants is also written down. (iv) The points (A,O) and (0,B) are plotted with suitable scales and joined to give a straight line. (v) Similar straight lines are drawn for values of A and B calculated from all the other experimental results. (vi) The values of K and K' are determined from the point of intersection and those of the desired constants are deduced from them. A few examples will make the procedure clear. EXAMPLE 1- determining the values of the two acid-dissociation constants A dibasic acid, H2A, has been titrated potentiometrically with alkali, with the object of and [H+l [A2-] K2 = [HA-] 'May, 19681 EXPERIMENTAL RESULTS THAT SHOULD FIT A LINEAR EQUATION 275 If the total amount of acid is known, CA = [&,A] + [HA-] + [A2-] and, if observations are made of the pH after adding various amounts of alkali, we have, from considerations of electro-neutrality, where a is the degree of neutralisation.The last term on the right-hand side, therefore, refers to the concentration of univalent cation (e.g., K+) introduced during the titration. By elimination we arrive at the general equation [OH-] + [HA-] + 2[A'-] = [H+] + aCA Comparison with the standard equation (2) shows that 1 K2 = K , K , =I K' ' and that Table I (rows 1 to 3) gives experimental data for the potentiometric titration of 1 O O m l of about 103 M anthranilic acid-NN'-diacetic acid, o-HOOC.C,H,.N(CH~COOH)~, with car- bonate-free 0.1 M potassium hydr0xide.l TABLE I THE POTENTIOMETRIC TITRATION OF ANTHRANILIC ACID-NN'-DIACETIC ACID 10'C~ .. .. 1.031 1.029 1.026 1.023 1.021 1.018 1.016 1.013 a .. .. .. 0 0.26 0.60 0.76 1.00 1.26 1.60 1.76 p H . . .. . . 2.886 2.966 3.032 3.123 3.228 3.369 3.561 3.886 lOyH+] . . 1.301 1.110 0.929 0.763 0.692 0.428 0-276 0.130 lWA (calcuiited) . . 4.616 6.436 6.207 7.122 8.169 8.707 9.241 9.386 lO-aB (calculated) - 1.596 - 2.229 - 3.097 -4.344 - 6.200 - 9.379 - 15-83 - 36.97 12 AxlC 0 Fig. 2. Data, from Table I relating to the titration of anthranilic acid-NN'-diacetic wid. Calculation of Kl and K,276 IRVING: A GENERAL GRAPHICAL METHOD FOR EVALUATING [Afla&St, VOl. 93 The various straight lines are plotted in Fig. 2. Apart from that for a = 1.75, the convergence to a single point is quite satisfactory and leads to the values K = 10.4 x 10-4 and K’ = 2.0 x lo2, from which we deduce K , = 5.00 x and K , = 1-04 x 10-3. The more tedious treatment of these experimental data by least-squares computation leads to However, as Schwarzenbach, Willi and Bach point out,l values of these constants obtained from a completely independent titration may show an absolute error, which exceeds that calculated from the same titration curve.For example, in a second titration of anthranilic acid-NN’-diacetic acid the values K , = 4.80 x and K , = 1.01 x 10-3 were obtained graphically and the values K, = 4.25 (+O-25) x and K , = 1-05 (+0*02) x 10-3 by computation. Greater accuracy in the determination of K , and K , can only be achieved by paying much greater attention to points of experimental technique (e.g., the more exact measurement of the concentration of the titrant and the constancy of the potential of the reference electrode).In the present instance the graphical method suggests visually that the results for a = 1-75 are inconsistent with the rest, and this reading is automatically omitted when locating the point (K, K’). Calculation shows that a pH reading of 3.876 (ie., only one hundredth of a unit higher) would have given the “correct” result. Of course, in the least- squares calculation the less reliable data for a = 1.75 will have been included with the same weighting as all the others. EXAMPLE 2- to the equation can often be obtained from spectrophotometric measurements of the optical density, E, of mixtures containing Merent initial concentrations, CM and CL, of the components, provided the molecular extinction coefficients, E ~ , cL and eNLL, of the three species are known.Clearly, for a 1-cm cell length and Kl = 5.06 (+0*05) x and K , = 1-04 (+0.01) x 10-3. The stability constant, KML, governing the formation of a weak complex ML according M+L+ML .. .. .. .. (4), E = [MIEM + [L]EL + LML]Em = (cM - [ML1)EM + (CL - EMLI)€L + [MLIEML** (5) (cM - pL1) (c - LML1) = K M L w L l * .. . . (6). When Km is small it is often necessary to add a considerable excess of one component (e.g., L) to give appreciable amounts of ML, and with CL > [ML] we find, on eliminating terns in rMLl, that This readily leads to values of KML (which should be identical within the limits of experimental error) on inserting values of E appropriate to different initial concentrations CM and CL.When using l-cm cells we can replace CMEM by E&, the optical density when CL = 0, and CLeL by EL’, the optical density when CM = 0. We can also write E, = CMEML for the limiting case, in which the whole of the component M has been transformed into MI, by a large excess of L. In practice, the actual value of E, may not be determinable by TABLE I1 SPECTROPHOTOMETRIC STUDIES OF ADDUCT FORMATION 1 03cL 0 39-70 79-41 119.2 159.1 199.0 278.7 398.5 598-3 798.1 E 0.388 0.431 0.460 0-482 0-497 0.513 0.644 0.568 0.600 0.618 A (calculated) - -0.397 - 0.507 -0.611 - 0.726 -0.756 - 0.972 - 1.258 - 1.693 -2.145 B (calculated) 0.388 0.43 1 0-460 0.482 0-497 0.513 0.644 0.568 0-600 0.618May, 19681 EXPERIMENTAL RESULTS THAT SHOULD FIT A LINEAR EQUATION 277 o'2 t Wavelength, nm Fig.3. Absorption spectra (1- cm cells) for mixtures of 2-5 x 1 0 - 3 ~ copper bisacetylacetonate with varying amounts of isoquino- line in benzene. Curves 1 to 6 correspond to 0.0, 0.12, 0.20, 0.40, 0-60 and 0.80 M isoquinoline, respec- tively ly-axis 1 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 x-axis Fig. 4. Data from Table I1 relating to adduct formation between copper bisacetylacetonate and isoquinoline in benzene. Calculation of K m and experiment, for this limiting value is only slowly approached asymptotically if the complex is weak. Indeed, the necessary large excess of the reagent L may so modify the nature of the solvent (specifically its dielectric constant and solvating power) as to yield spurious results.It now becomes necessary to treat both KML and cML as unknowns. Table I1 summarises data for mixtures of copper bisacetylacetonate, Cu(acac),, (=M) and quinoline (=L) in benzene when a weak 1 + 1 adduct, Cu(acac), . (quinoline), is formed.2 The absorption spectra of all the mixtures show an isosbestic point at 580 nm (Fig. 3), but the absorption caused by the 1 + 1 adduct and unchanged copper chelate has still not reached a constant value, even when the ratio of C, to CM is as high as 320: 1. Optical density measurements (1-cm cells) were made with a Unicam SP500 spectro- photometer at 670 mp, at which wavelength eL = 0, E& = 0.388, C, was always 2.5 x M and CL ranged from 0 to 0-8 M. By writing E~ = 0 in equation (7) and re-arranging we obtain and when this is compared with the standard form of the equation [equation (2)] we realise that KML = K , E, = K', and that the new variables A and B are to be calculated from A = ECL a n d B = E .. .... . . (9). @if - E) The calculated values of A and B are given in Table I1 and the corresponding lines are shown plotted in Fig. 4. The intersection leads to the values K = 0.27 f 0.01 and K' + 0-69, whence KML = 3-70 0.15 and E, = 0-690. The far more laborious method for handling such data is that suggested by Graddon and W a t t ~ n . ~ Here a trial value for E, is assumed and values for KML calculated for every experimental value of E. The average value of KML and its standard deviation is then calculated. The procedure is repeated for other assumed values of E,, and the approved value is taken as that for which the spread of values of K is least.The principle is illustrated by the calculations shown in Table 111.278 IRVING: A GENERAL GRAPHICAL METHOD FOR EVALUATING [hZa@!, VOl. 93 TABLE I11 CALCULATIONS OF THE STABILITY CONSTANT OF THE WEAK ADDUCT CU(ACAC)~. (QUINOLINE) E m .. . . . . 0.678 0.688 0.691 0.693 0.694 0-695 0.698 0-708 0.738 KML .. . . 4.200 3,870 3-810 3.760 3.714 3.700 3.630 3.400 3.860 Standard deviation 0-290 0-210 0.200 0.190 0.177 0.190 0.200 0-230 0.380 The close agreement between the graphical and the computed value KML = 3.71 0-18 is fortuitous in this instance. However, the graphical method quickly indicates a reasonable approximate value to use for E,.EXAMPLE 3- organic phase and water When a monobasic acid, HL, with acid-dissociation constant, Ke, partitions between an [HLI organic = [HL] + [L-] - Po [HL1orrranic , the partition coefficient for the undissociated species. where $0 = [HLI On re-arrangement, equation (10) gives ( $ - ) p o - ( & . . ) K a = l .. .. .. .. and comparison with the standard equation (2) shows that $0 = K , Ka = K’ and that we must set A = and B = [H+] as shown in Table IV, which summarises unpublished results for the partition of a weak acid between cyclohexane and buffer solutions adjusted to constant ionic strength (p = 0.1) with potassium chloride. TABLE IV PARTITION OF A WEAK BASE BETWEEN CYCLOHEXANE AND BUFFER (p = 0.1) pH .. .. . . 6.71 6.23 5.10 5-06 4.99 4.70 4.62 4-30 4.02 p ( = A ) ,.. . 0.360 0.745 0.880 0.940 1.01 1.32 1.48 1.60 1-80 1P [H+] (E B) . . 0.195 0.689 0.794 0.891 1.023 1.995 3.020 5.012 9.650 Fig. 5 shows the graphical solution, which leads to $0 = 1.93 & 0.03 and pKa = 5-02 & 0-02. Two points are especially noteworthy in these equations. Although the range of pH values covers only 1.7 units, [H+] varies 50-fold, and this makes it impossible to accommodate all the B-values along the ordinate axis without intolerably “crowding” the values for 0.2 < lOb[H+) < 2.0. Nevertheless, the results for values of B greater than 2 x lo4 can readily be incorporated by writing the equation to the line through the points (A,O) and (O,B), v ~ x . , y = (i) x + B, and calculating the value of x corresponding to the smallest accessible ordinate (in this example y = -2 x plotting this point and joining up to the point (A,O), as shown in Fig.5 for 106[H+] = 3.020, 5-012 and 99550. The same device has been used without explanation in Figs. 2 and 4. The second, and rather obvious, point is that this graphical procedure will break down if the partition coefficient $0 is more than an order of magnitude greater than the measured values of p. This imposes too lengthy an extrapolation to locate the point (K,K‘) with any accuracy. Indeed, it is just this condition which limits the value of the graphical procedure in the general case. The size of the area which, in practice, generally replaces the single unique point of intersection of all the lines through (A,O) and (0,B) gives an immediate indication of the quality of the experimental data and often shows whether the value of one or other of the constants K and K’ is particularly influenced by experimental error.May, 19681 EXPERIMENTAL RESULTS THAT SHOULD FIT A LINEAR EQUATION 279 Fig.5. Data from Table IV relating to the partition of a monobasic acid between cyclo- hexane and buffer solutions. Calculation of po and K, EXAMPLE A The stability of complex<s of EDTA and other complexones with iron(I1) and with iron(II1) ions can be determined from measurements of redox potential^.^ To take a simple example, the redox potential, EM, of the couple Fe(II1) - Fe(II), in the presence or absence of a complexone as desired, can be obtained from measurements of the potential E (relative to a standard hydrogen electrode) of a gold electrode dipping into V ml of a solution of an iron(I1) salt to which varying volumes, v, of a standardised oxidant such as iodine have been added.If Ve is the volume needed to oxidise the Fez+ completely we have V . . (12) .. .. .. RT E = EM + ( p) In (ve - V ) where the unknowns to be determined are the end-point volume, Ve, and the mid-point RT potential, EM. Writing (-) = s = 0*059517 at 25" C, and re-arranging we obtain Comparison with the standard equation (2) shows that Ve = K EM = S.lOg K', and that the variables A and B are to be calculated from A = v, and B = -loEls. Table V shows typical experimental data for the titration of 100 ml of about 8 X lo4 M iron(I1) sulphate containing 2 x 1 0 " ~ EDTA at a pH of 4.625 with a constant salt back- ground, p = 0.1, potassium chl~ride.~ The titrant was about 2.2 x M iodine.TABLE V TITRATION OF IRON(II) SULPHATE WITH IODINE IN THE PRESENCE OF EDTA E , mV .. . . 39.6 46.5 71.6 92.2 115.2 12443 v, ml .. .. 3-66 3-68 3.70 3-72 3.74 v, ml .. .. 1-40 1-70 2-60 3-20 3-52 3.60 lWP .. .. 4-671 6.109 15.02 36-19 88.59 128.7 E, mV .. . . 135.3 139.6 144-5 162-8 159.9 lWP .. . . 193.6 229.0 277.1 382.8 504.7280 IRVING The results are shown graphically in Fig. 6, which only displays the top right-hand portion of the whole large-scale graph. By inspection, V e = 3.8 and 10(EMls) = 7-5 whence EM = 0.0521. In this graph the precise location of the ordinate K’ is subject to considerable error and the uncertainty is about f0.002 volt, which is scarcely acceptable.More confidence could be attached to the result of plotting v-l against 10-EP for, as shown by equation (13), this gives a straight line of intercept vF1 and slope -v;llO(EM/s). Alternatively, a precise value of ve can be obtained by Gunnar Gran’s methoda; this involves plotting V.1017(k--E) against V (k is any arbitrary number such that 1017(k--E) forms a convenient range of positive numbers) and noting the values of v at which the resulting straight line cuts the x-axis. This value of ve can then be used in equation (12) to calculate the value of EM for all relevant values of E and v. Fig. 6. Data from Table V relaOing to the redox potentials of Fe(I1) - Fe(II1) in a buffer solution containing EDTA. Calculation of V , and EM In conclusion, I would again point out that the graphical procedures described above have their proper function. In some graphs difficulties of interpolation will not give results of acceptable precision, even when a large-scale graph is used; here more refined computational methods are essential. In others, the procedure will lead to numerical results entirely adequate for the precision of the calculations in train. There is little justification for using an elaborate programme and expensive computer time if the primary experimental results are insufficiently well defined, and the graphical procedure outlined above often provides a simple means of applying this criterion. REFERENCES 1. Schwarzenbach, G., Willi, A., and Bach, R. O., Heh. Ckim. Acta, 1947,30, 1303. 2. Al-Niaimi, N., D.Phi1. Thesis, Oxford, 1964. 3. Graddon, E. P., and Watton, E. C., J . Inorg. Nucl. Chem., 1961, 21, 4. 4. Schwarzenbach, G., and Heller, J., Heh. Chim. Acta, 1961, 34, 676. 5. Sharpe, K., Ph.D. Thesis, Leeds, 1968. 6. Gran, G., Analyst, 1952, 77, 661. Received October 27th, 1967