Analyst, July, 1979, Vol. 104, pp. 601-612 601 Automatic Titration by Stepwise Addition of Equal Volumes of Titrant Part IV.” Acid - Base Titrations General-purpose Program for Evaluating Potentiometric Axel Johansson and Sten Johansson Department of Analytical Chemistry, The Royal Institute of Technology, S-100 44 Stockholm 70, Sweden A computational procedure for evaluating titration results is described. It utilises non-linear regression techniques to solve the equations for titration graphs from acid - base titrations. The procedure is based on the fact that the titration of an n-protic acid can usually be treated as a titration of n monoprotic acids of the same molarity as the n-protic acid. In principle, the composition of an unknown mixture of acids can be deter- mined by non-linear regression if a sufficient number of measurements (millilitres of titrant veYsz4.s pH) are available.However, this assumes that the measured values are known with adequate accuracy, an assumption that is not correct in practice. All measured values (especially the pH values) are imperfect, hence a conventional non-linear regression can give a large number of mathematically equivalent solutions. This approach may there- fore, by chance, produce a result that is mathematically correct but chemically impossible. This problem is avoided by setting boundary conditions and carrying out calculations in an appropriate sequence. Dominating terms are dealt with first, then the effects of the remaining terms are included. A computer program TITRA, capable of handling such a calculation scheme, is presented.No preliminary estimates of the required concentrations or equivalence volumes are necessary in the program. The general equation for the titration graph derived is applicable to samples that may contain monoprotic acids (strong or weak), polyprotic acids, mixtures of acids, ampholytes, salts of weak acids and “abnormal” acids (acids with an abnormal sequence of stability constant values). By exchanging hydrogen-ion concentrations for hydroxide-ion concentrations and vice versa the equation is applicable to the titration of bases. Keywords : Acid - base titrimetry ; evaluation program In Part IIIl of this series we described a method for calculating equivalence volumes in the titration of monoprotic acids or bases with almost any value of the stability constants.This procedure, which is based upon solving a set of linear equations, has the following advantages : it requires only a few pairs of measurements (millilitres of titrant and e.m.f. or millilitres of titrant and pH) ; no preliminary estimate of equivalence volume is necessary; the stability constants of the acid or the base need not be known; and exact pH values are not required. Its main disadvantage is the fact that it does not allow calculation of the individual com- ponents in a mixture of acids or bases. However, in some instances it allows calculation of the total concentration of components. In this paper we discuss a more general computer program, TITRA, which is capable of handling results from acid - base titrations of almost any kind; it is an extended version of a program that has been in use for several years at this Institute.It has not previously been published, although applications of this program have been previously reported.2 Programs based on similar principles have also been described by us in collaboration with Pehrsson and Ir~grnan.~,~ These papers describe in detail several special cases, such as the titration of diprotic acids and of a mixture of two monoprotic acids. They also include a literature review of earlier work in this field. Since then, other similar computational procedures have been described.5-’ * For Part I11 of this series, see reference list, p. 612.602 JOHANSSON AND JOHANSSON : AUTOMATIC TITRATION BY STEPWISE Analyst, VoZ.104 A set of m equations is solved for r unknowns (m > r ) . The equations are no't linear, so the equation set is solved by estimating approximate values of the unknowns and then using a least-squares procedure to obtain more accurate values. The general equation for the titration graph of a polyprotic acid is complex and thus difficult to handle. However, the titration of an n-protic acid can usually be treated as a titration of n monoprotic acids of the same molarity as the rc-protic acid. It is then necessary to use so-called titration constants rather than conventional stability constants.8 All components in a mixture of acids are treated as monoprotic acids, thus reducing the equations to a simpler and more easily handled form. Note that treating an wprotic acid as a mixture of monoprotic acids does not introduce approximations (except for so-called abnormal acids, which simultaneously release -two or more protonsQ; however, these can be handled by a modification of the program).TITRA is based on classical mathematical procedures. Equations of Titration Graphs Mixtures of Acids and Polyprotic Acids Consider Vo ml of a solution containing n acids, HA(,,, HA(,,, . . ., HA(,,, . . ., HA,,,, each at concentration Col, C,,,, . ., Coi, . . ., Con, and with stability constants K,, K,, . . ., K,, . . ., K,. This solution is titrated with a strong base of concentration C,. After each addition of an aliquot of base, the e.m.f. is measured and the hydrogen-ion concentration, [HI, is calculated. A total of m pairs of measurements are obtained, i.e., at least (m - 1) aliquots of base are added.The hydrogen-ion concentra.tions [H'] measured often include a systematic error, f, such that [HI = f[H'], where f is an unknown, but constant, factor. This error factor f arises due to the fact that the electrode response is changed by the transfer from one solution to an0ther.l~~ At each titration point, the following set of equations applies,2 assuming that V ml of base are added: n Kk = Kwlf where K , is the ionic product of water; K; = Kif where Ki is the stability constant of acid HA,,, ; and Ki = [HA(,,]/{ [HI [A(,,] ). is the titration constant of one of the monoprotic acids that replaces the polyprotic acid, all acids having the same concentration.) (If the acid is polyprotic, Ampholytes For example, consider a solution of potassium hydrogen phthalate.It. is necessary to assume that phthalic acid is equivalent to two monoprotic acids, HA,,, and HA,,,, each a t concentration C, (C,, = C,, = Co) and with stability constants K, and K,, respectively; the initial potassium-ion concentration is C,. The charge balance gives With ampholytes, equation (1) takes a slightly different form. The ampholyte is titrated with sodium hydroxide solution. or Thus, equation (2) includes the term -VoCo, which is not in equation (1). Samples Mixed with a Known Amount of Strong Acid chloric acid, to the sample solution before the titration is carried out. It may sometimes be advantageous to add a known amount of strong acid, e.g., hydro- An increase in theJuly, 1979 ADDITION OF EQUAL VOLUMES OF TITRANT. PART IV 603 hydrogen-ion concentration permits calculation of better estimates of f (see below).Addition of a known amount of strong acid may also be useful when a salt of a weak acid or an ampholyte is titrated. All samples can then be titrated with standard sodium hydroxide solution. It is not necessary to add an excess of the acid iff is not to be calcu- lated (see Table 11). With the addition of strong acid a term Vo[C1] must be added to the titration equation, where Vo[C1] is the number of milliequivalents of strong (hydro- chloric) acid added. Acids that Simultaneously Release Several Protons The program TITRA treats the titration of an n-protic acid as being equivalent to the titration of n monoprotic acids, each of the acids having the same total concentration as the n-protic acid.The stability constants of these component acids will not be the same as those of the n-protic acid (KgA, K;+, etc., where KgnA is a conventional stability constant). The stability constants (or titration constants) of these component acids are related to the intrinsic constants of the individual groups in the rc-protic acid.* Titration constants are usually designated as Gi, but in this paper we have used the symbol Ki for all monoprotic acids (including component acids). The following discussion is limited to diprotic acids (H,A), so as to avoid excessively lengthy equations. A diprotic acid can be treated as being two monoprotic acids if KgA > 4Kz2,. Thus, acid H2A must be stronger than acid HA-, and this is almost always true.When such an acid is titrated, protons are released sequentially with increasing pH, forming first HA- and then A,-. Each of these ions is the predominant species over a certain pH range. In the special case KgA = 4Kg2A, the titration constants are equal, i.e., I<, = K, and equation (1) becomes 2T;r,GCO/(l + [HI K,) - (Vo + V ) [HI + (Vo + V)KW/[Hl - vc, = 0 This is the same as for a monoprotic acid of twice the concentration. The condition KgA < 4Kg2A is almost never found with diprotic acids, but it can apply to conditional constants if complex formation occurs.8s10 In this last instance, the titration graph has only one step, indicating that the species H2A and A,- predominate, with very low HA- concentrations during the entire titration.Vo ml of a diprotic acid, H,A, of total initial concentration Co, is titrated with C, M sodium hydroxide solution. Assuming that [HA-] is negligible, and after considering charge balance, we obtain The titration graph equation for such an abnormal acid can be derived as follows. [H+] + [Naf] - 2[A2-] - [OH-] = 0 . . . . * * (3) Following the addition of V ml of sodium hydroxide solution, "a+] = VC,/(V, + V ) . total concentration of acid, CHZA, is given by The and by combining this equation with the stability constant we obtain where Co(ab) is total initial concentration of the abnormal acid. yields Substitution in equation (3) 2 v o c o ( a b ) / ( l + [HI2Kab) - (Vo + V ) [HI -k ( v o + V)Kw/[H] - ~ C B = 0 (4) Equation (4) differs from equation (1) only in the first term.604 JOHANSSON AND JOHANSSON : AUTOMATIC TITRATION BY STEPWISE Analyst, Val.104 General Titration Graph Equation (2) and (4), adding V,[Cl] and considering that [HI = f[H’]. The following general formula for a titration graph is obtained by combining equations (1) , n where a is the number of neutralised steps in an ampholyte of concentration Co(am) [k, at least one Coi value is equal to Co(am)], b is the number of protons that are simultaneously released during an abnormal protolysis step of an acid of concentration Co(ab), and K& = Kabfb where Kab is the product of the stability constants for the abnormal step. VoJ V , C,, a, b, [Cl] and [H’] in equation (5) are known, while &, K’ , K& andf are unknown (or only approximately known).In principle, the composition of an unknown mixture of acids can be determined if a sufficient number of measurements are available. However, this assumes that the measured values are known with adequate accuracy, an assumption that is not correct in practice. Further, the e.m.f. values measured correspond1 to a variation in the hydrogen-ion concentra- tion over many orders of magnitude. The equations can therefore be characterised as being ill-conditioned.11 From a computational viewpoint this ill-conditioning means that there are a large number of nearly correct solutions to the equation set (5). The computational procedure used to solve these equations is important. For example, consider the term (V, + V ) [H’lf. If (V, + V ) = 100, the teim has the value 1 x f when log [H’] = -2 and the value 10-6 x f when log [H’] = -8.Thus, f cannot be determined precisely from titration results obtained at high pH values. The computational procedure must therefore be designed first to utilise dominating terms arid then to include the effects of the remaining terms. Also, chemically impossible values of the parameters, e.g., negative stability constants and negative concentrations, must be rejected. Coi are the required unknown concentrations. Detailed Calculation Procedure 1. 2. Measured e.m.f. or pH values are converted into [H’] values, which are then used in equation (5). If the values of all constants were known, leaving Coi values as the only unknowns, this would result in a set of linear equations thai could be solved by a least-squares procedure.In practice, this is usually not the case and the equation set is not linear. Instead, the problem is solved by substituting estimated values of the parameters Coi, Ki, f, K& and, if necessary , K&, and then obtaining improved. values by successive approximation. After substituting these estimated values plus a data pair (V, [H’])j, the left-hand sides of these equations have values U j ( j = 1, 2, . . ., nz), where Uj = 0. The estimated values of parameters are then corrected so as to minimise the sum C Uj2. m j = 1 3, For this purpose, a further set of equations is developed, with the corrections as the unknowns. The coefficients in these equations are obtained by differentiating equation (5) , thus obtaining a set of linear equations.Operations 2 and 3 above are carried out using the subroutine DPV. 4. The number of measurement points (equal to the number of equations developed in paragraph 2 above) must be at least equal to the number of unknowns. If more results are available a least-squares procedure can be used to compensate for random errors in the measurements. This compensation is achieved by reducing the equations to the corre- sponding normal equations; they are then solved by the Gauss elimination procedure, using an IBM routine SIMQ. The roots are added to the estimates and then new Uj values are calculated by using DPV. This procedure is repeated until 2 Uj2 meets certain conditions m j = l (see flow diaaam. Fin. 1).July, 1979 ADDITION OF EQUAL VOLUMES OF TITRANT. PART IV 605 START 1-2 +- 1CYK:l HDT vector for storing the values of the parameters during the variation UMINSIZ em f.DATA ph‘ IJATA J 1 I number of protolysis steps, parameters and refinement - DT HDT=DT I REPET = TRUE 4 - b l ’ 1CYK:O I C Y K = ICYKi1 I DAMP.1 card OT vector for storing the best values of the parameters @ repetition begins @ next cycle begins A r ‘ the number of points considered after the last equation point is reduced to max. 3 (if not otherwise stated) U1 and U2 are error square sums final output obs. ph, calc. ph, Ve;, Coi, log Ki, log K, and f HROT vector for storing the calcu I ated corrections of the parameter @ continuation on right Fig. 1. Flow diagram of program TITRA. The constant data consist of NCYK = maximum number of calculation cycles, IE = 0 if ph’ data are used, otherwise IE = 1, Cg, VO’, pWw, (temp., E&ja) and pairs of V and e.m.f.or ph’. If ph’ values are used it is not necessary to give the constants in parentheses. If a base is titrated poh’ is calculated from poh’ = P K ’ ~ - ph’ (here poh’ = - log[OH’] and ph’ = - log [H’]) . 5 . The values of Coi obtained are used to calculate the equivalence volumes (VeJ of the acids and all measurement points after the last equivalence point, except the first three, are omitted. The reason for this omission is that the quality of the experimental values decreases when a glass electrode is used at a high pH value.606 JOHANSSON AND JOHANSSON : AUTOMATIC TITRATION BY STEPWISE ApzaZyst, VoZ. 104 If the error factor f is calculated the [H’] and K; values are divided by f and the K& value is multiplied by f.This operation gives the values of [HI, Ki and K,. These values are then used to repeat the calculations from para,graph 4. Finally, the obtained values of the parameters are substituted into equation (5) and -log[H] or -log[H’] are calculated for each measurement point and compared with observed values. The calculation can be carried out in many ways; the program TITRA performs it as follows. The value of -log[H’] is assumed to be within the interval 0-14. The hydrogen-ion concentrations corresponding to the interval boundaries are substituted in the left-hand side of equation (5), yielding expressions with negative and positive values of U,. As equation (5) is a single-valued function of [H’], the po1arit:y of the new value of U, can be used to select the half interval that includes the true value of [H’].The other boundary of this half interval will have opposite polarity. The procedure is iterated until -log[H’] is deter- mined with an error of less than 0.001. 6. The pH interval is then halved and a new value of Ui is calculated. Which estimated values aye required? KI, Kk, K&, f and COi are unknown, but estimated values are required for the calculation. It is inconvenient to have to choose estimates before each calculation, and standard numbers, say 1 or 0, would be preferable as initial approximations. It is often, but not always, possible to obtain standard values (see examples below). The need for accurate estimates increases with increasing sample complexity. The factor f has a value of about 1, which can always be used as an initial estimate.If the maximum error in EA (see Experimental below) is known, it is possible to set limits on possible f values. For example, experimental conditions can be chosen such that -log[H’] has a maximum error of rt0.02 unit, i.e., the error in EL is less than 51.2 mV; f can then vary between 0.95 and 1.05. If the computed value off lie; outside this range the experimental results have not been sufficiently accurate to allow e::act calculation of f. As was mentioned above, the term (V, + V ) [H’] f, from which f is calculated, may be small compared with other terms in the equation. The errors in these other terms may be greater than the value of (V, + V ) [H’] f.Hence, direct application of the equations may give very large or very small values of f, which is chemically impossible. If the error in f is great, the iteration procedures either do not converge or converge to a completely wrong result. The program TITRA is designed so that if calculated f values ever lie outside tlhe range 0.95-1.05 they are corrected to the nearest range boundary. It is also possible to repeat the titration after adding a known amount of a strong acid to the sample solution. This increases the value of the term (V, + V ) [H‘] f so that f can be calculated satisfactorily. Log K’ generally lies between 13.5 and 14.0,12 but a more accurate value can be obtained by calibrating the electrode pair. If the only measurements available are from acidic solutions, no attempt should be made to obtain a better estimate of K& because the K& term has little effect in these instances.It is more difficult to estimate K; values. In many instances a log K; value of -10 for strong acids and 0 for other acids can be used. In other instances estimates can be obtained from tables, but these will probably not be exact values. K; values must be corrected for the ionic strength of the solution. Also, sample components may form complexes with anions of the acid, with marked effect on the .K; values. The stability constants affect only the term VoCoi/(l + [H’IK;) in equation (5), and errors in [H’] can be partly compensated for by using appropriately corrected values of Ki.Thus, it is usually desirable to calculate stability constants rather than to use pre-selected values, except with complex samples. For polyprotic acids it may be difficult to convert literature values of stability constants into the corresponding titration constants.8 However, calculations with TITRA yield the appropriate values directly. In difficult instances, the constants should be determined by titrating a known amount of acid using the appropriate electrode pair. TITRA can be used to make the calculations. If the stability constants are nearly equal, the calculations only yield acceptable results if This value is then used in the next calculation cycle. An approximate K& value is chosen according to the ionic strength of the solution.July, 1979 ADDITION OF EQUAL VOLUMES OF TITRANT.PART IV 607 the stability constants are accurately known. Examples are given in Table I and in reference 2. The Coi values are unknown, but as a first approximation they can be set as where Vm is the total volume of titrant added and 9 is the total number of protolysis steps. All acids are thus assumed to have the same initial concentration. As the quantities V,, CB, Vo and p must always be entered into the program, this calculation mode does not entail additional work. Experimental Selecting an Electrode Pair Potentiometric acid - base titrations are usually carried out by using glass and reference electrodes, with the electrode pair responding to hydrogen-ion activity rather than hydrogen-ion concentration. With conventional sigmoidal titration graphs this is not a problem, as equivalence volumes are determined by detecting a point of inflection.Differ- ences between hydrogen-ion activity and hydrogen-ion concentration will shift the inflection point along the pH axis, but will not affect the equivalence volume. Also, the behaviour of the electrode pair at values remote from the equivalence volume is not critical. With numerical evaluation, titration results measured over almost the entire titration graph are used and if the sample contains a mixture of acids the titration results may cover a wide pH range. This necessitates control of the electrode pair performance over almost the entire pH scale. The computer procedure is designed so that all measured values of hydrogen-ion concentra- tion can include an error factor f.This error factor must be constant over the pH range used. If this is not the case, the calculated composition of the sample may be wrong. The hydrogen-ion concentration in a solution can be calculated from measured e.m.f. values (E in mV) using the equation E = Ed + Qlog[H] + E; where EA stands for the conditional standard potential and includes the glass electrode's normal potential, E&, the reference half-cell potential and that part of the activity co- efficients and the liquid-junction potential that is independent of acidity, [HI is the hydrogen- ion concentration (not activity) and Q = RT x lnlO/F = 59.158 mV at 25 "C. E; = j,[H] + joHIOH] accounts for the acidity-dependent part of the activity coefficients and the liquid junction potential. The factors j , and joH are constants at constant ionic strength.Also, -log [HI is designated ph in the following text. The value of jH can be determined by titrating a known amount of 0.02-0.05 M hydro- chloric acid with a 0 . 1 - 0 . 4 ~ solution of sodium hydroxide. The base is added stepwise and E is measured after each addition of titrant. Plotting a graph of E - Q x ph versus [HI yields jH as the slope and EA as the intercept with the y-axis.13 The term j,, has such a small value that it is usually unnecessary to consider it if ph < 12. The value of jH for a given electrode pair is stable, but it should be checked if the electrode is used under extreme conditions. Eh values vary from day to day. A constant error AE& introduces an error, Aph, in all ph values, corresponding to the error factor f in all hydrogen-ion concentrations.However, a change in EA may take place in some glass electrodes going from the acid to the alkaline ph range. The value of EL in acidic solution, Eda, may differ from the value in alkaline solution, Edb, by several millivolts. According to Ciavatta14 the difference is to be ascribed to the change of the E,$ occurring in the ph range 5-8.5. He explains the change by assuming that the protective layer of hydrated silica formed in acidic solutions on the surface of the glass electrode will begin to dissolve as ph becomes greater than 7. Thereby, the surface structure of the glass membrane, which is in contact with the test solution, will be modified.In order to avoid any influence of the difference between the Ed values that are valid in acidic and alkaline solutions on the results of the titrations the results for e.m.f. obtained at ph less than 5 and ph greater than 8.5 are usually treated608 JOHANSSON AND JOHANSSON : AUTOMATIC TITRATION BY STEPWISE Analyst, VoZ. 104 separately.3J4 This treatment is not practical in routine analysis and therefore we have studied the influence of a change in Eh on the results. The effect was investigated by use of synthetic titrations in which 100.0 ml of a 0.01 M solution of an acid were titrated with 1.00-ml aliquots of 0.1 M strong base, Log K values of the acid varied between -10 and +lo. ph values were calculated and then altered SO as to simulate an electrode with Eda differing from E&.In one series of titrations ph values between 6.5 and 7.5 were increased by 0.01 per 0.1 unit while ph values above 7.5 were increased by 0.1. This is equivalent to a change in E, of 6 mV and of the hydrogen-ion concentration by a maximum of 21%. The equivalence volumes were calculated by different methods. When correct stability constants were used the greatest error was 5%. An error in the equivalence volume caused by an error introduced in changing [H'] can, as mentioned above, be reduced by using appropriate corrected values of Ki. Similarly, it is preferable to calculate K& for use in the term (V, + V ) K ' / [ H ] . At high ph values an error in [H'] can be compensated for by an equivalent change in K;. These procedures result in a maximum relative error of 1.1% for acids with 6.3 < log K' < 7.6.For other acids the relative errors were <0.1%. If the change in Ed is 3 mV the error never exceeds o.3y0, and if the change is 1.2 mV the maximum error is 0.2%. If a titration is performed rapidly it can be assumed that the change of E;, being rather slow, will occur over a wider pH range than if the electrode pair is allowed to attain equi- librium after each addition of titrant. If the change in EL appears, for example, in the pH range 6.5-9.5, the error will not exceed 0.1 yo. The optimum titration speed for a given glass electrode is the fastest speed that allows the concentration term Q x log[H] to reach equilibrium. From these synthetic titrations it can be concluded that the change in the conditional standard potential of a glass electrode pair must never be greater than 1.2 mV, corresponding to 0.02 ph unit if the greatest tolerable relative error in the equivalence volume is 0.2%.The difference Ed, - E,6 can be determined by comparing the glass electrode with a hydrogen electrode but this method of calibration is time consuming. It is sufficient to titrate a known solution consisting of a mixture of acids with log K evenly distributed over the ph range. An example of such a solution is hydrochloric acid (log K = -lo), phthalic acid (log K = 2.59 and 4.37), p-nitrophenol (log K = 6.88) and barbital (log K = 7.61). If the difference between calculated and experimentally determined ph values never exceeds 0.01 ph unit the electrode pair is acceptable.For each new electrode pair it is necessary first to determine jH by titration of a known 0.02-0.05 M solution of hydrochloric acid.l3 Then, an approximate value of EA should be obtained by measuring the e.m.f. of a 0.01 M hydrochloric acid solution at the appropriate ionic strength (2% = E + 2Q - O.OljH). It is not necessary to determine an exact value for J% as this value can change by about 0.1 mV when the electrode is transferred from one solution to another. Temperature measurements are required for determining the constant Q. The value of Q can be assumed to be sufficiently precise not to contribute significantly to the over-all error. Apparatus E.m.f. measurements were carried out with a micro-sized combination glass and silver - silver chloride electrode pair (AH-401 M5, Ingold, Zurich, Switzerland) and a digital volt- meter (Model S1016H, AB Systemteknik, Lidirtgo, Sweden).Titrant aliquots were added with a pneumatically driven pipette (AutoChem Instrument AB, Bromma, Sweden). The delivery volume of usually 1.001 1 & 0.0005 ml (mean & standard deviation) was determined by weighing ten consecutive aliquots of water. The titration vessel was a Metrohm glass vessel, E;A 875-200, equipped with an EA 880 cover and immersed in a thermostatically controlled bath maintained at 25.0 & 0.1 "C. A stream of nitrogen (purified by passing it through 2 M potassium hydroxide solution and pre-saturated by passing it through distilled water and the ionic medium) was passed over the titration solution in order to prevent absorption of atmospheric carbon dioxide.The titrations of monoprotic acids were carried out by using the simpler procedure described in Part 111.1July, 1979 ADDITION OF EQUAL VOLUMES OF TITRANT. PART IV 609 Hydrochloric acid was used as a primary standard and was prepared from Titrisol ampoules (Merck) and checked by gravimetric determination as silver chloride. The sodium hydroxide solution that was used as titrant was prepared by diluting a 50% solution and adding barium chloride. After separation of the barium carbonate formed from carbonate present as an impurity in the sodium hydroxide, the solution was diluted and the barium chloride content adjusted so that the ionic strength was 0.5. This solution was then standardised by titration against hydrochloric acid.There is no interference from carbonate formation with this titrant but many acids form complexes with barium ions. Thus, both titrant and titrand should have the same barium-ion concentration. The ionic strength of the titrand solution was adjusted to 0.5 by addition of sodium chloride. (In principle, higher ionic strengths are desirable but this requires the addition of large amounts of very pure neutral salts. When the use of barium chloride would result in precipitate formation with the sample, sodium chloride only was used as the ionic medium. Merck pro analysi grade barium chloride and sodium chloride were used in order to adjust ionic strengths. This procedure is impractical for routine analysis.) TABLE I TITRATION OF VARIOUS ACIDS, AMPHOLYTES AND BASES Figures in parentheses show the order in which estimates of parameters were adjusted: (0) no change in parameter value; (1) adjusted from first computation cycle; and (2) adjusted from second computation cycle.Sample 1. Hydrochloric acid 2. Picric acid . . 3. Sulphanilic acid 4. Acetic acid . . 5. Nitrophenol . . 6. Barbital Diprotic mi&- Monoprotic acids- 7. Adipic acid . . 8. Succinic acid . . 9. Phthalic acid . . 10. Malonic acid . . 11. Tartaric acid . . Triprotic acids- 12. Phosphoric acid 13. Boric acid . . Tetraprotic acids- 14. Diphosphoric acid 15. EDTA .. .. Pentaproiic acid- 16. Triphosphoric acid Mixtures of acids- 17. Hydrochloric acid + Phthalic acid +Barbital . . 18a. Phthalic acid . . I Nitrophenol +Barbital .. 1Sb. Phthalic acid + Nitrophenoi . +Barbital 19. Hydrochloric acid . . +Excess of acetic acid 20. Hydrochloric acid . . +Excess of acetic acid A mpholytes- 21. 22. Potassium hydrogen phthalate 24. Alanine 25. Disodiurn'EDTA' :: 1: Bases- 26. Sodium acetate . . . . . . 27. Sodiumhydroxide . _ .. +Sodium acetate . . .. Potassium hydrogen tartrate . . 23. Glycine .. .. .. Initial value of log K 3.1 (2), 7.2 (2), 12.7 (2) 9 (a), 12.5 (0), 14.0 (0) 1.0 (2), 2.5 (2), 6.1 (2), 8.5 (2) 2.3 (2), 2.3 (2), 0 (1) 0 (2), 2.0 (2), 2.2 (2), 5.0 (2), 7.0 (2) -10 (0) 2.5 ( l ) , 4.5 (1) 2.5 (2), 4.5 (2) 7.0 (2) 7.5 (1) 7.5 (2j 2.5 (2), 4.5 (2) 6.88 (0) 7.61 (0) -10 (0) 4.4 (0) -10 (0) 0 (1) f 1.0 (1) 1.0 (2) 1.0 (2) 1.0 (0) 1.0 (0) 1.0 (0) 1.0 (2) 1.0 (2) 1.0 (2) 1.0 (2) 1.0 (2) 1.0 (0) 1.0 (0) 1.0 (0) 1.0 (0) 1.0 (2) 1.0 (2) 1.0 (2) 1.0 (0) 1.0 (2) 1.0 (2) 1.0 (0) 1.0 (1) 1.0 (0) 1.0 (0) 1.0 (1) 1.0 (2) 1.0 (2) Amount added/ mmol 1.0000 0.5260 0.3950 0.685 3 1.0823 0.5046 0.9120 0.7706 0.9370 1.0289 0.483 0 1.0894 0.778 4 0.9155 0.9376 0.436 9 1.000 1.0025 1.001 1 1.023 4 0.4858 0.4934 1.0234 0.485 8 0.493 4 0.096 0.2400 1.815 7 0.7074 0.5167 1.2302 0.6074 0.9310 1.000 0.600 0.600 Amount found/ mmol 1.0004 0.5280 0.3955 0.6852 1.0778 0.5044 0.9130 0.7702 0.9374 1.026 4 0.4849 1.0877 0.7826 0.9124 0.9386 0.4349 1 .ooo 0.9992 1.0055 1.024 9 0.5295 0.4547 1.0249 0.4829 0.496 1 0.095 0.2381 1.5132 0.7076 0.5189 1.233 7 0.606 7 0.9308 1.001 0.598 0.602 Recovery, Yo 100.0 100.4 100.1 100.0 99.6 100.0 100.1 99.9 100.0 99.8 100.4 99.8 100.5 99.7 100.1 99.5 100.0 99.7 100.4 100.1 109.0 92.2 100.1 99.4 100.5 09.0 99.2 99.8 100.0 100.4 100.3 99.9 100.0 100.1 99.7 100.610 JOHANSSON AND JOHANSSON : AUTOMATIC TITRATION BY STEPWISE Analyst, VoZ.104 Evaluation of Program TITRA The program was tested by using results obtained from titrations of a considerable number of acids and mixtures of acids. A few titrations of bases and mixtures of bases were also used. Column 2 of Table I lists values of the stability constants that were used as initial approximate values, with the number in parentheses after each parameter referring to the order in which the parameters were adjusted. Adjustment from the first calculation cycle is designated by (1) and from the second cycle by (2), with no adjustment shown as (0).Approximate values of concentrations were never used and the program calculated all initial values by use of equation (6). An initial value of log K& = -13.61 for the parameter K' was used, and it was only adjusted when measurements at high pH were $available. The error factor f in hydrogen-ion concentration was assumed to be 1.00 and was adjusted only if two of the initial pH values were below 4. Also, f was not adjusted when Ei was determined in direct connection with the sample titration. The results are summarised in Tables I and 11. Comments on Table I The evaluation of titrations of single monoprotic acids is straightforward, particularly because knowledge of the initial values of stability constants and concentrations is unnecessary. (a) For strong acids, a low K value, typically 10-lo, should be selected and then not adjusted.All measurements after the equivalence volume has been reached should be rejected. If values beyond the equivalence volume are included, the sample should be treated as a mixture of a strong and a weak acid. Sample or titrant may contain small amounts of weak protolytes, e.g., carbonate. The contaminating acid can be assumed to have a log K value of 6-8. It is not necessary to correct this value during the calculations. (b) For weak acids, the error factor f should not be adjusted. ( c ) For acids of intermediate strength, Ir: should only be adjusted from the second The program EKVOL~ performs these operations automatically and is For diprotic acids, it is not necessary to provide initial values of stability constants unless For titrations of polyprotic acids, approximate values of the constants are required.Diphosphoric acid and triphosphoric acid were prepared by passing weighed amounts of the sodium salts through an ion-exchange coluinn (Dowex 50LvX8 [H+]) . The titration (No. 15) of EDTA (ethylenediaminetetraacetic acid) should be discussed in more detail. It was carried out in a barium chloride (0.133 M) and sodium chloride (0.100 M) solution, with the titrant having the same barium-ion concentration. Because of complex formation between the acid and the barium-ion, the conditional protonation constants for EDTA change, giving values of log K i y = 4 . 5 , log K&Y = 5.2 log KdSy = 2.4 and log KkdY = 1.9. The two equally strong carboxylic acid groups are titrated in the first step, followed by the remaining acid groups.(As the value of log KLy is less than log KL3. the two last groups are titrated simultaneously.) The titration is also affected by the fact that EDTA is so insoluble that it does not completely dissolve until about half of the total titrant has been added. In sample No. 25 the disodium salt of EDTA is titrated in the same ionic medium. In this instance the calculations will be further complicated by the salt being an ampholyte. Table I also summarises two different calculations (18a and 18b) of a titration of a mixture of phthalic acid, p-nitrophenol and barbital. The two last acids are of about equal strengths, with stability constants differing by only 0.8 logarithmic unit. A conventional titration graph of a similar sample (Fig.2) does not show a step between the two acids (point 3, Fig. 2), therefore graphical evaluation has little value. However, numerical evaluation can give good results if accurate values of the stability constants are available. Hence, the stability constants of the two acids, p-nitrophenol anid barbital, were determined by titration of known amounts of the separate acids. Correct concentrations were entered into the program TITRA, which was only used to calculate the constants. Use of the values obtained (log K = However, the following points should be noted. calculation cycle. preferred for these titrations. they are approximately equal (e.g., adipic acid). Under these conditions the titration graph shows only two steps.July, 1979 ADDITION OF EQUAL VOLUMES OF TITRANT.PART IV 61 1 9 - a - 7 - 0 2 4 6 8 10 12 14 16 18 20 2 2 k k k k Volume of titrant/ml ! Fig. 2. Titration of a mixture of phthalic acid, nitrophenol and barbital with sodium hydroxide solution. The graph was recorded with a Metrohm Potentiograph E336. The equivalence points are indicated by arrows. Numbers 1 and 2 refer to phthalic acid, number 3 to nitrophenol and number 4 to barbital. 6.88 for P-nitrophenol and 7.61 for barbital) gave a titration error of less than 0.5% for the mixture. An error of 0.1 logarithmic unit in the constants resulted in a titration error of about 10%. With both calculation procedures, there is excellent agreement between observed and calculated hydrogen-ion concentrations, showing that in this instance equation (5) has several mathematically equivalent solutions.Murtlow and Meites' reported on the potentiometric titration of solutions containing small amounts of hydrochloric acid together with much larger amounts of acetic acid. They titrated with a strong base, and used a non-linear regression technique in order to evaluate the titration graph equations. Titration of a solution containing 4.7 x lo-* M hydrochloric acid and 3 x 10-3 M acetic acid yielded errors of +3.3 and -1.3%, respectively. We have repeated Murtlow and Meites' titration using our experimental and evaluation techniques. Table I shows two sets of titration results. In the first set (No. 19), five 0.25-ml aliquots of 0.1 M sodium hydroxide solution were used for each 100-ml sample, so that only hydro- chloric acid was titrated.Duplicate analyses yielded hydrochloric acid recoveries of 99.0 and 101.5~0, respectively. In the second set of results (No. 20), a 500-ml sample was titrated with 40 0.5-ml aliquots of base. This titration yielded recoveries of 99.2 and 99.9% for hydrochloric and acetic acid , respectively. TABLE I1 TITRATIONS CARRIED OUT WITH ADDITION OF STRONG ACID Initial value of Amount Amount Recovery, Sample ph added/mmol found/mmol % Acetic acid .. .. .. 0.780 0 0.7775 99.7 0.7800 0.7787 99.8 0.780 0 0.780 7 100.1 Sodium acetate . . .. . . 2.32 0.551 7 0.550 2 99.7 2.40 0.625 1 0.621 2 99.4 3.14 0.993 7 0.992 6 99.9 3.27 1.0292 1.0307 100.1 4.13 1.5693 1.5720 99.8 Potassium hydrogen phthalate .. 2.15 0.437 1 0.435 7 99.7 2.16 0.435 4 0.4353 100.0 Log K 4.415 4.414 4.410 4.420 4.417 4.406 4.441 4.340 2.598, 4.371 2.593, 4.375612 JOHANSSON AND JOHANSSON Comments on Table 11 The titrations listed in Table I1 illustrate the importance of adding a known amount of strong acid to the sample solution. In all examples the volume V , was 100.0ml and the concentration of added hydrochloric acid was 0..01 M. The amounts (in mmol) of the samples are listed in column 3. These solutions were titrated with l-ml aliquots of 0.1 M sodium hydroxide solution until the last equivalence point was passed. In instances where the initial hydrogen-ion concentration was high f could be accurately calculated, which made it possible to calculate the stability constants. From the first five titrations in Table I1 the log K value of acetic acid was calculated as 4.415 & 0.003 (valid in 0.133 M barium chloride and 0.1 M sodium chloride solution).The last three titrations of sodium acetate give varying values of log K ; the initial ph values are too high to allow a correct evaluation off. Conclusions The results given in Tables I and I1 show t:hat the experimental and computational pro- cedures described in this paper can yield very accurate analytical results. A wide variety of analyses have been carried out: monoprotic to pentaprotic acids; mixtures of acids; ampholytes; acids with an abnormal sequence of stability constant values; and bases. The precision was determined by running duplicates (5-10 analyses) of hydrochloric acid, acetic acid and phthalic acid. The relative standard deviation was found to be less than 0.2%. The method of evaluation described in the present study is characterised as follows. (a) All normal acids are treated as if they were a mixture of monoprotic acids. This necessitates use of titration constants rather than conventional stability constants (or dissociation constants). (b) The procedure utilises non-linear regression techniques. (c) The parameter calculations are performed in an appropriate sequence. (d) Preliminary estimates of the required concentrations or equivalence volumes are not (e) Results from practically all types of acid - base titrations can be handled. necessary. The authors are indebted to Professor Folke Ingman for stimulating discussions and to Dr. Roland Ekelund for valuable assistance in preparing the computer program. They also thank Dr. Douglas Mitchell for translating the manuscript and Mrs. Karin Lindgren for skilful experiment a1 assistance. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. References Johansson, A., and Johansson, S., Analyst, 19713, 103, 305. Ingman, F., Johansson, A., Johansson, S., and Karlsson, R., Analytica Chim. Acta, 1973, 64, 113. Pehrsson, L., Ingman, F., and Johansson, A., Talanta, 1976, 23, 769. Pehrsson, L., Ingman, F., and Johansson, S., Talanta, 1976, 23, 781. Nowogrocki, G., Cannone, J., and Wozniak, M. Bull. SOC. Chim. Fr., 1976, 5. Bos, M., Analytica Chim. Acta, 1977, 90, 61. Murtlow, D., and Meites, L., Analytica Chim. Acta, 1977, 92, 286. Johansson, S., Analyst, 1979, 104, 593. Campbell, B. H., and Meites, L., Talanta, 1974, 21, 117. Schwarzenbach, G., and Ackermann, H., Helv. (Shim. Acta, 1947, 30, 1798. Sullivan, J . C., Kydberg, J., and Miller, W. F., Acta Chem. Scand., 1959, 13, 2023. Ilarned, H. S., and Gary, C. G., J . Am. Chem. SOC., 1937, 59, 2032. Biedermann, G., and SillCn, L. G., Ark. Kemi, 11963, 5, 425. Ciavatta, L., Ark. Kemi, 1963, 20, 417. 1L'oTE-Reference 1 is to Part 111 of this series. Received May 22nd, 1978 hmended December Sth, 1975 Accepted December 14tla, 1978