Determination of pK, Values of Acid Components in Atmospheric Condensates by Linearization of Segmented Titration Curves M. D. Seymour, J. W. Clayton, Jr., and Quintus Fernando* Department of Chemistry and Toxicology Program, University of Arizona, Tucson, Arizona 8572 I
An iterative calculation has been used to segment and linearlre a potentiometric titration curve that was obtalned for a mlxture of aclds. The slopes and x-intercepts of the h e a r plots (modified Gran plots) were used to calculate the acid dlssoclation constants and the concentratlons, respectively, of the acid components in the mixture.
Gran plots (1,2) and various forms of modified Gran plots have been used to locate the equivalence point in a potentiometric titration. This technique of end-point location has found wide application in acid-base titrimetry, particularly in the determination of very weak acids or bases or very dilute solutions of acids or mixtures of weak acids (3-6). Improved equations that are applicable to systems containing a monoprotic acid or polyprotic acids have been derived and tested successfully (7, 8). Computer programs based on modified Gran plots have been employed to process potentiometric data obtained in the course of the titration of a single acid (6) as well as mixtures of a c i z of nearly equal strengths (9). Computer calculations have also been used t o determine the accuracy of Gran plots (10) and t o examine their limitations (11).
Almost all of the published work has been concerned with the development of rapid and accurate methods for the analysis of weak acids or mixtures of acids (12-14). The use of Gran plots for the determination of acid dissociation constants has attracted relatively little attention (6, 7,15,16). We report below a method for the determination of the concentrations and pK, values of the acid components in a mixture of acids. The method is based on the use of modified Gran functions for carrying out the segmentation and linearization of a titration curve. The acid dissociation constants that are calculated from the slopes of the modified Gran plots serve to identify the acids, and their concentrations are given by the intercepts of the Gran plots with the abscissa. The method is of general applicability and can be computerized and used for the analysis of unknown mixtures of acids. The method has been verified by the analysis of synthetic mixtures of acids, that resemble the acid composition of atmospheric condensates, and rainwater samples that were collected in the Tucson area. We have found that most of these samples contain four acid components, one of which is a strong acid.
EXPERIMENTAL Reagents. A solution of NaOH (50% w/w) was prepared and filtered under nitrogen. Approximately 0.5 mL of the filtered solution was diluted with 1L of deionized water which had been boiled, and cooled under nitrogen. The NaOH solutionwas stored in a Teflon lined flask that was vented to the atmosphere through an Ascarite trap and a tube capped with a rubber septum was provided for removal of the NaOH solution. A syringe microburet was fiied with the carbonate free NaOH solution and standardized with potassium acid phthalate. Stock solutions (0.01M) of NaHC03 and NH4Cl were prepared from Analytical Reagent grade chemicals and standardized with
the NaOH solution. The NaHS04 stock solution was prepared from constant boiling H2S04,recrystallized Na2S04,and deionized water. These stock solutions were diluted a hundredfold just before use. The ionic strength was kept constant, in the course of a titration, with NaC1. A solution of 0.10 M NaCl was prepared in deionized, carbon dioxide-free water and stored under nitrogen. An appropriate volume of the NaCl solution was introduced directly into the titration vessel under nitrogen pressure, thereby preventing the NaCl solution from coming into contact with the carbon dioxide in the atmosphere. Potentiometric Titrations. All titrations were carried out in a jacketed titration vessel which was maintained a t 25 "C. During the titration, nitrogen gas, which was saturated with water vapor, was used to blanket the surface of the solution that was being titrated. Nitrogen gas was not bubbled through the solution during a titration. The required volume of the stock solution, or a mixture of the stock solutions, was introduced into the titration vessel and diluted with the stock solution of 0.10 M NaC1. The solution was stirred with a magnetic stirrer and any carbon dioxide that w a ~ introduced into the solution from the atmosphere was expelled by means of a stream of nitrogen gas presaturated with water vapor. The titration was carried out by placing the tip of the microburet below the surface of the solution in the titration vessel and adding increments of the NaOH solution. A glass-saturated calomel electrode pair, calibrated with NBS buffers at pH 4.01,6.86, and 9.18, together with an Orion Model 701 Digital pH meter were employed for the pH measurements. The NaHC03 stock solution was always added to the diluted solutions in the titration vessel to prevent the localized generation of an excess of carbon dioxide, which occured when concentrated solutions were mixed. Samples of atmospheric condensates or rainwater were made 0.10 M in NaCl by the addition of solid NaCl to 10 mL of the sample, and the potentiometric titration was carried out as described above. Derivation of Modified Gran Functions. All titrations were carried out in solutions in which the ionic strength was maintained a t a constant value of 0.10 with NaC1. The pH meter readings were converted into hydrogen ion concentrations, [H'], with the aid of Equations 1 and 2. n
.-.,
0.51 - log YH+ =
1 + (3.30 X 107)(9.0X 10-s)I"2
(2)
where y H + is the activity coefficient of the hydrogen ion; I , the ionic strength of the solution; and the ion size parameter of the hydrogen ion is assumed to be 9.0 X E represents the potential difference in millivolts at 25 "C between the glass electrode and the saturated calomel electrode. E'is a constant if the potential of the saturated calomel electrode and the liquid junction potential remain constant. If the behavior of the glass electrode is Nernstian and the measurements are carried out at 25 "C, In 10 R T I F is a constant and equal to 59.16. A synthetic mixture of KHS04, H2C03,and NHICl was prepared by adding an excess of KHS04 to a mixture of NaHC03 and NH4C1. The total volume of NaOH added at the successive ANALYTICAL CHEMISTRY, VOL. 49, NO. 9, AUGUST 1977
1420
equivalence points in the neutralization of each of the components, HS04-, H2C03, NH4', and HC03-, is V S A , V', Vw,, and V", respectively. Hence, the acid components in the mixture, HS04-, , VSA), (VWA HzC03, NH4+,and HC03- are neutralized by V ~ A(V'- V ? and (V" - Vw,) mL of standard NaOH solution. The analytical concentrations, (in moles/L), of each of the acid components, &so4-, C H ~ C O ~ , and C H C ~ are ~ - , given by the following mass balance equations:
(3) (the HS04- concentration is negligibly small because HS04- is considered to be completely dissociated in the solution under consideration). CH2C0
+
- ("-
(Vo+ V ) [co,2-]
=
[HzCO,]
+ [HCO;] (4) Le.,
= [Cl-]
(5)
K , V ' - K I V = F'
(6)
The linear plot of F'vs. V has a slope, -K1, and an intercept on the x-axis equal to V'. The volume of NaOH required to neutralize the weak acid, NH4+,with an acid dissociation constant of K N H ~is+( V ~ -AV'), and can be determined by extrapolation of the straight line plot of the modified Gran function FwA vs. V to intersect the abscissa at the point (VwA,0). The derivation of the function FWA follows the same pattern as before for the derivation of FSA and F', and is given by:
All the terms in square brackets are molar concentrations, CBthe molar concentration, and V the volume of RaOH added. The initial volume of the solution is Vo and the sodium ion concentration in solution is given by:
(7) The volume of NaOH required to neutralize the strong acid in solution is VSA, and can be determined experimentally. The electroneutrality condition is:
+ [K'] + [NH;] + [H'] = [OH-] + 2[S04'-] + [HCO;] + 2[C03'-] + [Cl-] (8)
KNH,+ VWA- KNH,+* V - (LH+l - [OH-] + 5)([H'1 + KNH,') CB + [H'] ( V - V') + ( 0 1 0 ~ 2- ~Q~Z3~ ~ ~ O ~ ) ( [ H + ]
+ KNH,+)(V'- VSA)
[Na']
where ( i.e.,
Substitution of the mass balance Equations 3-7 in Equation 8 and rearrangement of the terms gives:
K",+
VSACB - VCB = ([H'] - [OH-] )( V + Vo) - CB (VWA - V')&1"4+- CB (v' - vsA)(&1HzCo3
+ 2&zH2C03)
(9)
*
and ( & l H 2 " 3 ["I + KNH,+ - Kl[H'] + 2 k 1 K z [H']' + K,[H+] + K I K z =
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ANALYTICAL CHEMISTRY, VOL. 49, NO. 9, AUGUST 1977
= ~[H']2/([H']2 2 ~ ~
s
+ K1[Hf] + K1K2).
VWA - K N H 4 + V = F W A
(15)
- K2V
- (["I
[OH-I)(V+ VO)([H+l + K,) CB + [H'](V- VwA) + o(,-,"~+([H+] + K",+) (VWA- V') + ~ i o ~ , ~ O s ( [ H++2Kz)(V" ] - vWA)
+ 2azHzC03)
KNH4+represents the acid dissociation constant of NH4' and K1 and K 2 the successive acid dissociation constants of H2C03. The function F s A is a modified Gran function and a plot of FSAvs. V should give a straight line which has a slope of -CB.The point of intersection of this straight line and the abscissa is (Vs*,O). The volume of NaOH required to half-neutralize the H2C03 is ( V ' - V S A ) , and is determined experimentally with the aid of the modified Gran function F'which is derived as follows: the mass and charge balance Equations 3-8, together with the expression for K1, the first acid dissociation constant of HzC03give:
~
The slope and the x-intercept of the linear plot can be used to obtain the best estimates of KNH~+ and VWA, respectively. In the presence of excess strong acid, it is not necessary to determine a value for V" because, (V' - VSA)= (V" - VWA). The experimental determination of V" from a plot of the modified Gran function F"vs. V, will serve as a check on the validity of the method for the synthetic sample containing an excess of KHS04. The function F" is derived as before and is given by:
KzV" KNH,+
~
(14)
-
(16)
Le.,
K zV"
-
K2 V = F"
(17)
Values of K2 and V" can be obtained from the slope and x intercept of the linear plot of F" vs. V in the region V ~ CAV < V". Calculation of Approximate Values of the G r a n Functions. The experimental data set that is obtained consists of a series of measured pH values and the corresponding values of V ,
Table I. Determination and Identification of the Acid Components in Atmospheric Condensates and Rainwater Samples Collected in Tucson Condensate samples Rainwater samples Acid components No. 35A No. 36A No. 1 4 1 No. 142 pmol found in 1 0 mL Strong acid NH,+ H2CO3 HCO;
0.47 0.91 0.23 0.19
NH,' H*CO, HCO;
6.25 X lo-'' 7.69 X lo-' 1.58 X lo-''
0.47 0.10 0.68 0.60 0.18 0.14 0.18 0.07 Acid dissociation constants found 7.15
X
6.94 X
10.''
lo-''
8.33 x 10-7
3.33 x 10-7
1.92 X
1.72 X 10"'
the volume of NaOH added. The hydrogen ion concentration, [H'], can be readily calculated for each of the values of (pH)measured from Equations 1 and 2. The first two or three values of [H'] vs. V are sufficient to give an approximation of the straight line given by Equation 10 in which the values of F S A are calculated by using only the first term on the right hand side of Equation 9. A first estimate of VsA is obtained from the intersection of this straight line and the x-axis. The first two or three values of V > VsAare used next to obtain an approximate straight line given by Equation 13. The first and second terms on the right hand side of Equation 12 are used to calculate an approximate value of the function F'. The value of K1 in this term can either be ignored, since K1 < H+ in this region of the titration curve, or a reasonable estimate of K1 can be used. These approximate values of F' and V are used to establish a straight line that intersects the x-axis a t the point (V', 0). Values of V > V'are next used to establish a straight line described by Equation 15, and approximate values of the function FWA are calculated as before, by using the first two terms on the right hand side of Equation 14, either with or without a reasonable estimate of KN&++. After the straight line described by Equation 15 is established, and its point of intersection, ( V W ~0),, with x-axis is determined, the next several values of V > VWAare used to establish the succeeding straight line given by Equation 17. In this manner the titration curve is divided approximately into four segments from which initial estimates of V ~ Av', , VWA,v", K1,K N H a + , and Kz are obtained. These initial estimates are used to refine the modified Gran functions, F ~ AF', , FWA, and F". Refinement of the Modified Gran Functions. At this stage in the calculation, approximate values of VsA, V', VwA, and V" have been obtained together with initial estimates of K,, K N H a + , and Kz from the slopes of the straight lines given by the approximate Gran functions F', F W A , and F" vs. V. These initial estimates of the volumes and acid dissociation constants can be refined in the following manner. In the region 0 < V < VsA,the function FsAis given by Equation 9. The relative magnitude of each of the three terms on the right hand side of Equation 9 can be estimated and, if necessary, a better value of FSAcan be calculated for each value of V. The straight line plot of FSAvs. V should give an improved value of V ~ which A can be utilized in Equation 12. For values of V in the region VsA < V < V', the function F' can be calculated with the new value of VsA. The linear plot of F' vs. V in this region will give better values of V' and K1 which can be used to further refine the functions F'and FwA(Equations 13 and 15). In the region V' < V < VWA,the linear plot of FwAvs. V can be refined to give better values of K N H r + and VWAwhich are in turn employed to refine the function F" (Equation 17). This method of refinement will ultimately yield four straight lines (Figure 1) from which reliable values of the acid dissociation constants, K1, KNH~+, and Kz and the equivalence volumes VSA, V', VWA,and V" are obtained. The above equations are valid for the synthetic samples that contained four acid components, one of which was a strong acid. If any acidic impurities were introduced into the solution, there would be significant deviations from linearity in one or more linear segments. In the analysis of unknown samples of rainwater or
lo-''
I
0.03
0.65 0.19 0.15
6.25 5.00 1.67
X X
X
10"' 10"
lo-''
\
I
Figure 1. Linearized segments of the titration curve of a mixture of
1.84pmol HS0,-, 1.48pmol H,C03, and 1.31 pmol NH4+ vs. 1.540 X lo-*M NaOH. FSA, F', F W A , and F''are the modified Gran functions plotted vs. the volume of NaOH added (in pL). The slopes of F'and F"give the acid dissociation constants of H2C03and the slope of F w A gives the acid dissociation constant of NH4+. The intersections of these straight lines with the abscissa give the number of pL of NaOH required to neutralize each of the acid components in the mixture
atmospheric condensates, the presence of unexpected acidic species will become evident as the segmentation and linearization procedure that is described above is carried out. Simple modifications in the Gran functions that are derived above can be made t o take into account the presence of any acidic species that has a pK, value that is rt0.5 from the pK, value of another acid that is present in the solution.
RESULTS AND D I S C U S S I O N T h e segmentation and linearization of the titration curve of a synthetic mixture consisting of 1.84 pmol HS04-,1.48 pmol H2C03,and 1.3 pmol NH4+in a total volume of 10 mL is shown in Figure 1. The initial p H of the solution was 3.89. AB is the best straight line drawn through the experimental points calculated with the aid of Equation 10 in the segment of the titration curve from p H 3.93 t o 6.20. The slope of the straight line is C g , the concentration of the titrant. The straight line AI3 intersects the x-axis at the point ( V ~ A0), and the volume Vs, p L corresponds to 1.57 pmol of HS04-. CD ANALYTICAL CHEMISTRY, VOL. 49, NO. 9, AUGUST 1977
1431
is the linearization of the next segment of the titration curve from pH 5.86 to 9.04 (Equation 13). The slope of CD is 7.66 X which compares well with the expected value, 7.76 X the first acid dissociation constant of H2C03at an ionic strength of 0.10. The intersection of the straight line CD with the x-axis gave a value of ( V ' - VsA)which corresponded to 0.86 pmol of HzC03. The large discrepancy between this value and the expected value of 1.48 is caused by the loss of COz from the solution. This was verified by repeating the determination with the identical initial amounts of HS04-, HzC03,and NH4+. In the latter experiment, the solution was allowed to stand in a nitrogen atmosphere for 1 h before the titration was carried out. The value of ( V ' - VsA)obtained in this case corresponded to 0.15 pmol H2C03,which shows that there is a continuous loss of dissolved COz from the solution until an equilibrium concentration is reached. EF is the linearization of the segment of the titration curve from pH 8.71 to 9.82 (Equation 15). The slope of EF is 5.17 X which compares well with the expected value, 5.30 X the acid dissociation constant of NH4+ a t an ionic strength of 0.10. The intersection of EF with the abscissa gives ( V W A- V ? , which corresponds to 1.39 pmol of NH4+. The straight line GH represents the linearization of the segment of the titration curve from pH 9.45 to 10.07 (Equation 17). T h e slope of GH is 1.4 X the expected value for the a t an second dissociation constant of H & 0 3 is 1.40 X ionic strength of 0.10. Again, the number of pmoles the HCO; obtained experimentally (0.79) is much less than the expected value because COz is lost in the course of the titration. Replicate titrations were carried out to determine the reproducibility of the method. It was found that 1 to 2 pmol of HSO; and NH4+could be determined in 10 mL of solution with a relative standard deviation of 3 to 7%. All four acidic components could be readily identified from the slopes of the modified Gran plots. Weak acids with closely spaced acid dissociation constants, e.g., NH4+and HC03-, can be readily identified. T h e method was applied to two samples of atmospheric condensates and two samples of rainwater from Tucson. The condensate samples were collected on the roof of the Atmospheric Sciences building at the University of Arizona. A stainless steel plate cooled below the dew point was used to collect the condensate droplets. The results obtained are summarized in Table I. There was no evidence for the presence of any weak acids other than NH4+,HzCO3, and
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ANALYTICAL CHEMISTRY, VOL. 49, NO. 9,AUGUST 1977
HC03-. If any metal ions were present in the rainwater or condensate samples, their tendency to act as Bronsted acids was not detected by this method. Analysis of these samples by atomic absorption spectrophotometry showed that the only detectable cations besides the NH4+and H+ were K+ and Ca2+ which were present in the 0.02- to 0.07-ppm range. Very little strong acid was found in all the samples; the rainwater contained significantly smaller amounts of strong acid than the condensates. The strong acid can either be HS0,- or can be the dissociated protons from HNOB or HC1. It is significant that no HS03- was found in these samples. This weak acid at an species, with an acid dissociation constant of 1.9 X ionic strength of 0.10 would have affected the determination and identification of the H & 0 3 species. T h e concentration of HZCO3 in all the samples is approximately equal to the equilibrium value at ambient temperatures (0.1 to 0.2 bmol/lO mL). Therefore, it may be concluded that the concentration of HS03- in these samples is negligibly small. Moreover, we have found that in the titration of synthetic samples containing HS03-, rapid oxidation to HS04- occurs in solution in the presence of dissolved oxygen. The nature of the strong acid species that is present in rainwater has been a controversial issue for some time. We propose to employ our method in a nonaqueous system to distinguish between the various strong acids that may be present in rainwater.
LITERATURE CITED (1) G. Gran, Acta Chem. Scand., 4, 559 (1950).
(2) G. Gran, Analyst, (London), 77, 661 (1952). (3) F Ingman and E. Still, Talanta, 13, 1431 (1966). (4) A. Johansson, Analyst, (London), 95, 535 (1970). (5) A. Ivaska, Talanta, 21, 1167 (1974). (6) A. Ivaska and E. Wanninen, Anal. Lett., 6 ( I l ) , 961 (1973). (7) T. N. Briggs and J. E. Stuehr, Anal. Chem., 46, 1517 (1974). (8) D. Midgley and C. McCallum, Talanta, 21, 723 (1974). (9) F. Ingman, A. Johansson, S. Johansson, and R. Karlson, Anal. Chim. Acta, 64, 113 (1973). (IO) I. Hansson and D. Jagner, Anal. Chim. Acta, 65, 373 (1973). (1 1) C. McCallum and D. Midgley, Anal. Chim. Acta, 65, 155 (1973). (12) A. Ivaska, Taianta, 22, 995 (1975). (13) C. McCalium and D. Midgley, Anal. Chlm. Acta, 78, 171 (1975). (14) C. McCallum and D. Midgley. Anal. Chem., 46, 1232 (1976). (15) A. Ivaska and L. Harju, Talanta, 22, 1051 (1975). (16) M. D. Seymour and Q. Fernando, J. Chem. Educ., 54, 225 (1977).
RECEIVED for review April 4, 1977. Accepted June 3, 1977. Work supported by the Engineering Center, Metal Mining Division, Kennecott Copper Corporation, Salt Lake City, Utah.
