J Solution Chem (2012) 41:1224–1239 DOI 10.1007/s10953-012-9864-x
Liebig–Denigès Method of Cyanide Determination: A Comparative Study of Two Approaches Tadeusz Michałowski · Agustin G. Asuero · Maja Ponikvar-Svet · Marcin Toporek · Andrzej Pietrzyk · Maciej Rymanowski
Received: 3 May 2011 / Accepted: 18 October 2011 / Published online: 20 July 2012 © Springer Science+Business Media, LLC 2012
Abstract The paper compares two approaches to the simulated argentometric titration of cyanide with the use of the modified Liebig–Denigès (L-D) method, carried out according to GATES and applied with (1) classical and (2) pH-static titrations. Both approaches are discussed thoroughly and the results obtained from calculations are presented graphically. The calculations are performed with the use of an iterative computer program, based on charge and concentration balances and expressions for equilibrium constants, providing all physicochemical knowledge of the dynamic system in question. This way, physicochemical knowledge on complex electrolytic systems can be gained during the analytical procedure, i.e., the physicochemical and analytical knowledge are interrelated. The simulations follow the analytical procedures applied in experimental titrations and so provide an example of searching for the best a priori conditions for the quantitative analysis of cyanide. The computer program for pH-static titrations (described in this paper) enables one to carry out the simulation procedures with different preliminary data (concentrations, volumes). Keywords Solution thermodynamics · Computational chemistry · Titrimetric analysis 1 Introduction The pH-static titration, first reported in 2002 [1, 2], is a relatively new titrimetric method applicable to analyses of electrolytic systems [1–5]. The principle of this method resembles T. Michałowski () · M. Toporek · A. Pietrzyk · M. Rymanowski Faculty of Engineering and Chemical Technology, Cracow University of Technology, 31-155 Cracow, Poland e-mail:
[email protected] A.G. Asuero Department of Analytical Chemistry, The University of Seville, 41012 Seville, Spain M. Ponikvar-Svet Department of Inorganic Chemistry and Technology, Jožef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia
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Table 1 List of symbols used AD—adjusting titrant
AT—auxiliary titrant
D—titrand (solution titrated)
D + AD—solution obtained after addition of AD into D
VA —increment of AT
VAj —j th addition of AT;
VP —addition of PT
Φ = CP VP /(CKCN VD )—fraction titrated
pH0 —pre-assumed pH-value for D + AD
pr1 = AgI (precipitate)
PT—primary titrant
S-system—the system with H2 SO4 as AD
$-system—the system with NaOH as AD
VAD —volume of AD added
VA —volume of AT added
VD —volume of KCN + DEA + KI solution taken for titration
VDAD = VD + VAD
VP —volume of PT added
W = VD + VAD + VP + VA (Eq. 7)
the pH-stat that works on the inverse feedback principle, as other ‘-stats’ (potentiostat, thermostat, etc.). The main difference between pH-stat action and pH-static titration is that the small deflections δpH from a pre-assumed/starting pH0 value are compensated automatically, while in the case of pH-static titration, the intentional pH-changes (pH) are, in principle, greater than the δpH values, i.e., δpH pH. Nevertheless, this inequality is frequently not valid within all V -intervals covered by the pH-static titration curve [4, 5]. (See Table 1 for the list of symbols used.) The principles of pH-static titrations, considered as a kind of indicative analytical method, were discussed in [4, 5]. All physicochemical knowledge about the system in question can be included in the algorithm and resolved according to iterative computer programs, within the generalized approach to electrolytic systems (GATES) elaborated by Michałowski [6, 7]. Thus, the qualitative and quantitative properties of the system, such as a choice of buffer characteristics (concentration, complexing properties, starting pH0 value in pH titration) can be selected/pre-assumed. Great errors, resulting e.g., from improper choice of the pH0 value, can be thus avoided [4]. The GATES method includes all physicochemical knowledge about the system to be involved, and no simplifications are needed. An illustrative example, where pH-static titration can be applied, is the well-known Liebig–Denigès (L-D) (Liebig [8], Denigès [9]) method of determination of cyanide. In the (original) Liebig method [8], the precipitate of silver cyanide, formed locally, re-dissolves very slowly before the end point, making the titration tedious. In the L-D method, the temporary precipitate of AgI, formed locally after addition of AgNO3 solution, dissolves instantly on shaking, and the end point is very sharp [10]. The L-D method has been considered in different papers and specialized monographs [11–24]. However, up to 2005 [4, 5], the L-D method was not adequately discussed in the literature. In earlier approaches, some simplifying assumptions involving omission of some species were used, in order to obtain functional relationships between the variables considered. In order to avoid further complexity in calculations, the dilution effect has usually been omitted [11–15, 18]. More complex approaches [17, 19, 22, 24] referred only to the Liebig method. Apparently, the L-D method, considered in context with pH-static titration, puts the problem on much higher degree of complexity. However, when considered from the GATES viewpoint, this appears to be resolvable, with no detriment to the principles inherent in GATES. Particularly, all physicochemical knowledge is involved in the algorithm applied, i.e. no simplifying assumptions are needed. This way, the analytical procedure can be entirely reconstructed in detail. For this purpose, an iterative computer program has been
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Fig. 1 pH-static titration: VD + VAD mL of the solution is titrated, alternately, with successive portions [mL]: VPj of PT and VAj of AT (j = 1, . . . , N ); V0 , volume of titrand (D); VAD , volume of NaOH or H2 SO4 added to adjust pH to the desired (pre-assumed) pH0 value. Titrand (D) contains the analyte (X) and buffer species
developed, that is a part of the present paper. Further simulations obtained for this system within GATES are also illustrated and discussed.
2 Principles of L-D and Modified L-D Methods in the pH-Static Titration 2.1 Analytical Procedure As stated above, all steps of the analytical procedure involved in the L-D method, realized using a pH-static titration, can be simulated. The simulation involves the preparation ∗ mol·L−1 KCN solution, which is treated sucof titrand (D) by taking VKCN mL of CKCN ∗ cessively with additions of VDEA mL of CDEA mol·L−1 diethanolamine (DEA) and VKI mL ∗ mol·L−1 KI. The setup is presented in Fig. 1. The concentrations of the composing of CKI substances after dilution with water in the flask (VF mL) are: ∗ VKCN /VF CKCN = CKCN ∗ CDEA = CDEA VDEA /VF ∗ CKI = CKI VKI /VF
VD mL of the titrand (D) (VD ≤ VF ) is taken for the analysis. The pre-assumed pH0 value of the solution is adjusted by additions of VAD mL of NaOH (CAD mol·L−1 ) or H2 SO4 (CAD mol·L−1 ) as pH-adjusting titrant (AD). The resulting D + AD solution with volume VDAD = VD + VAD mL is treated first with addition of V P1 mL of CP mol·L−1 AgNO3 as the primary titrant (PT). The resulting solution, with pH1 = pH0 − pH1 , is adjusted to the initial pH0 value with VA1 mL of CA mol·L−1 NaOH, as auxiliary titrant (AT), added according to the normal titrimetric mode. Further portions of PT and AT are added repeatedly and alternately. After addition of the j th pair (j= 1, 2, . . . , N ) of both jtitrants, the total j volumes of PT and AT added are equal to VPj = i=1 VPi and VAj = i=1 VAi , respectively. Equal additions of VPi = VP1 = VP of PT are usually assumed; consequently VPj = j VP . The points (VPj , VAj ) (j = 1, 2, . . . , N ), plotted in co-ordinates (VP , VA ), are arranged along a line consisting of nearly linear segments. It is recommended to use as many experimental points N , such that N Vp ≈ 2Veq (Veq = equivalence volume). This way, the range of V -values applied in the titration is covered. The abscissa, corresponding to the point of intersection of linear parts, is considered as the end (e) point, VP = Ve of titration in which the solubility product of AgI precipitate is crossed. If the pH0 value is chosen correctly, the end volume in the titration is equal to the equivalence volume, i.e. Ve ∼ = Veq . /NH ) system used in original L-D method appears to be The ammonia buffer (NH+ 3 4 inapplicable to pH-static titration, due to the pH-changes resulting from the volatility
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Table 2 The species and the equilibrium data considered in the L-D system, precipitates are written in bold letters. The numbers in parentheses are equal to log10 Ki for stability constants Ki of complexes, or pKi for dissociation constants of protonated species, or pKs0 for solubility constants of precipitates (written in bold letters). pKw = 14.0 for the ionic product (Kw ) of water 2− Ag+ , AgOH (2.3), Ag(OH)− 2 (3.6), Ag(OH)3 (4.8),
2− 3− Ag(CN)− 2 (21.1), Ag(CN)3 (21.9), Ag(CN)4 (20.7),
2− 3− AgI (6.58), AgI− 2 (11.74), AgI3 (13.68), AgI4 (14.0),
AgDEA+ (3.48), Ag(DEA)+ 2 (5.60),
3− AgSO− 4 (0.23), Ag(SO4 )2 (0.28),
2− HCN (9.2), CN− ; HDEA+ (8.95), DEA; HSO− 4 (1.8), SO4 , − + − Na+ , NO− 3 , I , H , OH
AgI (16.08), AgCN (15.8), AgOH (7.84), Ag2 SO4 (4.83)
of NH3 . In the modified L-D method, NH3 was therefore replaced by less volatile diethanolamine (DEA), with acid–base properties similar to those of NH3 (pK1 = 9.35 for + + + NH+ 4 H + NH3 , pK1 = 8.95 for HDEA H + DEA). The stability constants for + + AgDEA and Ag(DEA)2 complexes (see Table 2) are close to those of the AgNH+ 3 and Ag(NH3 )+ complexes [25]. 2
3 Formulation of the Computer Program 3.1 General Remarks The program is designed to simulate the pH-static titration of cyanide according to the modified L-D method. The data related to the system consist of: (a) physicochemical data (Table 2); (b) analytical data (concentrations and volumes) and (c) fundamental variables (see Table 3). The stability constants for AgI+1−2i and Ag(SO4 )i+1−2i complexes were taken i from [25], and other equilibrium constants were taken from [26]. The analytical data and fundamental variables are chosen from the list in Table 3, depending on the defined stage of the titrimetric procedure. Numerical values for all the parameters, and initial values for all fundamental variables (Table 3) have to be specified in order to compile the program. The approximate initial values for the fundamental variables specified in the program refer to VAD = 0, i.e. for D itself. All the data specified in Tables 2 and 3 are presupposed at defined stages of the titrimetric procedure; some of them may not be used at certain initial conditions. The latter case occurs when NaOH (instead of H2 SO4 ) or no AD solution is used (see columns 1, 2, 4 and 5 in Table 3), i.e. at pH ≤ pH0 (pH refers to D). 3.2 Formulation of Balances Referring to the stage of pH-static titration in the S-system, i.e. one with H2 SO4 involved, and applying the notation used in Table 3 and in flowchart diagram (see Fig. 2), we write the following balances:
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Table 3 Analytical and fundamental data applicable to different stages of a pH-static titration: 1—calculation of pH for D; pH0 adjusting: 2—with NaOH, 3—with H2 SO4 (2 and 3 and then 5 and 6 are taken optionally); pH-static titration of the solution: 4—not adjusted (pH = pH0 for D), 5—adjusted with NaOH, 6—adjusted with H2 SO4 ; the sign “+” means that the corresponding fundamental variable is applied at defined stage of the titration pH for D
pH0 adjusting
pH-static titration
1
2
3
4
5
6
Volume of D
VD
VD
VD
VD
VD
VD
Conc. of KCN
CKCN
CKCN
CKCN
CKCN
CKCN
CKCN
Conc. of DEA
CDEA
CDEA
CDEA
CDEA
CDEA
CDEA
Conc. of KI
CKI
CKI
CKI
CKI
CKI
CKI
CAD
CAD
Analytical data
Conc. of AD Total volume of AD
CAD
CAD
VAD
VAD
Conc. of PT
CP
CP
CP
Portion of PT
VP
VP
VP
Conc. of AT
CA
CA
CA
Increment of AT
VA
VA
VA
Number of titration points
N
N
N
Fundamental data pH
+
+
+
+
+
+
log10 ([CN− ])
+
+
+
+
+
+
log10 ([DEA])
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
log10 ([SO2− 4 ]) log10 ([I− ])
+
log10 ([Ag(CN)− 2 ])
y1 < 0
log10 ([pr1 ])
y1 < 0
+
y1 = 0 y1 = 0
• charge balance 2− − Ag(CN)− F1 = H+ − OH− + Ag+ − Ag(OH)− 2 − 2 Ag(OH)3 2 − − + + − 2 Ag(CN)2− − 3 Ag(CN)3− − CN − NO3 + K + Na 3 4 − − 2− 3− − AgI2 − 2 AgI3 − 3 AgI4 − I + HDEA+ + AgDEA+ 2− − 3− + Ag(DEA)+ − HSO− 2 − AgSO4 − 3 Ag(SO4 )2 4 − 2 SO4 =0
(1) −
−
+
• concentration balances for CN , I , DEA, Ag , 2− + 4 Ag(CN)3− F2 = [HCN] + CN− + 2 Ag(CN)− 2 + 3 Ag(CN)3 4 SO2− 4 :
− CKCN VD /W = 0 (2) − − 2− 3− F3 = [pr1 ] + I + [AgI] + 2 AgI2 + 3 AgI3 + 4 AgI4 − CKI VD /W = 0 (3)
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Fig. 2 Flow diagram of the pH-static titration applied to the (modified) L-D method together with a simulation procedure involved; notations: S—refers to the system with H2 SO4 involved; $—refers to the system where NaOH (not H2 SO4 ) is added at the stage of pH0 adjusting; for further details (notation)—see text
F4 = HDEA+ + [DEA] + AgDEA+ + 2 Ag(DEA)+ 2 − CDEA VD /W = 0 2− + Ag(CN)− F5 = [pr1 ] + Ag+ + [AgOH] + Ag(OH)− 2 + Ag(OH)3 2 2− + Ag(CN)2− + Ag(CN)3− + [AgI] + AgI− + AgI3− 3 4 2 + AgI3 4 − 3− + AgDEA+ + Ag(DEA)+ − CP VP /W 2 + AgSO4 + Ag(SO4 )2 =0 − 3− + HSO− − CAD VAD /W = 0 F6 = SO2− 4 4 + AgSO4 + 2 Ag(SO4 )2
(4)
(5) (6)
where: W = VD + VAD + VP + VA − NO3 = CP VP /W + K = (CKCN + CKI ) · VD /W
(7) (8) (9)
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+ Na = CAD VAD /W
(10)
At the first step of the titration, the concentration of pr1 = AgI equals zero, [pr1 ] = 0, in Eqs. 3 and 5. 3.3 Fundamental Variables and Interrelations There are 26 species in the mono-phase S-system and 27 in the two-phase S-system that are fully described by n = 6 independent variables, considered as fundamental variables. The variables applied at the beginning of titration in the S-system are as follows: + − [DEA] = 10−x(2) H = 10−x(3) CN = 10−x(1) 2− −x(6) SO4 = 10−x(4) I− = 10−x(5) Ag(CN)− 2 = 10 where x(i) are the variables being optimized in the general purpose optimization algorithm. In the $-system (not involving H2 SO4 ), 22 (or 23) species are described by n = 5 fundamental variables; the variable x(4) = − log[SO2− 4 ] and the related sulfate species do not enter into the $-system. After crossing the solubility product for AgI (i.e. pr1 ), log10 [pr1 ] against the ‘old’ vari− − − able, i.e. log10 [Ag(CN)− 2 ] has to be applied as, among others, [CN ], [I ] and [Ag(CN)2 ] + are the independent variables at [pr1 ] = 0, i.e. at y1 < 0. In this region, [Ag ] is the depen21.1 × [CN− ]2 ). The [Ag+ ] and dent variable, due to the relation [Ag+ ] = [Ag(CN)− 2 ]/(10 [I− ] concentrations are not interrelated; the solubility product for pr1 is still not crossed. However, after crossing the solubility product for pr1 = AgI, we have the new species, pr1 , with concentration [pr1 ] > 0 and log10 [pr1 ] as the variable. In this case, [I− ] is not an independent variable, as it is interrelated with [Ag+ ] in the relation [I− ] = 10−16.08 /[Ag+ ], i.e. the subset: [CN− ], [I− ] and [Ag(CN)− 2 ] does not consist of independent variables; 21.1 × [Ag+ ][CN− ]2 = 1021.1 × (10−16.08 /[I− ])[CN− ]2 depends on the [Ag(CN)− 2 ] = 10 [CN− ] and [I− ] values. For this purpose, it is exchanged for [pr1 ] (more explicitly: for − log10 [pr1 ]), taken as the new independent variable, ‘filling the gap’ in the corresponding balances (i.e. for Ag and I). It is advisable to select the variables x(i) related to predominant species in the system + considered. For example, [Ag(CN)− 2 ], not [Ag ], is chosen as the variable involved with − −x(6) and [CN− ] = 10−x(1) we get silver species. From [Ag(CN)2 ] = 10 21.1 − 2 + × CN (11) Ag = Ag(CN)− 2 / 10 The variable V = VAD + VP + VA (see Eq. 7) is implicitly involved in the concentrations of all species entering the balances Eqs. 1–6. Equations 1–6 form a set of non-linear algebraic equations with fundamental variables x(i) = xi (V ) (i = 1, 2, . . . , n), where n = 6 (for S) or n = 5 (for $) and the variable V is incremented during each titration stage (see above). The steps for VP , VA (Table 3) imitate elementary portions of the titrants (PT and AT) added. Having the fundamental variables already defined, one can write (see Table 2): + H = 10−pH Kw = 10−14 − OH = Kw / H+ + 21.1 − 2 × CN Ag = Ag(CN)− 2 / 10
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[HCN] = 109.2 × H+ CN− [AgOH] = 102.3 × Ag+ OH− 2 3.6 × Ag+ OH− Ag(OH)− 2 = 10 3 Ag(OH)2− = 104.8 × Ag+ OH− 3 3 Ag(CN)2− = 1021.9 × Ag+ CN− 3 4 = 1020.7 × Ag+ CN− Ag(CN)3− 4 [HDEA] = 108.95 × H+ [DEA] 2 = 100.28 × Ag+ SO2− Ag(SO4 )3− 2 4
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[AgI] = 106.58 × Ag+ I− 2 11.74 × Ag+ I− AgI− 2 = 10 3 AgI2− = 1013.68 × Ag+ I− 3 4 AgI3− = 1014.0 × Ag+ I− 4 AgDEA+ = 103.48 × Ag+ [DEA] 5.60 × Ag+ [DEA]2 Ag(DEA)+ 2 = 10 0.23 × Ag+ SO2− AgSO− 4 = 10 4 1.8 × H+ SO2− HSO− 4 = 10 4
The physicochemical and analytical data, together with fundamental variables listed in Table 3, are taken automatically for particular purposes (at defined stages and/or options) realized during the optimization procedure. Moreover, the analytical parameters (concentrations and volumes) and initial values for the fundamental variables can be changed within reasonable limits. Numerical values for all parameters and fundamental variables must be specified; otherwise, the program cannot be compiled. In order to confirm whether or not the solubility products (Ks0i , i = 1, 2, 3, 4) for AgI, AgCN, Ag2 O and Ag2 SO4 are crossed, we ought to follow the yi (i = 1, 2, 3, 4) values expressed as (see Table 2 and Fig. 1): y1 = log10 Ag+ I− + 16.08 y2 = log10 Ag+ CN− + 15.8 (12) y3 = log10 Ag+ OH− + 7.84 2 + 4.83 y4 = log10 Ag+ SO2− 4 At y1 < 0, the solubility product for AgI is not crossed and its concentration [pr1 ] = 0. The point where y1 crosses zero is registered by the computer and the titration is continued at y1 = 0 (i.e. [Ag+ ][I− ] = Ks01 ) after putting pr1 for [Ag(CN)− 2 ] as the new variable, [pr1 ] = 10−x(5) and replacing Eq. 11 by [Ag+ ] = Ks01 /[I− ] (pKs01 = 16.08) and 21.1 × [Ag+ ][CN− ]2 ) = 105.02 × [CN− ]2 /[I− ]; all of the operations are [Ag(CN)− 2 ] = (10 realized automatically in the prepared program. The values yi < 0 for i ∈ 2, 3, 4 testify that another (AgCN, Ag2 O or Ag2 SO4 ) precipitate is not formed. However, when precipitates other than AgI are formed (yi ≥ 0 for i ∈ 2, 3, 4 ), the titration should be repeated at new initial values for CKI and/or CDEA ; the precipitate AgCN (not AgI) is formed at too low CKI values. Formation of other precipitates is not considered in the L-D method. The computer program enables the pH-static titration curve to be plotted in (VA , VP ) co-ordinates. Moreover, the pH versus VP + VA relationship, providing useful information concerning the system in question, is also obtained.
4 Short Description of the Program The program simulates the experimental titration of cyanide, as a pH-static titration, realized with use of the modified Liebig–Denigès method. In the modification, diethanolamine (DEA) is taken instead of ammonia as the buffering agent. This way, the source of errors resulting from the volatility of ammonia (when referred to real titration) are minimized. It is, of course, of no importance in the simulated titration.
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The simulating procedure consists of three stages, namely: (1) calculation of the starting pH-value; (2) the simulated “classical” titration with adjusting (AD) solution, up to a pre-assumed pH0 -value; and (3) the simulated pH-static titration. All the steps are realized automatically. The program is set up by introducing starting values for the variables, called fundamental variables. They are the values for: − − − log10 CN− , − log10 [DEA], pH, − log10 SO2− 4 , − log10 I , − log10 Ag(CN)2 Separate procedures were developed for each type of step in the titration whereby the relevant variables were varied (see Table 2) while the others remained unchanged. For example, the moment of crossing the solubility product (Ks01 ) of AgI, is realized without any user intervention. A more extended description of the pH-static titration is presented in the Manual which is available at Ref. [27]. 5 The Optimization Procedure The iterative computer program used for the pH-static titrations is described by the set of n independent scalar variables x(i) = xi (V ) (i = 1, . . . , n), related to different volumes V = VAD + VP + VA (Vj = VAD + VPj + VAj ) of titrants added and represented by the vector: T (13) x = x(V ) = x1 (V ), . . . , xn (V ) where superscript (T ) denotes the transposition sign. The variable V enters the set of n balances written in the generalized form: (14a) Fi = Fi x∗ (V ) = 0 (i = 1, . . . , n) (see Eqs. 1–6) or in a more compact form: F x∗ (V ) = 0 ∗
∗
(14b)
[x1∗ (V ), . . . , xn∗ (V )]T ∗
The vector x = x (V ) = zeroes, simultaneously, all the expressions comprising the vectorial function F(x (V )) = [F1 (x∗ (V )), . . . , Fn (x∗ (V ))]T . The x∗ (V ) values fulfill, simultaneously, the set of n balances expressed by Eqs. 14a and 14b. Generally, the x∗ (V ) values are calculated for different V -values taken from a defined V -interval in the simulated titration procedure applied to a titrand–titrant system. The iterative optimizing procedure consists of some operations performed according to simplex and gradient methods applied to the scalar function expressed by the sum of squares (see Eqs. 14a, 14b) SS = SS(V ) =
n n 2 (Fi )2 = FT x(V ) F x(V ) = Fi x(V ) i=1
(15)
i=1
considered as the response function; SS(V ) ≥ 0 for any V . Note that SS∗ (V ) = n ∗ 2 ∗ i=1 [Fi (x (V ))] = 0 (see Eq. 14a). Therefore, SS(V ) > 0 if x(V ) = x (V ). In the optimization procedure, the set of starting (s) values represented by the vector: T (16) xs (V ) = x1,s (V ), . . . , xn,s (V ) is chosen for the variables x1 (V ), . . . , xn (V ) searched at a defined V value (Eq. 13). If xs (V ) = x∗ (V ), then SSs (V ) =
n 2 Fi xs (V ) i=1
(17)
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is greater than zero, SSs (V ) > 0. The optimization tends to attain the minimal SS-value, min SS, for each V considered in the titration. This way, sufficiently accurate values for variables x1 (V ), . . . , xn (V ) at different V values are obtained. The value of the convergence criterion (Eq. 15) of max SS = 10−14 seems to be sufficient for practical purposes, i.e., it provides the numerical values for xi∗ (V ) with acceptable accuracy. The iterative computer program described in this paper, written in the DELPHI language and applied to the algorithm based on charge and concentration balances and interrelations resulting from expressions for equilibrium constants, has been applied over several years with excellent results, though it has been presented here for the first time. The program runs on all common PC computers, with Windows 95 and later versions. The choice of logarithms as variables (indices in decaying exponential functions) is very advantageous for optimization purposes, since xi = 10log10 xi = 10−pxi > 0 for any value assumed for log10 xi (concentrations are positive values). The iterative computer program works better (that is, the optimization procedure is improved) when this natural constraint is assumed for the variables. Commonly used constraints, such as (min xi (V ), max xi (V )) intervals are not needed.
6 Discussion The program supplied in File 3 [27] can easily be extended to graphical presentation of the results obtained, as are ones presented in Files 1 and 2 [27]. On this basis, one can obtain the results as presented in Figs. 3 and 4, where the following data were assumed: VD = 50 mL, CKCN = 0.002 mol·L−1 , CKI = 0.002 mol·L−1 , and CAD = 0.1 mol·L−1 . As results from Fig. 3a, the titration curve plotted in co-ordinates (VP , VA ) consists of a pair of rectilinear lines. The abscissa of the point of intersection of the lines defines the end volume, Ve , whose value depends on the pH0 value assumed in the pH-static titration. The Ve value corresponds to the first appearance of AgI precipitate (Fig. 3b) and occurred at the greatest concentration of cyanide species: CN− and HCN, not consumed up to the end point of titration (Fig. 3c). The Ve corresponding to the first appearance of AgI precipitate lies close to Veq at greater pH0 values, i.e., the systematic error of cyanide analysis, expressed by difference between Ve and Veq values, and then between the Φe and Φeq = 0.5 values for the fraction titrated Φ = CP VP /(CKCN VD ), grows with a lowering of pH0 (see Table 4). However, this advantageous a priori tendency is accompanied by flattening of the titration curve (Fig. 3a), which makes the location of Ve more difficult. Moreover, the effect of volatility of HCN from the mixture occurring at lower pH0 values, can also be taken into account since [HCN]/[CN− ] = 109.2−pH . As results from Fig. 3a and Table 4 (column B), the line obtained at pH0 ≈ 9 appears advantageous for the location of Φe close to Φeq . Then, further effects involved with the choice of CDEA and CKI for DEA and KI will be referred to pH0 = 9.0. Particularly, the effect of CKI on the shape of the pH-static titration curve is presented in Fig. 4a. Particularly, the curve obtained for CKI = 0.2 is more rounded in the vicinity of Ve . At low CKI , the precipitation of small quantities of I− (Fig. 4b) as AgI (Fig. 4c) is followed by precipitation of AgCN (Figs. 4d, 4e) + Ag(CN)− 2 + Ag → 2AgCN
from Ag(CN)− 2 ions formed in the preceding titration step. From the enlarged (Fig. 4f) fragment of Fig. 4c, drawn for CKI = 0.2 mol·L−1 , it results that AgI is formed at Φ exceeding
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Fig. 3 Simulated curves of (a) VA versus VP ; (b) log10 ([HCN] + [CN− ]) versus Φ; (c) log10 ([Ag+ ][I− ]) versus Φ; (d) log10 ([Ag+ ][CN− ]) versus Φ; relationships plotted for pH-static titrations at indicated pH0 -values; Φ = CP VP /(CKCN VD )
significantly Φeq = 0.5. One should notice here that, at higher [I− ] values, the concentrations (see Table 2) play a significant role in the related balances. of soluble complexes AgI+1−i i The pH-static titration related to L-D method can be compared with the L-D method carried out according to the classical titrimetric procedure. For this purpose, let us assume that VD mL of the solution containing KCN (CKCN = 0.002 mol·L−1 ) + DEA (CDEA = 0.02 mol·L−1 ) + KI (CKI = 0.002 mol·L−1 ) is adjusted with H2 SO4 (CAD = 0.1 mol·L−1 ) to the pH0 values indicated at the corresponding curves in Fig. 5a. The (buffered) solution thus
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Fig. 4 Simulated curves of relationships: (a) the VA versus VP relationship; (b) log10 [I− ] versus Φ; (c) log10 ([Ag+ ][I− ]) versus Φ; (d) log10 ([HCN] + [CN−1 ]) versus Φ; (e) log([Ag+ ][CN− ]) versus Φ; and (f) enlarged fragment of (c), all plotted for the solution (pH0 = 9) containing KCN (CKCN = 0.002 mol·L−1 ), DEA (CDEA = 0.02 mol·L−1 ) and KI (CKI ) with AgNO3 (CP = 0.01 mol·L−1 ) and NaOH (CA = 0.01 mol·L−1 ); Φ = CP VP /(CKCN VD )
1236 Table 4 pH0 and the related Φ = Φe values corresponding to the points where the solubility product for AgI is crossed [5]; Column A refers to the conventional L-D method and column B refers to the L-D method, realized according to pH-static titration; Φ = CP VP /(CKCN VD )
J Solution Chem (2012) 41:1224–1239 pH0
Φe A
B
7.5
0.1641
0.4473
8.0
0.4212
0.4820
8.5
0.4903
0.4936
9.0
0.4968
0.4973
9.5
0.4983
0.4985
10.0
0.4988
0.4988
10.5
0.4989
0.4990
formed is titrated with AgNO3 (CP = 0.01 mol·L−1 ) causing a further lowering of the pH. The break points in Fig. 5b indicate the first appearance of AgI precipitate. The Φe value, corresponding to the first appearance of AgI, decreases with the decrease in the pH0 value and is smaller than Φe referred to the pH-stated titration (Fig. 5c) made at the same pH0 value (compare columns A and B in Table 4). In the cases indicated, AgCN (Fig. 5d), Ag2 SO4 and Ag2 O are not precipitated. 7 Concluding Remarks The argentometric titration of cyanide according to the Liebig–Denigès (L-D) method is one of the most interesting processes in analytical chemistry [11]. An advantage of the L-D method compared with the Liebig method is the rapidity of analyses. In the L-D method, the temporary precipitates dissolve instantly on shaking and the end point is registered more correctly than in the Liebig method, where the locally precipitated silver cyanide dissolves slowly. However, the discussion of the mathematical relations involved with this process was unsatisfactory. The necessity to obtain the functional relationship, as in other approaches, generally required some simplifying assumptions involving the omission of some species. In this context, the pH-static titration applied to this method can be perceived as involving a further accumulation of difficulties. However, it is not the case from the viewpoint of the GATES approach [6, 7]. More detailed data involved with the L-D method, obtained according to GATES, are presented in [4, 5]. The idea of pH-static titration, suggested by Maccà et al. and applied for relatively simple electrolytic systems [1–3], was extended to systems of any degree of complexity [4, 5], resolved within GATES, with full reproduction of the analytical prescription and full use of all accessible (and pre-selected) physicochemical data. On this basis, one can find optimal a priori conditions for analysis and check all of the limitations involved. Particularly, the limitations of systematic error and flattening of pH-static titration curve, as indicated in Fig. 3a, can be controlled by the analyst in all aspects. The procedure of pH-static titrations enables one to avoid all effects involved with equivalence volume registration on the basis of inflection points on the titration curve [28]. The Liebig–Denigès titration is an interesting “touchstone” to apply the (any new) calculation method because of the interest and complexity of this reaction. A search of the literature shows that it is not frequent that textbooks or even monographs on chemical equilibria deal with the topic, at the level of the equilibria involved from a quantitative point of view. Nevertheless, approximations of various types are involved in order to tackle the topic. However, a definitive solution to the problem is given in this paper, with the aid of
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Fig. 5 The (a) pH versus Φ, (b) log10 ([HCN] + [CN− ]) versus Φ, (c) log10 ([Ag+ ][I− ]) versus Φ, and (d) log10 ([Ag+ ][CN− ]) versus Φ relationships plotted for conventional (modified) L-D method, at different starting pH0 values and other data indicated in the text; Φ = CP VP /(CKCN VD )
the iterative computer program developed, written in DELPHI, that runs on all common PC computers. The general skeleton of the program can also be applied to other complex analytical systems, without difficulty. The simulated titration procedure imitates a kind of a quasistatic (thermodynamic) process realized under isothermal conditions, where constant values for the equilibrium data are assumed. The simulation allows concentrations to be followed during the course of the
1238
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titration, e.g. pH, E and concentrations of different species (dynamic speciation, see e.g. [6, 29] and references cited therein). Similar possibilities are offered by simulated pH-static titrations, where some other plots, e.g. pH versus VP + VA , are also valuable. The related curves can be obtained from the computer program attached to this paper. Using this program, one has the opportunity to follow concentration changes (speciation curves) during the titration. The calculation procedure lasts several minutes. The iterative computer program, after some indispensable adaptations/modifications, can be used for plotting the related dynamic speciation curves and, in a more extended version, for resolution of any other electrolytic system, e.g. for complexometric titrations using the pH-static mode [4, 5]. The pH-static titration method can be applied to some systems, where complexation, precipitation or redox reactions are accompanied by acid–base reactions. A review of literature data published hitherto shows that the simulation procedures, applied to obtain the mathematical model of a process realized in titrimetric analyses, are still based on conditional stability constants [30] and exemplified by the papers [1, 31, 32]. This approach, which requires simplifications in order to formulate functional relationships, is ineffective and provides erroneous inferences, as indicated for example in [29, 33], where the data obtained with use of this method were compared with ones obtained according to GATES [6].
References 1. Maccà, C.: pH-stat techniques in titrimetric analysis: Part 3. The principles of pH-stat chelatometric titrations. Anal. Chim. Acta 456, 313–323 (2002) 2. Maccà, C., Soldà, L., Zancato, M.: pH-stat techniques in titrimetric analysis: IV. pH-stat monitoring of chelatometric titrations. Anal. Chim. Acta 470, 277–288 (2002) 3. Maccà, C., Soldà, L., Favaro, G., Pastore, P.: A pH-stat study of the reaction of some transition metal cations with disodium ethylenedinitrilotetraacetate (EDTA) and its analytical application. Talanta 72, 655–662 (2007) 4. Michałowski, T., Toporek, M., Rymanowski, M.: Overview on the Gran and other linearisation methods applied in titrimetric analyses. Talanta 65, 1241–1253 (2005) 5. Michałowski, T., Toporek, M., Rymanowski, M.: pH-static titration: a quasistatic approach. J. Chem. Educ. 84, 142–150 (2007) 6. Michałowski, T.: The generalized approach to electrolytic systems: I. Physicochemical and analytical implications. Crit. Rev. Anal. Chem. 40, 2–16 (2010) 7. Michałowski, T., Pietrzyk, A., Ponikvar-Svet, M., Rymanowski, M.: The generalized approach to electrolytic systems: II. The generalized equivalent mass (GEM) concept. Crit. Rev. Anal. Chem. 40, 17–29 (2010) 8. Liebig, J.: Titration of cyanide with silver nitrate. Justus Liebigs Ann. Chem. 77, 102–105 (1851) 9. Denigès, G.: Titration of cyanide with silver using potassium iodide as indicator. Compt. Rend. 117, 1078–1080 (1893) 10. Sharwood, W.J.: Notes on the estimation of cyanogen by silver nitrate, using potassium iodide and ammonia as indicators. J. Am. Chem. Soc. 19, 400–434 (1897) 11. Ricci, J.E.: Some mathematical relations involving the solubility of silver cyanide. J. Phys. Colloid Chem. 51, 1375–1394 (1947) 12. Ricci, J.E.: Titration curve for argentometric determination of cyanide. Anal. Chem. 25, 1650–1656 (1953) 13. Kolthoff, I.M., Furman, N.H.: Potentiometric Titrations: A Theoretical and Practical Treatise, 2nd edn. Wiley, New York (1947) 14. Kolthoff, I.M., Stock, J.T.: Equilibria in alkaline argentocyanide solutions. J. Am. Chem. Soc. 78, 2081– 2085 (1956) 15. Kolthoff, I.M., Sandell, E.B., Meehan, E.J., Bruckenstein, S.: Quantitative Chemical Analysis, 4th edn. Macmillan Co, New York (1969) 16. Charlot, G., Gauguin, R.: Les Méthodes d’Analyse des Reactions en Solution. Masson et Cie, Paris (1951) 17. Butler, J.N.: Ionic Equilibrium. A Mathematical Approach. Addison-Wesley, Reading (1964)
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18. Souchay, P.: Chimie Physique—Thermodynamique Chimique, 3rd edn. Masson et Cie, Paris (1968) 19. Garric, M.: Cours de Chimie. Dunod, Paris (1970) 20. Michalowski, T.: An error of Liebig–Denigès method of determination of the cyanide. Chem. Anal. 28, 313–320 (1983) 21. Rosset, R., Bauer, D., Desbarres, J.: Chimie analytique des solutions et informatique, 2nd edn. Masson, Paris (1991) 22. Butler, J.N., Cogley, D.R.: Ionic Equilibrium. Solubility and pH Calculation. Wiley, New York (1998) 23. Michałowski, T., Wajda, N., Janecki, D.: A unified quantitative approach to electrolytic systems. Chem. Anal. (Warsaw) 41, 667–685 (1996) 24. de Levie, R.: Principles of Quantitative Chemical Analysis. McGraw-Hill, New York (1997) 25. Lurie, Yu.Yu.: Handbook of Analytical Chemistry. Izd. Khimia, Moscow (1971) (in Russian) 26. Inczedy, J.: Analytical Applications of Complex Equilibria. Horwood, Chichester (1976) 27. http://www.farmacia.us.es/analisis/pHStat.exe (File 1); http://www.chemia.pk.edu.pl/~michalot/LIEBIG _DENIGES_FILE_2_MANUAL.doc (File 2); http://www.chemia.pk.edu.pl/~michalot/LIEBIG_ DENIGES_FILE_3_COMPUTER_PROGRAM.doc (File 3) 28. Asuero, A.G., Michałowski, T.: Comprehensive formulation of titration curves referred to complex acid– base systems and its analytical implications. Crit. Rev. Anal. Chem. 41, 151–187 (2011) 29. Michałowski, T.: Application of GATES and MATLAB for resolution of equilibrium, metastable and non-equilibrium electrolytic systems. In: Michałowski, T. (ed.) Applications of MATLAB in Science and Engineering. InTech—Open Access publisher in the fields of Science, Technology and Medicine, Rijeka (2011). http://www.intechopen.com/books/show/title/applications-of-matlab-in-science-andengineering 30. Ringbom, A.: Complexation in Analytical Chemistry. Wiley, New York (1963) 31. de Levie, R.: A simple expression for the redox titration curve. J. Electroanal. Chem. 323, 347–355 (1992) 32. Suzuki, T., Hioki, A., Kurahashi, M.: Development of a method for estimating an accurate equivalence point in nickel titration of cyanide ions. Anal. Chim. Acta 476, 159–165 (2003) 33. Michałowski, T., Lesiak, A.: Formulation of generalized equations for redox titration curves. Chem. Anal. (Warsaw) 39, 623–637 (1994)
