On How To Simulate Potentiometric Titration Curves without Making Any Approximation Daniel Rodrigues de Mwra ICEx-UFMG, Caixa Postal 702, 30.161-Belo Horizonte, MG, Brazil
Calculation of species concentration concerning the equilibrium n,Ox,
+ n,R,
n2R, + n,Ox,
2. At the equivalence point, EPE, the potential is given by the expression
(1)
derived from the two redox systems Ox,
+ n,e s R,; E,,
is no longer trivial when nl is different from nn, since this calculation depends on the resolution of a polynomial expression of a degree value up to (nl nz). The problem is related to the simulation of apotentiometric titration curve for these two redox systems since the calculation of the potential starts by the calculation of the species concentration a t any given point during the titration. To avoid solving the polynomial expression, approximations are usually made, which may induce biased or even erroneous results since the supposition is made that the reaction is 100% quantitative (IS).Besides, solving the polynomial may become more of a computatioual problem than a chemical one. Approximations are appropriate in an introductory analytical chemistry course, where only the situations in which approximations do not cause significant errors are considered. However, approximations cannot he made when one needs to understand the real behavior of the svstem, including under unfavorable conditions for the desiied reaction. We will show how to simulate the titration curve without making any approximation and also without solving the polynomial expression and even without necessarily using a computer. The method utilizes the same set of equations for aU the points of the titration curve, whichever the proposed situation may be. I t is based on the artifice of inverting the variables (6), and i t is so simple that it can be used in a classroom.
which can be derived from eqs 1 and 2. (Equations 2 and 3 are well known in the literature). 3. The analytical concentration of Oxl and Rz, C1 and Cq, respectively, are given by
+
Mathernatlcal Expressions During the titration of a solution of Rz (analytical concentration: C R ~volume: ; VRJ with a solution of Ox, (analytical concentration: Co.,), for each added volume (Vo,,) of solution Ox1, after equilibrium is established, the following applies: 1. The equilibrium potential, E, a t any point of the titration curve is given by
The values of E,, and E,,, besides incorporating the effects of the temperature and the ionic strength of the medium, also contains the perturbation due to the chemical reactions of Oxp, R2, OX,, and Rlr as well as the effects of auxiliary reagents present in the solution. 226
Journal of Chemical Education
where the terms under brackets are the equilibrium concentrations. On the other side, as a result of eq 1, the relationship between [Oxp] and [R1] will he:
4. By definition:
and T2 = [0~21I[R21
(9)
Consequently, one can arrive a t the following conclusions: 1. At the equivalence point, $ = 1. 2. Substituting in eq 7 the values of CI and Cz, given in eqs 4 and 5, gives:
3. From eqs 4 and 8, we obtain the result C, = IR,I(l+ TJ
(11)
and by using eqs 5,9,and 6, one obtains: Cp= (n,ln,)IR,l(l+ 11TJ
(12)
4. Association of eqs 11and 12 gives
where the values of TI and Tp are obtained from eq 2 as: (E-E,,,in,/0.059
T, = 10
IE-E,,Jn2/0.059
T, = 10
(14) (15)
late, for this given potential, the corresponding values of TI, Tz, +,and Vo,,, by using eqs 14,15, 16, and 17, respectively. (Using the calculated values of Vo, , TI, and Tz,calculate the values of C,, [RI],[Ox2].[Ox1]and [ R ~ )by , employing eqs 4,11,6,8, and 9, respectively. Test the validity of the results by calculating E by means of eq 2; the result should reproduce the proposed initial value.) 5. Repeat step 4 for all the desired values of E. This procedure has been followed t o obtain the data of the figure 1 showing the concentrations ( C ) of the four reactantslproducts @ C = -log C ) and the value of E, during titration of Ti3+ by U0z2+.The two redox systems involved (7) are: (Ox, + n,e
R,; E,,)
and
Accordingly, eq 1becomes equal to U0:+
pC(rigM wdinate)and E(1eflwdinate)asa hlnction of added volume of 0.10 M WZ2+ during titration of Ti3+ (0.10 M, 10.0 mL), pH = 0. (Data according to tea).
+ 2Ti3+
ri
U"
+ 2Ti02+
Even though the use of a computer is not indispensable, using one saves a considerable amount of time. We have developed a computer program, in BASIC, which for each given value of E calculates the corresponding values of Vo,,, +,and the concentrations of the chemical species in equilihrium. I t also prints the simulated titration curve. This program can be ohtained by writing t o the author. Concluslon
5. Finally, hy comparing eqs 7 and 13, one arrives at
+ = T,(l + T,)/(1 + T2)
(16)
and from eq 10 one obtains the value of Vo., as: vo,, = +n2c~~v&co.,
(17)
Slmulatlon of the Tltratlon Curve
The simulated titration curve may he either E = f(Vo,)
The method ~roooseddoesnot contain ao~roximations.I t employs the s a k e set of linear equations f G a l l the points in the titration curve. and it does not i m ~ o s eany restrictions regarding the values of nl, nz, E,, and E,,. he use of linear equations simplifies the calculations, which can easily be performed without a computer. Acknowledgment
The author thanks Eucler B. Paniago for English translation of this text and the referee for suggestions that helpedto clarify the text. Literature Clted
The data for the simulation is obtained according to the following sequence of steps: 1. Define the redox systems, i.e., the values of E,,, Ea, n ~and , n2. 2. Define the conditions of the titration, i.e., the values of CR~. VR~.
and Cox,.
3. Calculate the value of potential at the equivalence point, E m , by using eq 3.
4. Attribute a value to E (in the neighborhood of Em), and calm-
c.
GLnemle. Solutions
I. Chariot, C a m ds Chimie Anolytique Aqueuses; Masson: Paris, 1967;Tome I, p 182.
A q u e u m d "on
2. Vogel.A.1. Yogel's T~dbookolQuontitoiiu~lnorgonieAnolysislneludingI~lrum~nto1 Anulysb: Longmen: New York, 1978: p 288. 3. cuanter, W. B. Quaniitali"~ Chsmislry: Me.sur~m.nta and Equilibrium; AddinonWesley: Reading, MA, 1968; p 140. 4. Alereyev. V. N. Quoniiloliue Analysis; Mir: Morcow: p 327. 5. Ohlweiler, 0.A. Tsori. D Piatice do Anhlisa Quontitoliuo inorgbnic.;Editora univcrsidsde de Braailis: Brasilia, 1968: Vol. 2, 0489. 6 . Wi1lia.C. J. J. Chem.Educ. 1981,58.659463. 7. kitinen, H.A. Chrmicol Analysis a n Adonneed Test and Referenre: Mecraw-Hill: New York, 1960:p 280.
Volume 67
Number 3 March 1990
227
