Polarographic Determination of Composition and Thermodynamic Stability Constant of a Complex Metal Ion Dolores Marin and Francisco Mendicuti' Universidad de Alcala de Henares, Alcala de Henares, Madrid, Spain This laboratory experiment is designed to encourage lahoratory cooperation among individual undergraduate students or groups. Each student (or group) contributes results individually. Many times these results will be a single point on a straight line. Exchange of data is essential to obtain final results. Others experiments with similar motivation have been described previously (1.2). From a scientific point of view, the purpose of this experiment is to acquaint the students with the application of polarography (and the use of recording polarograph) as one of the most prominent tools in the investigation of complex metal ions. Theory (3, 4) The polarographic method for investigating complex metal ions is based on the fact that the characteristic half-wave potential of the complex (El& is more negative than that of the free ion (Ellz)f since that ion must he liberated from the complex (this requires acertainamount of energy). From the shift in the half-wave potential of the complex and from the concentration of the complex forming agent, i t is possible to obtain information concerning both the formula and the stability of the metal complex ion. A short mathematical derivation is eiven of the necessary relationships. Assume that the eauations for the reduction of a metallic ion and complex formation are O+ne=R and o+px=oxp
+ (RTInF) In([OlJ[Rl.)
(3)
where the subscript "0" denotes a t the surface on the electrode. The stability constant of complex is given by the equation: K, = IOXPIJIXI:[OI, Substituting for [O], from eq 4 in eq 3 E = ED+ (RTInF) InllOX~lJ[Xl~K~lRl,l
(4)
(5)
where all magnitudes are perfectly known. With the assumptions, (1)the concentration of the com-
'
Present address: Department of Polymer Science, University of Akron, Akron, OH 44325. 916
Journal of Chemical Education
(E,&
- (E,,Jf
= -(RTInF) lnK,
- p(RT1nF) I n [ a
(6)
Equation 6 shows that the dependence of the half-wave potential on the logarithm of the concentration of the complex-forming agent must he a straight line, from the slope of which p, i.e., the number of ligands, can he determined. I t also shows that the larger the value of&, i.e., the more stable complex, the more negative is the half-wave potential of the complex. However, the half-wave potential depends on the ionic strength of solution. It is easy to see that this dependence is linear with the squared root of ionic strength. I t would he desirable to extrapolate the results of eq 6 to an ionic strength of zero (5)a t each concentration. This permits the calculation of the thermodynamic stability constant of the complex by plotting the extrapolated values [(El& (Ellz)f],- against the logarithm of the ligand concentration, log [XI. To obtain eq 6 we assumed that the electrode process is reversihle. I t might be interesting to prove the reversibility of the system under investigation by carrying out the socalled logarithmic analysis of current-voltage curve. According to this analysis, the equation of the polarographic curve for a cathodic reduction may he written in the form (6, 7):
(2)
respectively, where X is the ligand capable of forming the metal comnlex. OXo of stoichiometw -D.with 0 , the oxidized form of the metal R. The electrode ootential for eu 1 is -eiven for the wellknown Nernst equation: E =E'
plex-forming agent is so large that [XI. = [XI, (2) that the value of K, is so high that [OXp] >> [0] and (3) that initially there is no reduced form, according to Cottrell's equation and considering the diffusion coefficients for the complex and free cation are approximately the same, eq 5 for the reversihle polarographic wave of a complex becomes
wheretis the intensity at each voltage E and ;is the limiting current.It follows from this equation that the dependence of log (:/(Id - I ] ) on the potential must he a straight line with slop? n!0.059 (at 25 OC). The potential, a t which the value of log (i/(id 31 is zero, gives the half-wave potentials. In fact, this is a good method to obtain these potentials when the measurements require precision.
-
Experlrnenlal Each undergraduate student (or group) will carry out five polarograms corresponding t o the solutions systematically represented in the table. KN03 was added for two reasons, (1) as supporting electrolyte and (2) to maintain the ionic strength constant for the different solutions (equal to 0.3, 0.4, . . .). In any case, these concentrations and ionic strength are representatives. The only requirements are a known a n d constant Ph2+ concentrations and ionic strengths. Previous to each recording, i t is necessary to bubble nitrogen through the polarographic cell for 5-10 min, to remove the oxygen from the solutions. I t is sufficient to record the
(?(c -
Figwe 1. Logarithmicanalysis of poiaragraphic curves: lag $1 against E S.C.E.) tor a 5 X lo-' M of Pb2+SOIutian~on addition of NaOH at different concentrations.
(YS.
Figure 3. Dependena,at (E,,,lc - (E,,,)f with the squared raat of ionicstren@h athiffsrent concentrationsa i ligsnd. [OH-].
-
Figure 2. Plots (El,&
- (E,,df vs. log [OH-] at differem ionic strengths.
SoIutl~nsUsed In the Exoerlmenisa Molar Concentrations (moi/L) MWNO~I~ MKNO* MUOH
Solution
'me
1
5x
2
5
x
lo-' 1
0
~
?
-
?
0.05
cmcenmtion of KNOS might be fined to obtain me ionic strengm cmrta*.
~ ( 1= " 112 T1 my:).
polarogram of the different solutions in the potential range -0.2 to -1.0 V vs. S.C.E. Results
One set of results for the testing of the polarographic waves for reversihility obtained hy a group of students is given in Figure 1.These results are for a ionic strength of 0.3. In all cases a slope of approximately 0.03 V a t 25 "C is obtained. The number of electrons transferred in the electrode reaction is n = 2. Usingeq 6 each group ohtained their own results of stoichiometry of the ionic complex and stability constants a t dif~~
~
~~~
~~
~
~
-
(EIIZ)tlp values enrapalated to zero ionic strength Figure 4. Plot [(El& vs. log [OH-] to obtain the thermodynamic stability constant of complex Pb(0H);.
ferent ionic strength. Their results are plotted inFigure 2. In allcases thevalue for p obtained from the slopes is close to 3, therefore the complex formulais Ph(0H);. The -pK,values of 12.57,12.39, 12.40, and 12.20for0.3,0.4,0.5,and0.6ionic strengths, respectively, can he determined from the intercepts. These results are in agreement with the literature, which gives a value of 12.8 for ionic strength 0.1 for the same ion complex (8). The next step of the experiment involves the collection of all the results for the half-wave potential a t each ligand concentration [OH-] and ionic strength and to extrapolate them to zero ionic strength. These plots are depicted in Figure 3 in form of (El& - (Ellz)f vs. p'bat each concentration. By extrapolation, intercepts, [(EIIZ)C - ( E ~ i z ) f-0, l ~ of -266 mV, -300 mV, -318 mV, and -346 mV at 0.05 M, 0.1 M, 0.15 M, and 0.3 M concentration of [OH-] ion, respectively, are ohtained. Finally, Figure 4 depicts the same plot given in Figure 1in which the influence of ions of the medium is removed. From the intercept log [OH-] = 0, a -pK,' = 13.56 was obtained. In this case K,' is the thermodynamic stability constant of complex Ph(0H);. This result is comparable with a -pK,' value of 13.95 i 0.08 found in the literature (4). A ~ossihleamlication of this thermodynamic constant is to ol;tain the standard oxidation-reduction potential of the I'b(0H); ion in equilibrium with the lead electrode. This Volume 65
Number 10
October 1988
917
corresponds to the reduction reaction Pb2+ + 30H- = Pb(OH);, E o p b c o ~ , r / p b n ,for which a value of -0.81 V was obtained. A published value for this is -0.82 V (9). Literature Cited 1. Marin. D.:Mendieufi, F. J. Cham. Educ., in prm. 2. Marin. D.: Mendicuti, F.J. Chsm. Educ., in press.
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Journal of Chemical Education
3. Berd,A.J.;Faulker,L.R.Elecfmehemi~olMefhods; Wi1ey:New York, 198(J:Chspter 5. 4. Heyravsky.J.: Kum, J. Principles ofPolorography: Academic: New York, 1956. Chap-
."-" . . a
0 ,,".
5. Vlcek, A. A. Collacfion Czechoslou.Chem. Cornmum. l955,20,400. 6. Tomes. J. Collection Czechoslou. C h m . Commum. 1937.9.12, 7. Lingme. J. J Am. Chem. Soe 1939.61.976. 8. Lingane. J. Cham.Rao. 1941.29. i. 9. Latimer, W. M. Ozidofion Potemtioh. The Oxidolion 01tho Elements and Thpir Potsnliolr in Aqueous So1uriana;Prsnfim-Hall: New York, 1952.
