V O L U M E 26, N O . 3, M A R C H 1 9 5 4 The method described has obvious limitations. I n the case where the ratio between the two K,‘s of the acids approached unity, the resulting error in the fraction of each acid in the mixture would become large. Furthermore, if the acids were present in strikingly different concentrations, the concentration of one being very small, the relative error for this minor conetituent would be large ACKKOWLEDGMEh-T
This work was supported in part by an E. I. du Pont de Nemours 8: Co. Grant-in-Aid and a Pure Oil Fellowship.
471 LITERATURE CITED
(1) Auerbach, F., and Smolczyk, E., 2. physiit. Chem., 110, 65 (1924). (2) Bjerrum, J., “Metal Ammine Formation in Aqueous Solution,” Copenhagen, P. Haase and Son, 1941. (3) Dole. &I.. “The Glass Electrode,” p. 251, Kew York, John Wiley 8: Sons, 1941. (4) hfichaelis. L., in “Physical Methods of Organic Chemistry,” A . Weissberger, ed., Vol. I, Part 11, p. 1746, Kew York, Interscience Publishers, 1949. (5) Steams, E. I., in “Analytical dbsorption Spectroscopy,” 11.G. hlellon, ed., p. 369, New York, John Wiley & Sons, 1950. RECEIVED f o r review August 6, 1953. Accepted Piorember 16, 1953.
Minimum Error Polarographic Analysis of Binary Mixtures ALVIN FRISQUE’, VlLLlERS W. MELOCHE, and IRVING SHAlN Department o f Chemistry, University o f Wisconsin, Madison, W i s .
In the polarographic analysis of a mixture of two reversibly reducible compounds whose half-wave potentials lie in close proximity to each other, polarograms do not exhibit sufficiently distinct changes in slope to enable the evaluation of the wave heights corresponding to each of the two components. A mathematical interpretation of snch a polarogram enables one to calculate the proportionate amounts of each of the components in a mixture. It is shown that to accomplish this, one need only determine the potential of minimum error, and measure the current at the potential of minimum error, and the current equivalent to the total diffusion current.
T
HE; polarographic analysis of a mixture containing two reducible compounds, in general, requires that the half-wave potentials of two compounds be fairly well separated from each other. If this condition is satisfied, the polarogram will exhibit two independent waves from which the concentration of each component in the mixture can be calculated. I n case the halfwave potential of the t n o components are close to each other, a polarogram of such a mixture ma\- exhibit two poorly defined naves. Holvever, if the half-wave potentials lie very close to each other, the polarogram may exhibit only one wave, that for the total reduction of both compounds in the mixture. Heyrovsh? ( 2 ) and others have used two closely synchronized dropping electrodes and constructed a differential plot, A i / A E us. E, to facilitate the calculation of concentration in cases where poor11 defined waves were obtained. The differential plot only accentuates the effect of changes in slope which occur in conventional polarograms. If the change in slope is not reasonably detectahle and, particularly, if the change in slope for each of two components in a mixture cannot be observed (last ease above), the differential plot will not suffice to provide analytical interpretation of the concentration of each component in the mixture. -4previous study ( 1 ) v a s concerned with the determination of a pH of minimum error and the titration of a mixture of two weak monobasic acids whose pK’s were in such close proximity that the pH curve exhihited only one inflection, that for total titration. It was shoivn that i t was possible to determine the proportionate amount of each acid present in the mixture by measuring the volume of titrant a t the pH of minimumerrorandattheequivalence point, that of the total titration. This study illustrates the application of a similar principle to the polarographic analysis of a mixture of two reversibly reducible Present address, Standard Oil Research Laboratories (Indiana), Whiting, Ind. 1
compounds whose half-wave potentials are in such close proximity that a polarogram of the misture exhibits only one wave, that for the reduction of both components. One can calculate the proportionate amounts of each substance in such a mixture by recording the current at a potential of minimum error, E, and the current at a potential corresponding t o that a t which the diffusion current would be measured for the complete wave. After the polarographic constants for a given system have been established it is no longer necessary to run complete polarograms in order to analyze a given mixture. THEORETICAL
If the fundamental equation of Heyrovsk? and Ilkovi; is valid for the polarographic wave of a single component, i t will likexise he valid for a mixture of txvo such components which do not interact. Designating the two components A and B , with the other symbols having their usual meaning, it follows from this that:
Rearranging Equation 1, the current from each component in the rising portion of the wave is seen to be: idA iA
= (1
+ 1O(E - EI,zA)TLA/O.O~S ) and iB (1
= idB
+ 1o(E - EI/zB)TLB/O.O~~ )
(2)
Since a t any one potential the denominators in Equation 2 are constant, these terms become: iA
=
KAidA and is = K B i d B
(3)
the constants K A and K B being the single potential constants defined in Equation 2. If the total current designated i A B is equal to the sum of the individual currents i A and i ~ it , follows from Equation 3 that: i a e = KAida
+ KBide
(4)
L41so, if the linear relationship between the diffusion current and the concentration holds for each component:
i d a = K a ‘ ( A ) and ide = KB‘(B)
(5)
in which the constants KA‘ and KB‘ relate the diffusion current
ANALYTICAL CHEMISTRY
472
to the concentration and in which ( A ) and ( B )are the concentrations of each component. If the two diffusion current terms in Equation 4 are substituted, in terms of their definition in Equation 5 this equation becomes: iAB
=
K A K A ’ ( A )f KBKB’(B)
(6)
In general, there will be no question as to which is the correct value as is illustrated later for the case of lead and thallium. When the same electron change is undergone by both components ( n =~ na), it follows from the relationship cosh ( - z ) = cosh (z), that the minimum error potential is defined b> :
+
E = ’/z(EI/zB E I / ~ A )
Bssuming additivity of the total diffusion current, designated adas, this reading will be: idAB
= KA‘(A)
+ KB’(B)
(7)
Equations 6 and 7 may be employed to solve for the concentrations ( A ) and ( B )from the current reading AB, which will be in the rising portion of the wave in the type of system under consideration, and i d A B , the total diffusion current, when the constants in the two equations are known. Solving for ( A ) and (R) between Equations 6 and 7, one obtains:
(14)
Even for cases when n~ does not equal nB, the value of E defined by Equation 14 is conveniently employed as a preliminary value for substituting into Equation 13 in order to obtain the exact value for the minimum error potential. In order to test the above development, it was necessary to sdect two compounds whose half-wave potentials were in such close proximity that a polarogram for the mixture gave a diffusion current which was the sum of the diffusion current for the two compounds. Thallium and lead in nonalkaline medium were chosen as satisfying this requirement. EXPERIMENTAL
If the potential dependent constants K A and K B can be determined with the same accuracy a t all applied potentials, the current AB should be measured a t a potential for which the difference between these two single potential constants is large. Since the concentrations ( A ) and ( B ) are constant in any one mixture of the two components, the right-hand term of each equation in 8 is likewise constant throughout the whole polarogram. I t is only the magnitude of the numerator and denominator which varies in 8, therefore, and not the ratio of the two. Since any small error in the determination of either Kd or K B will cause a relatively smaller error in the concentrations ( A ) and ( B ) if the difference between the two constants is large, the current AB should preferably be measured a t the potential a t which quantity ( K B - K A ) is a t a masimum. This will be called the potential of minimum error. Differentiation of the quantity ( K B - K A ) with respect to E gives:
Reagent grade chemicals were used in all experiments. Two stock solutions, one lO-3-W in lead nitrate and the other 10-3.W in thallium nitrate, were prepared. These stock solutions were both 0.10M in potassium chloride and 0.001M in hydrochloric acid. A solvent 0.10M in potassium chloride and 0.001.21 in hydrochloric acid was used to dilute the stock solutions to known volumes. Gelatin, equivalent to 0.002% was present in all solutions, Measurements were made with a Sargent Model I11 polarograph. Voltages were measured against a saturated calomel electrode and currents were corrected for residual current. Galvanometer deflections were used directly in all calculations.
Table I. Concent rations, 3 I i II i iiiole 0 99 0 73 0 .iO 0 2.3 0 10
Table 11.
Experimental F-alues for Lead Employed in Calculations on Synthetic Mixtures
RT/nF 0 031 0 033 0 032 0 031 0 031 .I\ 00322 .
El, z
390 399 397 398 396 0 3978
0 0 0 0 0
~~~~l DeHection (Currected) 1095 Ss!8 ~08 278 118
K’pb = Deflection Millimoles ( X IO) 111 114 112 111 118 1132
Experimental Yalues for Thallium Employed in Calci~lationson Synthetic Mixtures
Confentration, 3Iilliinole 0.99 0.75 0.50 0.25 0.10 .4r.
RT/iiF
E,/z
0.060 0 . 058 0.059 0 061 0.063
0.483 0,460 0.462 0,465 0.460
0 0602
0.4620
K’TI = ~ ~ t Deflection ~ l Deflection Millimoles (Corrected) ( X 10) 796 80 590 79 415 83 207 83 87 87 824
Table 111. Constanta for Each Component Present in 3lixtures and Difference between Constants as a Function of Applied Potential flpplied Potential, Volt -0.300 -0,317 -0 334 -0,350 -0.395 -0.415 -0.426 -0.428 -0,429 -0,440
a
a t the potential at which the quantity (KB - K A )is a maximum. When the two components undergo different electron changes ( n # ~ n ~ ) two , different values of E can satisfy Equation 13.
K B
0.0005 0.0019 0.007 0.023 0.442 0.791 0,899 0.913 0.918 0.964 -0.500 1.000 -0,600 1.000 Single potential constant: = + 1 0 ( ~- E ~ , W O . O B ~
,/rl
Ka
K7Pb +) K A = K(TI+) =
+
KA 0.0018 0.0036 0.007 0.012 0.068 0.138 0.197 0.210 0.216 0.298 0.815 0,995
E,/p(Pb) =’ -0.398 volt E1,2(Tl) = -0.462 volt
-
(KB RA) -0.0013 -0,0017 0.000 0.011 0.374 0.653
0.702
0.703 0.702 __ 0.666 0.185 0.005
473
V O L U M E 2 6 , NO. 3, M A R C H 1 9 5 4
All solutions were swept free of oxygen with deoxygenated nitrogen. The experimental conditions regarding the dropping mercury electrode and the temperature of the polarographic cell were the same for all solutions examined. Two sets of measurements were made. The first involved the polarographic examination of solutions of each of the two components alone; the second set involved the polarographic examination of synthetic mixtures of the two components. Polarographic data from the first set of measurements were used to establish the relation betvieen concentration and diffusion current for each of the pure components. These data also provided experimental values for E,, and RT/nF. The second set of measurements provided data from which one could calculate the proport,ionate amounts of each component present in the mixtures. Experimental Values for the Constants for Lead and Thallium. The values ior E ; ‘ 2 arid RT/nF were determined from a t least five measurein(wts on ea-nh of a series of solutions at varying concentrations. These experimental values, together with the total deflection of the galvanometer (corrected) as a function of concentration are shotvn in Table I for lead and Table I1 for thallium. Alt,hough it. might have been feasii ilc to employ literature values for the half-wave potentials of lead and thdlium and the theoretical values for the constant R T I n F , preliminary work with the individual solutions was in any case necessary in order to relat,e the diffusion current of each component, to its concentration. The determination of experimental values for the half-wave potentials and the RTInF values therefore did not’ involve the preparation of any additional solut~ions. The det’ermination of experimental values for all the constants used in subsequent calculations on the niixt.ures served, furthermore, to make the entire procedure a relative rather t’hsn an absolute one, and, therefore, this procedure was adopted. The espcrimental values for E,,? andRl‘/nFEor eachcomponent’ were employed to calculate the single potential constants defined in Equation 2. These single potential constants were in turn employed to calculate t,he concentration of each component in the synt,hetic mixtures according to Equation 8. Table I11 lists t’he calculated single potential constants for lead and thallium and the difference between the two constants as a function of the applied potent,ial. Since the theoretical value for Rl’/nF was employed in the derivation of the above equations, this t,heoretical value is used with the experimentally determined valurs for the half-wave potentials t’ocalculate t’hequantities listed in Table 111. The purpose of Table 111 is to illustrate the significance of the minimum error potential defined in Equation 13. The maximum negative and posit.ive values for the quantity ( R B- K a ) which appear in Table I11 occur a t the two applied potentials which are solutions to Equation 13. From t,he relative magnitudes of the two maxima, it is seen that only the positive quantity is of pract,ical significance. Analysis of Mixed Solutions of Lead and Thallium. Since ( K B - K a ) is a maximum when i a B is evaluated a t an applied potential of -0.428 volt, the difference between the theoretical and experimental values for the amount of lead and thallium present in each mixture should be a minimum when im is measured a t this potential. Synthetic mixtures of known concentrat,ions in lead and thallium were made as described above and the current AB was measured a t three different applied voltages. The total diffusion current i d A B was measured a t a single voltage. The theoretical and experimental values for the concentration of lead present in each mixtaureare listed in Table IV. The values for thallium appear in Table V. The solution numbers which appear in both tables refer to the same solutions. .4t the bottom of each column headed “relat,ive error” is the average relative error for the 10 mixtures based on calculat.ions made from the current readings a t the three applied voltages, -0.395, -0.428, and -0.465 volt. The total diffusion current was measured a t -0.600 volt. In view of the equations developed above, one would expect somewhat bett,er results when the concentrat’ions of the two com-
Table IV. Theoretical and Experimental Values for Lead in Lead-Thallium Mixtures Compared at Three Different Applied Potentials Solution To.
C0ncn.Q C0ncn.a ( - 0 . 3 9 5 (Theory) Volt)
Rel. Error,
%
Concn.a (-0.428
Volt)
Rel. Concn.a Error, (-0.465 % T’olt)
4
0 495
0.0
+5.9
0.886 0 . 137 0,115 0.625 0,503 0,242 0,376 0.116 0.246
+2.4 +9.6
+1
f2.5.6 -3.2 -2.9 +1.0 0.0 $2.1 -0.8 -2.0 4
-8.0 0.0 t0.6 -3.2
$0.3
-7.2 -2 0
3
0,480 0.883 0.113 0 092 0.6lfl 0.43: 0 21, 0.369
o inn
0 230
Rel. Error, % -3 0 +2.1 +9.6 -26.4 -1 4 -2.8
- 13
2 -1 6 -20 0 -8 0 9
3lillimoles per liter.
Table V. Theoretical and Experimental Values for Thallium in Lead-Thallium Mixtures Compared for Three Different iipplied Potentials Solution
Concn.”
(-0 395 Volt)
0.493 0,125 0.866 0.625 0,125 0.260
0.465
KO, (Theory) 1
2 3
4 J
6 7 8 9 10
0 .i00
0.125 0.375 0 250
0.115 0.801
Rel. Error,
‘70
C0ncn.O Rel. Concn.“ (-0 428 Error, ( - 0 465 Volt) “0 Volt)
-6.1 -8.0 -7.4 -4.6
0.596 0141 0.240 0.490 0.112 0.378 0,244
-1.5.2 -4
0 -2 0 -10.4 +!.8 --.4
Av. rel. error
0.474 0.121 0.835 b.604 0.119 0.242 0.501 0,121 0,387 0.245
-4.2 -3.2 -3.5 -3.4 -4.8 -8.2 f0.2
-3.2 +3.2 -2.0
3
6
0.496 0.124 0.868
0.636 0.132 0.264 0.538 0.132 0.411 0.261
Rel. Error
5
+0 2 -0 8 -0.3 +1.8
75.6 --A
6
-1.6 -,,6
+9 6 +4.4
4
Millimoles per liter.
ponents are solved from the total diffusion current reading and the current reading a t an applied potential of -0.428 volt than when the other two voltages are involved. That this is actually the case is seen from the fact the average relative error for lead for all mixtures is 4 5 at -0.305 volt, 3% a t -0.428 volt, and V; a t -0.465 volt. The average relative error for thallium for all mixtures is 6% at, -0 395 volt, 3% a t -0.428 volt and 4 5 a t -0.465 volt. .\ny method for the determination of binary mixtures employing a procedure such as is described above has obvious limitations -4s the ratio of the half-wave potentials of the two components approached unity the resulting error in the experimentally determined values for the amount of each component present becomes large. Furthcrmore, the minimum error potential defined in Equation 13 assumes a single clearly defined half-nave potential for each component in the applied potential region of interest. It will, therefore, not hold when a stepwise exchange of electrons occurs for one of components a t nearly similar voltages. Since the Heyrovskf-IlkoviE equation is of a basic Nernst form, the extension of the concepts developed above to the potentiometric titration of a binary mixture appears obvious. In the simplest case in which none of the oxidized or reduced form is present before the start of the titration, it should be possible t o use the terms “volume of titrant” interchangeably with “current reading” and the potential a t the “midpoint of the titration” interchangeably with the ‘*half-wavepotential,” in the devrlopnient above A C K Y O I LEDGMENT
This work was supported in part by an E. I. du Pont de Kernours & Co. Grant-in-Aid and a Pure Oil Fellowship. The authors arc grateful for the helpful criticisms of P. C. Hammer of the Numerical .4nalysis Laboratory of the University of Wisconein. LITERATURE CITED
(1) Frisque, Alvin, and hfeloche, V. W., ANAL. CHEM., 26, 468 (1954). (2) Heyrovskg, J., Analyst, 72, 229 (1947). RECEIVED for review August 24, 1953.
dccepted November 16, 1953.
