slowly changing signals. In Table I1 results are presented for rapidly changing synthetic signals. Resistor R I was changed to 15K and integrator times of 50 msec were used for measuring these slopes. Relative errors are again within 0.1 and relative standard deviations are about 0.2%. Results for the determinations of phosphate by the reaction rate procedure are shown in Table 111. These results were based on a single phosphate standard. In Table IV results of automatic determinations of glucose in the 5-20 ppm range are presented, In this case adjustments were made to obtain a direct digital readout of the glucose concentration. Rela-
tive errors and standard deviations of about 1% were obtained. Many of the rate curves recorded simultaneously indicated that inputs to the rate measuring system were often quite noisy. The results obtained here show the high noise immunity of the integration procedure. In addition, because of its versatility, it is possible to use the same readout system for reactions whose initial rates vary over a wide range, as described in the Instrumentation section. RECEIVED for review May 14,1968. Accepted July 3, 1968.
Reaction Rate Measurements with Fluoride Ion-Selective Membrane Electrode Formation Kinetics of Ferrous Fluoride and Aluminum Fluoride Complexes K. Srinivasan and G . A. Rechnitzl Department of Chemistry, State University of New York, Buffalo, N. Y. 1421 A study of the complex formation kinetics of FeF2+ and AIF2+ demonstrates the eminent suitability of the fluoride-selective membrane electrode for reaction rate measurements. Detailed experiments, carried out over a wide range of solution conditions, yielded the proposed mechanisms Fe3+ Fea+
+ F-
F?
FeF2+
+ H F a FeF2+ + H+
and
+ H F a A1F2++ H+ AIOH~++ HF A I ( H ~ O ) ~++ FA10H2++ H F A1F2++ H20 Al3+
-P
for the formation of FeF2+ and AIF2+, respectively. Values for rate constants of steps previously identified have been obtained with the membrane electrode and are in excellent agreement with the available literature data. Additional new rate data have also been obta ined. PREVIOUSLY, we reported ( I ) on the use of the fluoride ionselective membrane electrode in the study of solution equilibria between fluoride and hydrogen ions. The almost instantaneous response of the electrode (overall response rate limited by recorder response time of 0.5 sec.) and its Nernstian behavior with respect to free fluoride ions in acid solutions indicated that this electrode might be effectively employed in kinetic studies of reactions involving changes in fluoride ion concentration. The formation kinetics of monofluorocomplexes of metals were selected for detailed examination as representative dynamic systems, and the results of two such studies are presented here. The kinetics of the formation of FeF2+ were chosen for investigation because the relevant rate equation had been established by the spectrophotometric study of Pouli and Smith (2). Although different conditions 1
Alfred P. Sloan Fellow.
( 1 ) K. Srinivasan and G . A. Rechnitz, ANAL.CHEM., 40,509 (1968). (2) D.Pouli and W. MacF. Smith, Can. J . Chem., 38,567 (1960).
18 18
ANALYTICAL CHEMISTRY
were used, it was thought that a reasonable comparison could be made between the results of the present study at 25°C and those of the previous work. Once the reliability of the fluoride ion-selective electrode in monitoring a reasonably fast reaction such as the formation of FeF2+had been established, the electrode was employed with confidence to study the kinetics of the formation of AIF*+, a reaction known to be slow (3) but not previously studied from the mechanistic viewpoint, EXPERIMENTAL
Stock solutions of sodium fluoride were prepared by weight from the reagent grade salt after drying at 100 "C for 24 hours. Stock solutions of sodium perchlorate and perchloric acid were prepared by weighing anhydrous sodium perchlorate (supplied by the G . Frederick Smith Chemical Co.) and adding required amounts of perchloric acid to the solutions before final preparation. The concentration of the free perchloric acid in the sodium perchlorate solutions was determined by titration against standard sodium hydroxide. Stock solutions of ferric perchlorate were prepared from ferric perchlorate, Fe(ClO& * 6H20 (supplied by the G. Frederick Smith Chemical Co.), and calculated amounts of sodium perchlorate and perchloric acid were added to the solution to obtain the desired composition. The concentration of the ferric ion in the solution was estimated by titration with a standard solution of the sodium salt of EDTA using sulfosalicylic acid as the indicator. Stock solutions of aluminum nitrate were prepared by weighing reagent grade aluminum nitrate crystals, AI(NO&. 9Hz0, into solutions containing calculated amounts of sodium perchlorate and perchloric acid. The fluoride solution required for each kinetic run was prepared by mixing appropriate volumes of stock solutions of sodium fluoride and sodium perchlorate containing free perchloric acid, so as to obtain an ionic strength of 1.OM. In each experiment, 25 ml of the final solution was transferred to a dry polyethylene beaker thermostatted at 25 i 0.1 "C. The stock solution of ferric perchlorate or aluminum nitrate containing sufficient sodium perchlorate and perchloric acid to attain an ionic strength of 1.OM was also thermostatted at 25 & 0.1 "C. (3) C. Brosset and J. Orring, Suensk Kem. Tidskr., 55, 101 (1943).
A Beckman Expandomatic pH meter was used to measure the potential of the fluoride ion-selective electrode (supplied by Orion Research, Inc.). The potential was measured against a saturated calomel electrode connected to the experimental solution in the polyethylene beaker by means of an agar bridge containing 1 M sodium nitrate in a polyethylene tubing. The pH meter was connected to a Beckman 10-inch laboratory potentiometer recorder equipped with a voltage reference source (supplied by Heath Co.) to provide appropriate bucking voltages to the measured cell emf's; in this manner, the emf change could be monitored on a sensitive recorder scale. Each kinetic run was conducted as follows. The fluoride solution in the beaker was kept stirred by means of a Tefloncoated magnetic stirring bar, with the fluoride ion-selective electrode in the solution and the agar bridge connected to the calomel electrode. After about 15 minutes to ensure the attainment of thermal equilibrium, the recording of the cell emf was begun. The solution of ferric salt or aluminum salt, as the case may be, was withdrawn from the thermostatted bottle by means of a syringe and 1 or 2 ml of the solution (also 4 ml in the case of aluminum salt) was injected rapidly into the fluoride solution under conditions of effective mixing. All kinetic data were calculated from the resulting emf us. time curves and the known analytical compositions of the initial and final solutions.
11 7I
.45 mV
iim, seconds
Figure 1. Cell emf us. time curves at 25 "C and p = 1.OM for FeF2+formation [Fe3+Itotal= O.O01584M,[F-]t,a~ = 0.0009259M Curve 1. [H+] = 0.1316M Curve2. [H+] = 0.06887M Curve 3. [H+] = 0.03722M
RESULTS AND DISCUSSION
FeFZ+Formation. Figure 1 is typical of the cell emf us. time curves recorded during the reaction between ferric and fluoride ions. The potential of the fluoride ion-selective electrode becomes progressively more positive with respect to the calomel electrode indicating a decrease in the free fluoride ion concentration resulting from the formation of FeFZ+. By converting chart divisions to appropriate units and taking advantage of the Nernstian response ( I ) of the electrode to the free fluoride ions in acid medium, the concentration of the free fluoride ions at any instant ([F-1,) can be calculated from the relation
log [F-linitial
- log [F-],
+ F-
FeFZ+
KII
+
F*+ H F
and
2
K-II
FeF2+
+ H+
and thus formulated the rate equation, d[FeF2+1 = KI[Fe3+][F-] dt
+ KII[Fe3+l[HF] -
KI - [FeFZ+l- K1l [FeF2+l [H+] (6) Ke Ke * Ka
=
Increase of potential in millivolts (1) 59.16 where [F-]initial =
Fea+
[F-] total in the initial solution 1 f KHF[H+]
At the acid and fluoride concentrations employed in this study, the species HFz- does not make any significant contribution to the evaluation of the free fluoride ion concentration. The concentration of FeF2+ present at any time is then obtained from the equation
where
(7)
and After substitution for the concentration terms, Equation 6 can be rewritten as
where m = [Fe3+ltota~ = [Fe3+l
n
= [F-Itotal =
[F-1
+ [FeOH2+l + [FeF2+l
+ [HFI + [FeF2+l
x = [FeF2+] at time t
with Kh =
To ensure the validity of Equation 3, only those experiments were utilized in which the total fluoride concentration was less than the total ferric ion concentration. The average number of fluoride ions bound per ferric ion at equilibrium did not exceed 0.5 in such experiments. Pouli and Smith ( 2 ) interpreted their kinetic data for the formation of FeFZ+by means of the two paths
s =
[F-I
[FeOHz+][H+] [Fe
IF-I Ka Ka [H+l [HFI
+
+
Equation 9, on integration, yields
-.-1
1 rs
4;
q = K't + constant + b -+ ddq
2ax + a
In 2 ax
VOL. 40, NO. 12, OCTOBER 1968
(10) 1819
-1
/
12
10
-
8-
6-
4-
0.5'
2
t
9
4
6
8
4
I2
6
time, seconds
1 0 1 2
Figure 3. Test of Equation 10, conditions as in Figure 1 Figure 2. Test of Equation 14
wherea
-- l , b
=
1 - m - n - Ke r s,' c = mn,and q = bZ
- 4ac
The value of K , at 25" C and at an ionic strength of 1.OMis not available in literature and is, therefore, determined from the final equilibrium concentrations of FeFz+measured in the present study, using the data in Table I where A is defined as [F-] bound to ferric ions [Fe 3+1tots~
fi=
and In order to test Equation 10 with the present data, the values of K,, Kh, and K , at 25°C and at an ionic strength of 1.OMare required. The values of 1.26 X and 1.66 x for K, and Kh, respectively, are taken from literature ( I , 4).
(12)
With ii not exceeding 0.7, it is reasonable to suppose the existence of only FeF2+and FeF2+in the solutions under consideration. ii can, therefore, be expressed as [FeF2+l + 2[FeFz+l + [FeOH2+l+ [FeF2+l+ [FeFz+]
A=
(4) R. M. Milburn, J. Amer. Chem. SOC.,79,537 (1957).
(13)
[Fea+] leading to,
Table I. Average Number of Fluoride Ions Bound per Ferric Ion at Different Concentrations of Free Fluoride Ion
[F-]X 106M
ri
0.06731 0.06702 0.08919 0.1129 0.1005 0.1443 0.1393 0.2094 0.1862 0.2379 0.2057 0.3146 0.3035 0.2987 0.4453 0.4128 0.3887 0.4479 0.5458 0.5517 0.7366 0.5662 0.7187 0.9421 1.115 1.528
0.07215 0.07235 0.08557 0.09513
1820
0.1050
0.1367 0.1412 0.1633 0.1684 0.1880 0.2040 0.2621 0.2657 0.2689 0.3179 0.3225 0.3259 0.3612 0.3624 0.3906 0.4425 0.4434 0.4657 0.5535 0.5604 0.6552
ANALYTICAL CHEMISTRY
[H+I M 0.1316 0.1316 0.06887 0.03722 0.1316 0.06646 0.1316 0.03359 0.06887 0.03722 0.1316 0.06645 0.1316 0.1316 0.03359 0.06887 0.1316 0.1316 0.03722 0.06887 0.03722 0.1316 0.06646 0.06646 0.03359 0.03359
fi
( [&) If-
0.07306 0.07326 0.08763 0.09938 0.1063 0.1401
0.1430 0.1714 0.1724 0.1964 0.2065 0.2686 0.2690 0.2723 0.3336 0.3303 0.3299 0.3657 0.3785 0.4000 0.4622 0.4490 0.4773 0.5673 0.5878 0.6875
++2) (2
- i ~[F-]' )
= -(1. (2
-A)
- ii)
1 IF-]
-4
+ KiKz
(14)
where K l and Kz are the stability constants of FeF2+ and FeFz+, respectively. The values of ii given in Table I.
(
According to Equation 14,the plot of (1 - .- A)
1
(2
3
+-
are also
-,
- A ) [F-I2
should be a straight line with slope equal to (2 - A ) (F-) Kl. Such a plot is shown in Figure 2 from which the value of 1.14 X lo5is obtained for K I ( = Ke) at 25 "C and at an ionic strength of 1.OM. Equation 10 was then tested with the present data by making
- 2/4 against time. Time t 0 was + b+d q fixed in each case, at a convenient point on the curve, two plots of log 2ux
=
2ax
seconds after the commencement of the injection of the ferric solution into the fluoride solution. The values of m and n used for calculating b and were computed taking into account the [FeF2+]present at the time fixed as f = 0. Figure
4;
I
I
20
40
60
80
100
120
1- s -
Figure 5. Cell emf us. time curves for A1F2+formation (25 p = 1.OM)
"C,
S
Figure 4. Plot of k' against acidity function
-l-
+ + 2/4
us. time at different 2ax b d q acidities. The straight line plots that are obtained show that the data of the present study are consistent with the rate Equation 6. According to Equation 10, the slope of 3 gives typical plots of log
4 9 . The each straight line plot should be equal to K'rs ___ 2.303 values of K' calculated from the slopes of the plots are given in Table I1 and the average value of K' at each acidity is 1 -s given in Table 111. On plotting K' against __ , a straight S
line with an intercept of 3.73 X l o 3 and a slope of 67 is obtained (Figure 4). These can be compared with the values of K I and KII(4 X 103M-1sec-1and 11 M-Isec-', respectively) obtained by extrapolating the values of Pouli and Smith (2) at 0.11 "C,7.16 "C,and 12.10 "C. The agreement between the values of K I is good. The poorer agreement (although the order of magnitude is correct) between the values of KII is partly due to uncertainties inherent in the extraction of a small quantity by slope evaluation. The straight line plots in accordance with Equation 10, invariably obtained in this study, together with the consistency of the values of the rate constants K I and K I I with literature values point to the suitability of the fluoride ion-selective electrode for following rapid reactions involving changes in free fluoride ion concentration. Formation Kinetics of A1F2+. Typical potential us. time curves obtained on injecting aluminum nitrate solutions into fluoride solutions are given in Figure 5. In contrast to the FeF2+experiments, the decrease in fluoride concentration was found to be quite slow; at the highest concentrations of the reactants used (total concentrations of AI3+ and F- being 0.01 36 mole/liter and 0.003448 mole/liter, respectively) more than 1.5 minutes are required for the initial concentration of fluoride to be reduced by one half at an acidity of 0.1316M and about 1 minute at an acidity of 0.0408M. It thus seemed most appropriate to analyze the kinetic data using the initial rate method. The free fluoride ion concentrations at different times were calculated using Equation 1 and plotted against time on a convenient scale for drawing the tangents to the curves at t = 0. Typical curves with tangents drawn are given in Figure 6. The slopes of the tangents give the initial rates of
Curve 1. [H+]
=
0.06353M, [Al3+Itotsl= O.O1363M, [F-Jtotal=
0.003448M Curve 2. [H+] = 0.05849M, [Al3+ltOt,l= O.O07318M, [F-Jtotal=
0.003704M Curve3. [H+l = 0.05568A4, [Al3+Itota~ = 0.003800M, [F-Itotal = 0.003846M Curves recorded starting at arbitrary positions on the chart paper
Table 11. Values of Rate Constants K' for Formation of FeF2+ at 25 "C and at an Ionic Strength of 1.OM
[H+l M 0.1316 0.1316 0.1316 0.1316 0.1316 0.1316 0.1316 0.06887 0.06646 0.06887 0.06646 0.06887 0.06646 0.06887 0,03722 0.03359 0,03722 0.03359 0.03722 0.03359 0.03722
[Fe3+I1M
F-1,M
0.001584 0.0008223 0.001584 0.0008223 0.001584 0.0008223 0.001584 0.001584 0.0008223 0.001584 0.0008223 0.001584 0.0008223 0.001584 0.001584 0.0008223 0.001584 0.0008223 0.001584 0.0008223 0.001584
0.0009259 0.0007692 0.0007407 0.0003846 0.0003704 0.0001923 0.0001852 0.0009259 0.0007692 0.0007407 0.0003846 0.0003704 0.0001923 0.0001852 0.0009259 0.0007692 0.0007407 0.0003846 0.0003704 0.0001923 0.0001852
K' X 10-3 ( M - lsec- 1) 10.80, 10.90 11.21, 12.59 10.73, 10.40 11.19, 11.43 10.24, 10.05 11.07, 10.46 9.92, 9.52 7.24, 7.20 8.28, 8.84 7.62, 7.98 7.75, 8.30 6.71, 6.69 7.44, 8.48 6.63, 6.64 4.87, 5.75 6.47, 6.79 6.04, 5.22 5.98, 6.27 4.70, 4.81 5.24, 5.40 4:71, 4.60
Table 111. Average Values of Rate Constant K' for Formation of FeF2+at 25 "C and at an Ionic Strength of 1.OM 1-s K' X 10-3(M-1sec-1) S
104 53.7 28.1
10.75 7.55 5.49
free fluoride ion consumption. The rate of formation of A1F2+,however, is equal to the rate at which the total uncomplexed fluoride disappears since [A1F2+]= [F-ltota~ - ([F-I and
d[A1F2+l = dt
-d{[F-]
+ [HF]}
(15 )
+ [HF]) dt
VOL. 40, NO. 12, OCTOBER 1968
1821
I
gt
20
-
16
-
I 1
2
3
4
5
6
7
8
time, minuter
Figure 6. 5)
t
F-]free us. time plots (conditions as in Figure
0.09
0.06
[H']
0.12
0 15
LM.liter-'l
Figure 7. Plot of specific formation rate of A1F2+ as function of acidity (F-]= 3.45 X 10-5M)
F-ltobl being a constant. Assuming the equilibrium between [F-] and established rapidly, Equation 16 can be written as dt
0.03
[HF]to be
dt
Thus, the rate of formation of AIF2+ can be obtained by multiplying the absolute value of the initial slope of the [F-] us. time curve by the factor (1 KHF[H+]). Table IV gives the initial rates of formation of A1F2+at different concentra-
+
tions of aluminum, fluoride, and acid, calculated in this manner. Effect of Acidity on Reaction Rate. As has already been stated, for the same total fluoride and aluminum concentrations, the rate of formation of A1F2+increases with decreasing acidity. However, an examination of the data of Table IV shows that for a given free fluoride concentration, the rate increases with increasing acidity, the concentration of aluminum Rate remaining constant. Figure 7 shows the plot of __ against [Ala+]
Table IV. Initial Rates of Formation of Am2+at 25 "C and at an IoNc Strength of 1.OM
Acid conc 0.1316M 0.1316 0.1316 0.1316 0.1316 0.1316 0.1316 0.1316 0.1316 0.06353 0.06353 0.06353 0.05849 0.05849 0.05849 0.05568 0.05568 0.05568 0.0408 0,0408 0.0408 0.03412 0.03412 0.03412 0.0304 0.0304 0.0304 1822
0
ANALYTICAL CHEMISTRY
[AI 3+] 0.01357M 0.007245 0.01358 0.003790 0.007298 0.01360 0.003783 0. 007308 0.003794 0.01359 0.01362 0.01360 0.007214 0.007275 0. 007301 0.003736 0.003768 0.003791 0.01332 0.01360 0.01357 0.007222 0.007266 0.007206 0.003615 0.003636 0.003731
IF-I O.oooO3211M 0.oooO3445 O.oooO1587 0.oooO3640 O.oooO1738 0.000007917 O.oooO1809 0.000008687 0.000009068 0. oooO6627 0. oooO3340 O.oooO1624 0.oooO7591 O.oooO3815 O.oooO1917 O.oooO8371 O.oooO4186 O.oooO2109 0.oooO9414 0. oooO5063 O.oooO2414 0.0001285 O.oooO6411 0.oooO2900 0.0001457 0. oooO7OO1 O.oooO3554
Rate X
lo6,M sec-1 2.277 1.343 1.025 0.682 0.604 0.426 0.340 0.253 0.144 2.913 1.491 0.623 1.793 0.833 0.400 1.009 0.467 0.219 3.702 1.628 0.611 2.868 1.028 0.393 1.586 0.574 0.259
Corrected rate, M sec-1
1.913 1.129 0.882 0.561 0.517 0.363 0.293 0.2150.124 1.847 1.106 0.476 1.090 0.584 0.302 0.582 0.319 0.161 1.855 0.923 0.365 1.172 0.487 0.226 0.529 0.263 0.144
‘0
w
-
0.6
0.3
0.3
0.6
[Ai”’]
[F-]
1.2
x IO’ (moi8.1itei2)
Figure 8. Plot of A1F2+ formation rate [A13+]F-] product
VS.
([H+] = 0.1316M, [F-]< 10-5M) I 4
[H+] at [F-] values not differing by more than 3%. The simplest interpretation of the straight line plot is to assume that the rate equation may contain, among other terms, the term KIIAla+][H+] [F-1. Dividing the slope of the straight line by the average value of [F-1, a tentative value of 340 M-2 sec-1 is obtained for KI. It could be expected that this reaction path would predominate at high acidities and low free fluoride concentrations. A plot of the reaction rate against [A13+] [F-] at [H+] = 0.1316M and at low free fluoride concentrations is a straight line (Figure 8) passing through the origin and has a slope from which a value of 303M-2sec-1 is obtained for K I , in good agreement with the previous value. The Rate Equation. An examination of the data of Table IV shows that the rate of formation of A1F2+increases with increases in the concentration of free fluoride, the concentration of H F remaining more or less the same. This could indicate the presence of a fluoride concentration term or terms in the rate equation apart from HF. The non-zero Rate intercept at [H+] = 0 in the os. [H+] plot at constant [AI 3+] [F-] (Figure 7) also reveals the same possibility. The following procedure was, therefore, adopted to deduce the other term or terms in the rate equation. The value of KIIAla+][H+] [F-] was subtracted from the observed rate at each set of conobserved rate -KI[A13+][H+][F-] centrations and the quotient [Ala+][F-] was calculated. A plot of this quotient against [F-] was made (Figure 9) and gave a straight line with an intercept of 5 and a slope of 1.03 x lo5. It is thus reasonable to suppose that the rate equation contains the two additional terms [Ala+][F-I and [Ala+][F-I2, giving the complete rate equation
8
[FI
12 X
IO‘
16
(molr.litti’)
Figure 9. Test of rate Equation 18
~
+
[Al”i] [HF]
x IO5 ( m o l a 2 ~ l i t e ~ 2 )
dA1F2+1 = KIIH+][Ala+][F-] dt KII[AI~+][F-I KIII[A~~+I[F-]~ (18)
Figure 10. Plot of corrected A1F2+ formation rate US. [A13+] [HF] product
with KII = 5 M-’sec-’and KIII = 1.03 X 106M-2sec-1. Equation 18 was then tested over the entire range of concentrations investigated, after rearrangement to
The values of the left hand side of Equation 19 are also included in Table IV under the heading “corrected rate.” The plot of the left hand side of Equation 19 against [Ala+][HF] is given in Figure 10. The fact that this plot is a straight line passing through the origin can be taken as justifying the interpretation of the data and the formulation of the empirical rate equation as in Equation 18. The value of KIVis found to be
+
d[AIF2+] - KII[AI~+][F-]- KIII[AI~+][F-]~ = dt
VOL 40, NO. 12, OCTOBER 1968
1823
Table V. Half-Time Values for Formation of A1F2+ at Different Initial Concentrations at 25 "C and an Ionic Strength of 1.OM
[H+l 0.0304 M 0.03412 0.0408
0.0304 0.03412 0.0408 0.0304 0.03412 0.0408
0.05568 0.05849 0.06353 0.05568 0.05849 0.06353 0.05568 0.05849 0.06353 0.1316 0.1316 0.1316 0.1316 0.1316 0.1316 0.1316 0.1316 0.1316
tlh
t112,
calcd (min) 4.48 2.50 1.38 4.26 2.22 1.29 4.4 1.97 1.15 5.79 2.94 1.55 5.94 2.90 1.56
[Al3+1t
[F-1 t
obsd (min)
0.003800M 0.007318 0.01363 0.003800 0.007318 0.01363 0,003800 0.007318 0.01363 0.003800 0.007318 0.01363 0,003800 0.007318 0.01363 0.003800 0.007318 0.01363 0.003800 0.007318 0.01363 0. 003800 0.007318 0.01363 0,003800 0.007318 0.01363
0.0009615M 0.0009259 0. ooO8621 0.001923 0.001852 0.001724 0.003846 0.003704 0.003448 0.0009615 0.0009259 0.0008621 0.001923 0.001852 0.001724 0.003846 0.003704 0.003448 0.0009615 0. 0009259 0.0008621 0.001923 0.001852 0,001724 0,003846 0.003704 0.003448
4.20 2.40 1.56 3.75 2.11 1.30 3.2 1.71 1.04 6.03 3.24 1.84 5.36 2.87 1.62 6.00 2.94 1.62 9.70 4.60 2.48 8.15 3.99 2.15 8.13 3.95 1.97
2.86 1.48 6.82 3.41 1.80 7.32 3.50 1.81 9.16 3.71 1.84
+ (C- A)log(B - X ) +
( B - c)log ( A - x)
( ( B - C)BC
~
Ala+
7.01
for 0.435 M-kec-1 and, on dividing this by K, (1.26 X conversion to KI, we obtain a value of 346 M-%ec-l which is in excellent agreement with the provisional value employed in these calculations. Another test of Equation 18 can be made by calculating the half-times of the reaction for different initial concentrations making use of the rate constants KI, KII, and KIII already evaluated and comparing these values with the experimental values. Integration of Equation 18 yields
( A - B ) log (C - X ) =
At the acidities employed, the hydrolysis of Ala+ does not significantly affect the concentration of A13+and has therefore been neglected in writing Equation 20. The calculated and observed half-times are given in Table V. There is good agreement between the two sets of values except at high acidity and low fluoride concentration. Considering that the calculation of the half-time involves the use of three rate constants, the agreement can be taken as providing support for the formulation of the rate equation as in Equation 18. The Mechanism of A1F2+ Formation. The empirical rate law for the formation of A1F2+deduced in the present study consists of 3 terms, all containing [AI3+]to the first power while the fluoride ion appears along with hydrogen ion in one term, by itself in another, and raised to the second power in the third. At high acidities, when the concentration of the free fluoride is small, the predominant reaction could be assumed to be
-ts2K11i 2.303
+ (C - A)AC + ( A - B)AB} + constant
+ HF
+ H+
+ AIFZ+
(22)
As has been pointed out by Pouli and Smith (2), kinetic data cannot distinguish between Reaction 22 and a hydrogen ion catalyzed association, because [HFI = KE, [H+] [F-1. This still leaves the other two terms in the empirical rate equation to be interpreted. In accord with Eigen (5), the following scheme of reactions is advanced as a reasonable explanation of these two terms.
+ F- Kz A10H2+ + H F A10H2+ + H F 2 A1F2++ HzO
Al(HzO)*+
KI
(23) (24)
However, one should also consider the equilibrium that exists among AlOH *,H+ and A1a+,i.e., AlOH'+
+ H+ K4 Al(H20)a+
(25)
The rate law for the formation of AlF2+could be written as dt
+ K3[A10H*+][HF]
= KI[A13+]M+][F-]
(26)
assuming that Reaction 23 is faster than Reaction 24. Applying the principle of steady state to the concentration of AIOHZ+,
(20) and substituting for [A10H2+]in Equation 26, we get
where A =
B =
+
KIM+] KII C = KIIIS
4 A l F 2+] = K1[Ala+1[H+] [F-I dt
+ K3[HF] X
+
X =
s = Substituting X = B/2 after evaluating the constant in Equation 20, we obtain 2.303 tl/Z = KIIIS~ 1824
(B
( 5 ) Eigen, "Advances in the Chemistry of Coordination Compounds," S. Kirschner, Ed., Macmillan, New York, 1961, p 371.
A + (C - A ) log 2 + ( A - B ) log ___ - C ) log ___ C -B/2 A - B/2 ( B - C)BC + (C - A)AC + ( A - B)AB
ANALYTICAL CHEMISTRY
K4 can be expected to be greater than KZ or K3 and, because [F-] is of the order of loe5while KEF= 7.7 X lo2,Equation 28 can be simplified to
~-
- KI[A13+][H+] [F-1
dt
+ KHF[A13+l[F-I
X
which leads to
KEFKIK~ [Ala+][F-I2 (30) K4
The value compares favorably with the literature value of 8 x at 25 "Cand p = 0. Finally, Brosset (6), Biedermann (7)) and Sillen (8) are of the view that the hydrolysis of aquo aluminum ion yields polymeric hydroxy complexes. On the other hand, the work of Frink and Peech (9) favors the existence of the simple A10H2+ species in solution. In the mechanism proposed here, AIOHz+ is postulated only as a reaction intermediate and not necessarily as the final equilibrium product of the hydrolysis of aluminum ion. While the present method does not unambiguously distinguish among such mechanistic nuances, it should be apparent that the fluoride ion-selective electrode provides an effective and elegant means of studying the kinetics of even reasonably fast reactions involving changes in free fluoride concentration and should be useful for the continuous monitoring of fluoride activity in changing systems.
It will be noted that Equation 30 is of the same form as the experimental rate equation. On comparing the two equaKmKiKa KHFKS * K3 tions, we note that KII = and KIII = K4 K4 ' Assuming that KI and K4 are essentially diffusion controlled, we can estimate K3 as 1.3 X l o 2from the experimentally determined value of KIIIand inserting this in the equation for KII, K5 for the ratio - which should be we get a value of 4.9
x
K4
equal to the first hydrolysis constant of aquo aluminum ion.
RECEIVED for review May 8, 1968. Accepted July 12, 1968. Investigation supported by grants from the National Science Foundation, National Institutes of Health, and the Office of Saline Water. (6) C . Brosset, Acfa. Chern. Scand., 6,910 (1952). (7) C. Brosset, G. Biedermann, and L. G. Sillen, ibid., 8, 1917 (1954). (8) L. G . Sillen, Quarr. Rev. (London),13,146 (1959). (9) C. R. Frink and Michael Peech, Znorg. Clzern., 2,473 (1963). ~~~~~
~
Correction Atomic Fluorescence Flame Spectrometric Detection of Palladium, Titanium, Zirconium, Chromium, and Aluminum Using a Hot Hollow Cathode In this article by J. I. Dinnin [ANAL.CHEM., 39,1491 (1967)l the fluorescence responses attributed to titanium, zirconium, and aluminum are in error. Scattering of helium light by undissociated particles of the refractory oxides was mistaken for fluorescence. Additional investigation of the behavior of the vapors of these elements stems from questions raised by T. S. West and his associates ( I ) who point out that no resonance lines of these elements exist at the wavelengths indicated (387C-3895 A) and that it is unlikely that free atoms of the metals would be produced in the relatively cool, oxidizing, hydrogen-air flame used in this study. Although scattering of the 3888.6 A line was considered in the original study and was used to explain the apparent fluorescence signal encountered in the 3900 A region for water, scattering was not adequately taken into account in evaluating the strong responses in the 3900 A region encountered for aluminum, titanium, and zirconium. A simple means for testing for scatter has been suggested by Walter Slavin (2). In questioning the fluorescence attributed to titanium and zirconium, he recommended that concentrated solutions of other refractory elements be substituted for the element under study, under the same conditions of excitation. If the response was caused by fluorescence, no significant signal should be excited in vapors of
( 1 ) T. S. West, R. M. Dagnall, D. N. Hingle, E. F. Kirkbright, and
K. C. Thompson, Imperial College of Science and Technology, University of London, personal communication, January 1968. (2) W. Slavin, The Perkin-Elmer Corp., personal communication, December 1967.
other refractory elements; if the response was caused by scatter, a strong signal should be reflected from aspirated solutions of other refractory elements. The scattering test was applied at 3900 A, 2-mm slit, using the demountable hot hollow cathode lamp with helium atmosphere previously used for titanium, zirconium, and aluminum. The titanium lamp excited a signal of approximately the same magnitude in vapors of concentrated solutions of titanium, aluminum, and zirconium; aluminum and zirconium lamps excited similar signals from vapors of all three elements. The inescapable conclusion is that the signal observed in the 3900 A region is not caused by fluorescence but is the result of scattering of the 3888.6 A line of helium emitted by the hot hollow cathode. Additional proof is provided by the fact that no signal is observed at 3900 A with argon atmosphere in the hollow cathode. The sensitive fluorescence response reported for palladium in the 3450 A region is probably caused by the 3404.6 A line of palladium and is much more intense than that found in the 2500 A region. In correspondence ( I ) , the signal obtained at 2500 A where no resonance line of palladium is known to exist has been questioned. However, substitution of solutions of other refractory elements yields no scattering signal for either palladium line, thus indicating that the fluorescence may be real. The following corrections have also been noted ( I ) : In Table I, the wavelength noted for bismuth should be 3068 A; for cadmium, 2288 A; and for indium, 4102 A. I also extend my apology to Dr. K. C. Thompson for omitting his name from citations to reference ( 2 ) of the original article. VOL. 40, NO. 12, OCTOBER 1968
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