terminated prepolymer when this material was found to react with ethanol during the capping operation. This reaction caused a shift in elution volume of the material, confirming the presence of reactive end groups as opposed to unreacted polyol. The chromatogram of the prepolymer, shown as composite Peak C in Figure 4, was obtained by fractionation on the high permeability limit columns. The long tail in the high molecular weight region indicates that the prepolymer had an extremely broad molecular weight distribution. Although no suitable calibration of elution volume in terms of absolute molecular weight was obtained for this polyurethane, ratios of i V w / J T n and iVz/iRwcould be estimated from a polystyrene calibration. By this means it was estimated nw/nn to be 15
ivn
and Rz/Rw to be 4.6 where is the number average molecular weight, ,Vw is the weight average molecular weight and Rzis the “z” average molecular weight. ACKNOWLEDGEMENT
The authors acknowledge Hans W. Osterhoudt for his valuable guidance and the work of the Analytical Department of Armstrong Cork Company for its supplementary analyses of materials. RECEIVED for review April 23,1970. Accepted June 26,1970. Presented at the 156th National Meeting, American Chemical Society, Atlantic City, N. J., September 1968.
Dielectrometric Titrations : Nonquantitative Reactions Robert Megargle,l George L. Jones, Jr., and Donald RosenthaP Department of Chemistry, Clarkson College of Technology, Potsdam, N. Y . 13676 THEMAJOR PRODUCT of the reaction of an acid HA with a base B in a low dielectric constant solvent is usually an ion pair BH+A-: B 4-HA BH+A(1) The reactions between picric acid (PiOH) and triethylamine or N,N-dimethylbenzylamine have been shown ( I ) to proceed quantitatively from left to right. The dielectrometric and spectrophotometric titration curves showed sharp breaks at the equivalence point. In this study, the interactions of PiOH and pyridine (Py) or 3-iodopyridine (3-IPy) were selected as examples of titration reactions which do not proceed quantitatively to the salt and which can be followed dielectrometrically and spectrophotometrically. THEORETICAL TITRATION EQUATIONS
If a measurable property P of a solution depends linearly upon the equilibrium concentration of each solute, Ci,then
When Equation 4 is solved for CSanL substituted into Equation 3,
{CAT
+ CBT+ Kf-’ - [(CAT+ CBT+ Kl-l)* 4CATCBT]li2] (5)
The other root is extraneous because it requires CS to exceed C A T or CBT. If uA,U B , us, Kl, and Po are known, Equation 5 can be used to calculate a theoretical titration curve. For example, if acid is titrated with base, CATis the initial concentration of acid corrected for dilution and CBTis the number of moles of added titrant per liter of mixture. It was demonstrated ( I ) that the dielectric constant (e) of a dilute solution can be related to linear contributions from the various species by either of two equally applicable equations, both of which have the form of Equation 2 :
n
-€ =- -I
where Po is the constant contribution of the solvent, n is the number of solutes, and uf is the proportionality constant for the ith solute. In the usual case where acid, base, and salt coexist in solution, Equation 2 becomes
+
+
+
P = Po U A C A UBCB ~sCs. (3) CA,CB,and Csrepresent the equilibrium concentrations of acid, base, and salt, respectively. If CATand CBTare the analytical concentrations of acid and base, the formation constant Kl of the salt is given by K f = - =c s cs CACB (CAT- CS)(CBT- csi
(4)
Present address, Department of Chemistry, University of Missouri, Columbia, Mo. 65201. To whom reprint requests should be addressed at Clarkson College of Technology. (1) R. Megargle, G. L. Jones, Jr., and D. Rosenthal, ANAL.CHEM., 41, 1214 (1969).
e -
e,-1
+ 7 pici 1
or (7) where e, is the dielectric constant of the solvent and pi and Y~ are the proportionality constants for the ith species. Equation 7 was used to derive dielectrometric titration equations for this study, since it is the simpler of the two. Hereafter, when dielectric constant data are being discussed, P = e, Po = e,, and ut = Y ~ .Spectrophotometric titration equations can be derived in the same manner, since Beer’s law also has the form of Equation 2 when P = A/b (absorbance/cell path length), P, = 0, and uf = ai (molar absorptivity). Measurement of P for solutions containing various concentrations of added acid or base can be used to determine uA, and uBfrom Equation 2. When reaction 1 is not quantitative, us cannot be determined directly because the salt is not the only species present in solution at equilibrium. Since the
ANALYTICAL CHEMISTRY, VOL. 42, NO. 11, SEPTEMBER 1970
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Solutions. Standard solutions of PiOH, Py, and 3-IPy were prepared gravimetrically. Standard solutions of PiOH were checked as described previously ( I ) . Titrations were simulated by preparing a series of solutions containing a constant volume of one constituent and varying volumes of the other in 100-ml volumetric flasks and diluting to volume with benzene. Apparatus. Dielectric constants and spectrophotometric absorbances were obtained with equipment previously described ( I ) . 0
200 % Titrated
J
RESULTS AND DISCUSSION
400
Figure 1. Dielectrometric and spectrophotometric titrations of 1.0024 X 10-3MPy with PiOH (a, b respectively). Circles = experimental points; lines = Equation 5 with constants derived from calculation method I11 (- -. ), calculation method 1(-),and calculation method I1 (-)
----
Pyridine, 3-Iodopyridine, and Picric Acid. Dielectric constant measurements were made on solutions of Py and 3-IPy. A least squares fit of these data by Equation 7 gives Equations 10 and 11, the numbers after the f symbol are standard deviations. For Py:
formation constant is generally unknown also, the problem of establishing a theoretical titration curve becomes one of evaluating K f and u8, Two methods were employed for finding K f and us simultaneously. P was measured for a series of m solutions with differing total concentrations, CAT)^ and (C& In no case was acid or base present in such large excess that reaction 1 was driven to completion. An initial estimate of uswas made in order to use Equation 3 to calculate a series of salt concentrations, (CS),, and then Equation 4 to calculate a series of equilibrium constants, Kt. In one procedure (calculation method I), the value of us was varied to minimize the sum of the squares of the relative deviations of each Kt from the average K,,, (SSDRK). m
SSDRK=
i=l
[(Ki - Kav)/Kav12
(8)
Essentially, the value of us that produces the least variation in the relative value of Kf was determined. In the other procedure (Calculation method II), the estimate of u8 and the resulting K,, were used with a more comprehensive set of data in Equation 5 to calculate a series of P values, (P& These were compared to the measured values, P i , and QS was adjusted to minimize the sum of the squares of the deviations (SSDp) between (P,),and P i . 7n
SSDP =
i-1
[(Pc)t - Pi]'
(9)
Essentially, the last procedure gives the least squares best fit to the titration curve. EXPERIMENTAL
Materials. Benzene and picric acid were purified as previously described ( I ) . Pyridine (Py) was distilled from dry KOH. A chromatogram, obtained with a column of Apiezon L on Chromosorb W KOH, showed several impurities totalling less than 0.6 ppt. 3-Iodopyridine (3-IPy) was treated 3 times by dissolving in hot ethanol, filtering, adding water, and collecting the resultant crystals. The 3-IPy was dried by dissolving it in C6HBand distilling off most of the solvent and subsequent vacuum drying. A Karl Fischer (2) titration showed the 3-IPy contained less than 0.05 water. (2) R. Swensen and D. Keyworth, ANAL.CHEM., 35, 863 (1963). 1294
E =
(0.5966
&
0.00126)Cpy
+ (2.2739 f 0.00006)
(10)
&
O.O0453)C1p,
+ (2.2740 f 0.00021)
(11)
For 3-IPy: E =
(0.5099
Values of Y B = 0.597 for Py and Y B = 0.510 were used in later calculations. Neither base absorbs near 4000 A. Picric acid has yA = 0.495 ( I ) and was found ty have a molar absorptivity of U A = 164 in benzene at 4000 A. Picric acid conforms to Beer's law. Dielectrometric and Spectrophotometric Titrations. A. PYRIDINE AND PICRICACIDINVOLVING No LARGEEXCESS OF PYRIDINE.The circles in Figure 1 represent experimental results for the titrations of pyridine (Py) with PiOH. Since Py is a weak base, the titration curves are rounded to such an extent that quantitative analytical titrations are difficult. The lines in Figure 1 are plots of Equation 5 using appropriate values of Kfand us, Identical results for corresponding concentration ranges were obtained for titrations of PiOH with PY. To evaluate K f and r8 by calculation methods I and 11, it is necessary to measure P for solutions where no large excess of acid or base is present. The center portions of the titrations were pooled and used. Only solutions with an initial amount of acid to base or base to acid of less than 2:l were used. To evaluate SSD,, all available data were used. Table I reports the results of these calculations. The appropriate variances in Table I indicate that calculation method I gives constants that result in statistically poorer curve fits than those obtained by method 11. However, in both cases the standard error is less than the estimated experimental error, (Le., the normal errors of measurement of P can cause such differences). Tests with synthetic data showed calculation method I to be more sensitive to random errors. In general, calculation method I1 should produce the more reliable constants and the better curve fit. B. ~-IODOPYRIDME AND PICRICACID. The circles in Figure 2 represent experimental results for titrations of PiOH with 3-iodopyridine (3-IPy). The lines are calculated using Equation 5. The 3-IPy is such a weak base that excess amounts are necessary in order to form any measurable amount of picrate. It is not possible to add enough 3-IPy to a solution of 10-3M PiOH to force the reaction to completion. The results of measurements on nine solutions containing
ANALYTICAL CHEMISTRY, VOL. 42, NO. 11, SEPTEMBER 1970
Table I.
Results of Analysis of Data for Bases and Picric Acid
3-Iodopyiidine,calcn method
Pyridine, calcn method I
I1
I
18.38 2016 Z!Z 35 1.85 x 10-7 4.30 x 10-4
17.00 2686 i 53 7.72 X 2.78 x 10-4
18.05 15.69 f:0.81 2.69 x 10-7 5 . 1 9 x 10-4
6964 2313 f: 17 7.13 x 10-3 0.085
665 1 2741 i 26 1.67 x 10-3 0.041
I1
Dielectrometric data YS
K,, f: SD
Total variance of e Standard error Spectrophotometricdata as
K,, k SD Total variance of A/b Standard error
Table 11. Evaluation of
CP, ( M ) 0.05001 0.05001 0.05001 0.05001
( M x 10') 0.0000 1.9951 4,9878 9.9755
CPiOH
0.0000 1.9951 4.9878 9.9755
2.3035 2.3063 2.3104 2.3174
2.3336 2.3363 2.3404 2.3470 e =
6092 16.97 i 0.15 1.150 X 10-3 0.034
by Calculation Method 111 e
from Regression
using the measured value of P and predetermined values of Po, uA, and uB. P of solutions containing no large excess of acid or base can then be determined as before, Equation 3 used to find CS, and finally Equation 4 to find K,. If P is plotted against CATfor solutions containing a constant CBT,a straight line with a slope of (as - aB) and an intercept of (Po CTB CBT)should be obtained. Values of CTS determined by calculation method 111 are shown in Table 11. All of the solutions in the top set contain 0.05001M Py added to various smaller concentrations of PiOH ; an excess Py concentration of 0.10002M was used in the bottom set. When the data were plotted according to Equation 12, both sets yielded
A/b from regression
A/b (cm-l) 0.00 1.32 3.31 6.61
2.3035 2.3063 2.3104 2.3174 13.92 CAT 2.3035 ys = 14.52 2.3336 2.3363 2.3403 2.3470 13.44 CAT 2.3336 ys = 14.04
about 10-3M PiOH and concentrations of 3-IPy varying from 5.972 X 10-3M to 0.4897M were subjected to mathematical analysis. They are also presented in Table I. All of these results are in reasonable agreement, indicating that Equation 5 satisfactorily accounts for the 3-IPy-PiOH reaction. The inserts in Figure 2 show that Equation 5 still fits the data well at 3-IPy concentrations of OSM. C. COMPARISON AND CONCLUSION, The titration curves of Py with PiOH, involving no large excess of Py, and of 3IPy with PiOH can be satisfactorily accounted for by assuming Equations 1, 2, and 4 to be applicable to both the dielectrometric and spectrophotometric data. Effect of Excess Pyridine. Another approach to the evaluation of K , and U S for the case of the PiOH-Py titration is the addition of a large excess of Py to PiOH to force quantitative formation of the salt. Studies were performed to determine this effect upon both the dielectrometric and spectrophotometric measurements. In this case (calculation method 111) CSis known and as can be found from Equation 3 in the form
+
6106 16.86 i 0.15 1.147 X 0.034
Measured Measured e
c =
0.10002 0.10002 0.10002 0.10002
US
17.90 16.03 f: 0.84 2.69 x 10-7 5.19 x 10-4
0.00 1.32 3.31 6.61 A/b = 6.63 X IO3 CAT as = 6.63 x 103 0.00 0.00 1.35 1.35 3.34 3.37 6.77 6.74 Alb = 6.76 X IO3 CAT as = 6.76 x 103
+ +
dL
0
.-u 2.301 U
a
2.53 0.48 0.50
T o t a l Conc 3-IPy
(MI
i
0
0.05 T o t a l Conc 3-IPy
0.1
(M)
Figure 2. Dielectrometric and spectrophotometric measurements of 9.9669 X 10-4MPiOH with excess 3-IPy. Circles = experimental points; (-) = Equation 5 with constants from calculation method 11; (- -) = expected results if K = 0
--
straight lines with almost identical slopes, indicating that the assumption of complete reaction is probably justified. This treatment resulted in y s = 14.28, K,, = 7105 =t738, a total variance in E of 6.47 X and a standard error of 0.00805 for the dielectrometric determinations and as = 6695, K,, = 2672 f 24, a total variance in A/b of 1.72 X and a standard error of 0.042 for the spectrophotometric determinations. Figure 1 also shows plots of these results according to Equation 5. The spectrophotometric results here are in substantial agreement with those from calculation methods I and I1 shown in Table I for the Py-PiOH case. The much higher K , value from the dielectrometric method indicates one or more additional reactions become important in the presence of a large excess of pyridine. The excess pyridine presumably reacts with the salt to form a homoconjugate salt and perhaps promotes dissociation of the resulting species into ions. Similar
ANALYTICAL CHEMISTRY, VOL. 42, NO. 11, SEPTEMBER 1970
1295
reactions have been proposed for other systems (3-8). Such reactions could alter the dipole moment of the salt without (3) S. Bruckenstein and A. Saito, J. Amer. Chem. Soc., 87, 698 IlOhz;)
(4) J. Steigman and P. Lorenz, ibid., 88, 2083 (1966). (5) I. M. Kolthoff and S. Bruckenstein, ibid., 78, 1 (1956). (6) G. Barrow and E. Yerger, ibid., 76, 5211, 5248 (1954). (7) E. Yerger and G. Barrow, ibid., 77,4474, 6206 (1955). (8) E. K. Ralph and W. R. Gilkerson, ibid., 86,4783 (1964).
greatly affecting the UV-visible spectrum. This could qualitatively explain the apparent anomaly in the presence of a large excess of pyridine. It is to be noted that dielectrometric measurements are capable of revealing the presence of species which could not be detected by simple spectrophotometric measurements.
RECEIVED for review August 27, 1969. Accepted June 25, 1970.
CORRESPONDENCE Kinetic Parameters from Thermogravimetric Data SIR: In a recent article in this Journal (I), parameters derived from a thermogravimetric trace for the decomposition of calcium carbonate were compared using three different methods of kinetic analysis. These methods were labelled Method I, the difference-differential method of Freeman and Carroll ( 2 ) ; Method 11, the integral method of Coats and Redfern (3); and Method 111, the differential method of Achar, Brindley, and Sharp ( 4 ) . As a consequence of their findings Sharp and Wentworth ( I ) concluded that they could not recommend Method I because of an apparent poor precision for this method. Because Method I appears to be one of the more widely used thermogravimetric method for obtaining kinetic data (5) for solid-state chemical reactions, we would like to call attention to several points that may help to clarify this situation. One important advantage that attracts the chemists to Method I is that the order of a simple chemical reaction may be determined analytically, whereas in the case of Methods I1 and I11 an order has to be assumed. A trial and error procedure for the latter methods is then used, the criterion of acceptance of the numerical value for the order being that value which leads to the best fit for thermal trace over the entire range of the chemical reaction. Aside from the inconvenience of trial and error methods, the question arises whether the criterion of constancy of kinetic parameters is appropriate even for the case of a simple chemical reaction since a solid state reaction is a very complex process. In our present state of knowledge, constancy of kinetic parameters in general, for reactions in the solid state, is an unwarranted assumption (6, 7). Apparently in the case of the thermal decomposition of calcium carbonate there does not appear to be significant changes in the kinetic parameters during the course of the reaction. In this case all three methods should yield about (1) J. H. Sharp and S. A. Wentworth, ANAL.CHEM., 41, 2060
(1969). (2) E. S . Freeman and B. Carroll, J. Phys. Chem., 62, 394 (1958). (3) A. W. Coats and J. P. Redfern, Nature, 201,68 (1964). (4) B. N. N. Achar, G. W. Brindley, and J. H. Sharp, Proc. Znt. Clay C o n f , Jerusalem, 1, 67 (1966). (5) J. H. Flynn and L. A. Wall, J. Res. Nat. Bur. Stand., 70A,
487 (1966). (6) B. Carroll and E. P. Manche, J. Appl. Polym. Sci., 9, 1895 (1965). (7) I. A. Schneider, Makromol. Chem., 125, 201 (1969). 1296
the same results, provided a good guess is made for the order of the reaction for Methods I1 and 111. The results of Sharp and Wentworth demonstrate that this is the case when the initial portion of the reaction for Method I is omitted. Similarly in a previous publication (6),it was shown that both the integral method of Doyle and Method I yielded about the same results for the volatilization of a liquid. In this particular instance both order and activation energy were known, the order being obviously zero and the activation energy being that for the latent heat of vaporization. Again, whereas the order was assumed for the integral method, the value was determined analytically via Method I. As for the activation energy, Method I yielded a value that checked the thermodynamic value within a tenth of a kilocalorie. It should be noted that the latter results were obtained with the inclusion of the initial part of the process provided that the random errors in the slope measurements (weight us. temperature) were smoothed out, particularly for the initial and final stages of the process. The necessity for this is clear. Method I is based on the differences in the slopes of the thermogravimetric trace and in the initial and final regions of a reaction. These differences are usually extremely small, thus magnifying enormously the errors inherent in slope measurements. Precision can be enhanced considerably by the simple expedience of plotting the experimental slopes cs. the reciprocal of the temperature and smoothing out the random errors of the observed slopes. Further, the resulting curve may be subdivided into equal intervals of A(l/7'), thus simplifying the subsequent plot for yielding both order and activation energy (6).
The original thermogravimetric trace for the Sharp and Wentworth paper was not published, making it impossible to illustrate the effect of the above procedure for Method I. We are rather surprised at the valid results they obtained for Method I for the range of 25 to 8 4 x conversion of calcium carbonate, this being 0.55 for the order and 43 kcal/mole for the activation energy. Because the thermal decomposition of calcium carbonate is an endothermic reaction its activation energy should be equal to or greater than the thermodynamic value for the reaction. The thermodynamic value is 43.0 kcal/mole at the temperatures of the decomposition. The activation energy of the reverse reaction is known to be small and is probably in the neighborhood of one kilocalorie or so. Examining the Sharp and Wentworth values for Methods I1
ANALYTICAL CHEMISTRY, VOL. 42, NO. 11, SEPTEMBER 1970
