AN AMPEROMETRIC-KINETIC EXPERIMENT EMPHASIZING THE IMPORTANCE OF ERROR TREATMENT' JAY A. YOUNG and ROBERT J. ZET02 King's College, Wilkes-Barre, Pennsylvania
THIS experiment, proposed for use in the physical chemistry laboratory, is based upon work done by R. G. Pearson and L. H. Piette.8 It is a study of the ravid reaction. + OH-
C2HbN01
-
CnH4N0.-
+ HnO
From the data obtained, the student is required to determine the value of the rate constant. At reasonable concentrations this reaction is substantially complete within a few seconds (k = 350 l./mole sec. a t 25°C.4). Since direct determination of k is impractical, an indirect approach is required. This experiment has the additional value of showing the student that k can be determined by other than direct methods. The value of k which is obtained depends upon an exponential term (pH). Therefore, small, incorrigible, errors in the raw data have a marked effect upon the calculated value of lc. Thus the student is forced to recognize that small uncertainties in the values of empirically determined parameters can have an exaggerated effect upon the final result. He learns that undesirable though they may be, incorrigible errors are incorrigible, and to he distinguished from corrigible errors. The calculations which lead to these' couclusions, furthermore, are not complicated; that is, the student does not become so involved in mathematical detail that he loses the point. The use of calculus to demon-
' Presented a t the meeting of the Pennsylvania Association of College Chemistry Teachers, Wilson College, April 12-13, 1957. ' Present address: Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts. "EARSON,R. G., AND L. H. P I E ~ E J. , Am. Chem. Soc., 76, 3087-8 (1954). 'BELL, R. P., AND J. C. CLUNIE,Proe. Roy. Soe. London, AZlZ, 16-32 (1952).
strate this is recommended, but the calculations are simple enough so that they may be carried out arithmetically. In fact, for those students who are familiar with the mechanics of calculus but who remain unconvinced of its logic, an arithmetical evaluation of the large effect of the small uncertainty can be made with only a little extra effort and compared with the same result obtained by calculus, thus strengthening their faith in the reasonableness of calculus. But even more importantly, this experiment shows the student that the uncertainty he calculates by error treatment is an accurate estimate of the uncertainty. So often, the laboratory time available permits the accumulation of only enough data to calculate two or three results. The student is not convinced that the discordancy among these results agrees with the limits of uncertainty he has calculated by error treatment. I n this experiment it is reasonable t o require, in one afternoon, that sufficient data be obtained to permit the calculation of fifteen, or more, values of k. Therefore, the standard deviation or probable error of the average can be determined validly and compared with the uncertainty derived from error treatment. The two results will agree closely, thus demonstrating that error treatment is indeed a valid method, one that can be used with confidence to estimate the uncertainty of a result. THE EXPERIMENT
An aqueous solution is accurately prepared to contain 0.1 mole/liter of nitroethane and approximately 0.1 mole/l. of KC1. An accurately measured volume (approximately 250 ml.) is placed in a 400-ml. beaker. Electrodes from a pH meter and a cylindrical platinum gauze cathode are inserted. This assembly forms the cathode compartment of the electrolysis cell.
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The anode compartment consists of a glass tube, about 1.5 cm. in diameter and 12 cm. long, on one end of which is fixed a disc of cation exchange membrane, sodium form, by means of a rubber sleeve made from a finger cot. This vessel is filled about threequarters full with a slurry of granular cation exchange resin, sodium form, in 1 M KC1. A platinum wire anode is positioned within this slurry. "Nepton CR-61" cation exchange membrane (1 meq./dry gram capacity, 38% resin by weight) can be obtained from Ionics, Inc., 152 Sixth Street, Cambridge 42, Massachusetts, in 9- X 10-in. sheets. Unused portions should be stored, with a few ml. of water, in a sealed container. The membrane will crack, becoming useless, if allowed to dry. "Dowex AG50WX8" cation exchange resin, 20-50 mesh, analytical grade (5.1 meq./dry gram capacity, 44y0-50% resin by weight) can be obtained from BioRad Laboratories, 800 Delaware Avenue, Berkeley, California. This material also should be kept in a moist condition. Both the membrane and the resin are in the hydrogen form when received and must be converted to the sodium form before use. I n a suitable beaker, add a 40-fold excess, in terms of equivalents of sodium, of a 4 N sodium chloride solution to the resin and membrane to be converted. Stir continuously for about five minutes. Add methyl orange indicator and then add sufficient 0.5 M sodium hydroxide to make the supernatant liquid neutral. Stir for five minutes, add more sodium hydroxide to reattain neutrality. Repeat until the indicator color remains yellow after five minutes of stirring. Decant the solution and wash the solid until free of chloride. The whole assembly is then immersed in the catholyte, centering it, approximately, in the cylindrical cathode (figure). The catholyte is stirred with a magnetic stirrer. The cell can be thermostatted although this adds to the complexity of the apparatus. If the original materials are a t room temperature, the temperature of the cell contents will remain approximately constant for one hour, the probable maximum time required. A variable voltage, about 3 to 10 volts, is applied across the anode and cathode. Hydroxyl ions are produced within, and sodium ions pass into, the catholyte. The voltage is adjusted so that a milliammeter, which is in series with the cell, registers 20-80 ma., preferably nearer 20 ma. for the first determination. The pH meter reading will increase steadily. By adjusting the voltage, a condition is quickly reached where the current and p H are constant. These values are noted, the voltage varied slightly, readjusted a t this new potential so as to attain another steady condition. The pH and current values are recorded. This procedure may be repeated as often as desired; several runs may be made in a few minutes, and each run will yield a value for k, the rate constant of the reaction. The rate of disappearance of OHwill be: -d(OH-)/at
= k (CnH&?Os) (OH-)
All concentrations are expressed in mole/liter. If nitroethane is present in large excess, the product, k X (CaH6N02),is independent of time; the reaction VOLUME 35, NO. 3, MARCH, 1958
Slurry of cotion exchoqe resin in 1 M KC1
electrodes
Rubber r1enr
Cation archonge
Stirrer bar,
is pseudo-first order. Since the concentration of hydroxyl ions is held steady (pH does not change during a run), the rate of (electrolytic) addition of hydroxyl d(0H-) * must equal the rate a t which these ions,
[+
ions are chemically consumed.
[+
1-'
= k (GHJYO,) (OH-)
Hence, k
=
[+ ~ ] ' / ( c , H , N o , )
(OH-)
Since all of the terms on the right side of (1) can be known, the value of k can be determined. (OH-) = 10 (pH - 14)
where pH = the steady reading of the pH meter I = the steady current, in amperes, a t the steady value of the pH V= the volume of the catholyte solution, in liters F = the Faraday constant, in coulombs g= the weight of nitroethene, in grams, used to make up the stock solution of nitroethane in aqueous KC1 L = the volume of the nitroethane-KC1 stock solution, in liters M = the molecular weight of nitroethene s = the number of seconds in one minute
Therefore, from equation (I),and the foregoing, k = IsLMIO"/FVg(lWB)
(2)
The uncertainty in k due to uncertainties in the parameters of equation (2) can be determined by the usual meth0d.l Thus, for example: A
hk
= - 7, =
bI
k- rr l./mole min I
C R ~ P L BT.R B., AND J. H. YOE, "Chemical Computations and Errors," John Wiley & Sons, Inc., New York, 1940, Chap. X.
Where *.A, is the uncertainty in k due to the uncertainty, *T,, in the measured value of I. Similarly, corresponding values may be calculated for the other parameters. Typical results are summarized in the table. The Uncertainty in k Due to Uncertainties in the Parameters in Equation (2) Uncertaintu in k due lo
Parameter Symbol
I
L
V 9
aH
Typical value 0 050 0.5000 0.2500 3.754 8.54
un&&inty *r= 0.002 0 0005 0.0005 0.001 0.05
Dimension
amperes liters liters grams
.. .
l./mole mn. 14.0 0.5 1.0 0.1 41.0
The over-all uncertainty in the value of k, the square root of the sum of the squares of the individual un-
certainties, is approximately *42 l./mole min. Clearly, a small uncertainty in the observed value of the pH over-rides all the others. In a series of typical runs the average value for k was found to be 351 * 49 (standard deviation) I./mole min. The agreement between the comparable uncertainties, *42 aud *49, is satisfying. Similar results were obtained in other series of runs. In summary, this experiment is easy to set up. It illustrates a unique method for the determination of k , particularly applicable to moderately rapid reactions. It demonstrates in a clear manner, unobscured by mathematical detail, that a small uncertainty can have marked effectupon the calculated result; that there is a difference between corrigible and incorrigible error. In one laboratory period the student can obtain several values of the rate constant. From these he can see that the uncertainty predicted by error treatment closely approximates the standard deviation actually obtained and that error treatment is therefore a valid method for the estimation of errors.
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