Boris Musulin
Southern Illinois University Carbondale, 62901
I
I
A Computer Program for Chemical ~inetic; by Titration
T h e chemistry instructor in a modern undergraduate chemistry lahoratory must take full advantage of the accessibility of computer facilities. One purposc of the present paper is to discuss a computer program which updates a common undergraduate physical chemistry experiment. A second purpose is t o illustrate how interpretation of computer results may be used to find faults in the lahorat,ory procedure and to suggest corrections t o that procedure. I n particular, removal of calculation t,edium allows the instructor to incorporate into the experiment multiple methods of data treatment. Ideally, each method would he expected to yield the same result. If this ideal is not uniformly attained by the class, eithcr (1) the experimental procedure is defective or (2) each student is uniformly poor in experimental technique. Experimental
The basic experiment was a revision ( I ) , "Chemical Kinetics (By Titration)," of an experiment originally designed by Daniels, el al. ( 2 ) . T o update the expcriment, a program1 was writt,eri in FORTRAD: I1 (5) for execution on the I B l I 1G"O. Each student keypunched their own data cards and printed a listing of their answers with an I B J I 407. A majority of the students performed the intermediate manual computer operations during s scheduled laboratory hour. The final program incorporated seven different calculations; five calculations assumed that an alkyl acetate (either ethyl or propyl) saponification was a simple second-order reaction and two assumed that the reaction involved a chemical equilibrium. Three of the five calculations involved averaging a set of individually calculated rate constants while the other two and the two equilibrium calculations utilized the slope of a linear function. Ea.ch of the seven calculations was performed a total of three times with each subcalculation differing in the number of data points. For the student whose data included N (time, volume) points, the Nth, (N - l)st, and (N - 2)nd were sncccssively discarded. (If required, the N t h point was considered the reading a t infinite time.) The present numerical experimentation mas an attempt t o find a non-graphical technique for predicting the onset of a tailing-off phenomenon obtained by students in preparing the usual simple secondorder linear plot. Each of the duplicate determinations (one exception) is treated separately. One data set which yielded
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Program is available from tho repository of chemical expe1.iments xl Sottthern Illinois Uuiveriiity. Complete versions of Tables 1 through 6 m e available upon requesl from the author.
imaginary logarithms was discarded prior to analysis. Several othcr students' data were as poor but the conclusion to discard must be drawn from an analysis of the results, not from a calculation brcalidowrr. Consequently, these other extremely poor results werc included in the analysis and sampled in the tables if for no other reason than to discourage the overly optimi~tic.~ The tables also cont,ain select,ed lit,erat,ure values including some rcsults obtained in mixed solvents whose values do not differ greatly from values obtained in pure watcr. Jiterature values reported by Evans, et al. (4),Smith and 1,evinson ( 5 ) ,and Davies and Evans ( 6 ) obtain the init,ialester concentration directly. Averaging Methods Single Point Colculotion with fhe Initial Ester Concentration Determined at Infinite Time
The rate constant, lc, was calculated a t every data point from the appropriat,e integrated equation for a simple second-order react,ion. The initial base concentration, a, was obtained from a standard stjock solution; t,he initial ester concent,ration, b, was assumed t o he t,he reaction variable value for the Nth reading. Columns 4, 5 , and 6 of Table 1 present sample average lc values with their average ahsolut,e deviation from the mean of data set,s containing (N - I), (N - 2), and (N - 3) values, respectively. Student values differ numerically from the literature values by more than solvent error. An examinat~ionof t,he single point 16 values in the program output showed that, in all but two duplicate sets, the initial rate constants were above the mean value and decreased (essentially monotonically in twelve of sixteen cases), as the reaction proceeded, to a value below the mcan value. Of the two exceptions, Student 1,'s values were physically unacceptable (negative values) while Student K's values were randomly distributed. Of the typical cases, the rate of decrease decreased with increasing h e , thus, placing t,he mean k value closer to the value a t the reaction initiation. If this were a research project a conclusion that the reaction obeyed simple second-order kinetics would he doubtful. The best precision was 13YGwhile most students had precisions in t,he rangc of 20-307& Even a liberal supposition that these particular students might have experiment,al errors of 10% is insufficient t o account for the results. Consequently, the lahoratory procedures must be questioned. A comparison of all lc values indicates that successive elimination of t,he termirial points usually increases the meau and the mean absolute deviation from the mean Volume 46, Number 2, February 1969
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109
T&te
1.
Sample Averrrge R a h Constants Calculated from lndividuol Constants Determined with Ester ConcRllrotion Assumed to b e the Final Amount of Bose Reacted t (OCI
Ethyl Aeetwte Student 1) Sludettt I) Rylartder & Tarbell (10)" Myers, el a / . ( 8 ) &&nt E Student E Pmpgl Acdrtte Student L Stu&nt L Radent N Rtndent N Myen, el a1 ( 8 ) Stu&nt Q At,&"t Q Stdent R Student R
m
9 9
20.3 25 28 28
8 8
20
k X IOP (M-I
N
1 4 . 3 f 11.1 16.4 f 12.6 2.92 11.095 f 0.045 5 . 4 5 f 2.12 16.4 f 6.12
k X 102 1M-' \ceCL)
k X 101 ( M - I see-li 15.7 f 12.0 18.2 f 1 3 . 4
1 . 5 & 12.8 2 0 . 4 f 14.2
5.86 & 2.06 17.6 zt 6.38
6.40 f I .83 19.2 zt 6 . 3 8
62% Acetone solvent.
To$le 2.
Sempk A w w Rate Constants Calculated from.lndividual Constants Determined with Ester Comeenhetion Assumed to b e Known from Dilution of Standard Solution
Ethyl Acetste *,nb,!t
I)
Evana, el a/. (4). Student E Student E Smith & Levimun (5)" Smith & Levinson (5).
h p y i Acetate Student N
a
85% Ethsnd &vent.
7% Acetone solvent. (as expected from the treud analysis) but does not change the percentage deviation The data of a typical student, Student N, illustrates the expected performance of a data set containing usual terminal reactiuu errors. Since individual k values are less time dependent a t low temperatures, termiual point discard improves the per ceut deviat~onm e a t higher temperatures. Comparison to the hterature values and use of Chauvenet's criteriou (7) both iodicated that initial data points rather than t&miunl points should bc discarded. Smith and Levirison (5) and Myers, et al. (8) (20 aIld both initial and terminal data 25% each, respectively). Such exc~usionsin the pres. ent laboratory procedure would eliminate a majority of data. poillts but, with ioelTasing availahilit,y of puter facilities, an increase in the llumher of measured points may be substituted for a decrease in calculatior~
+:-LlLlLG.
%gJe Beinf Colrvlotisn wifh the EniPIaJ Ester Concentrofion Determined horn h e c t Meoavrement
-
This ealrulation is identical to the ~recedineealculation except that the initial ester mncentration is assumed to be 0.005 M with no attempt made to ascer110
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Journal o f C h e k a l Education
tain the correctness. For each student the relationship of the h value a t infinite time to 0.005 ni was eompared to the relationship of the k value of the prcceding calculation to the corresponding lc value of this calculation. The comparison agreed with conclusions drawn from an inspection of the series, obtained by expansion of logarithmic terms, viz., A
I
=1 . n +. h . .-.
.
I n 23 of the 35 data sets, low b values a t infinite time might indicate that the reaction had not gone to completion. NO correlation to temperature existed although the b values a t O°C were the lowest. Precision in thc preceding calculation is about the same as in this ca1cu1ation, but the accuracy is slightly better. Comparison of Tables 1and 2 affords an excellent example of computer analysis to help asecrtain student errors. Both studcnts D and E had diverse b values as determined from infinite time, Usage of nf with Student D's data produced almost two identical sets of k values while the same calculation for Student E yielded k values which were still very poor. Thus, the instructor can surmise that Student D had a laboratory
Table 3.
Sample Average Rote Constants Calculated from Individual Constants Using the Formula of Reicher (Pott and Amis (9)) )
iV
0 0 5 5 5 5 0.8 19.1 20 20
9
(
Ethyl Acetate Student A P o t k & Amis (9) Student B Student B Student C Student C Potts & Amis (8) Potts & Amis ( 9 ) Student 1) Student 1) Propyl Acetate Student N Student N Student Q Student Q
7 7 30 30
9 0 9 9 9 9 9 9 9
9
k X 10Z (M-I seer1)
k X 10% (ACLseer')
k X 102 sec-')
2.71 1 1 . 1 2 1.82 zt 0.017 3.70 f 1.33 4.29 i 0.686 4.08 zt 0.972 4.02 iz 0.955 3.90 7.23 3.75 + 1.01 4.75 zt 1.71
2.83 f 1.17
3.00
3.78 f 1.41 4.15 zt 0.703 4.27 & 0.754 4.20 + 0.705
3.75 4.15 4.30 4.21
3.60 iz 1.02 4.90 f 1.72
3.62 f 1.20 5.15 i 1.84
4.12 zt 1.12 4.07f1.10 6.75 i 1.77 7.13 zt 1.76
4.34 f 0.892 4.3010.!149 7.03 zt 1.71 7.34 +c 1.74
4.43 zt 0.890 4.40iz0.952 7.10 f 1.96 7.35 f 2.08
+ 1.18 f 1.68 f 0.830 10.859
zt 0.845
Table 4. Sample Rate Constants Determined from the Slope of o Typical Second-Order Plot with Ester Concentration Assumed to b e the Final Amount of Bose Reacted
t ("c)
k X 10'
AT
(M-'
k X 1OP k X loZ ( M - I secc1) (At-'seer1)
(N - 4) points) were in closest agreement to the literature values (see Table 3). I n general the student results were smaller, with poorer precision, than the literature values. The hetter agreement probably results from the fact that the procedure emphasizes relative measurements which tend to average out expcrimental error. Again t,he additional calculation tcdium is of no import if an instructor adonts comnuter
Slope Methods Slope Only Cokulafion
A least squares (7) suhroutine was used to calcu0 9 1.77 1.82 2.07 late the slope and intercept 0.6 0.675 of the best straight line 5 9 3.21 3.98 4.15 through a set of points. 5 9 5.20 4.05 4.15 5 9 2.89 4.13 4.28 Table 4 gives sample rate 5 9 2.89 4.15 4.19 constants which were calcu10.9 1.51 20 0 4.76 3.39 3.19 lated from the slope, the b 20 9 3.06 3.93 4.08 value which was obtained 20.3 2.92 from the Nth point (infinite 7 9 2.76 4.18 4.86 time), and the known a 7 9 2.61 3.90 4.47 value. Literature values of 30 9 4.97 6.72 7.13 Rylander and Tarbell (10) 30 0 5.81 7.42 7.79 were determined by this " 62% Acetone solvent. method. The results, for data sets containing (N - 1) points, Table 5. Somple Rote Constants Determined from the Slope and Intercept were smaller than those of a Typical Second-Order Plot from thc preceding calculation which, in turn, were smaller than the literature values. I n 25 data sets, the 0 0 1.57 1.62 1.85 rate constant increased Rylander & Tarbell (lop 0.6 0.675 Student B 5 9 2.99 3.88 4.08 monotonically as the terStudent B 5 9 5.71 4.02 4.14 minal points were discarded. Student C 5 9 2.58 3.92 4.09 If, arbitrarily, one estabStudent C 5 9 2.5!1 3.98 4.02 Itylander & Tarbell ( l o p lo.$] 1.51 lishes a criterion that linearStudent D 20 9 4.02 2.64 2.47 ity is obeyed if, upon disStudent 1) 20 9 2.27 3.01 3.14 card of a point, the rate 20.3 2.92 Rvlmder & Tarbell 110)" Propyl Acetate constant does not change by Student N more than lo%, then, after Student N Student Q discard of the penultimate Student Q reading, there was linearity a 62% Acetone solvent. in only 21 data sets. The remaining sets, representing error involvcd with the Nth point while Student E had nonlinearity by this crude criterion, were a reflection many more problems. of the large mean absolute deviation from the mean values found in the first t,hree calculations. The presThree Point Calculation ent student data were too varied to draw any firm conclusion on the elimination of terminal point,s. This calculation follows the procedure of Potts and Amis (9)which uses two fixed reference points (in the Slope-Intercept Cokulotion present example, the first reading after solution mixing and the Nth reading) to evaluate k at each data point. This calculation is identical to the preceding calcnli~. The results (from averages over (N - 2 ) , (N- 3), and tion except that the intercept from the least squares sec-1)
Ethyl Acebate Student A Itylander & Tabell (10p St,udent B Student B Student C Student C Itylander & Tmbell ( l o p Student D Sludent L, Rylander & Tmbell ( l o p Propyl Acetate Student N Student N Student Q Student Q
Volume 46, Number 2, February 1 9 6 9
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11 1
subroutine is used as an independent variable in the calculation of the rate constant in place of the b value from infinite time. The results were further, except for five data sets, from the literature values than the corresponding results of the preceding calculation (Table 5 ) . Equilibrium Calculation
Table 6.
Sample Forward and Reverse Rate Constants Calculated from the Slope of a Typical Second-Order Equilibrium Plot
.
c.c' , Ethyl Acetate Student. R Student R Student E Student E Student J St,udent J Propyl Acetate Student 4 Student N Sbudent Q St~ldent1,) Student S Student S
43 7
Immeasurable tha~llrsare accorded to the Data Processing and Computing Center of Southern Illinois University for student utilization of their facilities and for post-course analysis. The technical assistance of Jcss Thompson and Shelba Jean Choate hIusulin is acIr~iowledged. Appreciation is expressed for comments by Professor Ralph G. Pearson.
Journal of Chemical Education
k X lo8 (M-' sec-I)
43
Acknowledgment
/
Reverse k X lo8 (M-,I sec )
5 6 28 28
Finally, assuming that the 7 :10 products reacted according t o 30 simple second-order kinetics, 40 40 the techniques given by Frost and Pearson (11) were utilized t o derive an integrated rate expression which i a n be cast into linear form. The rate constant of the forward reaction was then calculated from the slope of the linear form, the initial concentration, a and b, and the equilibrium value of the reaction variable. The program also was designed to yield the rate constant for the reverse reaction and thc equilibrium constant, K, for the reaction (Table 6). Since the literature clearly indicates that saporiification of alliyl esters obeys simple second-order lrinet,ics, t,his calculation is included for pedagogical purposes only wit,h the hope that the results would approximat,e the known facts. The forward reaction k values were much smaller in magnitude than the values predicted from simple kinetics. The calcnlat~edK values, 10" t o lo1,were much too small.
112
F o r w a r d k X 10' k X 10' k X 10' k X lo8 ( A P ( 1 (wl ser-1) sec-1) iec-1) see-')
Literature Cited (1) YANLKRTI,:,K. A,, A N D MUGULIN, B., T h e m i c d Kinetics," I)itt,oed Material, Southern Illinois Universit,y, Carhondale, Ill., 1960. (2) DANIWS,F., M.~THI(H.s, J. H., A N D WILLIAMS,J. W. "EXperimental Physical Chemistry" (3rd ed.), McGmw-Hill Book Company, Inr., New York, 1941, p. 167. (3) IBhl, "IBM 16'20 FORTIIAN I1 Specifications," Pamphlet, International Business hlachines Corporation, San Jose, Cnl., 1962. (4) EVANS,1). P., GORDON, j. J., .AND WATSON, 11. B., J . Chem. Soc., 143!1 (1038). (.>) SMITII, H. A,, . i N D LI':VINSON, H. S.,J . .4m. Ch
