May, 1954
NOTE
dieted T , cannot be reliable. A 3% difference in T, is roughly equivalent to a 5% difference in a. Acknowledgment.-This work was carried out
under a grant established a t the University of Tennessee by the Fulton-Sylphon Company of Knoxville, Tennessee.
447
NOTE THE EVALUATION OF CONSTANTS I N FIRST-ORDER CONSECUTIVE IRREVERSIBLE REACTIONS BY ELIASKLEINAND T. F. FAGLEY Hichardson Chemistry Lab., Tulane University, New Orleans, La. Received January SO, 1964
Swain' has treated consecutive irreversible firstorder kinetics for the case in which both products Pairs of graphs Wa) W/a)
0.3 .3 .3 .4 .4
0.4 .6 .7 .6 .7
Offset
log K
0.100 .274 .359 .173 .258
-0.285 .300 - .302 - .301 - .301
-
B and C in the reaction A + B + C are analyzed as a single chemical species. A study of a first-order consecutive reaction in this Laboratory reveals a reaction in which the only species which can be measured with any experimental precision is the species k¶ C represented in the equation A -%- B + C. If lcl and kz are to be evaluated from this measurement alone, a series of tedious approximation8 has been the usual procedure in the past. K
log m
0.476 ,562 .475 .561
kz = K
ki
kit
Tangency of curves 0.502 2.99 ,499 3.65 .500 2.99 ,500 3.69 Av. 0.50026 ki = 0.0500
0.0997 .1007 .0997 ,1004 0,1001~~
TABLE O F K us. ni VALUES 7
K
y/a = 0.3
0.01000 .01053 .01111 ,01176 .01250 ,01333 .01429 .01539 ,01667 .01818 ,02000 .02222 ,02500 .02857 .03333 .04000 .05000 .06667 ,10000 ,11111 ,12500 .I4290 .16667 .20000 ,25000 .33333 ,50000 ,66667
36. 6700 34.8935 33.1020 31.3225 29,5360 27.7575 25,9770 24,1930 22.4100 20.6250 18.8457 17.0640 15.2800 13.4995 11.7180 9.9375 8,1600 6.3825 4.6040 4.2471 3.8880 3.5273 3.1620 2.7905 2.4096 2.0130 1.5868 1.3542
-m y/a = 0.4 u / a = 0.G
52.0900 49.5330 46.9260 44.4295 41 ..7600 39,3150 36,7640 34.2095 31.6560 29.1060 26.5500 23.9985 21.4440 18,8930 16,3410 13.7900 11.2420 8.6970 6.1580 5.6502 5.1432 4.6361 4.1268 3.6140 3.0944 2.5617 2.0016 1,7019
92.6300 88.0460 83.4750 78.8885 74.3120 69.7275 65.1490 60.5670 55.9860 51.4030 46.8250 42.2460 37.6640 33.0855 28.5060 23.9275 19,3520 14.7795 10.2160 9.3069 8.3976 7.4928 6.5874 5.6865 4.7884 3.8895 2.9798 2.5113
(1) C. G. Swain, J . -4m. Chem. Soc., 6 6 , 1600 (194.1).
y/a = 0.7
K
121.4000 115.3870 109.3590 103.3430 97.3280 91.3050 85.2880 79.2675 73.2480 67.2265 61.2100 55,1880 49.1720 43,1550 37,1370 31.1200 25,1060 19,0950 13.0930 11.8962 10,7000 9.5067 8.3166 7.1325 5.9552 4.7868 3 . 6236 3.0378
1.5000 2.0000 3.0000 4.0000 5.0000 6.0000 7.0000 8.0000 9.0000 10.000 15.000 20.000 25.000 30,000 35.000 40.000 45.000 50,000 55.000 GO. 000 65.000 70.000 75.000 80.000 85.000 90.000 95.000 100.000
u/a
=
0.3
0.9028 ,7925 .6710 ,6024 .5581 ,5270 ,5039 ,4860 ,4719 ,4604 ,4255 .4080 .3975 .3906 .3857 .3820 ,3792 ,3769 ,3750 .3735 ,3722 ,3711 .3701 .3692 ,3685 .3678 ,3073 ,3667
y/a
= 0.4
I . 1346 1.0008 0.8539 ,7736 ,7228 ,6878 ,6623 ,6429 .6278 .6158 ,5798 ,5621 .5516 .5447 ,5398 .5361 ,5333 ,5310 .5292 .5276 .5263 ,5252 .5242 ,5234 ,5227 .5220 .5214 ,5209
m
y / a = 0.G
u/a = 0 . 7
1,6742 1.4899 I . 2965 1.1971 1.1373 1.0979 1.0704 1.0497 1.0341 1.0216 0.9853 ,9676 ,9571 .9502 .9453 .9416 ,9388 .9365 ,9346 ,9331 ,9318 .9307 .9297 ,9289 ,9281 .9275 ,9268 ,9263
2.0252 1.8118 1 ,5956 I .4888 1.4265 1.3861 1.3581 1.3375 1.3218 1.3093 1,2730 1.2553 1.2448 1.2379 1.2330 1.2293 1,2264 1.2242 1.2223 1.2208 1.2195 1.2184 1.2174 1.2166 1.2158 1.2151 1,2146 1.2140
If ( a - x), (x - y) and y represent the amount or
NOTE
448
VOl. 58
concentration of A, B and C, respectively, where u is the amount or concentration of A at time t = 0, the rates may be written dxjdt = kl(a - x) (1)
corresponds to the time where y / u completion has occurred. The intersection of the two graphs must then give the unique solution for K. This generally applicable solution then resolves itself into a tabulation of K versus m values for three or four dl//dt k i ( ~- l/) (2) values of y/a. The values of v / a = 0.3, 0.4, 0.6 and 0.7 would allow several determinations of K and the integration leads to at various stages of completion of the kinetic procx = a(1 - e - k l l ) (3) ess. From equation 5 it can be seen that there is y = a ( 1 - [l/(h - kz)](kie-R~t- k l e - b z i ) ) (4) no algebraic method for expressing m explicitly and that only a series of approximations would lead to If the following substitutions are made in equation a value of m for a given set of values y / a and K . 4, one obtains Solutions were obtained by using successive apK = k z / k l ; m = krt proximations with the aid of mathematical tables.2 The values of m, K and y / a are given in Table I. ( y / a - 1)(1 - K ) = K e - m - e - m K (5) The method was tested by applying it to the hypoThe value y / u represents the degree of completion thetical reaction for which IC; = 0.1 min.-’ and of the reaction at any time, If for, say, two values, ICz = 0.05 min.-l for which the times for y / u = y / a and y’/a, the solution of ( 5 ) is tabulated within 0.3,0.4,0.6and 0.7 are 15.87,20.02,29.80and 36.24 the range of K = 0.01 t o K = 100.0, the specific minutes, respectively. Graphs constructed on reaction rate constants can be evaluated in the fol- sufficiently large paper to permit a precision of four lowing way. significant figures in log t and three in log m were For each value, y / a and y ’ / a , plot a graph of K offset on one another by log t - log t’ and K read off versus log:m. This. from the definition of m,is seen directly. to have7he abscissa log kt. If the graph paper is (2) Tables of Exponential Functions, Federal Works Agency, transparent, the abscissas of the two graphs may Works Project Administration (Under Sponsorship of Natl. Bur. be equated by off-setting one graph from the other Standards), 1939; “A New Manual of Logarithms,” K. C. Bruhns. by an amount equal to log t - log t’, where t Rev. ed., The Charles T. Powner Co., Chicago, Illinois, 1942. =i
’i
. \
