Arrhenius Kinetics Solutions

Arrhenius kinetics Solutions

The data analyses are shown below. To download a worked spreadsheet, go to:
http://webserver.lemoyne.edu/faculty/giunta/classicalcs/arrkinans.xls
The necessary manipulations are to convert the Celsius temperature to an absolute temperature (Kelvin). We need the natural log of the rates for all of the exercises, the reciprocal of the absolute temperature for the Arrhenius plots (exercise 1) and the natural log of the absolute temperature for exercise 3.

a) Hood KClO3 + FeSO4

T(°C) / relative rate / T(K) / 1/T(K) / ln rate / ln T(K)
10 / 1.00 / 283 / 0.003534 / 0.00 / 5.65
12 / 1.21 / 285 / 0.003509 / 0.19 / 5.65
14 / 1.46 / 287 / 0.003484 / 0.38 / 5.66
16 / 1.73 / 289 / 0.003460 / 0.55 / 5.67
18 / 2.11 / 291 / 0.003436 / 0.75 / 5.67
20 / 2.51 / 293 / 0.003413 / 0.92 / 5.68
22 / 2.96 / 295 / 0.003390 / 1.09 / 5.69
24 / 3.59 / 297 / 0.003367 / 1.28 / 5.69
28 / 5.08 / 301 / 0.003322 / 1.63 / 5.71
30 / 6.04 / 303 / 0.003300 / 1.80 / 5.71
32 / 7.15 / 305 / 0.003279 / 1.97 / 5.72

b) Warder sodium lye + ethyl acetate

T(°C) / rate / T(K) / 1/T(K) / ln rate / ln T(K)
3.6 / 1.42 / 276.6 / 0.003615 / 0.35 / 5.62
5.5 / 1.68 / 278.5 / 0.003591 / 0.52 / 5.63
7.2 / 1.92 / 280.2 / 0.003569 / 0.65 / 5.64
11 / 2.56 / 284 / 0.003521 / 0.94 / 5.65
12.7 / 2.87 / 285.7 / 0.003500 / 1.05 / 5.65
19.3 / 4.57 / 292.3 / 0.003421 / 1.52 / 5.68
20.9 / 4.99 / 293.9 / 0.003403 / 1.61 / 5.68
23.6 / 6.01 / 296.6 / 0.003372 / 1.79 / 5.69
27 / 7.24 / 300 / 0.003333 / 1.98 / 5.70
28.4 / 8.03 / 301.4 / 0.003318 / 2.08 / 5.71
30.4 / 8.88 / 303.4 / 0.003296 / 2.18 / 5.72
32.9 / 9.87 / 305.9 / 0.003269 / 2.29 / 5.72
34 / 10.92 / 307 / 0.003257 / 2.39 / 5.73
35 / 11.69 / 308 / 0.003247 / 2.46 / 5.73
37.7 / 13.41 / 310.7 / 0.003219 / 2.60 / 5.74

c) Urech inversion of sucrose

T(°C) / relative rate / T(K) / 1/T(K) / ln rate / ln T(K)
1 / 0.434 / 274 / 0.003650 / -0.83 / 5.61
10 / 0.217 / 283 / 0.003534 / -1.53 / 5.65
20 / 1.00 / 293 / 0.003413 / 0.00 / 5.68
30 / 4.3 / 303 / 0.003300 / 1.46 / 5.71
40 / 14.6 / 313 / 0.003195 / 2.68 / 5.75

d) Spohr inversion of sucrose

T(°C) / rate / T(K) / 1/T(K) / ln rate / ln T(K)
25 / 9.7 / 298 / 0.003356 / 2.27 / 5.70
40 / 73.4 / 313 / 0.003195 / 4.30 / 5.75
45 / 139 / 318 / 0.003145 / 4.93 / 5.76
50 / 268 / 323 / 0.003096 / 5.59 / 5.78
55 / 491 / 328 / 0.003049 / 6.20 / 5.79

e) Hecht & Conrad ethoxide + methyl iodide

T(°C) / rate / T(K) / 1/T(K) / ln rate / ln T(K)
0 / 168 / 273 / 0.003663 / 5.12 / 5.61
6 / 354 / 279 / 0.003584 / 5.87 / 5.63
12 / 735 / 285 / 0.003509 / 6.60 / 5.65
18 / 1463 / 291 / 0.003436 / 7.29 / 5.67
24 / 3010 / 297 / 0.003367 / 8.01 / 5.69
30 / 6250 / 303 / 0.003300 / 8.74 / 5.71

f) van't Hoff chloroacetic acid decomposition

T(°C) / rate / T(K) / 1/T(K) / ln rate / ln T(K)
80 / 2.22 / 353 / 0.002833 / 0.80 / 5.87
90 / 6.03 / 363 / 0.002755 / 1.80 / 5.89
100 / 17.3 / 373 / 0.002681 / 2.85 / 5.92
110 / 43.6 / 383 / 0.002611 / 3.78 / 5.95
120 / 105 / 393 / 0.002545 / 4.65 / 5.97
130 / 237 / 403 / 0.002481 / 5.47 / 6.00

g) van't Hoff sodium chloroacetate + NaOH

T(°C) / rate / T(K) / 1/T(K) / ln rate / ln T(K)
70 / 8.22 / 343 / 0.002915 / 2.11 / 5.84
80 / 19.8 / 353 / 0.002833 / 2.99 / 5.87
90 / 49.9 / 363 / 0.002755 / 3.91 / 5.89
100 / 128 / 373 / 0.002681 / 4.85 / 5.92
110 / 305 / 383 / 0.002611 / 5.72 / 5.95
120 / 857 / 393 / 0.002545 / 6.75 / 5.97
130 / 2170 / 403 / 0.002481 / 7.68 / 6.00

h) van't Hoff dibromosuccinic acid + H2O

T(°C) / rate / T(K) / 1/T(K) / ln rate / ln T(K)
15 / 4.2 / 288 / 0.003472 / 1.44 / 5.66
40 / 37.5 / 313 / 0.003195 / 3.62 / 5.75
50 / 108 / 323 / 0.003096 / 4.68 / 5.78
60 / 284 / 333 / 0.003003 / 5.65 / 5.81
70 / 734 / 343 / 0.002915 / 6.60 / 5.84
80 / 2000 / 353 / 0.002833 / 7.60 / 5.87
89.4 / 4540 / 362.4 / 0.002759 / 8.42 / 5.89
101 / 13800 / 374 / 0.002674 / 9.53 / 5.92

1) Taking the natural log of the Arrhenius equation,
rate = Ae–E/RT ,
gives ln rate = ln A – (E/R)(1/T) .
Plotting ln rate on the y-axis vs. 1/T on the x-axis should result in a straight line with slope m equal to –E/R; so the activation energy is
E = –mR ,
where R = 8.3145 J mol–1 K–1 = 8.3145x10–3 kJ mol–1 K–1 .
The Arrhenius plot is satisfactorily linear in all cases except the Urech data. (See exercise 2 for more on that set.) Plots and activation energies follow.

E = 64.2 kJ mol–1

E = 46.6 kJ mol–1 .

E = 72.6 kJ mol–1 .
It is easy to see that the five points here do not define a very straight line.

E = 106 kJ mol–1 .

E = 82.5 kJ mol–1 .

E = 111 kJ mol–1 .

E = 107 kJ mol–1 .

E = 85.1 kJ mol–1 .

2a) In the plot of uncorrected Urech data above, four points appear to line up rather straight, but the lowest-temperature point does not line up with them. The fact that all points but one line up is enough to suspect that one point. (Note: four points are enough to be suggestive, at least. That is, if one point falls off a line defined by nine other points, then one has very good reason to suspect the point; one point falling off a line defined by four others is suspicious, but not definitely erroneous.) Changing the rate of that one point from 0.434 to 0.0434 puts that point on the line defined by the other points. That the point is put on line by correcting a putative mistake that is easy to make (as opposed to some arbitrary or contrived change) strengthens our suspicions that the point was erroneous.

b) E = 106 kJ mol–1 .
The fact that the corrected activation energy agrees with the value obtained from Spohr’s data for the same reaction is an additional piece of evidence that the point in question was erroneous.

c) For the reasons mentioned already in parts (a) and (b), the change is justified. This differs markedly from simply making the questionable point fit. The change was made on the basis of a plausible hypothesis (namely that a decimal place was dropped) that might or might not have improved the fit. The hypothesis was tested and would have been rejected if the point had not lined up. I.e., we made a plausible hypothesis that happened to fix the point; we did not simply determine the value that the point had to have to fall on the line.

3) A power law temperature dependence would look like
rate = BTc,
where B and c are constants. Taking the natural log of this relationship yields
ln rate = ln B + c ln T ,
so a plot of ln rate vs. ln T would be a straight line whose slope is c, the exponent in the power law. Over the relatively small temperature ranges employed in these data sets, a power law fits just as well as the Arrhenius equation, as the following plots illustrate.

A simple exponential temperature dependence would look like
rate = Ce–DT ,
where C and D are constants. Taking the natural log of this relationship yields
ln rate = ln C – DT ,
so a plot of ln rate vs. T itself would be a straight line whose slope is D, the constant in the exponent. Over the relatively small temperature ranges employed in these data sets, this simple exponential fits just as well as the Arrhenius equation.

As Keith Laidler pointed out in an article on the origin of the Arrhenius equation, T, 1/T, and ln T are linear functions of each other over relatively small temperature ranges (as can be seen from power series expansions), and all of these temperature ranges are fairly small compared to the absolute temperatures themselves. So a quantity (ln rate, in this case) that can be fit to a linear function of one can be fit to a linear function of any of them. The eventual emergence of the Arrhenius equation can be attributed to its theoretical fruitfulness rather than an empirically objective superiority.[1]

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[1]Keith J. Laidler, “The Development of the Arrhenius Equation,” J. Chem. Educ. 61, 494-8 (1984).