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Steven D. Gammon University of Idaho Moscow, ID 83844
Sensitivity to Experimental Parameters via Spreadsheets B. R. Sundheim Department of Chemistry, New York University, New York, NY 10003 In the treatment of experimental data it is important to examine the ways in which various experimental errors affect the calculated results. When there is a simple, clearcut functional relation between measured and calculated quantities, the theory of propagation of errors as described, e.g., by Sime (1) can be used. If the relationship is not simple and/or the student is not familiar with calculus it is still possible to deal with this problem by use of a spreadsheet. The method consists simply of setting up all the relevant calculations in a spreadsheet using formulas that refer back to the experimental quantities. Then adding or subtracting
estimated experimental errors in turn to the entries for observed quantities immediately yields the corresponding changes in the calculated quantities. This procedure can be helpful in assessing the relative importance of various errors and thus identifying where efforts at improvement may be best expended. As an example we describe the calculation associated with a calorimeter experiment used in our physical chemistry laboratory course. The experiment entails comparison of heat changes associated with carrying out a chemical change by two different paths. In the first, calcium oxide is dissolved in water and then reacted with a sulfuric acid solution. In the second, sulfuric acid solution is diluted appropriately and then reacted directly with calcium oxide. The experimental data consist of thermistor calibrations, time/temperature curves, and weights, volumes, and concentrations of stock solutions. The final calculation is of the observed difference in the heats corresponding to the two paths; that is, a test of Hess’ law that the first law of thermodynamics assures that the enthalpy changes are the same for different but equivalent paths. A representative spreadsheet for this experiment is shown in Figure 1. When some of the relevant experimental results are varied as indicated, the corresponding changes in the calculated heats and net differences are shown in Table 1. One unexpected conclusion is that the effects of errors in the water equivalent of the calorimeter are relatively unimportant, whereas temperature changes occasioned by reactions are quite important. It can also be seen that the results are unlikely to be improved by changing to a more sensitive balance, but that a volumetric pipet might well be substituted for a graduated cylinder in measuring the acid. Literature Cited 1. Sime, R. J. Physical Chemistry, Methods, Techniques, and Experiments; Saunders, Philadelphia, 1990; Chapter 7.
Table 1. Sensitivities to Some Experimental Variablesa
328
Change (cal)
Where
How Much
wt CaO
IA
+10 mg
1231
{39
wt CaO
II B
+10 mg
1387
+117
T2
II B
0.1°
818
{452
T1
II B
0.1°
1722
{452
vol acid
IB
+0.1 mL
900
{270
vol acid
II A
+0.1 mL
1359
+189
T2
W. EQ.
+.01°
1236
+66
conc acid
IB
+ 0.01 M
1223
+53
a Changes
Figure 1. Spreadsheet for Hess’ law experiment. The relevant formulas are indicated by text adjacent to the numerical values.
Result (cal)
What
produced in the result (bottom line in Fig. 1) calculated in the spreadsheet (Fig. I) by the indicated modifications to some of the experimental variables. Temperature changes associated with the other steps lead to similar results. “What” refers to the experimental parameter or variable; “Where”, to its location in the worksheet; “How Much”, to the amount by which the variable was modified.
Journal of Chemical Education • Vol. 74 No. 3 March 1997
