Propagation of Error

Now that you have found the experimental value for the strain in cyclopropane, and predicted it based on a semiempirical calculation, you need to assign your experimental data confidence limits to describe the quality of your experimental finding.  Since there are only two trials for each ester, you will not be able to provide a meaningful analysis based on a standard deviation.

However, your random error should be primarily a function of the measurements you took (mass, temperature) and errors in the constants you might have used in calculations.  We can therefore see how these propagate into the final result.  A full treatment is given on pp. 56-61 of Shoemaker, Garland and Nibler's "Experiments in Physical Chemistry."

In general, there are two slightly different formulas for propagating errors, each based on the form of the equation used for computing a new property.

If terms are added or subtracted:

F = a + b - c + ...
D2(F) = D2a + D2b + D2c + ...

F is the function; a, b, c, etc. are the measured components; D2(F) is the square of the (absolute) uncertainty in F, and D2a, etc. are the squares of the uncertainties in any measure.  Note that whether a quantity is added or subtracted, the error is always additive.

Alternatively, if we have a multiplicative function (including division):

F = ab/c
D2(F)  = D2a + D2b + D2c
   F2         a2       b2     c2

Here we are dealing with relative error; to get the absolute error value (like the additive function above), multiply the right side by F after taking the square root.  Again, whether an element is multiplied or divided, its treatment in the error function is the same.

If a computed value is a complex function containing both additive and multiplicative elements, calculate the error in the multiplicative elements first, then apply the additive methodology.

Uncertainties in your measurements should be:

Mass: 0.0003 g (if you used the analytical balance)
Temperature: 0.001C

Assume an uncertainty in any constant of 1 unit in the last significant figure.  For example, the uncertainty in a molecular weight reported to the hundredth of a mass unit is 0.01 g/mol.



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Last updated:  8/5/97
Comments to K. Gable