Error analysis exercise solution

We'll first take a look at any measurement errors and put some percentages on them.

The copper(II) sulfate solution was measured out in a 25 cm³ measuring cylinder. This has an error of ± 0.3 cm³. The percentage error is 0.3/25 × 100 = 1.2%

The temperature change error is 8.1 ± 0.1 ºC. This gives a percentage error of 0.1/8.1 × 100 = 1.2%.

Did you calculate a percentage error for the mass of zinc? We don't need to as the zinc is an excess, and so the mass is not involved in the calculation.

We now need to make a list of any other errors which cannot easily be given a percentage error.

The concentration of the copper(II) sulfate may not be precise. Note that it was given as 0.2 M. this is quite common, but it is unlikely that it is an indication of the actual accuracy of the solution concentration. Technically, a 0.15 M solution could be classified as 0.2M (rounded to one significant figure) - this would be a 25% error. The solution concentration should really have been given to an appropriate number of significant figures which indicated the accuracy of the solution, such as 0.200M. If this had been done we could give a percentage error for this factor. A 0.200 M solution could have an error of ± 0.0005 M, giving a percentage error of 0.0005/0.2 × 100 = 0.25%. You may have to ask for more information if your concentration is given to an unlikely, small number of significant figures.

Heat loss is always a problem with exothermic calorimetry experiments. Heat may be lost to the surroundings (by evaporation, radiation and a little conduction) and into the cup, thermometer and zinc by conduction. We cannot put a figure on this. There is an improvement to the method that takes into account some of the heat loss - you can read about it here.

There is a small error in taking the specific heat capacity of the solution to be that of water and assuming that the solution had a density of 1 g cm^-3.

Some of my past students suggested that there is an error due to energy being provided to the solution through the stirring. Technically, this is correct, but it will be very small. Try raising the temperature of some water in a cup simply by stirring it - you won't have much success!

We have assumed that the reaction has gone to completion and all the copper(II) sulfate solution has reacted. This may not be the case.

If we try to identify the largest error it is clear it is either one of the 1.2% errors we calculated or one of the errors which we did not assign a percentage to. It is useful to see how far away from the accepted answer we are. Our answer of - 169 kJ mol^-1 differs from the accepted value of -218.7 kJ mol^-1 by 49.7 kJ mol^-1. This is a percentage of 40.7/218.7 × 100 = 22.7%. Don't confuse this with the error. This is just how far out our one result is. The actual error could be larger than this, but it can't be less. Bear in mind that an error of ± 40% includes 0. In other words, a single result could, by chance, be spot on. Adding up our percentage errors gives a value of 2.7%. This means that the other errors must be at least 20%. This means that the largest error is one of these other errors. At this point any reasonable suggestion as to which is the largest error should gain credit.

It is likely that in this case the largest error is either the failure of the reaction to go to completion or heat loss. These are systematic errors which would lead to our answer being smaller than the actual value, which agrees with the observation. There was also a clue in the description where it stated that the blue colour of the copper(II) sulfate faded. The zinc was in excess, so if the reaction went to completion then all the copper(II) sulfate should have been used up, so the colour should have disappeared completely. In this case the zinc solid will become coated with copper in the displacement reaction and this may limit further reaction with the solution. Perhaps a finer zinc dust would have given a better result.

Some displaced copper - but is it hiding any zinc underneath?

This is a very detailed analysis for an "A" Level answer, but it should give you some idea of the sort of things you should be thinking about when you have to consider errors.