Experimental Errors (part 2)

The previous post (Experimental Errors, https://physbang.com/2023/11/12/experimental-errors/) ended with an invitation to “consider whether there are any other, previously unidentified, sources of uncertainty” in the method used to derive the specific heat capacity of an iron block. As it happens, there are (at least) three possibilities that weren’t covered in the original article.

Firstly, there is the fact that a 250 ml measuring cylinder was used to determine the water volume, which was 660 ml. This clearly cannot be done in a single stage and a minimum of three separate volumes are required to achieve this total. That means the total error in the volume measurement is not simply +/- 5 ml, as was stated in the original article, but rather three errors of +/- 5 ml in a volume that was obtained by summing the individual measures. This gives a total error that is root-three times the single-measurement error, which equates to +/- 8.7 ml.

Alternatively, the total volume could have been obtained in a single step using a 1000 ml measuring cylinder but if this had a scale marked in 20 ml intervals, giving an error of +/- 10 ml, it would have introduced greater uncertainty than was obtained by using the 250 ml measuring cylinder three times!

Of course, there are both 250 ml and 1000 ml measuring cylinders that have smaller gradations than those listed above but it is interesting to note that the finest common scales are 5 ml and 10 ml respectively, which still makes repeated use of the smaller measuring cylinder a better option. Add in the availability of 250 ml measuring cylinders with 2 ml divisions and it becomes clear that multiple uses of higher-resolution apparatus might well be a better choice than measuring a large volume in a single step.

Secondly, there is the magnetic stirrer in the left-hand beaker containing the room-temperature water. Even though the stirrer was enclosed within a rubber shell, which should offer a good level of insulation, it is nevertheless the case that the iron block would transfer some of its thermal energy to the stirrer but this was not included in the calculation. This omission means some of the energy was unaccounted and the calculated specific heat capacity of the iron block would therefore have been smaller than the true value.

As it happens, this particular problem occurred to me only when the iron block was lowered into the water – and the magnetic stirrer attached itself to the base of the iron! The block was therefore lifted and the stirrer was removed. This would have resulted in a slight loss of water (probably no more than 1 ml) but it also introduced a delay before the iron block was fully immersed in the left-hand beaker.

Thirdly, although there is little doubt that the iron block was heated to the boiling point of the water used (which may not have been exactly 100 ⁰C depending on impurities and atmospheric pressure) it is possible that the block might have cooled slightly while being transferred from the hot-water beaker to the room-temperature beaker. This possibility may well have been compounded by the additional delay caused by removing the stirrer. It is therefore interesting to consider the extent of cooling that would have been necessary to account for the discrepancy between the measured heat capacity of the iron block (398 J kg^-1 ⁰C^-1) and the reference value (449 J kg^-1 ⁰C^-1).

Using the equation given in the original article and rearranging to solve for the “hot temperature”, HT, of the iron block as it enters the room-temperature water gives;

Inserting the known values, as listed in the original article, results in a value for HT of 96.6 ⁰C. This figure may well be very surprising as you might have expected a much lower temperature to account for an (approximately) 10% discrepancy between the originally-calculated specific heat capacity and the reference value. But if you plug the new value for HT into the original equation in place of 100 ⁰C, you will indeed find that the calculated heat capacity now matches the reference value.

Is it reasonable to suggest that the iron block could have lost about 4 ⁰C while being transferred between the beakers? This certainly sounds believable to me but I would love to have access to a real-time thermometric camera, together with a thermocouple inserted into the middle of the iron block, to record the temperatures throughout the brief transfer process so that this explanation can be confirmed (or refuted). And, of course, it would be useful to repeat the entire experiment without using a magnetic stirrer!