Written by Chris; Chris is available for private tutoring.

In this blog we are going to look at how to identify errors and propagate uncertainties present in practical work. Not only is this understanding examinable but it is a vital part of writing a strong piece of coursework for internal assessment.

Uncertainty:

Uncertainty is inevitable in any data collection exercise that requires the use of measuring devices. In school experiments this refers to apparatus such as electronic balances, measuring cylinders, burettes, pipettes and thermometers to name just a few. The inherent uncertainty of experimental apparatus is a measure of the precision with which data can be recorded; it does not affect accuracy (see later).

Experimental uncertainty is often used interchangeably with the term random error. This term indicates that the error or uncertainty can occur in either direction from the measured quantity, i.e., the actual value may lie slightly below or above the recorded measurement. This variation is shown by using the ± symbol.

For digital apparatus, the uncertainty is often quoted as being equal to the smallest increment measurable; for example, a two decimal place balance would have an uncertainty of ± 0.01 g whereas a four decimal place balance would have an uncertainty of ± 0.0001 g. For analogue apparatus, the uncertainty is typically quoted as half the smallest increment measurable. So, for a 100 cm³ measuring cylinder with gradations for every 1 cm³, the uncertainty is given as ± 0.5 cm³.

Confusion tends to arise when half the smallest increment yields an uncertainty with two significant figures. Take a thermometer which has gradations every 0.5 °C; half this value produces an uncertainty of ± 0.25 °C. However, by their very nature, uncertainties are…uncertain, and so quoting one with more decimal places than the original measurement is problematic. For this reason, uncertainties are always quoted to one significant figure and so our thermometer would have an uncertainty of ± 0.3 °C.

Once a sensible uncertainty on a piece of apparatus has been established, the recorded measurements generated by it should be quoted to the same level of precision; you can think of this as ensuring the final significant figure of your recorded quantity is in the same ‘tens column’ are the uncertainty. So, if your uncertainty is ± 0.1 cm³ then a quantity of 21.12 cm³ becomes 21.1 ±0.1 cm³ and a quantity of 19 cm³ becomes 19.0 ± 0.1 cm³; this ensures all values quoted are in the 10^-1 column and are of equal precision.

Propagating uncertainties:

This is an area that students find a real challenge irrespective of background. The reason for propagating uncertainties is so that we can determine a sensible range within which our final calculated quantity might lie once all our raw data has been processed. The more processing experimental data undergoes, the more its quality is degraded such that we will be far more uncertain about the final value than any of the original data that was collected.

There are two basic guidelines to follow when propagating uncertainties:

1. When adding or subtracting quantities with the same unit, the uncertainty of the result is the sum of the original uncertainties.
2. When multiplying or dividing quantities with different units, each uncertainty must be converted to a percentage and then these percentages are summated.

Example 1:

In an experiment to record the gain in mass of some copper after it has been heated in air, the initial mass of the copper was measured and the final mass was also recorded.

Initial mass of copper  = 3.45 ± 0.01 g

Final mass of copper   = 3.78 ± 0.01 g

In each case, the raw data is quoted with a one significant figure uncertainty and the measured quantity is quoted to the same level of precision, i.e., two decimal places.

The mass gain is the difference between the two values and the new uncertainty is the sum of the two uncertainties.

Calculated mass gain  = 3.78 – 3.45 = 0.33 ± 0.02 g

Example 2:

In an effort to determine the enthalpy change of combustion of ethanol, the following data were recorded.

The volume of water can be converted to mass of water using an approximate density of water of 1 g cm^-3 and so the mass of water heated is 100 ± 1 g.

The equations required to determine the enthalpy change are:

1. Q = m × c × ΔT where Q = heat energy absorbed by the water in J

m = mass of water heated in g

c = specific heat capacity of water (4.18 J °C ^-1 g^-1)

ΔT = temperature change of water in °C

2. ΔH_c = (-Q) ÷ moles of ethanol burned

After some initial data processing the following data is obtained:

The uncertainty of each processed value is the sum of the individual uncertainties used to calculate it, i.e., 0.5 + 0.5 and 0.01 + 0.01. Also notice how the single decimal place present on the raw temperature readings disappears as the new uncertainty is now in the units column.

The moles of ethanol burned can be calculated as:

n_ethanol = m_ethanol ÷ M_r

= 1.36 ÷ 46.08 = 2.95 × 10^-2 mol

These new data can be used to calculate both Q and ΔH_c as follows:

Q = 100 × 4.18 × 36 = 15048 J

ΔH_c = -15048 ÷ (2.95 × 10^-2) = –509862 J mol^-1

In order to determine the uncertainty on this final value we will need to find the uncertainties on the values that we used to calculate it, i.e., the uncertainties on Q and moles of ethanol burned. As both these values were calculated using multiplication or division it is necessary to work with percentage uncertainties.

In the first step, you need to convert the absolute uncertainties (that is those in the same unit as the measured quantity) into percentages of the recorded values.

For 36 ± 1 °C we need to express 1 as a percentage of 36. Similarly, for mass of ethanol burned we need to express 0.02 as a percentage of 1.36. The same applies for the mass of water heated.

• % uncertainty for temperature change = (1 ÷ 36) × 100 = ± 2.8%
• % uncertainty for mass of water heated = (1 ÷ 100) × 100 = ± 1.0%
• % uncertainty for mass of ethanol burned = (0.02 ÷ 1.36) × 100 = ± 1.5%

Once we have these percentage we can find the percentage uncertainties for Q and moles of ethanol burned. The uncertainty on a value calculated by multiplication and/or division is the sum of the percentage uncertainties of the values used in the calculation. So, for Q, we sum the percentage uncertainties for m, c and ΔT. For the moles of ethanol burned we sum the percentage uncertainties for the mass of ethanol burned and the M_r of ethanol.

To simplify matters, we can assume the values for c and M_r have no uncertainty (though they would in reality). Thus the uncertainties can be calculated as:

• % uncertainty for Q = 2.8 + 1.0 = ± 3.8%
• % uncertainty for moles of ethanol burned = ± 1.5%

The uncertainty of ΔH_c is now the sum of these two values, i.e., 5.3%. We can now express our calculated value for ΔH_c­ as:

ΔH_c = -509862 ± 5.3%

However, this is not a useful or appropriate way to quote the final value. First, the percentage uncertainty must be converted back into an absolute uncertainty in the same unit as the quantity, in the case joules.

Absolute uncertainty on ΔH_c = (5.3 ÷ 100) × 509862 = ± 27023 J mol^-1

We must now remember that uncertainties are always quoted to one significant figure and so our uncertainty is rounded to ± 30000 J. Finally, the calculated quantity must be rounded to the same level of precision (in this case to the 10⁴ column). After this final processing we can quote our value for the enthalpy change of combustion of ethanol as:

ΔH_c = -510000 ± 30000 J mol^-1

or more conveniently as

ΔH_c = -510 ± 30 kJ mol^-1

We can now have confidence that our recorded data gives us a value that lies within the range -480 to -540 kJ mol^-1. However, uncertainties only take into account issues regarding the precision of the data collected not the accuracy of the method used to collect them.

A quick glance at a data booklet tells us that the enthalpy change of combustion of ethanol is around -1370 kJ mol^-1. We can calculate the percentage error on our value as follows:

Percentage error = (|Literature value – Experimental value| ÷ |literature value|) × 100

In our case this is: ((1370 – 510)) ÷ 1370 × 100 = 62.8%

This is a significant level of inaccuracy and tells us that our experimental method was not suitable for collecting data capable of producing a result close to the accepted value. Our apparatus was good enough to yield a relatively precise value (one with a low uncertainty) but it wasn’t used in a manner suitable for getting an accurate result.

Problems with a method that lead to a reduction in accuracy are termed systematic errors.

Systematic errors:

We have already seen that imprecision or uncertainty is inherent in all experiments that require the recording of measurements. Precision can be improved by using apparatus with less ‘in-built’ uncertainty. Whether you are accurate or not is typically down to the quality of the method employed; a poor working procedure cannot be overcome simply by using advanced and expensive equipment.

At school level, one of the best ways to identify sources of systematic error is to make detailed observations about every process carried out. That does not simply mean that part of the method where the dependent variable is measured but all the preparatory stages as well. A lack of rigour or poor design in any part of a procedure can reduce the overall accuracy of the experiment.

In the example above, the value for the enthalpy change of combustion of ethanol was only about one-third of that expected. This indicates severe limitations with the method used and suggests it was not fit-for-purpose. As scientists, we need to try and make sensible suggestions as to what the sources of error might have been in order to make necessary improvements to the method before performing it again.

Typical observation for this experiment might be:

• Orange flame
• Flame was very small
• Flame flickered
• Soot deposited on base of calorimeter
• Calorimeter got hot
• Air around experiment got hot
• Water vapour rose from top of calorimeter
• Lid left of spirit burner for a few minutes before and after flame was lit/extinguished

Our task now, is to determine is any or all of these might have an impact on the data recorded and in what way (magnitude and direction) that impact might influence our final value for the enthalpy change.

One essential thing to do is ensure that our errors and the discussion of the errors fits with the observed result. In this case we got a value far less exothermic than expected and so any errors that run counter to this, e.g., volume of water being reduced must be of little to no significance. Judging the significance of the main errors isn’t straightforward but some educated guesswork will normally get you to the correct identification of the major culprits. In this case, heat loss to the surroundings has the largest impact on the data collected.

The systematic errors identified will vary from experiment to experiment but almost always come from making good, detailed observations. Also, observations should not be generalised or grouped. For example, if you were burning different alcohols to compare enthalpies of combustion then observations should be made for each alcohol being burned individually. Some errors may have only occurred for some of the experiments; perhaps the lid was only left off for two of the five being tested, perhaps water vapour was observed in only the propan-1-ol experiment, perhaps methanol burned with a blue flame. Being this specific allows you to explain why some data points might not fit a trendline. We couldn’t use incomplete combustion as a catch-all error as there was no evidence for this error in our methanol experiment but we could explain that the enthalpy of combustion of propan-1-ol was less exothermic than expected due, in part, to evaporation of water.

Other common errors in prescribed practicals include:

Incomplete transfer of liquids from beakers and measuring cylinders

Use of non-standardised solutions, i.e., the concentration isn’t what it says on the bottle

Inconsistent stirring

Determining colour changes by eye

Loss of gas, e.g., not replacing rubber stopper quickly

Air bubble in measuring cylinder when collecting gas over water

Reaction is exothermic and speeds up reactions (introduces new variable)

Parallax errors, i.e., not having burettes, etc upright or looking at gradations at an angle

Impurities in chemicals used (check detail on the bottle)

When discussing systematic (and random) errors, an attempt should be made to provide an improvement for each error that is justified scientifically and would, ideally, make a significant improvement.

Summary:

After reading this blog you should be able to:

• Describe the difference between an uncertainty (random errors) and a systematic error
• Appreciate that uncertainty is inevitable and caused by inherent imprecision in measuring devices
• Describe the difference between precision and accuracy
• Compound uncertainties in the same units
• Propagate uncertainties in different units
• Determine the range of values your calculated quantity lies within
• Appreciate the importance of keeping detailed observations
• Understand the link between observations and systematic errors
• Relate the magnitude and direction of systematic errors to the data collected
• Understand the importance of suggesting improvements to the method

Be sure to give us a shout if you have any questions!