Uncertainty & % Error.

Maximum experimental error and percentage error.

When determining the maximum experimental error and the percentage error for a titration using a grade A burette and a 25.00 cm³ pipette, one should consider the tolerance associated with each piece of equipment. The tolerance values for grade A equipment are often found on the equipment or in the manufacturer’s documentation. However, for the sake of this discussion, let’s use commonly accepted tolerance values:

1. Maximum Error for Grade A Equipment:

Burette: Typically, for a $50.00 cm 3$ grade A burette, the tolerance is ± 0.05 cm³.
25.00 cm³ Bulb Pipette: For a $25.00 cm 3$ grade A pipette, the tolerance is often ± 0.03 cm³.

2. Determining Maximum Experimental Error:

The maximum error in a titration is the sum of the errors associated with the burette and the pipette.

Burette error: Since you take two readings (initial and final) on a burette, the error is doubled. Therefore, the error associated with the burette is $2 \times 0.05 cm^{3} = 0.10 cm 3$ .
Pipette error: ± 0.03 cm³.

Thus, the total maximum experimental error for the titration is:
$0.10 cm 3 (b u re tt e) + 0.03 cm 3 (p i p e tt e) = 0.13 cm 3$

3. Percentage Error:

The percentage error is calculated based on the actual volume of NaOH used in the titration to reach the endpoint, which we take it as 23.75 cm³ in our example.

Percentage error is given by:

Percentage error= (Maximum experimental error x 100)/Actual Volume Used
= (0.13 x 100)/23.75 = 0.547 %.

Thus, the percentage error based on a titration of 23.75 cm³ is approximately 0.547%.

This percentage error takes into account the uncertainties associated with the measurement equipment used. A smaller percentage error indicates a high precision in the experimental procedure. However, remember that this doesn’t necessarily equate to accuracy, as systematic errors or mistakes in the procedure can still affect the results.

Above left, is a 10 cm³ graduated pipette.
Its diameter is larger than that found in the stem of a bulb pipette.
The maximum error for a graduated pipette is greater as a result.
For the calibrated pipette you can see the maximum error is ± 0.1 cm³.
In the case of a grade A 25 cm³ bulb pipette it is only ± 0.03 cm³.
The grade A burette above shows that the smallest divisions are 0.1 ml (cm³) and the maximum error is ± 0.05 cm³.

Systematic errors.

In the context of experimental procedures and measurements, errors can be broadly categorized into two types: random errors and systematic errors.

Systematic Errors: Systematic errors are consistent, repeatable errors associated with faulty equipment or a flawed experimental design. They cause measurements to be consistently biased in one direction, either too high or too low. They are not related to the precision of the measurement but to its accuracy. Key points about systematic errors include:

Cause: Systematic errors can be caused by various factors such as calibration errors in the equipment, environmental conditions not accounted for, or procedural errors in the experiment.
Consistency: These errors are consistent in magnitude and/or direction. This means that if you repeat the experiment under the same conditions, the error will remain.
Correction: Systematic errors can often be corrected when their source is identified. For example, if a piece of equipment is known to measure consistently too high by a certain amount, results can be adjusted by this amount to account for the error.
Examples:
- Using a burette that always delivers slightly more liquid than it should because of a manufacturing defect.
- An electronic balance that has not been zeroed properly and always reads 0.02g too high.
- A thermometer that hasn’t been calibrated correctly and always reads 2°C above the actual temperature.

Random errors.

Contrast with Random Errors: While systematic errors cause results to consistently deviate in one direction, random errors cause results to scatter in an unpredictable manner. Random errors are inherent in all measurements and are caused by unpredictable fluctuations in readings of a measurement apparatus or in the experimenter’s interpretation of the instrumental reading. These errors can often be reduced by increasing the number of observations and averaging the results.

In the Discussion Context: In the earlier discussion on percentage error, the focus was primarily on precision (how repeatable the results are), which is often affected by random errors. Systematic errors, on the other hand, relate to how close the experimental value is to the true or accepted value. When I mentioned systematic errors, I was highlighting that even if an experiment is precise (low random errors), it might still be inaccurate if there are uncorrected systematic errors. For example, if the burette always delivers slightly more solution than it should, every titration will use more titrant than necessary, leading to consistently biased results.

QUESTIONS.

A Class B 250 cm³ volumetric flask has a maximum
error of ±0.2 cm³
Calculate the percentage error when it is used.
A 25 cm ³ pipette had a maximum error of ±0.06 cm ³
Calculate the percentage error when it is used to measure 25 cm ³
A top-pan balance reading to two-decimal places had a maximum error of ±0.005 g. For a mass measurement of 2.66 g, what would be the maximum percentage error?
The smaller the titre, the larger the % error in using the burette. How can the titration be changed to reduce this percentage error?
A student carried out an experiment to determine the concentration of ethanoic acid in a solution of vinegar.

• A 25 cm ³ measuring cylinder was used to measure out 25.0 cm³
of the vinegar solution.
• The solution was then transferred to a 250 cm³ volumetric flask
and made up to the mark with distilled water.
• A pipette was used to transfer 25.0 cm³ portions of the acidic
solution to a conical flask.
• The solution was then titrated with 0.100 moldm ^–3 sodium
hydroxide solution, concentration, using phenolphthalein.

The average titre was 23.5 cm³
a) Suggest, with reasons, how the student’s method of preparing
the diluted solution could be improved.
b) The maximum total errors for the measuring cylinder, pipette and the burette in the titration are:
measuring cylinder ± 0.50 cm³; pipette ± 0.10 cm³; burette ± 0.05 cm³

NOTE: The burette requires 2 readings to record a titre, but the pipette is read just once (when full).

Estimate the combined maximum percentage error in using both of these pieces of apparatus. Give your answer to 2 decimal places.

ANSWERS.

0.2 ÷ 250 x 100 = 0.08 %.
0.06 ÷ 25 x 100 = 0.24 %.
0.005 ÷ 2.66 x 100 = 0.19 %
Dilute the titrant. For example, make 10.0 cm ³ of the titrant up to 100 cm ³ using a volumetric flask. This will multiply the titre by 10 and reduce the % error ten-fold.
(a) Use a 25 cm³ pipette instead of a measuring cyclinder
as its maximum error is much less ( ± 0.10 cm ³ vs with ± 0.50 cm³)
(b) Measuring Cylinder = 0.50 ÷ 25 x 100 = 2.0 %.
Pipette = 0.10 ÷ 25 x 100 = 0.40 %.
Burette = 2(0.05) ÷ 23.5 x 100 = 0.43 %.
Total 5 error = 2.0 + 0.40 + 0.43 = 2.83 %