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Propagation of Uncertainty and Titrations

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  • Learning Objectives

    Following this activity, students should be able to:

    • Determine an unknown quantity using titration data.
    • Propagate uncertainty for common mathematical operations including:
      • Addition/subtractions
      • Multiplication/division
    • Perform the appropriate propagation of uncertainty in the context of volumetric chemical analysis.
    • Use error analysis to determine the least accurate part of an analytical procedure.

    In the lab…

    You will become familiar with 50-mL Class A volumetric burets and how to use them to perform chemical analyses. The tolerance on the burets you will use is 0.05 mL, meaning that when you record a reading from the buret, the volume you record (to two decimal places!) may be “off” by 0.05 mL above or below the true value. What is the relative error associated with each volume as measured using a buret?

    Initial volume

    Finial reading

    Total volume transferred

    Relative uncertainty

    0.04 mL

    7.43 mL



    1.27 mL

    15.93 mL



    0.00 mL

    38.29 mL



    0.00 mL

    50.00 mL

    83.54 mL *


    0.03 mL

    33.57 mL

    *Note: because this volume is greater than the maximum capacity of the buret, you will have to refill it. Discuss with your group a strategy for applying the equations for the propagation of uncertainty, then use the overall uncertainty to find the relative uncertainty.

    Good Titrations

    The basic set-up of a titration usually includes the following:

    • A buret containing a titrant of a known concentration
    • A flask containing an analyte of unknown concentration or quantity

    For any titration the titrant and analyte must react both quickly and to completion. This allows us to relate the quantity of one unknown reactant to that which is known using stoichiometry. The chemical reaction that shows the molar relationship between the titratrant and analyte is the analytical reaction. If the titrant can’t be accurately measured itself, the solution is standardized – that is, its exact concentration is determined (usually also through a titration!) through the reaction with something that can be accurately weighed on an analytical balance.

    The equivalence point of a titration is the volume one would expect to theoretically add so that the moles of titrant added is stoichiometrically equivalent to the moles of analyte according the to analytical reaction. In reality, we observe the end point in the lab by some sudden change indicating the end of the chemical reaction – usally either a color change or a sudden change in a reading by some instrument or meter (like a pH electrode).  You can think of the relationship between the end point and equivalence point as being analogous to the relationship between actual yield and theoretical yield, respectively. A well-executed titration should yield an end point that is in close agreement with the equivalence point.

    Consider if two acid solutions of unknown concentrations, one containing hydrochloric and one containing sulfuric acid, were both titrated with a standardized solution of NaOH. Complete and balance the neutralization reactions below.

    \[\ce{HCl\:\:\:\:  +  \:\:\:\:NaOH \:\:\:\: → }\nonumber\]

    \[\ce{H2SO4\:\:\:\:  +  \:\:\:\:NaOH \:\:\:\: →}\nonumber\]

    The standardized NaOH had a concentration of 0.1456 M and the table below summarizes the titration data for the two acid solutions.


    Initial acid solution volume


    Concentration (M)


    100.0 mL

    42.36 mL



    75.00 mL

    19.42 mL


    Complete the last column of the table by determining the concentration of each acid.







    Make a simple flow chart laying out how to perform the calculation that could be applied to either titration.






    Some students mistakenly use M1V1 = M2V2 for titrations. Although it is not correct to do so, it sometimes results in the correct answer. When will it work and when will it not? Would it have worked for either titration above?





    Now we will examine how uncertainty is used in the context of chemical analysis using the standardization of NaOH with a reagent called KHP (potassium hydrogen phthalate).

    Before we get started, it’s worth asking ourselves: why this is necessary in the first place?! Why can’t we just weigh out pellets of NaOH? The pictures below show, from left to right, how NaOH changes over time when exposed to the air.


    What is occurring to the NaOH? How would this affect your ability to accurately prepare the solution?




    If you did not notice how NaOH behaved in air and you went on to use this NaOH solution in your acid analysis, would your final result for the acid concentration be too high or too low?






    Would you describe this as random or systematic error? Explain your reasoning among your groupmates and come to an agreement. Describe how the other type of error might occur in this analysis.



    Use the rules for the propagation of uncertainty for the standardization of the NaOH with KHP, which can be represented as:

    \[\ce{NaOH  +  KHP  →  KNaP  +  H2O}\mathit{\tag{balanced}}\]

    Mass KHP

    1.0613(±0.0001) g

    Molar mass of KHP

    204.22 g/mol

    Water for KHP solution

    100 mL

    Initial buret reading

    0.000(±0.05) mL

    End point

    35.89(±0.05) mL


    Find the concentration of NaOH based on the results of the titration above, including the uncertainty.








    Note that the volume of water used to dissolve the KHP and the molar mass of KHP do not have uncertainty included. Why not?







    Based on your calculation, which measured value in the lab introduces the most error in the overall final result?





    Repeating an analysis is a common way to help ensure accuracy. Two additional trials yielded the results in the table below.

    Assuming everything was done correctly, what explains the variation between trials?


    [NaOH] (M)

    Trial 2

    0.1451 (±0.0003) M

    Trial 3

    0.1453 (±0.0004) M








    When performing analyses in the lab, you will routinely complete multiple trials, but you usually will not need to consider the uncertainty arising from every single measurement. Why do you think this is acceptable?





    Titrations: Best Practices

    • 50-mL Class A burets should be read to two decimal places (e.g., 26.38 mL).
    • It is best to avoid refilling a buret during the course of a titration.
    • You should always know the balanced analytical reaction for a titration.
    • Make sure all titrant reaches the sample solution – it is okay to rinse the inner walls of the receiving flask with water.
    • Performing a “pilot” titration is a good way to get a ballpark value for your endpoint.

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