Intensive General Chemistry
Uncertainty Analysis
Wet Techniques
Luis Avila
Isabelle Vu Trieu
avila@chem.columbia.edu
ilv2@columbia.edu
Introduction
• Measurement and Uncertainty
• Qualitative Analysis
• What is in the unknown?
• Quantitative Analysis
• How much of it is in the unknown?
Uncertainty in Measurement
• Measurements always involve a comparison
• The comparison always involve some uncertainty
Length of the beetle’s body?
-between 0 and 2 in
-between 1 and 2 in
-between 1.5 and 1.6 in
-between 1.54 and 1.56 in
-between 1.543 and 1.548 in
Convention:
Read 1/10 of the distance between the smallest scale division
Significant Figures
• Definition:
– all digits up to and including the first uncertain digit
• Ex: Beetle’s length is
1.55 in (3 sig fig)
4.0 cm (2 sig fig)
0.04 m (1 sig fig)
– The more significant figures, the more reproducible
a measurement is (ex: ∏)
– Counts and defines numbers are exact - They have
no uncertain digits!
Counting significant figures in a
series of measurements
• Compute the average
• Identify the first uncertain digit
• Round the average so that the last digit is the first
uncertain digit
Ex: Beetle’s length
» Measurement 1: 3.98 cm
» Measurement 2: 4.01 cm
» Measurement 3: 4.00 cm
AVERAGE = 4.00 cm or 4.00 x 10-2 m
Precision of Calculated Results
• calculated results are never more reliable than the
measurements they are built from
• multistep calculation: never round intermediate results!
• Sums and differences: round result to the same number of
fraction digits as the poorest measurement
Ex: 4.01+ 22.2222 = 26.23
• Products and quotients: round result to the same number of
significant figures as the poorest measurement
Ex: 4.01 x 22.2222 = 89.1
Precision versus Accuracy
good precision &
good accuracy
poor precision but
good accuracy
good precision but
poor accuracy
poor precision &
poor accuracy
Precision versus Accuracy
Random Errors versus Systematic Errors
• Precision
– Reproducibility
– Check by repeating
measurements
– Poor precision results from
poor technique
• Random Errors
– Random sign
– Varying magnitude
• Accuracy
– Correctness
– Check by using a different
method
– Poor accuracy results from
procedurial or equipment defects
• Systematic Errors
– Reproducible sign
– Reproducible magnitude
Estimating Precision
Standard Deviation
the ith value
sample mean
standard deviation
total number of measurements
Expressing Experimental Error
• Absolute error
= Magnitude of the random error
Ex: Beetle’s length = 4.00 ± 0.02 cm
• Relative error
= Ratio of the absolute error to the measurement
Ex: 0.02/4.00 = .005 = 5%
Beetle’s length = 4.00 ± 5% cm
All your final experimental results must be
reported with absolute error.
Propagation of Errors
Result obtained by adding or subtracting experimental quantities
absolute error = sum of the absolute errors in the exp quantities
Result obtained by multiplying or dividing exp quantities
relative error = sum of the relative errors in the exp quantities
Absolute error = Relative error x Measurement
4.00 ± 0.02 cm
Perimeter? 12.00 ± 0.08 cm
Area?
cm2
8.00 ± (1% + 0.5%)
8.0 ± 0.1 cm2
Propagation of Errors
Result obtained by multiplying or dividing an exp qty by a constant
Absolute error = same constant x absolute error in the exp quantity
Logarithmic expression
Relative error = 0.434 x relative error in the exp quantity
Average
Absolute error = greatest absolute error in exp quantities being averaged
Only absolute errors can be used for final results
Wet Techniques
• Experiments:
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Calibrating Glassware
Preparation of standards
Titration
Qualitative Analysis for Cations
• Collaborative/Cooperative work necessary!
Calibrating Glassware
• Volumetric glassware:
– “to contain” (TC)
– “to deliver” (TD)
• Objectives:
– Estimate precision of volumetric glassware
– Compare with manufacturer’s uncertainty
Gravimetric Calibration
• Determine:
– Mass of water in the Measured volume
– Temperature of water
• Calculate:
– Volume of water (using the density of water)
• Compare:
– Calculated and Measured Vwater.
Qualitative Analysis for Cations
• Objectives
– Design a Cation Analysis Scheme
– Identify and Separate Cations in a mixture
Cation Analysis Scheme
Volumetry
• Objectives
– Prepare solutions of known concentration from primary
and non-primary standards
– Perform titrations
Pre-lab questions
• E2 - 6: Retrieve the MSDS of KHP and
NaOH. Calculate the mass of NaOH and
KHP needed in order to prepare the
solutions.
• E2 - 12: Sketch an alternative analysis
scheme starting with precipitation with
NaOH instead of HCl.