Luis Avila avila@chem.columbia.eduIsabelle Vu Trieu ilv2@columbia.edu

Intensive General Chemistry

Uncertainty Analysis

Wet Techniques

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

good precision but poor accuracy

poor precision & poor accuracy

poor precision but good 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 PrecisionStandard Deviation

standard deviationsample mean

total number of measurements

the ith value

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 quantitiesabsolute error = sum of the absolute errors in the exp quantities

Result obtained by multiplying or dividing exp quantitiesrelative error = sum of the relative errors in the exp quantities

Absolute error = Relative error x Measurement

4.00 ± 0.02 cm

2.00

± 0

.02

cm Perimeter? 12.00 ± 0.08 cm

Area? 8.00 ± (1% + 0.5%) cm2

8.0 ± 0.1 cm2

Propagation of Errors

Result obtained by multiplying or dividing an exp qty by a constantAbsolute error = same constant x absolute error in the exp quantity

Logarithmic expressionRelative error = 0.434 x relative error in the exp quantity

AverageAbsolute error = greatest absolute error in exp quantities being averaged

Only absolute errors can be used for final results

Wet Techniques

• Experiments:– 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.