Topic 11: Measurement and data collectionSECTION 11.1 – UNCERTAINTIES AND ERRORS IN MEASUREMENT AND RESULTS

Qualitative vs. Quantitative Qualitative data: all non-numerical information obtained from observations not from measurement

Quantitative data: all data obtained from measurements◦ Always associated with random errors/uncertainties determined by the

apparatus and by human limitations, such as reaction times.

Uncertainty in measurement Data can be divided into two groups: exact numbers and inexact numbers.

Any experiment involving measurements will always have some uncertainty associated with the measured data.

Precision and accuracy Precision – the closeness of agreement between independent test results. The smaller the random part of the experimental errors which affect the results, the more precise the procedure.

Accuracy – the closeness of the agreement between the result of a measurement and a true value of the measurand (the particular quantity to be measured).

measured value

Significant Figures (SF)The number of digits that reflect the precision of a given measurement.

◦ The greater the number of SFs, the greater the certainty of the measured value

Scientific notation helps to determine the number of SFs.◦ Ex: 6.02 x 1023 atoms

Rules for counting SFsWhat is counted:

1. Every non-zero digit.

2. Every zero between two non-zero digits.

3. Every zero after a decimal point that follows a non-zero digit (even if the non-zero digit is before the decimal point).

What is not counted:

1. Zeroes before a decimal for numbers less than 1.

2. Every digit between the decimal point and a non-zero digit for numbers less than 1.

3. Every zero that follows a non-zero digit in a number expressed with no decimal.

PRACTICE:Rewrite the following measurements in scientific notation and determine the number of significant figures.

1. 135.680 g2. 0.00620 dm3

3. 6.00 kg4. 2.0600 m3

5. 0.2 mg6. 300 kg

1.35680 x 102 g6.20 x 10-3 dm3

6.00 kg2.0600 m3

2 x 10-1 mg3 x 102 kg

SixThreeThreeFiveOneOne

Reporting SFs in calculations For addition/subtraction: The result should be expressed based on the measurement with smallest number of decimal places.

Ex: (6.702 g) + (5.2 g) = (11.9 g)

For multiplication/division: The result should be expressed based on the measurement with the smallest number of significant figures.

Ex: (5.4376 g)/(2.31 cm3)=(12.6 g/cm3)

3 decimal places

1 decimal place

1 decimalplace

5 SFs 3 SFs 3 SFs

PRACTICE:The mass of a sample bottle and a piece of aluminum metal is 35.4200 g. The mass of the empty sample bottle is 28.9200 g. If the aluminum displaces 2.41 cm3 of water, calculate the density of aluminum, in g cm-3.

Special case: logarithms Log of a number is made of two parts: characteristic (integer part) and mantissa (decimal part).

Ex: log10(2.7) = 0.43 = 4.3 x 10-1

Ex: ln(6.28) = 1.837

two SFs mantissa (decimal part) two SFs

three SFs mantissa (decimal part) three SFs

Experimental errors•Systematic errors: flaws in the actual experimental design or with the instrument used.• Instrumentation errors (ex: leaky gas syringes, faulty calibration of

instruments)• Experimental methodology errors (ex: poorly insulated

calorimeter, measuring from the top of a meniscus instead of the bottom, occurrence of a side reaction)• Personal errors (ex: the exact color of a solution at its end point,

reading a graduated cylinder incorrectly)

Experimental errors•Random errors occur because of uncontrolled variables in an experiment and hence cannot be eliminated. •Examples: •Estimating a quantity which lies between marked

graduations of an instrument•Not being able to read an instrument due to fluctuations in

reading (temperature variations, airflow, changes in pressure)•Reaction time

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Percentage error

Examiner’s hints 1. You should compare your data to the literature value where appropriate.

2. When evaluating investigations (or experiment design/results), distinguish between systematic and random errors.

3. When evaluating procedures, you should discuss the precision and accuracy of the measurements. You should specifically look at the procedure and use of equipment.

Absolute and relative uncertainty

Every single measurement has a degree of uncertainty. That uncertainty needs to be recorded with all measurements.

Analog instruments: ±½ the smallest division◦ Ex: 10 mL graduated cylinder: ±0.5 mL

Digital instruments: ± the smallest scale division◦ Ex: top-pan balance: ±0.01 g

Absolute and relative uncertainty

Experimental results should be reported in the following form: Experimental result = (A ± ΔA) unit

Where A represents the measured result and ΔA is the absolute uncertainty.

Ex: experimental result = (3.56 ± 0.1) g

Absolute and relative uncertainty

Absolute uncertainty (ΔA) is the margin of uncertainty given for a particular instrument (usually found in a handbook).

Relative uncertainty is the ratio comparing the size of the absolute uncertainty to the size of the measure experimental result, A, and has no units.

DO NOT CONFUSE UNCERTAINTY WITH ERROR!

Examiner’s hint The calculated uncertainty is generally quoted to not more than one SF if it is greater or equal to 2% of the answer and to not more than 2 SFs if it is less than 2%. Intermediate values in calculations should not be rounded off to avoid unnecessary imprecision.

EXAMPLE:A calibrated burette has an absolute uncertainty of ±0.02 cm3. During a titration, the volume of a 0.15 mol dm-3 solution of hydrochloric acid at the end point was recorded as 12.25 cm3. Calculate the associated relative uncertainty.

Relative uncertainty is often expressed as percentage uncertainty, so in this case, it would be 0.2%.

PRACTICE:A thermometer has an uncertainty of ±0.1oC. The boiling point of an unknown substance was measured to be 36.1oC. Calculate the percentage relative uncertainty.

Propagation of uncertainty The process of determining how different uncertainties combine to give the resultant uncertainty.

Rule 1: when adding or subtracting measurements, the absolute uncertainties are added.

Rule 2: when multiplying or dividing measurements, the percentage relative uncertainties are added.

EXAMPLE: During a titration the following titres were recorded for a 0.10 mol dm-3 solution of hydrochloric acid from a burette:

initial titre = (5.00±0.02) cm3

final titre = (21.35±0.02) cm3

Calculate the volume delivered, in cm3, and the uncertainty of this volume.

Volume delivered = 21.35 – 5.00 = 16.35 cm3

For uncertainty, use rule 1 since subtraction is used:(±0.02 cm3)+(±0.02 cm3) = (±0.04 cm3)

So volume would be reported as (16.35±0.04) cm3.

EXAMPLE:(13.37±0.1) g of sodium chloride salt is dissolved in a flask containing (2.0±0.1) dm3 of water. Calculate the concentration of sodium chloride, in g dm-3, in the solution and the percentage relative uncertainty of this concentration. Assume that the salt is fully dissolved in the solution.

For uncertainty, use rule 2 since division is involved.

(Cont on next slide)

Percentage relative uncertainty in mass =0.7%

Percentage relative uncertainty in volume =

Percentage relative uncertainty = (0.7%) + (5%) = (5.7%)

Concentration = 6.7 g dm-3 (±6%)Absolute uncertainty can be calculated from this:

So concentration = (6.7±0.4) g dm-3

PRACTICE:During a thermodynamic experiment, a change in temperature was measured using a thermometer with an absolute uncertainty of ±0.05oC.

Temperature (initial): 24.32oCTemperature (final): 78.49oC

Calculate the change in temperature and the uncertainty for this measurement.

PRACTICE:The mass of a sample bottle and piece of titanium metal is (33.29±0.005) g. The mass of the empty sample bottle is (26.35±0.005) g. If the metal displaced 1.5±0.05 cm3 of water, what is the density of the metal. What is the percent uncertainty?

Special case: averages For a calculated average, the propagated uncertainty for the component numbers can be used in the final answer.

Ex: ΔHmean = [+100 kJ mol-1 (±10%) + 110 kJ mole-1 (±10%) + 108 kJ mol-1 (±10%) / 3

ΔHmean = +106 kJ mol-1 (±10%)

Graphical AnalysisIndependent Variable Dependent VariableCause EffectPlotted on horizontal axis (X) Plotted on vertical axis (Y)

Correlation Correlation – a measure of the extent to which the two variables change with one another.Positive correlation – the two variables increase or decrease in parallel to one another.

Negative correlation – one variable increases while the second variable decreases or vice versa.

Checklist for graphingGive the graph a titleLabel axes with both quantities and unitsUse space effectivelyUse sensible linear scales – no uneven jumpsPlot points correctlyLine of best fit drawn smoothly/clearly – it should show overall trendIdentify points which do not agree with trendThink about inclusion of origin – may not always be necessary

‘Best fit’ line This is a line that attempted to demonstrate the correlation between the IV and DV. Keep in mind that this line may not include all the data points.

Equation of line: y = mx + c◦ Where m is the gradient (slope) and c is the y-intercept

Extrapolation – extension of line beyond range of measurements Interpolation – process of assuming that the trend line applies between two points.

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Video: Richard Thornley – Drawing Best Fit Lines

Graph the following data. Include an extrapolated best-fit line to show absolute zero. Find the slope and overall line equation of your best-fit line (y=mx+c).

Independent Variable Dependent Variable

Temp (oC) Volume (cm3)

10.0 50

50.0 57

100.0 66