measurement and data processing - weebly

Topic 11: Measurement and data processing 11.1 Uncertainty and error in measurement 11.2 Uncertainties in calculated results 11.3 Graphical techniques -later!

Thursday, September 13, 2012

From the syllabus


Precision v. AccuracyThe accuracy of a measurement is an expression of how close the measured value is to the “correct” or “true” value.Expressed as a Percentage Difference

The precision of a set of measurements refers to how closely the individual measurements agree with one another. Precision is a measure of the reproducibility or consistency of a result.


Imagine a dartboard!


Significant Figures★ What is 22.1 x 13.7 ?★ It’s NOT 302.77! ★ It’s 303

★ Your answer cannot be more certain than the data you started

with!


Remember! All non-zero digits count as significant figures: 4Leading zeros are those that precede the first non-zero digit. Leading zeros are not counted as significant figures. 0.003

Captive zeros are zeros between non-zero integers. Captive zeros always count as significant figures. 406Trailing zeros are zeros to the right of a number. Trailing zeros are significant if the number contains a decimal point.If there is no decimal point, the number of significant figures may be unclear. 1000


Scientific Notation• 400 vs. 400. • To avoid this confusion the number should

be written in standard form or scientific notation.

• Rewrite as 4 x102

• Rewrite as 4.00 x102


A Caveat Sig figs are only applied to measurements and calculations involving measurements. They do not apply to quantities that are inherently integers or fractions

a stoichiometric ratio such as 2 or ½ mole, defined quantities (for example, one metre equals 100 centimetres), orconversion factors (multiplying by 100 to get a percentage or adding 273 to convert °C to K).


Multiplication and division

The result should have the same number of significant figures as the factor with the least number of significant figures.

2.54 × 2.6 = 6.604.


Addition and SubtractionThe result should have the same number of decimal places as the number used with the fewest decimal places. Note that when adding and subtracting we are interested in decimal places, not significant figures.

3.467 + 4.5 + 3.66 = 11.627


Exercise2.568 x 5.8 ÷4.186

5.41 – 0.398

3.38 – 3.01

4.18 – 58.16 x (3.38 – 3.01)


Uncertainty of a Digital Reading

+/- the last measurable digitIf a measured mass is read on a digital scale as 18.34 g, then it’s expressed as: 18.34+/- 0.01 gIf a measure mass is read on a digital scale as 18.3g, then it’s expressed as 18.3 +/- 0.1 g

Imagine when you’re on the scale and the last digit flickers between two...


Uncertainty of an Analog Reading

+/- 0.5 of the last measurable digit.

The volume of solution in this burette is 48.80 cm3.

The burette reading can be recorded as 48.80 ± 0.05 cm3. The burette reading is between 48.75 cm3 and 48.85 cm3.


Reading a MeniscusWhen reading a meniscus you should always have your eyes level with the meniscus, and for aqueous solutions the volume is read from the bottom of the meniscus.


MISTAKES Sometimes, during an experimental investigation, a student may make a mistake:misreading a scale or a digital readingusing different balances for a number of related measurementswrongly transferring raw data to a table of resultspressing the wrong buttons on a calculator or making arithmetical errors in mental calculationsfailing to carry out a procedure as described in the method.

MISTAKES CAN BE AVOIDED!!! ERRORS CANNOT


Systematic ErrorPoor accuracy in measurements is usually associated with an error in the system—a systematic error.Using a balance that has been incorrectly zeroed (for example, so that the zero reading is in fact a mass of 0.1 g) will produce measurements that are inaccurate (below their true value).Systematic errors are always biased in the same direction. The incorrectly zeroed balance will always produce measurements that are below their true value. Repeating the measurements will not improve the accuracy of the result.


Systematic errors can be the result of:

poorly calibrated instrumentsinstrument parallax error (reading a scale from a position that is not directly in front of the scale)badly made instrumentspoorly timed actions (such as the reaction time involved in clicking a stopwatch).


Random uncertaintyPoor precision in measurements is associated with random uncertainty.These are the minor uncertainties inherent in any measurement. Error associated with estimating the last digit of a reading is a random uncertainty.

How can we reduce random uncertainty???


Repeat, repeat, repeat! Repeating the measurement a number of times and averaging the results reduces the effect of random uncertainty.


Percent UncertaintyThe absolute uncertainty is the size of an uncertainty, including its units. 20.00 ± 0.05 cm3

The percentage uncertainty changes with the amount of material that you are measuring. Percentage uncertainty is found by dividing the absolute uncertainty by the measurement that is being made.

Percentage uncertainty = absolute uncertainty x 100 m measurement

0.05/20 x100 =


Processing Data

Determine the uncertainties in results

When adding or subtracting absolute uncertainties can be added together

For multiplication/division, percentage uncertainties can be added together


Example

CHEMISTRY: FOR USE WITH THE IB DIPLOMA PROGRAMME STANDARD LEVEL

CH

APT

ER 5

M

EAS

UR

EMEN

T A

ND

DA

TA P

RO

CES

SIN

G

173

Worked exampleCalculate the percentage uncertainty in each of the following calculations.

a 12.5 ± 0.16 cm3 of HCl is added from a burette to 4.0 ± 0.5 cm3 of water that had been measured in a measuring cylinder. Calculate the absolute uncertainty and hence the percentage uncertainty in the fi nal solution.

b A sample of copper has been produced during a multi-stage experiment. The 250 cm3 beaker was weighed at the start of the experiment and found to have a mass of 180.15 ± 0.02 g. Several days later the beaker and copper was found to have a mass of 183.58 ± 0.02 g. Calculate the absolute uncertainty and hence the percentage uncertainty in the fi nal mass of copper.

c Calculate the percentage uncertainty, and hence the absolute uncertainty, in the concentration of a solution of sodium carbonate, Na2CO3, that has been prepared by the dissolution of 2.346 ± 0.002 g of sodium carbonate in a 250.0 ± 0.15 cm3 volumetric fl ask.

Solutiona Total volume of solution = 12.5 + 4.0 = 16.5 cm3

Absolute uncertainty = ±(0.16 + 0.5) = ±0.66 cm3

Percentage uncertainty = 0 6616 5

1001

..

× = ±4%

b Mass of copper = 183.58 − 180.15 = 3.43 g

Absolute uncertainty = ±(0.02 + 0.02) = ±0.04 g


1001

.

.× = ±1%

c n(Na2CO3) = mM

(Na CO )(Na CO )

2 3

2 3

= 2 346

2 22 99 12 01 3 16 002 346

105 99.

. . ..

.× + + ×=

= 0.022 13 mol (to 4 signifi cant fi gures)

Absolute uncertainty in n(Na2CO3) = 0 002

105 99.

. = 2 × 10−5 mol

Percentage uncertainty in n(Na2CO3) = 2 100 02213

1001

5× ×−

. = 0.09%

c(Na2CO3) = 0 022130 250..

= 0.0885 mol dm−3

Absolute uncertainty in V(solution) = 0.15 cm3

Percentage uncertainty in V(solution) = 0 15250

1001

. × = 0.06%

Percentage uncertainty in c(Na2CO3) = 0.09% + 0.06% = 0.15%

Absolute uncertainty in c(Na2CO3) = 0.15% of 0.0885 = 0.0003 mol dm−3

The concentration of the sodium carbonate solution is 0.0885 ± 0.0003 mol dm−3.

WORKSHEET 5.2 Treatment of uncertainties

CHEMISTRY: FOR USE WITH THE IB DIPLOMA PROGRAMME STANDARD LEVELC

HA

PTER

5

MEA

SU

REM

ENT

AN

D D

ATA

PR

OC

ESS

ING

173








1001

..

× = ±4%

b Mass of copper = 183.58 − 180.15 = 3.43 g



1001

.

.× = ±1%

c n(Na2CO3) = mM

(Na CO )(Na CO )

2 3

2 3

= 2 346

2 22 99 12 01 3 16 002 346

105 99.

. . ..

.× + + ×=



105 99.

. = 2 × 10−5 mol


1001

5× ×−

. = 0.09%

c(Na2CO3) = 0 022130 250..

= 0.0885 mol dm−3



1001

. × = 0.06%






Example


CH

APT

ER 5

M

EAS

UR

EMEN

T A

ND

DA

TA P

RO

CES

SIN

G

173








1001

..

× = ±4%

b Mass of copper = 183.58 − 180.15 = 3.43 g



1001

.

.× = ±1%

c n(Na2CO3) = mM

(Na CO )(Na CO )

2 3

2 3

= 2 346

2 22 99 12 01 3 16 002 346

105 99.

. . ..

.× + + ×=



105 99.

. = 2 × 10−5 mol


1001

5× ×−

. = 0.09%

c(Na2CO3) = 0 022130 250..

= 0.0885 mol dm−3



1001

. × = 0.06%






CH

APT

ER 5

M

EAS

UR

EMEN

T A

ND

DA

TA P

RO

CES

SIN

G

173








1001

..

× = ±4%

b Mass of copper = 183.58 − 180.15 = 3.43 g



1001

.

.× = ±1%

c n(Na2CO3) = mM

(Na CO )(Na CO )

2 3

2 3

= 2 346

2 22 99 12 01 3 16 002 346

105 99.

. . ..

.× + + ×=



105 99.

. = 2 × 10−5 mol


1001

5× ×−

. = 0.09%

c(Na2CO3) = 0 022130 250..

= 0.0885 mol dm−3



1001

. × = 0.06%






Homework

Complete sig fig worksheet


measurement and data processing - weebly

Documents