Fall 2006

CH 421/521 Analytical Chemistry

Daniel C. HarrisQuantitative Chemical Analysis7th edition

Lecture 1

Experimental Error

Reading: Ch. 3

No measurement made is ever exact.

Outline

1. Significant figures (3-1)

2. Significance arithmetic (3-2)

3. Types of experimental error (3-3)

4. Propagation of uncertainty from experimental error (3-4, 3-6)

Significant figures

Significant figures = significant digitsThe number of significant figures is the minimum number of digits needed towrite a given value in a scientific notation without loss of accuracy.

Number Number of significant figures

426

3.14

40.078

1.234 x 10-9

52000

Use scientific notation to clearly show the number of significant figures.

Significance arithmetic (3-2)

Addition and subtractionThe number of significant figures may be less or exceed than that in the original data. In general, you are limited to the less certain one.

Rules for rounding off numbers:If the insignificant figures are more than halfway to the next higher digit , we round it up.If the insignificant figures are less than halfway, we round it down.If the number is exactly halfway, round to the nearest even digit.Multiplication and division

The result is limited to the number of digits contained in the number with the fewest significant figures

The power of 10 has no influence on the resulting number of significant figures

Significance arithmetic (3-2)

Logarithms and antilogarithms

log n = a

Log (5.356 x 10-9) = -8.2711 -8.2711

Characteristic Mantissa

Number of digits in mantissa of log (x) = number of significant figures in x

n = 10a

10 -3.567 = 2.71 x 100

Number of digits in antilog (x) = number of significant figures in mantissa of x

Significance arithmetic (3-2)

Graphs

Types of experimental error (3-3)

RESULT The final value reported for a measured or computed quantity, after performing a measuring procedure including all subprocedures and evaluations. {IUPAC-1994}

SYSTEMATIC ERROR (determinate error)This error is reproducible and can be corrected1. Analyze a known sample2. Analyze blank sample3. Use different analytical methods performed by different people in different

laboratories and compare the results.

RANDOM ERRORAlways present and can not be corrected

PRECISION The reproducibility of a result. The closeness of agreement between independent test results obtained under stipulated conditions. {ISO 3534-1}Precision depends only on the distribution of random errors and does not relate to the true value or to the specified value.

ACCURACYThe closeness of agreement between a reported result and the accepted referencevalue (“true value”). {ISO 3534-1}

Types of experimental error (3-3)

ABSOLUTE UNCERTAINTY

RELATIVE UNCERTAINTY = = ABSOLUTE UNCERTAINTY / MAGNITUDE OF MEASUREMENT

PERCENT RELATIVE UNCERTAINTY = = RELATIVE UNCERTAINTY x 100

Function Uncertainty Function Uncertainty

21 xxy +=

22

21 xxy eee +=

axy =

xy eae %% ⋅=

21 xxy −=

22

21 xxy eee +=

xy log=

xe

xee xx

y 43429.010ln1

≈=

21 xxy ⋅=

22

2 %%%1 xxy eee +=

xy ln=

xee x

y =

2

1

xxy =

22

2 %%%1 xxy eee +=

xy 10=

xxy eeye

3026.2)10(ln ≈=

xey =

xy eye

=

x represenr a variable and a represent a constant that has no uncertainty

Propagation of uncertainty from random error (3-4)

The first digit of absolute uncertainty is the last significant digit in the result

Propagation of uncertainty from systematic error (3-5)

82Pb207.2(1)

Values found for atomic weight of lead from different sources.

Lead from North Carolina uraninite 206.40Lead from Joachimsthal pitchblende 206.57Lead from Colorado carnotite 206.59Lead from Ceylonese thorianite 206.82Lead from English pitchblende 206.86Common lead 207.15

Isotope Atomic mass Natural abundance (atom %)

204Pb 203.973020 (5) 1.4 206Pb 205.974440 (4) 24.1 207Pb 206.975872 (4) 22.1 208Pb 207.976627 (4) 52.4

For systematic uncertainty, we add the uncertainties of each term in a sum or difference.